Content-Type: multipart/mixed; boundary="-------------0110241808200" This is a multi-part message in MIME format. ---------------0110241808200 Content-Type: text/plain; name="01-397.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-397.comments" AMS-Code: 81Q20 PACS-Code: 03.65.-w, 02.30.Lt Submitted to Communications in Mathematical Physics ---------------0110241808200 Content-Type: text/plain; name="01-397.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-397.keywords" semiclassical limit, optimal truncation, schrodinger equation, exponentially accurate estimates ---------------0110241808200 Content-Type: application/postscript; name="draft-final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="draft-final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: draft-final.dvi %%Pages: 37 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments 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Fq(l)r Fw(.)44 b(That)33 b(is,)991 5028 y Fv(X)974 5244 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(l)1169 5123 y Fw(1)27 b(=)1359 5055 y(\()p Fq(l)d Fw(+)e Fq(d)g Fu(\000)g Fw(1\)!)p 1359 5100 476 4 v 1405 5191 a Fq(l)r Fw(!\()p Fq(d)g Fu(\000)h Fw(1\)!)1872 5123 y Fu(\024)2125 5055 y Fw(1)p 1987 5100 325 4 v 1987 5191 a(\()p Fq(d)f Fu(\000)g Fw(1\)!)2321 5123 y(\()p Fq(l)i Fw(+)e Fq(d)g Fu(\000)h Fw(1\))2770 5081 y Fr(d)p Fm(\000)p Fs(1)1913 5470 y Fw(3)p eop %%Page: 4 4 4 3 bop -71 123 a Fw(Therefore,)34 b(w)m(e)g(ha)m(v)m(e)1409 198 y Fv(X)1392 414 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(l)1597 226 y Fu(j)o Fq(D)1708 190 y Fr(m)1775 226 y Fq(V)21 b Fw(\(0\))p Fu(j)p 1597 270 409 4 v 1745 361 a Fq(m)p Fw(!)2015 293 y Fq(\016)2062 252 y Fm(j)p Fr(m)p Fm(j)2196 293 y Fu(\024)28 b Fq(C)7 b(Y)2456 252 y Fr(l)-71 570 y Fw(for)33 b(some)h(constan)m(ts)h Fq(C)41 b Fw(and)34 b Fq(Y)21 b Fw(,)34 b(whic)m(h)g(dep)s(end)h(only)e(on)h(the)g (dimensional)e(constan)m(t)i Fq(d)p Fw(.)47 b(Without)33 b(loss)-71 690 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y(co)s(ordinates,)32 b(the)h(eigenfunctions)g(of)f Fq(H)1472 1779 y Fs(0)1544 1764 y Fw(can)g(b)s(e)h(written)g(as)713 2066 y(\010)783 2081 y Fr(j)820 2066 y Fw(\()p Fq(x)p Fw(\))28 b(=)f Fq(\031)1141 2024 y Fm(\000)1206 1997 y Fk(1)p 1206 2009 31 3 v 1206 2050 a(4)1267 1985 y Fv(\000)1313 2066 y Fw(2)1362 2024 y Fm(j)p Fr(j)t Fm(j)1437 2066 y Fq(j)6 b Fw(!)1510 1985 y Fv(\001)1556 2001 y Fm(\000)1621 1974 y Fk(1)p 1621 1986 V 1621 2027 a(2)1682 2066 y Fw(exp)1848 1895 y Fv( )1927 2066 y Fu(\000)2014 1998 y Fw(1)p 2014 2043 49 4 v 2014 2134 a(2)2143 1941 y Fr(d)2090 1971 y Fv(X)2104 2181 y Fr(i)p Fs(=1)2250 2066 y Fq(!)2311 2081 y Fr(i)2339 2066 y Fq(x)2394 2024 y Fs(2)2394 2090 y Fr(i)2434 1895 y Fv(!)2575 1941 y Fr(d)2529 1971 y Fv(Y)2536 2181 y Fr(i)p Fs(=1)2673 2066 y Fq(H)2754 2081 y Fr(j)2783 2091 y Fl(i)2830 2066 y Fw(\()2868 1996 y Fu(p)p 2951 1996 89 4 v 70 x Fq(!)3012 2081 y Fr(i)3040 2066 y Fq(x)3095 2081 y Fr(i)3123 2066 y Fw(\))659 b(\(5\))-71 2367 y(where)33 b Fq(j)h Fw(=)27 b(\()p Fq(j)465 2382 y Fs(1)505 2367 y Fq(;)17 b(:)g(:)g(:)f(;)h(j)764 2382 y Fr(d)804 2367 y Fw(\))32 b(and)g Fq(H)1144 2382 y Fr(j)1173 2392 y Fl(i)1203 2367 y Fw(\()p Fq(y)t Fw(\))f(are)h(the)g(Hermite)f(p) s(olynomials.)40 b(The)33 b(corresp)s(onding)f(eigen)m(v)-5 b(alues)-71 2498 y(are)33 b Fq(e)137 2513 y Fr(j)201 2498 y Fw(=)305 2423 y Fv(P)410 2449 y Fr(d)410 2527 y(i)p Fs(=1)545 2498 y Fq(!)606 2513 y Fr(i)634 2498 y Fq(j)674 2513 y Fr(i)724 2498 y Fw(+)22 b Fq(d=)p Fw(2.)75 2618 y(In)37 b(the)f(semiclassical)e(limit)f(w)m(e)k(ma)m(y)f(consider) g Fq(W)14 b Fw(\()p Ft(~)p Fq(;)j(x)p Fw(\))36 b(as)g(a)g(p)s (erturbation)f(of)h Fq(H)3273 2633 y Fs(0)3312 2618 y Fw(.)54 b(Then)37 b(w)m(e)h(can)-71 2738 y(prop)s(ose)33 b(formal)d(Ra)m(yleigh-Sc)m(hr\177)-49 b(odinger)31 b(series)j(for)e(b) s(oth)g Fq(E)6 b Fw(\()p Ft(~)p Fw(\))32 b(and)2617 2713 y(~)2603 2738 y(\011\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))557 2940 y(~)544 2965 y(\011\()p Fq(x)p Fw(\))83 b Fu(\030)1015 2939 y Fw(~)995 2965 y Fq( )1058 2980 y Fs(0)1097 2965 y Fw(\()p Fq(x)p Fw(\))23 b(+)f Ft(~)1413 2897 y Fk(1)p 1413 2909 31 3 v 1413 2950 a(2)1477 2939 y Fw(~)1457 2965 y Fq( )1520 2980 y Fs(1)1560 2965 y Fw(\()p Fq(x)p Fw(\))h(+)f Ft(~)1876 2897 y Fk(2)p 1875 2909 V 1875 2950 a(2)1940 2939 y Fw(~)1920 2965 y Fq( )1983 2980 y Fs(2)2023 2965 y Fw(\()p Fq(x)p Fw(\))g(+)g Ft(~)2338 2897 y Fk(3)p 2338 2909 V 2338 2950 a(2)2403 2939 y Fw(~)2383 2965 y Fq( )2446 2980 y Fs(3)2485 2965 y Fw(\()p Fq(x)p Fw(\))h(+)f Ft(~)2801 2897 y Fk(4)p 2801 2909 V 2801 2950 a(2)2865 2939 y Fw(~)2845 2965 y Fq( )2908 2980 y Fs(4)2948 2965 y Fw(\()p Fq(x)p Fw(\))h(+)f Fq(:)17 b(:)g(:)505 b Fw(\(6\))543 3194 y Fq(E)6 b Fw(\()p Ft(~)p Fw(\))83 b Fu(\030)h Fq(e)22 b Fw(+)g Ft(~)1224 3126 y Fk(1)p 1224 3138 V 1224 3179 a(2)1268 3194 y Fu(E)1321 3209 y Fs(1)1382 3194 y Fw(+)h Ft(~)1545 3126 y Fk(2)p 1544 3138 V 1544 3179 a(2)1589 3194 y Fu(E)1642 3209 y Fs(2)1703 3194 y Fw(+)f Ft(~)1865 3126 y Fk(3)p 1865 3138 V 1865 3179 a(2)1909 3194 y Fu(E)1962 3209 y Fs(3)2024 3194 y Fw(+)g Ft(~)2186 3126 y Fk(4)p 2185 3138 V 2185 3179 a(2)2230 3194 y Fu(E)2283 3209 y Fs(4)2344 3194 y Fw(+)g Fq(:)17 b(:)g(:)f(;)1220 b Fw(\(7\))-71 3408 y(where)35 b Fq(e)g Fw(is)e(an)h(eigen)m(v)-5 b(alue)34 b(of)f Fq(H)1189 3423 y Fs(0)1263 3408 y Fw(with)g(m)m(ultiplicit)m(y)e Fq(g)37 b Fw(and)d(asso)s(ciated)g(eigenspace)h Fq(B)5 b Fw(.)48 b(Later)34 b(w)m(e)h(will)-71 3528 y(\014nd)e(that)356 3502 y(~)336 3528 y Fq( )399 3543 y Fs(0)438 3528 y Fw(\()p Fq(x)p Fw(\))g(m)m(ust)g(b)s(e)g(a)f(v)m(ector)i(in)e Fq(B)5 b Fw(,)32 b(as)h(exp)s(ected.)75 3649 y(Before)g(going)f(in)m (to)g(the)h(standard)g(ph)m(ysicist-lik)m(e)f(p)s(erturbation)g(pro)s (cedure,)i(let)e(us)h(mak)m(e)g(a)g(further)-71 3769 y(transformation)h(of)h(\(4\).)53 b(Let)36 b Fu(f)p Fw(\010)1214 3784 y Fr(j)1251 3769 y Fw(\()p Fq(x)p Fw(\))p Fu(g)g Fw(b)s(e)g(a)g(basis)g(of)f(eigen)m(v)m(ectors)j(of)d Fq(H)2790 3784 y Fs(0)2829 3769 y Fw(.)54 b(F)-8 b(or)35 b(a)g(giv)m(en)h(eigen)m(v)-5 b(alue)36 b Fq(e)-71 3889 y Fw(of)c Fq(H)121 3904 y Fs(0)193 3889 y Fw(de\014ne)i(a)e(new)i(op)s (erator)e Fq(A)1223 3904 y Fr(e)1292 3889 y Fw(b)m(y)936 4214 y Fq(A)1009 4229 y Fr(e)1046 4214 y Fw(\010)1116 4229 y Fr(j)1153 4214 y Fw(\()p Fq(x)p Fw(\))61 b(=)1481 4010 y Fv(8)1481 4100 y(<)1481 4279 y(:)1611 4106 y Fw(\010)1681 4121 y Fr(j)1718 4106 y Fw(\()p Fq(x)p Fw(\))513 b(if)27 b(\010)2517 4121 y Fr(j)2554 4106 y Fw(\()p Fq(x)p Fw(\))h Fu(2)g Fq(B)1611 4321 y Fu(j)p Fq(e)22 b Fu(\000)h Fq(e)1851 4336 y Fr(j)1887 4321 y Fu(j)1915 4272 y Fm(\000)1980 4245 y Fk(1)p 1980 4257 V 1980 4298 a(2)2041 4321 y Fw(\010)2111 4336 y Fr(j)2148 4321 y Fw(\()p Fq(x)p Fw(\))83 b(otherwise.)-71 4545 y(where)37 b Fq(e)259 4560 y Fr(j)331 4545 y Fw(is)e(the)h(eigen)m (v)-5 b(alue)35 b(asso)s(ciated)g(with)h(\010)1835 4560 y Fr(j)1872 4545 y Fw(\()p Fq(x)p Fw(\).)52 b(Then)37 b(extend)g Fq(A)2733 4560 y Fr(e)2805 4545 y Fw(to)f(the)f(whole)h (Hilb)s(ert)e(space)-71 4665 y Fu(H)43 b Fw(b)m(y)h(linearit)m(y)-8 b(.)71 b(So)42 b(de\014ned,)k Fq(A)1239 4680 y Fr(e)1318 4665 y Fw(is)c(a)g(b)s(ounded)h(op)s(erator)f(with)g(un)m(b)s(ounded)i (in)m(v)m(erse.)74 b(No)m(w)43 b(de\014ne)-71 4785 y Fq(H)10 4800 y Fr(e)75 4785 y Fw(=)27 b Fq(A)251 4800 y Fr(e)288 4785 y Fw(\()p Fq(H)407 4800 y Fs(0)468 4785 y Fu(\000)c Fq(e)p Fw(\))p Fq(A)724 4800 y Fr(e)761 4785 y Fw(.)44 b(This)32 b(op)s(erator)g(satis\014es)1042 5116 y Fq(H)1123 5131 y Fr(e)1159 5116 y Fw(\010)1229 5131 y Fr(j)1266 5116 y Fw(\()p Fq(x)p Fw(\))61 b(=)1594 4912 y Fv(8)1594 5001 y(<)1594 5181 y(:)1724 5007 y Fw(0)484 b(if)27 b(\010)2412 5022 y Fr(j)2449 5007 y Fw(\()p Fq(x)p Fw(\))h Fu(2)g Fq(B)1753 5154 y Fr(e)p Fm(\000)p Fr(e)1874 5164 y Fl(j)p 1734 5182 193 4 v 1734 5239 a Fm(j)o Fr(e)p Fm(\000)p Fr(e)1874 5249 y Fl(j)1906 5239 y Fm(j)1936 5205 y Fw(\010)2006 5220 y Fr(j)2043 5205 y Fw(\()p Fq(x)p Fw(\))83 b(otherwise.)1913 5470 y(4)p eop %%Page: 5 5 5 4 bop -71 123 a Fw(on)33 b(an)f(eigen)m(basis)h(of)f Fq(H)846 138 y Fs(0)885 123 y Fw(.)44 b(With)31 b(these)j (de\014nitions,)e(\(4\))h(transforms)f(in)m(to)954 343 y([)p Fq(H)1062 358 y Fr(e)1121 343 y Fw(+)22 b Fq(A)1292 358 y Fr(e)1329 343 y Fq(W)14 b Fw(\()p Ft(~)p Fw(;)j Fq(x)p Fw(\))p Fq(A)1737 358 y Fr(e)1773 343 y Fw(])g(\011\()p Fq(x)p Fw(\))28 b(=)f([)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)h Fq(e)p Fw(])17 b Fq(A)2674 302 y Fs(2)2674 368 y Fr(e)2713 343 y Fw(\011\()p Fq(x)p Fw(\))900 b(\(8\))-71 563 y(and)33 b(instead)f(of)g(\(6\))h(w)m(e)g(ha)m(v)m(e)607 797 y(\011\()p Fq(x)p Fw(\))28 b Fu(\030)g Fq( )1010 812 y Fs(0)1050 797 y Fw(\()p Fq(x)p Fw(\))23 b(+)f Ft(~)1366 728 y Fk(1)p 1365 740 31 3 v 1365 781 a(2)1410 797 y Fq( )1473 812 y Fs(1)1513 797 y Fw(\()p Fq(x)p Fw(\))g(+)g Ft(~)1828 728 y Fk(2)p 1828 740 V 1828 781 a(2)1873 797 y Fq( )1936 812 y Fs(2)1975 797 y Fw(\()p Fq(x)p Fw(\))h(+)f Ft(~)2291 728 y Fk(3)p 2291 740 V 2291 781 a(2)2335 797 y Fq( )2398 812 y Fs(3)2438 797 y Fw(\()p Fq(x)p Fw(\))h(+)f Ft(~)2754 728 y Fk(4)p 2753 740 V 2753 781 a(2)2798 797 y Fq( )2861 812 y Fs(4)2901 797 y Fw(\()p Fq(x)p Fw(\))g(+)g Fq(:)17 b(:)g(:)553 b Fw(\(9\))-71 1017 y(It)37 b(is)e(easy)j(to)e(pro) m(v)m(e)h(that)f(eac)m(h)1207 990 y(~)1187 1017 y Fq( )1250 1032 y Fr(n)1333 1017 y Fw(lies)g(in)f(a)h(\014nite-dimensional)d (subspace)38 b(that)f(is)e(in)h(the)h(domain)d(of)-71 1137 y Fq(A)2 1101 y Fm(\000)p Fs(1)2 1162 y Fr(e)97 1137 y Fw(.)41 b(Th)m(us)29 b(eac)m(h)g(term)e(of)g(\(9\))g(is)g (related)h(to)f(those)h(in)f(\(6\))g(through)h(the)g(iden)m(tit)m(y) 3049 1111 y(~)3029 1137 y Fq( )3092 1152 y Fr(n)3167 1137 y Fw(=)g Fq(A)3344 1152 y Fr(e)3381 1137 y Fq( )3444 1152 y Fr(n)3491 1137 y Fw(.)42 b(W)-8 b(e)28 b(mak)m(e)-71 1257 y(this)43 b(transformation)e(b)s(ecause)j(it)e(yields)h(b)s(etter) g(estimates)g(to)f(the)i(gro)m(wth)f(of)g(the)g(R-S)g(co)s(e\016cien)m (ts.)-71 1378 y(F)-8 b(urthermore,)32 b(w)m(e)h(need)g(only)f(to)g (transform)f(bac)m(k)i(the)g(solution)e(for)g(the)i(eigen)m(v)m(ector)g (via)f(the)g(op)s(erator)-71 1498 y Fq(A)2 1513 y Fr(e)39 1498 y Fw(.)75 1619 y(No)m(w)i(insert)f(\(9\))g(and)g(\(7\))g(in)m(to)f (\(8\))h(and)g(equate)h(p)s(o)m(w)m(ers)h(of)d Ft(~)2433 1555 y Fk(1)p 2433 1567 V 2433 1608 a(2)2478 1619 y Fw(.)45 b(The)34 b(zeroth-order)f(equation)g(yields)-71 1739 y Fq(H)10 1754 y Fr(e)47 1739 y Fq( )110 1754 y Fs(0)177 1739 y Fw(=)28 b(0)k(so)h Fq( )545 1754 y Fs(0)613 1739 y Fu(2)28 b Fq(B)5 b Fw(.)43 b(F)-8 b(or)32 b Fq(n)c Fw(=)f(1)p Fq(;)17 b Fw(2)p Fq(;)g(:)g(:)g(:)e(;)33 b Fw(w)m(e)h(ha)m(v)m(e)1133 2029 y Fq(H)1214 2044 y Fr(e)1251 2029 y Fq( )1314 2044 y Fr(n)1383 2029 y Fw(+)1532 1905 y Fr(n)1481 1935 y Fv(X)1497 2147 y Fr(l)q Fs(=1)1642 2029 y Fq(T)1713 1988 y Fs(\()p Fr(l)q Fs(+2\))1884 2029 y Fq( )1947 2044 y Fr(n)p Fm(\000)p Fr(l)2099 2029 y Fw(=)2253 1905 y Fr(n)2202 1935 y Fv(X)2218 2147 y Fr(l)q Fs(=1)2363 2029 y Fu(E)2416 2044 y Fr(l)2441 2029 y Fq(A)2514 1988 y Fs(2)2514 2054 y Fr(e)2554 2029 y Fq( )2617 2044 y Fr(n)p Fm(\000)p Fr(l)3771 2029 y Fw(\(10\))-71 2334 y(where)1271 2474 y Fq(T)1342 2433 y Fs(\()p Fr(l)q Fs(\))1450 2474 y Fw(:=)1598 2379 y Fv(X)1581 2595 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(l)1817 2406 y Fw(1)p 1785 2451 113 4 v 1785 2542 a Fq(m)p Fw(!)1908 2474 y Fq(D)1992 2433 y Fr(m)2058 2474 y Fq(V)22 b Fw(\(0\))p Fq(A)2335 2489 y Fr(e)2371 2474 y Fq(x)2426 2433 y Fr(m)2493 2474 y Fq(A)2566 2489 y Fr(e)3771 2474 y Fw(\(11\))-71 2753 y(Dep)s(ending)30 b(on)g(what)g(the)h(eigenfrequencies)h(of)d Fq(H)1859 2768 y Fs(0)1929 2753 y Fw(actually)f(are,)j(w)m(e)g(migh)m (t)e(need)i(to)f(deal)g(with)f(degen-)-71 2873 y(erate)36 b(p)s(erturbation)e(theory)-8 b(.)51 b(F)-8 b(urthermore)35 b(in)f(that)h(case,)i(the)e(degeneracy)i(ma)m(y)e(b)s(e)h(partly)e(or)h (totally)-71 2994 y(preserv)m(ed)k(b)m(y)e(the)g(p)s(erturbation)f (term)f Fq(W)14 b Fw(\()p Ft(~)p Fq(;)j(x)p Fw(\),)37 b(as)g(happ)s(ens)g(when)h Fq(W)14 b Fw(\()p Ft(~)p Fq(;)j(x)p Fw(\))35 b(has)i(some)f(of)g(the)h(sym-)-71 3114 y(metries)d(of)g Fq(H)467 3129 y Fs(0)506 3114 y Fw(.)49 b(Because)36 b(of)d(the)i(m)m(ultiple)d(p)s(ossibilities,)g(the)j(computation)d(of)i (the)h(sev)m(eral)g(correction)-71 3235 y(terms)e(for)f(b)s(oth)g (\(9\))g(and)h(\(7\))f(is)g(rather)h(in)m(v)m(olv)m(ed;)g(the)g(next)g (section)g(is)f(dev)m(oted)i(to)f(it.)75 3355 y(Before)38 b(concluding)f(this)g(section,)i(let)d(us)i(state)g(some)f(tec)m (hnical)g(results.)59 b(First,)37 b(de\014ne)i Fq(a)d Fw(=)g Fq(a)3784 3370 y Fr(e)3858 3355 y Fw(to)-71 3475 y(b)s(e)d(the)g(smallest)e(non-negativ)m(e)h(in)m(teger)h(suc)m(h)h (that)e Fq(B)h Fu(\022)28 b Fw(Ran)2336 3395 y Fv(\000)2382 3475 y Fq(P)2445 3491 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2613 3395 y Fv(\001)2659 3475 y Fw(,)k(where)i Fq(P)3063 3491 y Fm(j)p Fr(j)t Fm(j\024)p Fr(l)3248 3475 y Fw(is)e(the)h(pro)5 b(jection)-71 3596 y(on)m(to)33 b(the)g(subspace)h(spanned)g(b)m(y)g Fu(f)17 b Fw(\010)1377 3611 y Fr(j)1457 3596 y Fw(:)45 b Fu(j)p Fq(j)6 b Fu(j)27 b(\024)h Fq(l)19 b Fu(g)p Fw(.)-71 3824 y Fn(Lemma)37 b(1)49 b Fh(F)-7 b(or)34 b Fq(n)28 b Fu(\025)g Fw(1)p Fh(,)35 b Fq( )981 3839 y Fr(n)1063 3824 y Fh(exists)f(and)h Fq( )1582 3839 y Fr(n)1657 3824 y Fu(2)28 b Fh(R)-5 b(an)1938 3743 y Fv(\000)1984 3824 y Fq(P)2047 3839 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)2348 3743 y Fv(\001)-71 4076 y Fh(Pr)g(o)g(of.)51 b Fw(First,)35 b(decomp)s(ose)g Fq( )1061 4091 y Fr(n)1140 4076 y Fw(=)d Fq(P)1311 4091 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1479 4076 y Fq( )1542 4091 y Fr(n)1614 4076 y Fw(+)1713 3995 y Fv(\000)1759 4076 y Fw(1)22 b Fu(\000)h Fq(P)1993 4091 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2160 3995 y Fv(\001)2223 4076 y Fq( )2286 4091 y Fr(n)2365 4076 y Fw(=:)32 b Fq( )2567 4025 y Fs(\(1\))2563 4086 y Fr(n)2685 4076 y Fw(+)24 b Fq( )2852 4025 y Fs(\(2\))2848 4086 y Fr(n)2946 4076 y Fw(.)51 b(W)-8 b(e)36 b(ha)m(v)m(e)g(to)f(pro)m(v)m(e)h(the)-71 4220 y(statemen)m(t)d(only)f(for)g Fq( )812 4169 y Fs(\(2\))808 4230 y Fr(n)907 4220 y Fw(.)43 b(Equation)32 b(\(10\))g(yields)701 4527 y Fq( )768 4486 y Fs(\(2\))764 4552 y Fr(n)890 4527 y Fw(=)27 b(\()p Fq(H)1112 4542 y Fr(e)1149 4527 y Fw(\))1187 4486 y Fm(\000)p Fs(1)1187 4552 y Fr(r)1298 4446 y Fv(\000)1343 4527 y Fw(1)22 b Fu(\000)h Fq(P)1577 4542 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1745 4446 y Fv(\001)1807 4357 y(")1916 4402 y Fr(n)1865 4432 y Fv(X)1881 4644 y Fr(l)q Fs(=1)2026 4527 y Fu(E)2079 4542 y Fr(l)2104 4527 y Fq(A)2177 4486 y Fs(2)2177 4552 y Fr(e)2217 4527 y Fq( )2280 4542 y Fr(n)p Fm(\000)p Fr(l)2426 4527 y Fu(\000)2576 4402 y Fr(n)2526 4432 y Fv(X)2542 4644 y Fr(l)q Fs(=1)2686 4527 y Fq(T)2757 4486 y Fs(\()p Fr(l)q Fs(+2\))2743 4552 y Fr(e)2928 4527 y Fq( )2991 4542 y Fr(n)p Fm(\000)p Fr(l)3115 4357 y Fv(#)-71 4847 y Fw(where)34 b(\()p Fq(H)330 4862 y Fr(e)367 4847 y Fw(\))405 4811 y Fm(\000)p Fs(1)405 4872 y Fr(r)531 4847 y Fw(is)e(the)h(in)m(v)m(erse)h(of)f(the)g (restriction)e(of)h Fq(H)2058 4862 y Fr(e)2128 4847 y Fw(to)g(Ran)2438 4766 y Fv(\000)2484 4847 y Fw(1)22 b Fu(\000)g Fq(P)2717 4863 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2885 4766 y Fv(\001)2931 4847 y Fw(.)43 b(Since)795 5067 y(Ran)986 4986 y Fv(\000)1032 5067 y Fw(\()p Fq(H)1151 5082 y Fr(e)1188 5067 y Fw(\))1226 5026 y Fm(\000)p Fs(1)1226 5092 y Fr(r)1337 4986 y Fv(\000)1382 5067 y Fw(1)22 b Fu(\000)h Fq(P)1616 5083 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1784 4986 y Fv(\001)1846 5067 y Fq(P)1909 5083 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)2210 4986 y Fv(\001)2283 5067 y Fu(\032)29 b Fw(Ran)2580 4986 y Fv(\000)2626 5067 y Fq(P)2689 5083 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)2990 4986 y Fv(\001)3052 5067 y Fq(;)1913 5470 y Fw(5)p eop %%Page: 6 6 6 5 bop -71 123 a Fw(it)32 b(is)g(su\016cien)m(t)h(to)g(sho)m(w)g(that) 990 235 y Fv( )1120 281 y Fr(n)1069 311 y Fv(X)1085 523 y Fr(l)q Fs(=1)1230 406 y Fu(E)1283 421 y Fr(l)1308 406 y Fq(A)1381 364 y Fs(2)1381 430 y Fr(e)1421 406 y Fq( )1484 421 y Fr(n)p Fm(\000)p Fr(l)1630 406 y Fu(\000)1780 281 y Fr(n)1730 311 y Fv(X)1746 523 y Fr(l)q Fs(=1)1890 406 y Fq(T)1961 364 y Fs(\()p Fr(l)q Fs(+2\))1947 430 y Fr(e)2132 406 y Fq( )2195 421 y Fr(n)p Fm(\000)p Fr(l)2319 235 y Fv(!)2426 406 y Fu(2)28 b Fq(P)2583 421 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)75 710 y Fw(No)m(w)41 b(use)f(induction.)64 b(The)41 b(\014rst)f(term)g(ab)s(o)m(v)m(e)g(is)f (trivial.)63 b(F)-8 b(or)39 b(the)h(second)h(one,)h(let)d(us)h(note)g (that)-71 831 y(Ran)120 750 y Fv(\000)166 831 y Fq(A)239 846 y Fr(e)276 831 y Fq(P)339 846 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)640 750 y Fv(\001)724 831 y Fw(=)e(Ran)1030 750 y Fv(\000)1075 831 y Fq(P)1138 846 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)1439 750 y Fv(\001)1485 831 y Fw(.)62 b(Also,)40 b(a)f(simple)e(calculation)g(with)h(ladder)h (op)s(erators)f(sho)m(ws)-71 951 y(that)h Fq(x)202 915 y Fr(m)269 951 y Fq(')f Fu(2)g Fw(Ran)667 870 y Fv(\000)712 951 y Fq(P)775 966 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fr(l)q Fs(\)+)p Fm(j)p Fr(m)p Fm(j)1364 870 y Fv(\001)1449 951 y Fw(whenev)m(er)j Fq(')d Fu(2)h Fw(Ran)2281 870 y Fv(\000)2327 951 y Fq(P)2390 966 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fr(l)q Fs(\))2822 870 y Fv(\001)2868 951 y Fw(.)62 b(Finally)-8 b(,)37 b(w)m(e)j(ha)m(v)m(e)g(3\()p Fq(n)26 b Fu(\000)-71 1071 y Fq(l)r Fw(\))c(+)g(2)g(+)g Fq(l)30 b Fw(=)e(3)p Fq(n)22 b Fw(+)g(2\(1)g Fu(\000)g Fq(l)r Fw(\))28 b Fu(\024)g Fw(3)p Fq(n)33 b Fw(for)f Fq(l)e Fw(=)d(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(n)p Fw(.)1930 b Fg(2)-71 1300 y Fn(Lemma)37 b(2)49 b Fh(F)-7 b(or)34 b Fu(j)p Fq(m)p Fu(j)27 b(\025)h Fw(2)p Fh(,)35 b Fq(n)28 b Fu(\025)g Fw(0)35 b Fh(and)f(some)g(c)-5 b(onstant)35 b Fq(\015)2153 1315 y Fs(1)2219 1300 y Fq(>)28 b Fw(0)p Fh(,)962 1525 y Fv(\015)962 1585 y(\015)1018 1610 y Fq(A)1091 1625 y Fr(e)1128 1610 y Fq(x)1183 1569 y Fr(m)1250 1610 y Fq(A)1323 1625 y Fr(e)1360 1610 y Fq(P)1423 1625 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)1596 1525 y Fv(\015)1596 1585 y(\015)1679 1610 y Fu(\024)g Fq(\015)1840 1569 y Fs(2)1835 1635 y(1)1880 1610 y Fw(2)1939 1535 y Fi(j)p Fl(m)p Fi(j\000)p Fk(2)p 1938 1554 171 3 v 2008 1595 a(2)2140 1469 y Fv(\024)2202 1543 y Fw(\()p Fq(n)23 b Fw(+)f Fu(j)p Fq(m)p Fu(j)f(\000)i Fw(1\)!)p 2202 1587 593 4 v 2333 1678 a(\()p Fq(n)g Fw(+)f(1\)!)2805 1469 y Fv(\025)2867 1465 y Fk(1)p 2867 1477 31 3 v 2867 1518 a(2)-71 1881 y Fh(As)35 b(a)g(c)-5 b(onse)g(quenc)g(e,)1164 2001 y Fv(\015)1164 2061 y(\015)1220 2086 y Fq(T)1291 2045 y Fs(\()p Fr(l)q Fs(\))1371 2086 y Fq(P)1434 2101 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)1608 2001 y Fv(\015)1608 2061 y(\015)1691 2086 y Fu(\024)28 b Fq(\015)5 b(\024)1918 2017 y Fl(l)p Fi(\000)p Fk(2)p 1918 2030 99 3 v 1952 2071 a(2)2047 1945 y Fv(\024)2110 2018 y Fw(\()p Fq(n)22 b Fw(+)g Fq(l)i Fu(\000)f Fw(1\)!)p 2110 2063 484 4 v 2186 2154 a(\()p Fq(n)f Fw(+)g(1\)!)2603 1945 y Fv(\025)2665 1940 y Fk(1)p 2665 1952 31 3 v 2665 1994 a(2)-71 2312 y Fh(for)35 b(some)f Fq(\015)f(>)27 b Fw(0)35 b Fh(and)f Fq(\024)28 b Fu(\025)g Fw(2)p Fh(.)-71 2540 y(Pr)-5 b(o)g(of.)43 b Fw(F)-8 b(or)32 b(a)g(single)g(co)s(ordinate)f Fq(x)1291 2555 y Fr(\013)1341 2540 y Fw(,)i(w)m(e)g(ha)m(v)m(e)1481 2799 y Fq(x)1536 2814 y Fr(\013)1614 2799 y Fw(=)1824 2732 y(1)p 1727 2776 243 4 v 1727 2796 a Fu(p)p 1810 2796 159 4 v 75 x Fw(2)p Fq(!)1920 2886 y Fr(\013)1996 2799 y Fw(\()p Fq(a)2085 2814 y Fr(\013)2156 2799 y Fw(+)22 b Fq(a)2305 2758 y Fm(\003)2305 2824 y Fr(\013)2355 2799 y Fw(\))-71 3069 y(where)34 b Fq(a)262 3084 y Fr(\013)344 3069 y Fw(and)f Fq(a)585 3033 y Fm(\003)585 3093 y Fr(\013)667 3069 y Fw(are)g(the)g(asso)s(ciated)f(ladder)g(op)s(erators.)75 3189 y(Consider)h(an)m(y)h Fq(')27 b Fw(=)862 3114 y Fv(P)967 3218 y Fr(j)1020 3189 y Fq(d)1071 3204 y Fr(j)1107 3189 y Fw(\010)1177 3204 y Fr(j)1214 3189 y Fw(,)1274 3114 y Fv(P)1379 3218 y Fr(j)1432 3189 y Fu(j)p Fq(d)1511 3204 y Fr(j)1547 3189 y Fu(j)1575 3141 y Fs(2)1642 3189 y Fq(<)g Fu(1)p Fw(.)43 b(De\014ne)34 b Fq(J)2272 3204 y Fr(B)2360 3189 y Fw(:=)27 b Fu(f)p Fw(m)m(ulti-indices)14 b Fq(j)34 b Fw(:)28 b(\010)3301 3204 y Fr(j)3365 3189 y Fu(2)g Fq(B)5 b Fu(g)p Fw(.)44 b(Then)537 3444 y Fq(a)588 3403 y Fm(\003)588 3468 y Fr(\013)637 3444 y Fq(A)710 3459 y Fr(e)747 3444 y Fq(')83 b Fw(=)1067 3349 y Fv(X)1053 3560 y Fr(j)t Fm(2)p Fr(J)1172 3571 y Fl(B)1240 3444 y Fq(d)1291 3459 y Fr(j)1328 3444 y Fq(a)1379 3403 y Fm(\003)1379 3468 y Fr(\013)1428 3444 y Fw(\010)1498 3459 y Fr(j)1557 3444 y Fw(+)1669 3349 y Fv(X)1655 3561 y Fr(j)t Fm(62)p Fr(J)1774 3572 y Fl(B)1843 3444 y Fq(d)1894 3459 y Fr(j)1930 3444 y Fu(j)p Fq(e)22 b Fu(\000)h Fq(e)2170 3459 y Fr(j)2206 3444 y Fu(j)2234 3403 y Fm(\000)2299 3375 y Fk(1)p 2299 3387 31 3 v 2299 3429 a(2)2343 3444 y Fq(a)2394 3403 y Fm(\003)2394 3468 y Fr(\013)2444 3444 y Fw(\010)2514 3459 y Fr(j)894 3722 y Fw(=)1067 3627 y Fv(X)1053 3839 y Fr(j)t Fm(2)p Fr(J)1172 3850 y Fl(B)1240 3722 y Fq(d)1291 3737 y Fr(j)1328 3634 y Fv(p)p 1427 3634 259 4 v 1427 3722 a Fq(j)1467 3737 y Fr(\013)1539 3722 y Fw(+)f(1\010)1756 3737 y Fr(j)t Fs(+1)p Fr(\013)1951 3722 y Fw(+)2062 3627 y Fv(X)2049 3839 y Fr(j)t Fm(62)p Fr(J)2168 3850 y Fl(B)2236 3722 y Fq(d)2287 3737 y Fr(j)2323 3722 y Fu(j)p Fq(e)g Fu(\000)h Fq(e)2563 3737 y Fr(j)2599 3722 y Fu(j)2627 3681 y Fm(\000)2692 3654 y Fk(1)p 2692 3666 31 3 v 2692 3707 a(2)2737 3634 y Fv(p)p 2836 3634 259 4 v 2836 3722 a Fq(j)2876 3737 y Fr(\013)2948 3722 y Fw(+)f(1\010)3165 3737 y Fr(j)t Fs(+1)p Fr(\013)-71 4045 y Fw(where)34 b Fq(j)28 b Fw(+)22 b(1)p Fq(\013)28 b Fw(=)g(\()p Fq(j)698 4060 y Fs(1)737 4045 y Fq(;)17 b(:)g(:)g(:)f(;)h(j)996 4060 y Fr(\013)1068 4045 y Fw(+)22 b(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(j)1474 4060 y Fr(d)1514 4045 y Fw(\).)43 b(Th)m(us,)612 4282 y Fu(k)p Fq(a)713 4241 y Fm(\003)713 4307 y Fr(\013)763 4282 y Fq(A)836 4297 y Fr(e)873 4282 y Fq(')p Fu(k)986 4234 y Fs(2)1110 4282 y Fw(=)1283 4188 y Fv(X)1269 4399 y Fr(j)t Fm(2)p Fr(J)1388 4410 y Fl(B)1457 4282 y Fu(j)o Fq(d)1535 4297 y Fr(j)1571 4282 y Fu(j)1599 4234 y Fs(2)1655 4282 y Fw(\()p Fq(j)1733 4297 y Fr(\013)1805 4282 y Fw(+)22 b(1\))g(+)2123 4188 y Fv(X)2110 4400 y Fr(j)t Fm(62)p Fr(J)2229 4411 y Fl(B)2297 4282 y Fu(j)p Fq(d)2376 4297 y Fr(j)2412 4282 y Fu(j)2440 4234 y Fs(2)2496 4282 y Fu(j)p Fq(e)g Fu(\000)g Fq(e)2735 4297 y Fr(j)2772 4282 y Fu(j)2800 4241 y Fm(\000)p Fs(1)2911 4282 y Fw(\()p Fq(j)2989 4297 y Fr(\013)3061 4282 y Fw(+)g(1\))1109 4561 y Fu(\024)83 b Fw(\(1)22 b(+)g Fq(a)p Fw(\))1595 4466 y Fv(X)1582 4677 y Fr(j)t Fm(2)p Fr(J)1701 4688 y Fl(B)1769 4561 y Fu(j)p Fq(d)1848 4576 y Fr(j)1884 4561 y Fu(j)1912 4512 y Fs(2)1973 4561 y Fw(+)2085 4466 y Fv(X)2071 4678 y Fr(j)t Fm(62)p Fr(J)2190 4689 y Fl(B)2259 4561 y Fu(j)o Fq(d)2337 4576 y Fr(j)2373 4561 y Fu(j)2401 4512 y Fs(2)2457 4561 y Fu(j)p Fq(e)g Fu(\000)h Fq(e)2697 4576 y Fr(j)2733 4561 y Fu(j)2761 4519 y Fm(\000)p Fs(1)2872 4561 y Fw(\()p Fq(j)2950 4576 y Fr(\013)3022 4561 y Fw(+)f(1\))-71 4884 y(b)s(ecause)34 b Fq(j)g Fu(2)28 b Fq(J)512 4899 y Fr(B)600 4884 y Fu(\))f Fq(j)767 4899 y Fr(\013)845 4884 y Fu(\024)h(j)p Fq(j)6 b Fu(j)27 b(\024)h Fq(a)p Fw(.)44 b(Also,)658 5091 y Fq(j)698 5106 y Fr(\013)770 5091 y Fw(+)22 b(1)p 636 5136 305 4 v 636 5227 a Fu(j)p Fq(e)g Fu(\000)g Fq(e)875 5242 y Fr(j)912 5227 y Fu(j)977 5159 y Fw(=)1121 5091 y(1)p 1091 5136 111 4 v 1091 5227 a Fq(!)1152 5242 y Fr(\013)1221 5091 y Fq(!)1282 5106 y Fr(\013)1331 5091 y Fw(\()p Fq(j)1409 5106 y Fr(\013)1481 5091 y Fw(+)g(1)p Fq(=)p Fw(2\))p 1221 5136 543 4 v 1340 5227 a Fu(j)p Fq(e)g Fu(\000)h Fq(e)1580 5242 y Fr(j)1616 5227 y Fu(j)1795 5159 y Fw(+)1982 5091 y(1)p Fq(=)p Fw(2)p 1903 5136 305 4 v 1903 5227 a Fu(j)p Fq(e)f Fu(\000)h Fq(e)2143 5242 y Fr(j)2180 5227 y Fu(j)2245 5159 y(\024)2391 5091 y Fw(1)p 2360 5136 111 4 v 2360 5227 a Fq(!)2421 5242 y Fr(\013)2601 5091 y Fq(e)2646 5106 y Fr(j)p 2490 5136 305 4 v 2490 5227 a Fu(j)p Fq(e)f Fu(\000)h Fq(e)2730 5242 y Fr(j)2767 5227 y Fu(j)2826 5159 y Fw(+)3013 5091 y(1)p Fq(=)p Fw(2)p 2934 5136 V 2934 5227 a Fu(j)p Fq(e)f Fu(\000)h Fq(e)3174 5242 y Fr(j)3211 5227 y Fu(j)1913 5470 y Fw(6)p eop %%Page: 7 7 7 6 bop -71 123 a Fw(Since)36 b Fq(\033)t Fw(\()p Fq(H)365 138 y Fs(0)404 123 y Fw(\))g(has)g(no)f(accum)m(ulation)f(p)s(oin)m(ts) h(and)h Fq(e)1929 138 y Fr(j)1999 123 y Fu(6)p Fw(=)c Fq(e)k Fw(for)f(all)f Fq(j)39 b Fu(62)33 b Fq(J)2711 138 y Fr(B)2772 123 y Fw(,)j(inf)2953 138 y Fr(j)t Fm(62)p Fr(J)3072 149 y Fl(B)3144 123 y Fu(j)p Fq(e)25 b Fu(\000)f Fq(e)3388 138 y Fr(j)3425 123 y Fu(j)33 b Fq(>)f Fw(0.)53 b(Since)-71 244 y(furthermore)32 b(lim)604 259 y Fm(j)p Fr(j)t Fm(j!1)837 244 y Fq(e)882 259 y Fr(j)919 244 y Fu(j)p Fq(e)22 b Fu(\000)h Fq(e)1159 259 y Fr(j)1195 244 y Fu(j)1223 208 y Fm(\000)p Fs(1)1345 244 y Fw(=)28 b(1,)k(sup)1704 267 y Fr(j)t Fm(62)p Fr(J)1823 278 y Fl(B)1895 244 y Fq(e)1940 259 y Fr(j)1977 244 y Fu(j)p Fq(e)22 b Fu(\000)h Fq(e)2217 259 y Fr(j)2253 244 y Fu(j)2281 208 y Fm(\000)p Fs(1)2403 244 y Fq(<)28 b Fu(1)p Fw(.)43 b(Th)m(us,)437 520 y Fu(j)p Fq(e)22 b Fu(\000)h Fq(e)677 535 y Fr(j)714 520 y Fu(j)742 479 y Fm(\000)p Fs(1)852 520 y Fw(\()p Fq(j)930 535 y Fr(\013)1002 520 y Fw(+)f(1\))28 b Fu(\024)1361 452 y Fw(1)p 1330 497 111 4 v 1330 588 a Fq(!)1391 603 y Fr(\013)1478 520 y Fw(sup)1467 602 y Fr(j)t Fm(62)p Fr(J)1586 613 y Fl(B)1654 520 y Fq(e)1699 535 y Fr(j)1736 520 y Fu(j)p Fq(e)22 b Fu(\000)g Fq(e)1975 535 y Fr(j)2012 520 y Fu(j)2040 479 y Fm(\000)p Fs(1)2156 520 y Fw(+)2264 452 y(1)p 2264 497 49 4 v 2264 588 a(2)2351 520 y(sup)2339 602 y Fr(j)t Fm(62)p Fr(J)2458 613 y Fl(B)2527 520 y Fu(j)p Fq(e)g Fu(\000)g Fq(e)2766 535 y Fr(j)2803 520 y Fu(j)2831 479 y Fm(\000)p Fs(1)2953 520 y Fw(=:)27 b Fq(K)3166 535 y Fs(1)3233 520 y Fq(<)h Fu(1)-71 802 y Fw(whic)m(h)33 b(implies)787 922 y Fu(k)p Fq(a)888 881 y Fm(\003)888 947 y Fr(\013)937 922 y Fq(A)1010 937 y Fr(e)1047 922 y Fu(k)1097 874 y Fs(2)1164 922 y Fu(\024)28 b Fw(max)p Fu(f)p Fw(\(1)22 b(+)g Fq(a)p Fw(\))p Fq(;)17 b(K)1924 937 y Fs(1)1963 922 y Fu(g)27 b(\024)i Fw(max)2166 989 y Fm(f)p Fr(!)2245 999 y Fl(i)2272 989 y Fm(g)2344 922 y Fw(max)o Fu(f)p Fw(\(1)22 b(+)g Fq(a)p Fw(\))p Fq(;)17 b(K)2998 937 y Fs(1)3037 922 y Fu(g)684 b Fw(\(12\))-71 1147 y(A)33 b(similar)c(calculation)h(yields,)806 1367 y Fu(k)p Fq(a)907 1382 y Fr(\013)956 1367 y Fq(A)1029 1382 y Fr(e)1066 1367 y Fu(k)1116 1318 y Fs(2)1183 1367 y Fu(\024)e Fw(max)p Fu(fj)p Fw(1)21 b Fu(\000)i Fq(a)p Fu(j)p Fq(;)17 b(K)1924 1382 y Fs(2)1963 1367 y Fu(g)27 b(\024)i Fw(max)2166 1433 y Fm(f)p Fr(!)2245 1443 y Fl(i)2272 1433 y Fm(g)2344 1367 y Fw(max)o Fu(fj)p Fw(1)22 b Fu(\000)g Fq(a)p Fu(j)p Fq(;)17 b(K)2979 1382 y Fs(2)3018 1367 y Fu(g)703 b Fw(\(13\))-71 1637 y(for)32 b(some)h Fq(K)406 1652 y Fs(2)473 1637 y Fq(<)27 b Fu(1)p Fw(.)43 b(Therefore,)959 1896 y Fu(k)p Fq(x)1064 1911 y Fr(\013)1114 1896 y Fq(A)1187 1911 y Fr(e)1224 1896 y Fu(k)28 b(\024)1510 1828 y Fw(1)p 1417 1873 235 4 v 1417 1893 a Fu(p)p 1500 1893 152 4 v 75 x Fw(2)p Fq(!)1610 1983 y Fr(a)1678 1896 y Fu(k)o Fq(a)1778 1911 y Fr(\013)1828 1896 y Fq(A)1901 1911 y Fr(e)1938 1896 y Fu(k)22 b Fw(+)2211 1828 y(1)p 2118 1873 235 4 v 2118 1893 a Fu(p)p 2201 1893 152 4 v 75 x Fw(2)p Fq(!)2311 1983 y Fr(a)2379 1896 y Fu(k)p Fq(a)2480 1911 y Fr(\013)2529 1896 y Fq(A)2602 1854 y Fm(\003)2602 1920 y Fr(e)2642 1896 y Fu(k)27 b(\024)h Fq(\015)2875 1911 y Fs(1)-71 2171 y Fw(where)34 b Fq(\015)262 2186 y Fs(1)333 2171 y Fw(is)f(the)g(sum)f(of)g(the)h(righ)m(t-hand)f(sides) h(of)f(\(12\))g(and)h(\(13\).)43 b(T)-8 b(aking)32 b(the)h(adjoin)m(t)e (yields)1668 2391 y Fu(k)p Fq(A)1791 2406 y Fr(e)1828 2391 y Fq(x)1883 2406 y Fr(\013)1933 2391 y Fu(k)d(\024)g Fq(\015)2167 2406 y Fs(1)75 2611 y Fw(Since)33 b Fu(j)p Fq(m)p Fu(j)28 b(\025)g Fw(2,)k(w)m(e)i(can)f(write)f Fq(x)1339 2575 y Fr(m)1434 2611 y Fw(=)27 b Fq(x)1592 2626 y Fr(\013)1642 2611 y Fq(x)1697 2575 y Fr(m)1759 2551 y Fi(0)1786 2611 y Fq(x)1841 2626 y Fr(\014)1921 2611 y Fw(for)32 b(some)h Fq(x)2370 2626 y Fr(\013)2420 2611 y Fw(,)f Fq(x)2534 2626 y Fr(\014)2582 2611 y Fw(,)h(with)f Fu(j)p Fq(m)2977 2575 y Fm(0)3000 2611 y Fu(j)27 b Fw(=)h Fu(j)p Fq(m)p Fu(j)22 b(\000)g Fw(2.)44 b(Then)840 2777 y Fv(\015)840 2836 y(\015)895 2861 y Fq(A)968 2876 y Fr(e)1005 2861 y Fq(x)1060 2820 y Fr(m)1127 2861 y Fq(A)1200 2876 y Fr(e)1237 2861 y Fq(P)1300 2877 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)1473 2777 y Fv(\015)1473 2836 y(\015)1612 2861 y Fu(\024)1772 2747 y Fv(\015)1772 2806 y(\015)1772 2866 y(\015)1828 2861 y Fq(A)1901 2876 y Fr(e)1938 2861 y Fq(x)1993 2876 y Fr(\013)2043 2861 y Fq(x)2098 2820 y Fr(m)2160 2797 y Fi(0)2187 2861 y Fq(P)2250 2877 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)p Fs(+1)2513 2861 y Fq(x)2568 2876 y Fr(\014)2616 2861 y Fq(A)2689 2876 y Fr(e)2726 2861 y Fq(P)2789 2877 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)2962 2747 y Fv(\015)2962 2806 y(\015)2962 2866 y(\015)1612 3074 y Fu(\024)83 b(k)p Fq(A)1895 3089 y Fr(e)1932 3074 y Fq(x)1987 3089 y Fr(\013)2037 3074 y Fu(k)16 b(k)p Fq(x)2208 3089 y Fr(\014)2256 3074 y Fq(A)2329 3089 y Fr(e)2366 3074 y Fu(k)2432 2959 y Fv(\015)2432 3019 y(\015)2432 3079 y(\015)2488 3074 y Fq(x)2543 3033 y Fr(m)2605 3009 y Fi(0)2632 3074 y Fq(P)2695 3089 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)p Fs(+1)2958 2959 y Fv(\015)2958 3019 y(\015)2958 3079 y(\015)1612 3349 y Fu(\024)83 b Fq(\015)1828 3308 y Fs(2)1823 3374 y(1)1868 3349 y Fw(2)1927 3275 y Fi(j)p Fl(m)2000 3254 y Fi(0)2022 3275 y(j)p 1926 3293 115 3 v 1969 3334 a Fk(2)2072 3209 y Fv(\024)2134 3282 y Fw(\()p Fq(n)23 b Fw(+)f Fu(j)p Fq(m)2464 3246 y Fm(0)2487 3282 y Fu(j)g Fw(+)g(1\)!)p 2134 3326 615 4 v 2276 3418 a(\()p Fq(n)h Fw(+)f(1\)!)2759 3209 y Fv(\025)2821 3204 y Fk(1)p 2821 3216 31 3 v 2821 3257 a(2)1612 3654 y Fu(\024)83 b Fq(\015)1828 3613 y Fs(2)1823 3679 y(1)1868 3654 y Fw(2)1927 3580 y Fi(j)p Fl(m)p Fi(j\000)p Fk(2)p 1926 3598 171 3 v 1997 3639 a(2)2128 3514 y Fv(\024)2190 3587 y Fw(\()p Fq(n)23 b Fw(+)f Fu(j)p Fq(m)p Fu(j)g(\000)g Fw(1\)!)p 2190 3632 593 4 v 2322 3723 a(\()p Fq(n)g Fw(+)g(1\)!)2793 3514 y Fv(\025)2856 3509 y Fk(1)p 2856 3521 31 3 v 2856 3562 a(2)3771 3654 y Fw(\(14\))-71 3944 y(where)36 b(w)m(e)g(use)f (Lemma)f(5.1)g(of)g([3])h(to)f(b)s(ound)1714 3860 y Fv(\015)1714 3919 y(\015)1769 3944 y Fq(x)1824 3908 y Fr(m)1886 3885 y Fi(0)1914 3944 y Fq(P)1977 3960 y Fm(j)p Fr(j)t Fm(j\024)p Fr(n)p Fs(+1)2240 3860 y Fv(\015)2240 3919 y(\015)2295 3944 y Fw(.)50 b(The)36 b(last)e(statemen)m(t)h(follo)m(ws)e(from)g (\(11\))-71 4065 y(and)g(\(14\),)f(along)f(with)h(the)h(de\014nitions)f Fq(\015)h Fw(=)27 b Fq(C)7 b(\015)1794 4029 y Fs(2)1789 4089 y(1)1834 4065 y Fq(\016)1881 4029 y Fm(\000)p Fs(2)1975 4065 y Fq(Y)2053 4029 y Fs(2)2125 4065 y Fw(and)33 b Fq(\024)28 b Fw(=)f(2)p Fq(\016)2598 4029 y Fm(\000)p Fs(2)2692 4065 y Fq(Y)2771 4029 y Fs(2)2810 4065 y Fw(.)1033 b Fg(2)-71 4397 y Fx(3)161 b(Computation)52 b(of)i(R-S)g(co)t (e\016cien)l(ts)-71 4617 y Fw(W)-8 b(e)37 b(essen)m(tially)e(follo)m(w) f(the)j(standard,)g(formal)d(metho)s(d)h(to)h(compute)g(R-S)g(co)s (e\016cien)m(ts)h(\(see)g(e.)f(g.,)h([10,)-71 4737 y(Chapter)44 b(XVI],\))g(although)e(alternativ)m(ely)g(one)h(could)g(use)h(Kato)e ([9].)75 b(Ho)m(w)m(ev)m(er,)48 b(this)43 b(last)f(approac)m(h)-71 4857 y(seems)g(rather)f(di\016cult)f(to)g(implemen)m(t)f(here,)44 b(in)c(particular)f(when)j(degeneracy)h(o)s(ccurs.)69 b(Concerning)-71 4978 y(asymptotics)39 b(in)f(degenerate)i(p)s (erturbation)e(theory)-8 b(,)41 b(w)m(e)f(m)m(ust)f(men)m(tion)f(the)i (approac)m(h)f(dev)m(elop)s(ed)h(b)m(y)-71 5098 y(Hunzik)m(er-Pillet)32 b([7].)48 b(See)34 b(also)f([8])h(for)f(a)h(more)f(sp)s(eci\014c)i (treatmen)m(t)e(for)h(Sc)m(hr\177)-49 b(odinger)34 b(eigen)m(v)-5 b(alue)33 b(prob-)-71 5218 y(lems.)1913 5470 y(7)p eop %%Page: 8 8 8 7 bop 75 123 a Fw(Let)51 b(us)f(assume)h(that)f(the)g(zeroth-order)h (eigen)m(v)-5 b(alue)49 b Fq(e)h Fw(is)g Fq(g)t Fw(-fold)e(degenerate,) 55 b(with)50 b(asso)s(ciated)-71 244 y(eigenspace)34 b Fq(B)5 b Fw(.)43 b(W)-8 b(e)33 b(allo)m(w)e Fq(g)36 b Fw(to)c(b)s(e)h(equal)f(to)h(1.)43 b(Let)33 b(P)f(b)s(e)h(the)g(pro)5 b(jector)33 b(on)m(to)g Fq(B)k Fw(and)c Fq(Q)28 b Fw(:=)g(1)22 b Fu(\000)g Fq(P)14 b Fw(.)75 364 y(Up)41 b(to)f(zeroth-order,)i Fq( )1022 379 y Fs(0)1102 364 y Fw(can)f(b)s(e)f(an)m(y)h(v)m(ector)h (in)d Fq(B)5 b Fw(,)43 b(whic)m(h)d(w)m(e)i(ma)m(y)e(require)g(to)g(b)s (e)h(normalized,)-71 484 y Fu(k)p Fq( )42 499 y Fs(0)82 484 y Fu(k)j Fw(=)g(1.)72 b(Tw)m(o)43 b(cases)g(ma)m(y)f(arise)g(from)f (solving)g(\(10\))h(at)g(higher)f(order.)73 b(Either)42 b(the)h(zeroth-order)-71 605 y(degeneracy)38 b(is)d(preserv)m(ed)k(at)d (all)d(orders,)38 b(or)e(it)f(is)g(remo)m(v)m(ed)i(to)f(some)g(exten)m (t)h(at)f(higher)f(order.)54 b(Let)36 b(us)-71 725 y(start)d(b)m(y)g (discussing)g(the)g(former)f(case,)h(whic)m(h)g(trivially)d(includes)i (the)h(non-degenerate)h(one.)-71 1014 y Ff(3.1)135 b(Degeneracy)47 b(is)e(preserv)l(ed.)-71 1199 y Fw(Fix)32 b Fq( )167 1214 y Fs(0)234 1199 y Fu(2)d Fq(B)5 b Fw(,)32 b(with)g Fu(k)p Fq( )802 1214 y Fs(0)842 1199 y Fu(k)c Fw(=)f(1.)43 b(The)34 b(\014rst-order)f(equation)f(is)1413 1419 y Fq(H)1494 1434 y Fr(e)1531 1419 y Fq( )1594 1434 y Fs(1)1656 1419 y Fw(+)22 b Fq(T)1825 1378 y Fs(\(3\))1919 1419 y Fq( )1982 1434 y Fs(0)2050 1419 y Fw(=)27 b Fu(E)2206 1434 y Fs(1)2245 1419 y Fq(A)2318 1378 y Fs(2)2318 1443 y Fr(e)2358 1419 y Fq( )2421 1434 y Fs(0)3771 1419 y Fw(\(15\))-71 1639 y(Let)33 b(us)g(m)m(ultiply)d(b)m(y)k Fq(P)14 b Fw(.)43 b(Noting)31 b(that)h Fq(P)14 b(H)1593 1654 y Fr(e)1657 1639 y Fw(=)28 b(0)k(and)h Fq(P)14 b(A)2182 1603 y Fs(2)2182 1663 y Fr(e)2221 1639 y Fq( )2284 1654 y Fs(0)2351 1639 y Fw(=)28 b Fq( )2518 1654 y Fs(0)2558 1639 y Fw(,)k(w)m(e)i(obtain)1564 1859 y Fq(P)14 b(T)1712 1818 y Fs(\(3\))1805 1859 y Fq(P)g( )1945 1874 y Fs(0)2012 1859 y Fw(=)28 b Fu(E)2169 1874 y Fs(1)2208 1859 y Fq( )2271 1874 y Fs(0)-71 2079 y Fw(This)k(is)f(the)h(eigen)m(v)-5 b(alue)31 b(problem)g(for)g(the)h(\014nite-dimensional,)d(self-adjoin)m (t)h(op)s(erator)h(\003)3348 2043 y Fs(\(1\))3469 2079 y Fw(:=)d Fq(P)14 b(T)3748 2043 y Fs(\(3\))3841 2079 y Fq(P)g Fw(.)-71 2199 y(Since)36 b(w)m(e)g(assume)g(that)f(the)h (zeroth-order)f(degeneracy)i(is)e(not)g(brok)m(en)i(at)e(an)m(y)h (order,)g(\003)3379 2163 y Fs(\(1\))3508 2199 y Fw(m)m(ust)f(ha)m(v)m (e)-71 2320 y(only)d(one)h(eigen)m(v)-5 b(alue.)43 b(Let)33 b(us)g(call)e(it)g Fq(\013)1464 2335 y Fs(1)1504 2320 y Fw(.)43 b(Then)34 b Fu(E)1882 2335 y Fs(1)1949 2320 y Fw(=)27 b Fq(\013)2114 2335 y Fs(1)2154 2320 y Fw(.)43 b(No)m(w)33 b(m)m(ultiply)e(\(15\))h(b)m(y)h Fq(Q)p Fw(.)44 b(W)-8 b(e)33 b(obtain)1511 2540 y Fq(H)1592 2555 y Fr(e)1629 2540 y Fq(Q )1769 2555 y Fs(1)1837 2540 y Fw(=)27 b Fu(\000)p Fq(QT)2165 2498 y Fs(\(3\))2260 2540 y Fq( )2323 2555 y Fs(0)-71 2760 y Fw(Let)33 b(us)g(in)m(tro)s(duce)f(more)g(notation.) 42 b(F)-8 b(or)32 b(an)m(y)h(v)m(ector)h Fq( )e Fu(2)c(H)q Fw(,)k(de\014ne)i Fq( )2668 2723 y Fm(k)2735 2760 y Fw(:=)28 b Fq(P)14 b( )36 b Fw(and)c Fq( )3298 2723 y Fm(?)3385 2760 y Fw(:=)c Fq(Q )t Fw(.)43 b(Also,)-71 2880 y(let)32 b(\()p Fq(H)189 2895 y Fr(e)226 2880 y Fw(\))264 2909 y Fm(?)355 2880 y Fw(b)s(e)h(the)g(restriction)f(of)g Fq(H)1315 2895 y Fr(e)1384 2880 y Fw(to)g(Ran\()p Fq(Q)p Fw(\).)43 b(So)33 b(de\014ned,)h(\()p Fq(H)2519 2895 y Fr(e)2556 2880 y Fw(\))2594 2909 y Fm(?)2685 2880 y Fw(is)e(in)m(v)m(ertible.)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)1640 3100 y Fq( )1707 3059 y Fm(?)1703 3125 y Fs(1)1794 3100 y Fw(=)27 b(\004)1962 3059 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2131 3100 y Fq( )2194 3115 y Fs(0)-71 3333 y Fw(where)34 b(\004)276 3297 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))472 3333 y Fw(:=)28 b(\()p Fq(H)722 3348 y Fr(e)759 3333 y Fw(\))797 3285 y Fm(\000)p Fs(1)797 3362 y Fm(?)908 3253 y Fv(\000)953 3333 y Fu(\000)p Fq(QT)1178 3297 y Fs(\(3\))1273 3253 y Fv(\001)1319 3333 y Fw(.)43 b(So)33 b(far)f Fq( )1741 3282 y Fm(k)1737 3358 y Fs(1)1813 3333 y Fw(remains)f(unde\014ned.)75 3525 y(The)j(second-order)g(equation)e(is)1001 3745 y Fq(H)1082 3760 y Fr(e)1119 3745 y Fq( )1182 3760 y Fs(2)1244 3745 y Fw(+)22 b Fq(T)1413 3703 y Fs(\(3\))1507 3745 y Fq( )1570 3760 y Fs(1)1632 3745 y Fw(+)g Fq(T)1801 3703 y Fs(\(4\))1895 3745 y Fq( )1958 3760 y Fs(0)2025 3745 y Fw(=)28 b Fu(E)2182 3760 y Fs(2)2221 3745 y Fq(A)2294 3703 y Fs(2)2294 3769 y Fr(e)2333 3745 y Fq( )2396 3760 y Fs(0)2458 3745 y Fw(+)22 b Fq(\013)2618 3760 y Fs(1)2658 3745 y Fq(A)2731 3703 y Fs(2)2731 3769 y Fr(e)2770 3745 y Fq( )2833 3760 y Fs(1)3771 3745 y Fw(\(16\))-71 3965 y(Multiply)31 b(\(16\))h(b)m(y)i Fq(P)14 b Fw(.)42 b(After)33 b(some)g(algebra)e(in)m(v)m(olving)g(the)i(de\014nitions)f(of)g(\003) 2895 3928 y Fs(\(1\))3022 3965 y Fw(and)h(\004)3277 3928 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3445 3965 y Fw(,)g(w)m(e)h(obtain)1174 4104 y Fv(\000)1219 4185 y Fq(P)14 b(T)1367 4143 y Fs(\(3\))1461 4185 y Fw(\004)1526 4143 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1695 4185 y Fq(P)35 b Fw(+)22 b Fq(P)14 b(T)2039 4143 y Fs(\(4\))2133 4185 y Fq(P)2210 4104 y Fv(\001)2271 4185 y Fq( )2334 4200 y Fs(0)2402 4185 y Fw(=)27 b Fu(E)2558 4200 y Fs(2)2597 4185 y Fq( )2660 4200 y Fs(0)-71 4413 y Fw(Then)38 b Fu(E)241 4428 y Fs(2)317 4413 y Fw(has)g(to)f(b)s(e)g(equal)g(to)g(the) g(unique)h(eigen)m(v)-5 b(alue)36 b(of)h(\003)2291 4377 y Fs(\(2\))2420 4413 y Fw(:=)e Fq(P)2651 4333 y Fv(\000)2697 4413 y Fq(T)2768 4377 y Fs(\(3\))2862 4413 y Fw(\004)2927 4377 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3118 4413 y Fw(+)22 b Fq(T)3287 4377 y Fs(\(4\))3381 4333 y Fv(\001)3443 4413 y Fq(P)14 b Fw(.)57 b(That)37 b(is,)-71 4534 y Fu(E)-18 4549 y Fs(2)49 4534 y Fw(=)27 b Fq(\013)214 4549 y Fs(2)254 4534 y Fw(.)43 b(No)m(w)33 b(m)m(ultiply)e(\(16\))h(b)m(y)h Fq(Q)g Fw(to)f(obtain)921 4781 y Fq(H)1002 4796 y Fr(e)1039 4781 y Fq( )1106 4740 y Fm(?)1102 4806 y Fs(2)1187 4781 y Fw(+)22 b Fq(QT)1433 4740 y Fs(\(3\))1544 4671 y Fv(\020)1604 4781 y Fq( )1671 4730 y Fm(k)1667 4806 y Fs(1)1732 4781 y Fw(+)g Fq( )1897 4740 y Fm(?)1893 4806 y Fs(1)1956 4671 y Fv(\021)2038 4781 y Fw(+)g Fq(QT)2284 4740 y Fs(\(4\))2378 4781 y Fq( )2441 4796 y Fs(0)2509 4781 y Fw(=)27 b Fq(\013)2674 4796 y Fs(1)2714 4781 y Fq(A)2787 4740 y Fs(2)2787 4806 y Fr(e)2826 4781 y Fq( )2893 4740 y Fm(?)2889 4806 y Fs(1)-71 5023 y Fw(whic)m(h)33 b(yields)1410 5143 y Fq( )1477 5102 y Fm(?)1473 5168 y Fs(2)1564 5143 y Fw(=)27 b(\004)1732 5102 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1901 5143 y Fq( )1964 5158 y Fs(0)2026 5143 y Fw(+)22 b(\004)2189 5102 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2358 5143 y Fq( )2425 5092 y Fm(k)2421 5168 y Fs(1)1913 5470 y Fw(8)p eop %%Page: 9 9 9 8 bop -71 123 a Fw(where)34 b(w)m(e)g(de\014ne)906 244 y(\004)971 203 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1168 244 y Fw(:=)27 b(\()p Fq(H)1417 259 y Fr(e)1454 244 y Fw(\))1492 202 y Fm(\000)p Fs(1)1492 271 y Fm(?)1603 163 y Fv(\002)o(\000)1690 244 y Fq(\013)1752 259 y Fs(1)1791 244 y Fq(A)1864 203 y Fs(2)1864 268 y Fr(e)1926 244 y Fu(\000)c Fq(QT)2174 203 y Fs(\(3\))2268 163 y Fv(\001)2330 244 y Fw(\004)2395 203 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2586 244 y Fw(+)f Fq(QT)2832 203 y Fs(\(4\))2927 163 y Fv(\003)-71 442 y Fw(and)33 b(no)f(requiremen)m(t)h(is)f(imp)s(osed)g(on)g(either)h Fq( )1756 391 y Fm(k)1752 466 y Fs(2)1828 442 y Fw(or)f Fq( )2014 391 y Fm(k)2010 466 y Fs(1)2053 442 y Fw(.)75 633 y(The)i(third-order)e(equation)g(is)588 853 y Fq(H)669 868 y Fr(e)706 853 y Fq( )769 868 y Fs(3)831 853 y Fw(+)22 b Fq(T)1000 812 y Fs(\(3\))1094 853 y Fq( )1157 868 y Fs(2)1219 853 y Fw(+)g Fq(T)1388 812 y Fs(\(4\))1482 853 y Fq( )1545 868 y Fs(1)1607 853 y Fw(+)g Fq(T)1776 812 y Fs(\(5\))1870 853 y Fq( )1933 868 y Fs(0)2001 853 y Fw(=)27 b Fu(E)2157 868 y Fs(3)2196 853 y Fq(A)2269 812 y Fs(2)2269 877 y Fr(e)2309 853 y Fq( )2372 868 y Fs(0)2434 853 y Fw(+)22 b Fq(\013)2594 868 y Fs(2)2633 853 y Fq(A)2706 812 y Fs(2)2706 877 y Fr(e)2746 853 y Fq( )2809 868 y Fs(1)2871 853 y Fw(+)g Fq(\013)3031 868 y Fs(1)3070 853 y Fq(A)3143 812 y Fs(2)3143 877 y Fr(e)3183 853 y Fq( )3246 868 y Fs(2)-71 1073 y Fw(F)-8 b(ollo)m(wing)30 b(the)j(pro)s(cedure)g(already)f(describ)s(ed,)i(w)m(e)f(obtain)1642 1293 y(\003)1710 1252 y Fs(\(3\))1804 1293 y Fq( )1867 1308 y Fs(0)1934 1293 y Fw(=)28 b Fu(E)2091 1308 y Fs(3)2130 1293 y Fq( )2193 1308 y Fs(0)-71 1513 y Fw(where)1046 1633 y(\003)1114 1592 y Fs(\(3\))1236 1633 y Fw(:=)g Fq(P)1460 1553 y Fv(\000)1505 1633 y Fq(T)1576 1592 y Fs(\(3\))1670 1633 y Fw(\004)1735 1592 y Fs(\(2)p Fm(?)p Fs(\))1907 1633 y Fw(+)22 b Fq(T)2076 1592 y Fs(\(4\))2170 1633 y Fw(\004)2235 1592 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2426 1633 y Fw(+)g Fq(T)2595 1592 y Fs(\(5\))2689 1553 y Fv(\001)2751 1633 y Fq(P)-71 1808 y Fw(has)33 b(only)f(one)h(eigen)m(v)-5 b(alue)32 b Fq(\013)1024 1823 y Fs(3)1064 1808 y Fw(.)43 b(Th)m(us)34 b Fu(E)1434 1823 y Fs(3)1501 1808 y Fw(=)27 b Fq(\013)1666 1823 y Fs(3)1706 1808 y Fw(.)43 b(Also)1180 2028 y Fq( )1247 1986 y Fm(?)1243 2052 y Fs(3)1333 2028 y Fw(=)28 b(\004)1502 1986 y Fs(\(3)p Fr(;)p Fm(?)p Fs(\))1671 2028 y Fq( )1734 2043 y Fs(0)1796 2028 y Fw(+)22 b(\004)1959 1986 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2128 2028 y Fq( )2195 1977 y Fm(k)2191 2052 y Fs(1)2256 2028 y Fw(+)g(\004)2419 1986 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2588 2028 y Fq( )2655 1977 y Fm(k)2651 2052 y Fs(2)-71 2248 y Fw(where)385 2368 y(\004)450 2327 y Fs(\(3)p Fr(;)p Fm(?)p Fs(\))647 2368 y Fw(:=)27 b(\()p Fq(H)896 2383 y Fr(e)933 2368 y Fw(\))971 2327 y Fm(\000)p Fs(1)971 2396 y Fm(?)1082 2287 y Fv(\002)o(\000)1169 2368 y Fq(\013)1231 2383 y Fs(1)1270 2368 y Fq(A)1343 2327 y Fs(2)1343 2393 y Fr(e)1405 2368 y Fu(\000)c Fq(QT)1653 2327 y Fs(\(3\))1747 2287 y Fv(\001)1809 2368 y Fw(\004)1874 2327 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2065 2368 y Fw(+)2163 2287 y Fv(\000)2209 2368 y Fq(\013)2271 2383 y Fs(2)2311 2368 y Fq(A)2384 2327 y Fs(2)2384 2393 y Fr(e)2445 2368 y Fu(\000)g Fq(QT)2693 2327 y Fs(\(4\))2787 2287 y Fv(\001)2849 2368 y Fw(\004)2914 2327 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3105 2368 y Fu(\000)g Fq(QT)3353 2327 y Fs(\(5\))3447 2287 y Fv(\003)-71 2566 y Fw(and)33 b(nothing)e(is)i(said)f(ab)s(out)g Fq( )1122 2515 y Fm(k)1118 2590 y Fs(3)1161 2566 y Fw(,)h Fq( )1288 2515 y Fm(k)1284 2590 y Fs(2)1360 2566 y Fw(or)f Fq( )1546 2515 y Fm(k)1542 2590 y Fs(1)1585 2566 y Fw(.)75 2757 y(As)24 b(one)f(can)h(see,)i Fu(E)779 2772 y Fr(n)848 2757 y Fw(and)e Fq( )1096 2721 y Fm(?)1092 2782 y Fr(n)1178 2757 y Fw(can)f(b)s(e)g(calculated)f(through)h(recursiv)m(e)i (de\014nition)d(of)g(certain)h(op)s(erators.)-71 2878 y(The)34 b(form)d(of)h(these)i(op)s(erators)e(is)g(no)m(w)i(easy)f(to)f (guess:)-71 3106 y Fn(Prop)s(osition)k(1)48 b Fh(F)-7 b(or)34 b Fq(n)28 b Fw(=)g(1)p Fq(;)17 b Fw(2)p Fq(;)g(:)g(:)g(:)n Fh(,)35 b(r)-5 b(e)g(cursively)35 b(de\014ne)565 3326 y Fw(\004)630 3285 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))882 3326 y Fw(:=)83 b Fu(\000)p Fw(\()p Fq(H)1264 3341 y Fr(e)1301 3326 y Fw(\))1339 3285 y Fm(\000)p Fs(1)1339 3354 y Fm(?)1433 3326 y Fq(QT)1581 3285 y Fs(\(3\))557 3588 y Fw(\004)622 3547 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))882 3588 y Fw(:=)g(\()p Fq(H)1187 3603 y Fr(e)1223 3588 y Fw(\))1261 3546 y Fm(\000)p Fs(1)1261 3615 y Fm(?)1372 3417 y Fv(")1436 3463 y Fr(n)p Fm(\000)p Fs(1)1430 3493 y Fv(X)1439 3702 y Fr(p)p Fs(=1)1591 3507 y Fv(\000)1636 3588 y Fq(\013)1698 3603 y Fr(n)p Fm(\000)p Fr(p)1836 3588 y Fq(A)1909 3547 y Fs(2)1909 3612 y Fr(e)1971 3588 y Fu(\000)22 b Fq(QT)2218 3547 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(p)p Fs(\))2501 3507 y Fv(\001)2563 3588 y Fw(\004)2628 3547 y Fs(\()p Fr(p;)p Fm(?)p Fs(\))2819 3588 y Fu(\000)h Fq(QT)3067 3547 y Fs(\()p Fr(n)p Fs(+2\))3259 3417 y Fv(#)-71 3902 y Fh(wher)-5 b(e)34 b Fq(\013)266 3917 y Fr(l)327 3902 y Fh(is,)h(by)g(assumption,)f(the)h(unique)f (eigenvalue)g(of)1086 4207 y Fw(\003)1154 4166 y Fs(\()p Fr(l)q Fs(\))1263 4207 y Fw(:=)27 b Fq(P)14 b(T)1541 4166 y Fs(\()p Fr(l)q Fs(+2\))1711 4207 y Fq(P)36 b Fw(+)1913 4082 y Fr(n)p Fm(\000)p Fs(1)1908 4112 y Fv(X)1917 4321 y Fr(p)p Fs(=1)2069 4207 y Fq(P)14 b(T)2217 4166 y Fs(\()p Fr(l)q Fs(+2)p Fm(\000)p Fr(p)p Fs(\))2477 4207 y Fw(\004)2542 4166 y Fs(\()p Fr(p;)p Fm(?)p Fs(\))2711 4207 y Fq(P)-71 4521 y Fh(Then,)34 b(given)g Fq( )530 4536 y Fs(0)598 4521 y Fu(2)28 b Fq(B)5 b Fh(,)35 b Fu(E)889 4536 y Fr(n)963 4521 y Fw(=)27 b Fq(\013)1128 4536 y Fr(n)1210 4521 y Fh(and)1169 4826 y Fq( )1232 4841 y Fr(n)1307 4826 y Fw(=)h(\004)1476 4785 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))1652 4826 y Fq( )1715 4841 y Fs(0)1777 4826 y Fw(+)1881 4701 y Fr(n)p Fm(\000)p Fs(1)1875 4731 y Fv(X)1884 4941 y Fr(p)p Fs(=1)2036 4826 y Fw(\004)2101 4785 y Fs(\()p Fr(n)p Fm(\000)p Fr(p;)p Fm(?)p Fs(\))2368 4826 y Fq( )2435 4785 y Fm(k)2431 4851 y Fr(p)2496 4826 y Fw(+)22 b Fq( )2661 4785 y Fm(k)2657 4851 y Fr(n)-71 5172 y Fh(wher)-5 b(e)34 b Fq( )271 5121 y Fm(k)267 5197 y Fs(1)311 5172 y Fq(;)17 b(:)g(:)g(:)f(;)h( )597 5121 y Fm(k)593 5182 y Fr(n)675 5172 y Fh(ar)-5 b(e)34 b(ve)-5 b(ctors)35 b(arbitr)-5 b(arily)35 b(chosen)f(fr)-5 b(om)34 b Fq(B)5 b Fh(.)1913 5470 y Fw(9)p eop %%Page: 10 10 10 9 bop 75 123 a Fw(This)26 b(construction)g(will)d(b)s(e)j (generalized)f(in)g(Prop)s(osition)e(2,)k(from)d(whic)m(h)j(the)f(pro)s (of)e(of)h(Prop)s(osition)f(1)-71 244 y(can)i(b)s(e)g(easily)f(read)h (out.)41 b(T)-8 b(o)26 b(rule)g(out)f(arbitrariness,)i(w)m(e)g(set)f Fq( )2336 193 y Fm(k)2332 254 y Fr(n)2407 244 y Fw(=)i(0)d(for)h(all)d Fq(n)28 b Fu(\025)g Fw(1,)f(whic)m(h)g(is)e(equiv)-5 b(alen)m(t)-71 364 y(to)32 b(absorbing)g(those)i(v)m(ectors)g(in)m(to)d Fq( )1340 379 y Fs(0)1380 364 y Fw(.)75 484 y(The)j(recursiv)m(e)f (expressions)h(for)e(the)g(op)s(erators)g(\003)2005 448 y Fs(\()p Fr(n)p Fs(\))2139 484 y Fw(and)g(\004)2393 448 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))2602 484 y Fw(can)g(b)s(e)g (translated)g(in)m(to)f(recursiv)m(e)-71 605 y(expressions)j(for)f Fu(E)640 620 y Fr(n)719 605 y Fw(and)f Fq( )971 620 y Fr(n)1019 605 y Fw(.)43 b(The)34 b(result)e(is)666 909 y Fu(E)719 924 y Fr(n)848 909 y Fw(=)1013 785 y Fr(n)p Fm(\000)p Fs(1)1007 815 y Fv(X)1016 1024 y Fr(p)p Fs(=0)1168 829 y Fv(\012)1215 909 y Fq(T)1286 868 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(p)p Fs(\))1568 909 y Fq(P)1631 925 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1799 909 y Fq( )1862 924 y Fs(0)1902 909 y Fq(;)17 b( )2009 924 y Fr(p)2049 829 y Fv(\013)655 1257 y Fq( )718 1272 y Fr(n)848 1257 y Fw(=)83 b(\()p Fq(H)1126 1272 y Fr(e)1163 1257 y Fw(\))1201 1216 y Fm(\000)p Fs(1)1201 1285 y Fm(?)1312 1087 y Fv(")1375 1133 y Fr(n)p Fm(\000)p Fs(1)1370 1163 y Fv(X)1379 1372 y Fr(p)p Fs(=1)1530 1177 y Fv(\000)1576 1257 y Fu(E)1629 1272 y Fr(n)p Fm(\000)p Fr(p)1766 1257 y Fq(A)1839 1216 y Fs(2)1839 1282 y Fr(e)1901 1257 y Fu(\000)22 b Fq(QT)2148 1216 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(p)p Fs(\))2431 1177 y Fv(\001)2493 1257 y Fq( )2556 1272 y Fr(p)2618 1257 y Fu(\000)h Fq(QT)2866 1216 y Fs(\()p Fr(n)p Fs(+2\))3058 1257 y Fq( )3121 1272 y Fs(0)3161 1087 y Fv(#)-71 1572 y Fw(F)-8 b(urthermore)24 b(w)m(e)h(can)g(obtain)e(the)i(follo)m(wing)d (inequalities,)i(whose)h(deriv)-5 b(ation)23 b(is)h(presen)m(ted)j (after)d(Prop)s(o-)-71 1692 y(sition)31 b(2:)706 1963 y Fu(kE)809 1978 y Fr(n)855 1963 y Fu(k)83 b(\024)1199 1839 y Fr(n)1149 1869 y Fv(X)1165 2081 y Fr(l)q Fs(=1)1309 1879 y Fv(\015)1309 1938 y(\015)1365 1963 y Fq(T)1436 1922 y Fs(\()p Fr(l)q Fs(+2\))1606 1963 y Fq(P)1669 1979 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1837 1879 y Fv(\015)1837 1938 y(\015)1909 1963 y Fu(k)p Fq( )2022 1978 y Fr(n)p Fm(\000)p Fr(l)2146 1963 y Fu(k)695 2301 y(k)p Fq( )808 2316 y Fr(n)855 2301 y Fu(k)g(\024)1154 2177 y Fr(n)p Fm(\000)p Fs(1)1149 2207 y Fv(X)1165 2419 y Fr(l)q Fs(=1)1309 2301 y Fu(jE)1390 2316 y Fr(l)1416 2301 y Fu(j)16 b(k)p Fq( )1573 2316 y Fr(n)p Fm(\000)p Fr(l)1697 2301 y Fu(k)22 b Fw(+)1917 2177 y Fr(n)1867 2207 y Fv(X)1883 2419 y Fr(l)q Fs(=1)2027 2217 y Fv(\015)2027 2276 y(\015)2083 2301 y Fq(T)2154 2260 y Fs(\()p Fr(l)q Fs(+2\))2324 2301 y Fq(P)2387 2317 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fr(l)q Fs(\))2820 2217 y Fv(\015)2820 2276 y(\015)2892 2301 y Fu(k)p Fq( )3005 2316 y Fr(n)p Fm(\000)p Fr(l)3129 2301 y Fu(k)-71 2606 y Fw(By)33 b(resorting)f(to)g (Lemma)g(2,)g(w)m(e)i(\014nally)d(obtain)564 2911 y Fu(kE)667 2926 y Fr(n)714 2911 y Fu(k)83 b(\024)g Fq(\015)1130 2786 y Fr(n)1080 2816 y Fv(X)1096 3028 y Fr(l)q Fs(=1)1240 2911 y Fq(\024)1311 2842 y Fl(l)p 1306 2854 31 3 v 1306 2896 a Fk(2)1368 2770 y Fv(\024)1430 2843 y Fw(\(1)22 b(+)g Fq(a)g Fw(+)g Fq(l)r Fw(\)!)p 1430 2888 475 4 v 1506 2979 a(\(1)g(+)g Fq(a)p Fw(\)!)1915 2770 y Fv(\025)1977 2765 y Fk(1)p 1977 2777 31 3 v 1977 2819 a(2)2038 2911 y Fu(k)p Fq( )2151 2926 y Fr(n)p Fm(\000)p Fr(l)2275 2911 y Fu(k)554 3249 y(k)o Fq( )666 3264 y Fr(n)714 3249 y Fu(k)83 b(\024)1013 3124 y Fr(n)p Fm(\000)p Fs(1)1007 3154 y Fv(X)1023 3366 y Fr(l)q Fs(=1)1168 3249 y Fu(j)o(E)1248 3264 y Fr(l)1274 3249 y Fu(j)16 b(k)p Fq( )1431 3264 y Fr(n)p Fm(\000)p Fr(l)1555 3249 y Fu(k)22 b Fw(+)g Fq(\015)1848 3124 y Fr(n)1798 3154 y Fv(X)1814 3366 y Fr(l)q Fs(=1)1958 3249 y Fq(\024)2029 3181 y Fl(l)p 2024 3193 V 2024 3234 a Fk(2)2085 3108 y Fv(\024)2148 3181 y Fw(\(1)g(+)g Fq(a)g Fw(+)g(3)p Fq(n)g Fu(\000)h Fw(2)p Fq(l)r Fw(\)!)p 2148 3226 753 4 v 2148 3317 a(\(1)f(+)g Fq(a)g Fw(+)g(3)p Fq(n)g Fu(\000)h Fw(3)p Fq(l)r Fw(\)!)2910 3108 y Fv(\025)2973 3104 y Fk(1)p 2973 3116 31 3 v 2973 3157 a(2)3034 3249 y Fu(k)p Fq( )3147 3264 y Fr(n)p Fm(\000)p Fr(l)3271 3249 y Fu(k)-71 3559 y Fw(These)h(inequalities)d(are)i(essen) m(tially)f(iden)m(tical)f(to)h(\(14\))g(and)h(\(15\))f(in)f([15].)40 b(As)23 b(an)g(inmediate)e(consequence,)-71 3679 y(w)m(e)34 b(ha)m(v)m(e)-71 3908 y Fn(Theorem)j(1)49 b Fh(F)-7 b(or)34 b(e)-5 b(ach)34 b Fq(a)28 b Fu(\025)g Fw(0)p Fh(,)35 b(ther)-5 b(e)35 b(is)f Fq(b)28 b(>)g Fw(0)34 b Fh(so)h(that)1303 4128 y Fu(k)o(E)1405 4143 y Fr(n)1452 4128 y Fu(k)83 b(\024)g Fq(\024)1801 4087 y Fs(3)p Fr(n)1884 4128 y Fq(b)1925 4087 y Fr(n)1972 4128 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2537 4059 y Fk(1)p 2538 4071 V 2538 4113 a(2)1292 4286 y Fu(k)p Fq( )1405 4301 y Fr(n)1452 4286 y Fu(k)83 b(\024)g Fq(\024)1801 4245 y Fs(3)p Fr(n)1884 4286 y Fq(b)1925 4245 y Fr(n)1972 4286 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2537 4218 y Fk(1)p 2538 4230 V 2538 4271 a(2)-71 4506 y Fh(for)35 b(al)5 b(l)34 b Fq(n)28 b Fu(\025)g Fw(1)p Fh(.)75 4734 y Fw(This)34 b(theorem)f(is)g(sho)m(wn)h(in)e([15].)46 b(Alternativ)m(ely)-8 b(,)32 b(one)h(can)h(use)g(the)g(pro)s(of)e(of)h(Theorem)g(2)g(b)s(elo) m(w)g(to)-71 4855 y(get)g(somewhat)f(tigh)m(ter)h(b)s(ounds.)1888 5470 y(10)p eop %%Page: 11 11 11 10 bop -71 123 a Ff(3.2)135 b(Degeneracy)47 b(is)e(remo)l(v)l(ed.) -71 308 y Fw(Let)38 b(us)h(examine)e(the)i(case)f(where)i(the)e (zeroth-order)g(degeneracy)i(is)d(partially)e(remo)m(v)m(ed)k(only)f (at)f(\014rst)-71 428 y(order.)-71 620 y Fh(First-or)-5 b(der:)65 b Fw(No)m(w)45 b(the)f(op)s(erator)g(\003)1374 584 y Fs(\(1\))1515 620 y Fw(=)j Fq(P)14 b(T)1786 584 y Fs(\(3\))1879 620 y Fq(P)58 b Fw(has)44 b Fq(k)50 b Fu(\025)e Fw(2)43 b(distinct)g(eigen)m(v)-5 b(alues)44 b Fq(\013)3445 635 y Fs(1)p Fr(;)p Fs(1)3540 620 y Fq(;)17 b(:)g(:)g(:)e(;)i(\013)3820 635 y Fs(1)p Fr(;k)3918 620 y Fw(.)-71 740 y(Let)46 b Fq(B)191 755 y Fs(1)230 740 y Fq(;)17 b(:)g(:)g(:)f(;)h(B)523 755 y Fr(k)610 740 y Fw(b)s(e)46 b(the)f(corresp)s(onding)h(eigenspaces,)j(and)d(let)e Fq(P)2567 704 y Fs(\(1\))2661 740 y Fq(;)17 b(:)g(:)g(:)f(;)h(P)2957 704 y Fs(\()p Fr(k)r Fs(\))3099 740 y Fw(b)s(e)45 b(their)g(orthogonal) -71 860 y(pro)5 b(jections.)59 b(Set)39 b Fu(E)711 875 y Fs(1)786 860 y Fw(=)d Fq(\013)960 875 y Fs(1)p Fr(;i)1044 860 y Fw(.)59 b(Then)38 b Fq( )1452 875 y Fs(0)1530 860 y Fw(m)m(ust)g(lie)e(in)h Fq(B)2105 875 y Fr(i)2133 860 y Fw(.)59 b(As)39 b(b)s(efore,)g Fq( )2762 824 y Fm(?)2758 885 y Fs(1)2857 860 y Fw(=)e(\004)3035 824 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3204 860 y Fq( )3267 875 y Fs(0)3344 860 y Fw(with)h(\004)3637 824 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3842 860 y Fw(:=)-71 981 y(\()p Fq(H)48 996 y Fr(e)85 981 y Fw(\))123 932 y Fm(\000)p Fs(1)123 1010 y Fm(?)234 900 y Fv(\000)279 981 y Fu(\000)p Fq(QT)504 945 y Fs(\(3\))599 900 y Fv(\001)645 981 y Fw(.)-71 1172 y Fh(Se)-5 b(c)g(ond-or)g(der:)42 b Fw(Because)34 b(of)e(the)h(c)m(hoice)g(for)f Fu(E)1692 1187 y Fs(1)1764 1172 y Fw(w)m(e)h(ha)m(v)m(e)979 1392 y Fq(H)1060 1407 y Fr(e)1097 1392 y Fq( )1160 1407 y Fs(2)1222 1392 y Fw(+)22 b Fq(T)1391 1351 y Fs(\(3\))1485 1392 y Fq( )1548 1407 y Fs(1)1610 1392 y Fw(+)g Fq(T)1779 1351 y Fs(\(4\))1873 1392 y Fq( )1936 1407 y Fs(0)2003 1392 y Fw(=)28 b Fu(E)2160 1407 y Fs(2)2199 1392 y Fq(A)2272 1351 y Fs(2)2272 1417 y Fr(e)2311 1392 y Fq( )2374 1407 y Fs(0)2436 1392 y Fw(+)22 b Fq(\013)2596 1407 y Fs(1)p Fr(;i)2680 1392 y Fq(A)2753 1351 y Fs(2)2753 1417 y Fr(e)2792 1392 y Fq( )2855 1407 y Fs(1)3771 1392 y Fw(\(17\))-71 1612 y(Multiply)31 b(\(17\))h(b)m(y)i Fq(P)749 1576 y Fs(\()p Fr(j)t Fs(\))926 1832 y Fq(P)1003 1791 y Fs(\()p Fr(j)t Fs(\))1094 1832 y Fq(T)1165 1791 y Fs(\(3\))1259 1832 y Fq( )1322 1847 y Fs(1)1384 1832 y Fw(+)22 b Fq(P)1559 1791 y Fs(\()p Fr(j)t Fs(\))1650 1832 y Fq(T)1721 1791 y Fs(\(4\))1815 1832 y Fq( )1878 1847 y Fs(0)1946 1832 y Fw(=)27 b Fu(E)2102 1847 y Fs(2)2141 1832 y Fq(P)2218 1791 y Fs(\()p Fr(j)t Fs(\))2309 1832 y Fq( )2372 1847 y Fs(0)2434 1832 y Fw(+)22 b Fq(\013)2594 1847 y Fs(1)p Fr(;i)2677 1832 y Fq(P)2754 1791 y Fs(\()p Fr(j)t Fs(\))2845 1832 y Fq( )2908 1847 y Fs(1)3771 1832 y Fw(\(18\))-71 2052 y(Note)33 b(that)f(for)g(an)m(y)h(v)m(ector)h Fq( )t Fw(,)1141 2360 y Fq(P)1218 2319 y Fs(\()p Fr(j)t Fs(\))1309 2360 y Fq(T)1380 2319 y Fs(\(3\))1474 2360 y Fq( )1541 2319 y Fm(k)1664 2360 y Fw(=)1875 2236 y Fr(k)1823 2265 y Fv(X)1839 2478 y Fr(l)q Fs(=1)1983 2360 y Fq(P)2060 2319 y Fs(\()p Fr(j)t Fs(\))2151 2360 y Fq(P)14 b(T)2299 2319 y Fs(\(3\))2392 2360 y Fq(P)g(P)2546 2319 y Fs(\()p Fr(l)q Fs(\))2626 2360 y Fq( )2693 2319 y Fm(k)1664 2701 y Fw(=)1875 2577 y Fr(k)1823 2607 y Fv(X)1839 2819 y Fr(l)q Fs(=1)1983 2701 y Fq(P)2060 2660 y Fs(\()p Fr(j)t Fs(\))2151 2701 y Fw(\003)2219 2660 y Fs(\(1\))2313 2701 y Fq(P)2390 2660 y Fs(\()p Fr(l)q Fs(\))2470 2701 y Fq( )2537 2660 y Fm(k)1664 3043 y Fw(=)1875 2918 y Fr(k)1823 2948 y Fv(X)1839 3160 y Fr(l)q Fs(=1)1983 3043 y Fq(\013)2045 3058 y Fs(1)p Fr(;l)2126 3043 y Fq(P)2203 3002 y Fs(\()p Fr(j)t Fs(\))2294 3043 y Fq(P)2371 3002 y Fs(\()p Fr(l)q Fs(\))2451 3043 y Fq( )2518 3002 y Fm(k)1664 3294 y Fw(=)83 b Fq(\013)1885 3309 y Fs(1)p Fr(;j)1976 3294 y Fq( )2043 3253 y Fs(\()p Fr(j)t Fs(\))3771 3294 y Fw(\(19\))-71 3525 y(where)35 b Fq( )279 3489 y Fm(k)349 3525 y Fw(=)29 b Fq(P)14 b( )38 b Fw(as)c(b)s(efore.)47 b(De\014ne)35 b Fq( )1458 3489 y Fs(\()p Fr(j)t Fs(\))1579 3525 y Fw(=)30 b Fq(P)1762 3489 y Fs(\()p Fr(j)t Fs(\))1853 3525 y Fq( )t Fw(.)47 b(Then)35 b Fq( )2317 3489 y Fm(k)2387 3525 y Fw(=)2492 3450 y Fv(P)2598 3476 y Fr(k)2598 3554 y(j)t Fs(=1)2741 3525 y Fq( )2808 3489 y Fs(\()p Fr(j)t Fs(\))2899 3525 y Fw(.)48 b(With)33 b(these)i(de\014nitions,)-71 3645 y(\(19\))d(yields)1089 3879 y Fq(P)1166 3837 y Fs(\()p Fr(j)t Fs(\))1257 3879 y Fq(T)1328 3837 y Fs(\(3\))1422 3879 y Fq( )1485 3894 y Fs(1)1608 3879 y Fw(=)83 b Fq(P)1844 3837 y Fs(\()p Fr(j)t Fs(\))1934 3879 y Fq(T)2005 3837 y Fs(\(3\))2100 3879 y Fq( )2167 3828 y Fm(k)2163 3903 y Fs(1)2228 3879 y Fw(+)22 b Fq(P)2403 3837 y Fs(\()p Fr(j)t Fs(\))2494 3879 y Fq(T)2565 3837 y Fs(\(3\))2659 3879 y Fq( )2726 3837 y Fm(?)2722 3903 y Fs(1)1608 4037 y Fw(=)83 b Fq(\013)1829 4052 y Fs(1)p Fr(;j)1920 4037 y Fq( )1987 3986 y Fs(\()p Fr(j)t Fs(\))1983 4062 y(1)2101 4037 y Fw(+)22 b Fq(P)2276 3996 y Fs(\()p Fr(j)t Fs(\))2366 4037 y Fq(T)2437 3996 y Fs(\(3\))2532 4037 y Fq( )2599 3996 y Fm(?)2595 4062 y Fs(1)3771 4037 y Fw(\(20\))-71 4257 y(No)m(w)33 b(insert)g(\(20\))f(in)m(to)g(\(18\).)42 b(F)-8 b(or)32 b Fq(j)i Fw(=)27 b Fq(i)33 b Fw(w)m(e)h(ha)m(v)m(e)1275 4477 y Fq(P)1352 4436 y Fs(\()p Fr(i)p Fs(\))1434 4477 y Fq(T)1505 4436 y Fs(\(4\))1599 4477 y Fq( )1662 4492 y Fs(0)1724 4477 y Fw(+)22 b Fq(P)1899 4436 y Fs(\()p Fr(i)p Fs(\))1982 4477 y Fq(T)2053 4436 y Fs(\(3\))2147 4477 y Fq( )2214 4436 y Fm(?)2210 4502 y Fs(1)2301 4477 y Fw(=)27 b Fu(E)2457 4492 y Fs(2)2496 4477 y Fq( )2559 4492 y Fs(0)-71 4697 y Fw(De\014ne)1191 4818 y(\003)1259 4776 y Fs(\(2)p Fr(;i)p Fs(\))1425 4818 y Fw(:=)g Fq(P)1632 4776 y Fs(\()p Fr(i)p Fs(\))1731 4737 y Fv(\000)1777 4818 y Fq(T)1848 4776 y Fs(\(4\))1964 4818 y Fw(+)22 b Fq(T)2133 4776 y Fs(\(3\))2227 4818 y Fw(\004)2292 4776 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2461 4737 y Fv(\001)2523 4818 y Fq(P)2600 4776 y Fs(\()p Fr(i)p Fs(\))-71 5001 y Fw(Then)36 b(w)m(e)g(obtain)e(\003)706 4965 y Fs(\(2)p Fr(;i)p Fs(\))843 5001 y Fq( )906 5016 y Fs(0)978 5001 y Fw(=)d Fu(E)1138 5016 y Fs(2)1177 5001 y Fq( )1240 5016 y Fs(0)1280 5001 y Fw(.)50 b(By)36 b(assumption)e(\003)2099 4965 y Fs(\(2)p Fr(;i)p Fs(\))2271 5001 y Fw(has)i(only)e(one)h(eigen)m (v)-5 b(alue)34 b Fq(\013)3375 5016 y Fs(2)p Fr(;i)3458 5001 y Fw(.)51 b(Therefore)-71 5121 y Fu(E)-18 5136 y Fs(2)49 5121 y Fw(=)27 b Fq(\013)214 5136 y Fs(2)p Fr(;i)297 5121 y Fw(.)1888 5470 y(11)p eop %%Page: 12 12 12 11 bop 75 123 a Fw(F)-8 b(or)32 b Fq(j)i Fu(6)p Fw(=)27 b Fq(i)33 b Fw(w)m(e)h(ha)m(v)m(e)940 323 y Fq(P)1017 282 y Fs(\()p Fr(j)t Fs(\))1108 323 y Fq(T)1179 282 y Fs(\(4\))1273 323 y Fq( )1336 338 y Fs(0)1398 323 y Fw(+)22 b Fq(P)1573 282 y Fs(\()p Fr(j)t Fs(\))1664 323 y Fq(T)1735 282 y Fs(\(3\))1829 323 y Fq( )1896 282 y Fm(?)1892 348 y Fs(1)1977 323 y Fw(+)g Fq(\013)2137 338 y Fs(1)p Fr(;j)2228 323 y Fq( )2295 272 y Fs(\()p Fr(j)t Fs(\))2291 348 y(1)2415 323 y Fw(=)27 b Fq(\013)2580 338 y Fs(1)p Fr(;i)2663 323 y Fq(P)2740 282 y Fs(\()p Fr(j)t Fs(\))2831 323 y Fq( )2894 338 y Fs(1)-71 540 y Fw(b)s(ecause)32 b Fq(P)365 503 y Fs(\()p Fr(j)t Fs(\))456 540 y Fq( )519 555 y Fs(0)586 540 y Fw(=)c(0)i(whenev)m(er)j Fq(j)h Fu(6)p Fw(=)28 b Fq(i)p Fw(.)43 b(Rearranging)29 b(terms)h(w)m(e)i(\014nally)d(obtain) h Fq( )3101 489 y Fs(\()p Fr(j)t Fs(\))3097 564 y(1)3220 540 y Fw(=)d(\004)3388 503 y Fs(\(1)p Fr(;j)t Fs(\))3535 540 y Fq( )3598 555 y Fs(0)3638 540 y Fw(,)k(where)-71 660 y(w)m(e)j(de\014ne)881 780 y(\004)946 739 y Fs(\(1)p Fr(;j)t Fs(\))1120 780 y Fw(:=)27 b(\()p Fq(\013)1350 795 y Fs(1)p Fr(;i)1455 780 y Fu(\000)c Fq(\013)1617 795 y Fs(1)p Fr(;j)1709 780 y Fw(\))1746 732 y Fm(\000)p Fs(1)1857 780 y Fq(P)1934 739 y Fs(\()p Fr(j)t Fs(\))2042 700 y Fv(\000)2088 780 y Fq(T)2159 739 y Fs(\(4\))2275 780 y Fw(+)f Fq(T)2444 739 y Fs(\(3\))2538 780 y Fw(\004)2603 739 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2772 700 y Fv(\001)2834 780 y Fq(P)2911 739 y Fs(\()p Fr(i)p Fs(\))3771 780 y Fw(\(21\))-71 970 y(So)33 b(far)f(no)g(requiremen)m(t)h(is)f(imp)s (osed)g(to)g Fq( )1558 919 y Fs(\()p Fr(i)p Fs(\))1554 994 y(1)1641 970 y Fw(.)75 1090 y(No)m(w)i(m)m(ultiply)c(\(17\))i(b)m (y)h Fq(Q)p Fw(,)1092 1290 y Fq(H)1173 1305 y Fr(e)1210 1290 y Fq( )1277 1249 y Fm(?)1273 1314 y Fs(2)1358 1290 y Fw(+)22 b Fq(QT)1604 1249 y Fs(\(4\))1699 1290 y Fq( )1762 1305 y Fs(0)1824 1290 y Fw(+)g Fq(QT)2070 1249 y Fs(\(3\))2164 1290 y Fq( )2227 1305 y Fs(1)2294 1290 y Fw(=)28 b Fq(\013)2460 1305 y Fs(1)p Fr(;i)2543 1290 y Fq(A)2616 1249 y Fs(2)2616 1314 y Fr(e)2656 1290 y Fq( )2723 1249 y Fm(?)2719 1314 y Fs(1)3771 1290 y Fw(\(22\))-71 1490 y(Since)684 1689 y Fq(QT)832 1648 y Fs(\(3\))927 1689 y Fq( )990 1704 y Fs(1)1113 1689 y Fw(=)82 b Fq(QT)1419 1648 y Fs(\(3\))1514 1689 y Fq( )1581 1648 y Fm(?)1577 1714 y Fs(1)1662 1689 y Fw(+)1760 1595 y Fv(X)1782 1807 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1920 1689 y Fq(QT)2068 1648 y Fs(\(3\))2163 1689 y Fq( )2230 1639 y Fs(\()p Fr(l)q Fs(\))2226 1714 y(1)2333 1689 y Fw(+)22 b Fq(QT)2579 1648 y Fs(\(3\))2673 1689 y Fq( )2740 1639 y Fs(\()p Fr(i)p Fs(\))2736 1714 y(1)1113 1968 y Fw(=)82 b Fq(QT)1419 1927 y Fs(\(3\))1514 1968 y Fw(\004)1579 1927 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1748 1968 y Fq( )1811 1983 y Fs(0)1873 1968 y Fw(+)1971 1873 y Fv(X)1992 2085 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2131 1968 y Fq(QT)2279 1927 y Fs(\(3\))2373 1968 y Fw(\004)2438 1927 y Fs(\(1)p Fr(;l)q Fs(\))2574 1968 y Fq( )2637 1983 y Fs(0)2699 1968 y Fw(+)22 b Fq(QT)2945 1927 y Fs(\(3\))3040 1968 y Fq( )3107 1917 y Fs(\()p Fr(i)p Fs(\))3103 1992 y(1)-71 2271 y Fw(\(22\))32 b(yields)159 2488 y Fq(H)240 2503 y Fr(e)277 2488 y Fq( )344 2447 y Fm(?)340 2512 y Fs(2)431 2488 y Fw(=)27 b Fu(\000)p Fq(QT)759 2447 y Fs(\(4\))854 2488 y Fq( )917 2503 y Fs(0)979 2488 y Fw(+)22 b Fq(\013)1139 2503 y Fs(1)p Fr(;i)1222 2488 y Fq(A)1295 2447 y Fs(2)1295 2512 y Fr(e)1335 2488 y Fw(\004)1400 2447 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1569 2488 y Fq( )1632 2503 y Fs(0)1694 2488 y Fu(\000)g Fq(QT)1941 2447 y Fs(\(3\))2036 2488 y Fw(\004)2101 2447 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2269 2488 y Fq( )2332 2503 y Fs(0)2394 2488 y Fu(\000)2494 2393 y Fv(X)2516 2605 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2655 2488 y Fq(QT)2803 2447 y Fs(\(3\))2897 2488 y Fw(\004)2962 2447 y Fs(\(1)p Fr(;l)q Fs(\))3098 2488 y Fq( )3161 2503 y Fs(0)3223 2488 y Fu(\000)g Fq(QT)3470 2447 y Fs(\(3\))3565 2488 y Fq( )3632 2437 y Fs(\()p Fr(i)p Fs(\))3628 2512 y(1)-71 2785 y Fw(Therefrom)33 b(w)m(e)g(obtain)1388 2906 y Fq( )1455 2864 y Fm(?)1451 2930 y Fs(2)1542 2906 y Fw(=)27 b(\004)1710 2864 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1879 2906 y Fq( )1942 2921 y Fs(0)2004 2906 y Fw(+)22 b(\004)2167 2864 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2336 2906 y Fq( )2403 2855 y Fs(\()p Fr(i)p Fs(\))2399 2930 y(1)-71 3071 y Fw(where)476 3254 y(\004)541 3213 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))737 3254 y Fw(:=)28 b(\()p Fq(H)987 3269 y Fr(e)1024 3254 y Fw(\))1062 3213 y Fm(\000)p Fs(1)1062 3282 y Fm(?)1173 3084 y Fv(")1231 3254 y Fq(\013)1293 3269 y Fs(1)p Fr(;i)1376 3254 y Fw(\004)1441 3213 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1610 3254 y Fq(A)1683 3213 y Fs(2)1683 3279 y Fr(e)1744 3254 y Fu(\000)23 b Fq(QT)1992 3213 y Fs(\(3\))2103 3084 y Fv( )2182 3254 y Fw(\004)2247 3213 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2438 3254 y Fw(+)2536 3159 y Fv(X)2557 3371 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2696 3254 y Fw(\004)2761 3213 y Fs(\(1)p Fr(;l)q Fs(\))2897 3084 y Fv(!)2998 3254 y Fu(\000)g Fq(QT)3246 3213 y Fs(\(4\))3340 3084 y Fv(#)-71 3588 y Fh(Thir)-5 b(d-or)g(der:)545 3788 y Fq(H)626 3803 y Fr(e)662 3788 y Fq( )725 3803 y Fs(3)787 3788 y Fw(+)22 b Fq(T)956 3747 y Fs(\(3\))1050 3788 y Fq( )1113 3803 y Fs(2)1175 3788 y Fw(+)g Fq(T)1344 3747 y Fs(\(4\))1438 3788 y Fq( )1501 3803 y Fs(1)1563 3788 y Fw(+)g Fq(T)1732 3747 y Fs(\(5\))1827 3788 y Fq( )1890 3803 y Fs(0)1957 3788 y Fw(=)28 b Fu(E)2114 3803 y Fs(3)2153 3788 y Fq(A)2226 3747 y Fs(2)2226 3812 y Fr(e)2265 3788 y Fq( )2328 3803 y Fs(0)2390 3788 y Fw(+)22 b Fq(\013)2550 3803 y Fs(2)p Fr(;i)2633 3788 y Fq(A)2706 3747 y Fs(2)2706 3812 y Fr(e)2746 3788 y Fq( )2809 3803 y Fs(1)2871 3788 y Fw(+)g Fq(\013)3031 3803 y Fs(1)p Fr(;i)3114 3788 y Fq(A)3187 3747 y Fs(2)3187 3812 y Fr(e)3227 3788 y Fq( )3290 3803 y Fs(2)3771 3788 y Fw(\(23\))-71 3988 y(Multiply)31 b(b)m(y)j Fq(P)543 3951 y Fs(\()p Fr(j)t Fs(\))633 3988 y Fw(,)f(rearrange)g(terms,)f(and)h(use)g(\(19\))f(to)h(obtain)436 4201 y Fu(E)489 4216 y Fs(3)528 4201 y Fq(P)605 4160 y Fs(\()p Fr(j)t Fs(\))696 4201 y Fq( )759 4216 y Fs(0)881 4201 y Fw(=)83 b Fq(P)1117 4160 y Fs(\()p Fr(j)t Fs(\))1208 4201 y Fq(T)1279 4160 y Fs(\(3\))1373 4201 y Fq( )1436 4216 y Fs(2)1498 4201 y Fw(+)22 b Fq(P)1673 4160 y Fs(\()p Fr(j)t Fs(\))1764 4201 y Fq(T)1835 4160 y Fs(\(4\))1929 4201 y Fq( )1992 4216 y Fs(1)2054 4201 y Fw(+)g Fq(P)2229 4160 y Fs(\()p Fr(j)t Fs(\))2320 4201 y Fq(T)2391 4160 y Fs(\(5\))2485 4201 y Fq( )2548 4216 y Fs(0)2610 4201 y Fu(\000)h Fq(\013)2772 4216 y Fs(2)p Fr(;i)2855 4201 y Fq( )2922 4150 y Fs(\()p Fr(j)t Fs(\))2918 4225 y(1)3035 4201 y Fu(\000)g Fq(\013)3197 4216 y Fs(1)p Fr(;i)3280 4201 y Fq( )3347 4150 y Fs(\()p Fr(j)t Fs(\))3343 4225 y(2)881 4436 y Fw(=)83 b Fq(P)1117 4395 y Fs(\()p Fr(j)t Fs(\))1208 4436 y Fq(T)1279 4395 y Fs(\(3\))1390 4326 y Fv(\020)1449 4436 y Fq( )1516 4395 y Fm(?)1512 4461 y Fs(2)1597 4436 y Fw(+)22 b Fq( )1762 4386 y Fm(k)1758 4461 y Fs(2)1802 4326 y Fv(\021)1884 4436 y Fw(+)g Fq(P)2059 4395 y Fs(\()p Fr(j)t Fs(\))2149 4436 y Fq(T)2220 4395 y Fs(\(4\))2331 4266 y Fv( )2410 4436 y Fq( )2477 4395 y Fm(?)2473 4461 y Fs(1)2558 4436 y Fw(+)2656 4342 y Fv(X)2678 4554 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2817 4436 y Fq( )2884 4386 y Fs(\()p Fr(l)q Fs(\))2880 4461 y(1)2987 4436 y Fw(+)g Fq( )3152 4386 y Fs(\()p Fr(i)p Fs(\))3148 4461 y(1)3235 4266 y Fv(!)1040 4711 y Fw(+)p Fq(P)1193 4670 y Fs(\()p Fr(j)t Fs(\))1284 4711 y Fq(T)1355 4670 y Fs(\(5\))1449 4711 y Fq( )1512 4726 y Fs(0)1574 4711 y Fu(\000)h Fq(\013)1736 4726 y Fs(2)p Fr(;i)1819 4711 y Fq( )1886 4660 y Fs(\()p Fr(j)t Fs(\))1882 4735 y(1)1999 4711 y Fu(\000)g Fq(\013)2161 4726 y Fs(1)p Fr(;i)2244 4711 y Fq( )2311 4660 y Fs(\()p Fr(j)t Fs(\))2307 4735 y(2)881 4946 y Fw(=)83 b Fq(P)1117 4905 y Fs(\()p Fr(j)t Fs(\))1208 4946 y Fq(T)1279 4905 y Fs(\(3\))1373 4946 y Fq( )1440 4905 y Fm(?)1436 4971 y Fs(2)1521 4946 y Fw(+)22 b Fq(P)1696 4905 y Fs(\()p Fr(j)t Fs(\))1787 4946 y Fq(T)1858 4905 y Fs(\(4\))1969 4776 y Fv( )2048 4946 y Fq( )2115 4905 y Fm(?)2111 4971 y Fs(1)2196 4946 y Fw(+)2294 4852 y Fv(X)2315 5064 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2454 4946 y Fq( )2521 4895 y Fs(\()p Fr(i)p Fs(\))2517 4971 y(1)2627 4946 y Fw(+)g Fq( )2792 4895 y Fs(\()p Fr(i)p Fs(\))2788 4971 y(1)2875 4776 y Fv(!)1040 5221 y Fw(+)p Fq(P)1193 5179 y Fs(\()p Fr(j)t Fs(\))1284 5221 y Fq(T)1355 5179 y Fs(\(5\))1449 5221 y Fq( )1512 5236 y Fs(0)1574 5221 y Fu(\000)h Fw(\()p Fq(\013)1774 5236 y Fs(1)p Fr(;i)1879 5221 y Fu(\000)f Fq(\013)2040 5236 y Fs(1)p Fr(;j)2132 5221 y Fw(\))16 b Fq( )2253 5170 y Fs(\()p Fr(j)t Fs(\))2249 5245 y(2)2367 5221 y Fu(\000)23 b Fq(\013)2529 5236 y Fs(2)p Fr(;i)2612 5221 y Fq( )2679 5170 y Fs(\()p Fr(j)t Fs(\))2675 5245 y(1)3771 5221 y Fw(\(24\))1888 5470 y(12)p eop %%Page: 13 13 13 12 bop -71 123 a Fw(F)-8 b(or)32 b Fq(j)i Fw(=)27 b Fq(i)33 b Fw(w)m(e)g(ha)m(v)m(e)387 386 y Fu(E)440 401 y Fs(3)479 386 y Fq( )542 401 y Fs(0)665 386 y Fw(=)83 b Fq(P)901 345 y Fs(\()p Fr(i)p Fs(\))984 386 y Fq(T)1055 345 y Fs(\(3\))1149 386 y Fw(\004)1214 345 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1382 386 y Fq( )1445 401 y Fs(0)1507 386 y Fw(+)22 b Fq(P)1682 345 y Fs(\()p Fr(i)p Fs(\))1765 386 y Fq(T)1836 345 y Fs(\(4\))1947 216 y Fv( )2026 386 y Fw(\004)2091 345 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2281 386 y Fw(+)2380 292 y Fv(X)2401 504 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2540 386 y Fw(\004)2605 345 y Fs(\(1)p Fr(;l)q Fs(\))2741 216 y Fv(!)2836 386 y Fq( )2899 401 y Fs(0)2961 386 y Fw(+)g Fq(P)3136 345 y Fs(\()p Fr(i)p Fs(\))3219 386 y Fq(T)3290 345 y Fs(\(5\))3384 386 y Fq( )3447 401 y Fs(0)824 661 y Fw(+)p Fq(P)977 620 y Fs(\()p Fr(i)p Fs(\))1059 661 y Fq(T)1130 620 y Fs(\(3\))1225 661 y Fw(\004)1290 620 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1458 661 y Fq( )1525 610 y Fs(\()p Fr(i)p Fs(\))1521 685 y(1)1631 661 y Fw(+)g Fq(P)1806 620 y Fs(\()p Fr(i)p Fs(\))1888 661 y Fq(T)1959 620 y Fs(\(4\))2053 661 y Fq( )2120 610 y Fs(\()p Fr(i)p Fs(\))2116 685 y(1)2225 661 y Fu(\000)h Fq(\013)2387 676 y Fs(2)p Fr(;i)2470 661 y Fq( )2537 610 y Fs(\()p Fr(i)p Fs(\))2533 685 y(1)-71 842 y Fw(Let)33 b(us)g(note)g(that)829 962 y Fq(P)906 921 y Fs(\()p Fr(i)p Fs(\))988 962 y Fq(T)1059 921 y Fs(\(4\))1153 962 y Fq( )1220 921 y Fs(\()p Fr(i)p Fs(\))1325 962 y Fw(+)22 b Fq(P)1500 921 y Fs(\()p Fr(i)p Fs(\))1583 962 y Fq(T)1654 921 y Fs(\(3\))1748 962 y Fw(\004)1813 921 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1982 962 y Fq( )2049 921 y Fs(\()p Fr(i)p Fs(\))2160 962 y Fw(=)27 b(\003)2331 921 y Fs(\(2)p Fr(;i)p Fs(\))2469 962 y Fq( )2536 921 y Fs(\()p Fr(i)p Fs(\))2647 962 y Fw(=)g Fq(\013)2812 977 y Fs(2)p Fr(;i)2895 962 y Fq( )2962 921 y Fs(\()p Fr(i)p Fs(\))-71 1120 y Fw(Th)m(us)34 b(w)m(e)g(obtain)d Fu(E)676 1135 y Fs(3)715 1120 y Fq( )778 1135 y Fs(0)846 1120 y Fw(=)c(\003)1017 1084 y Fs(\(3)p Fr(;i)p Fs(\))1155 1120 y Fq( )1218 1135 y Fs(0)1258 1120 y Fw(,)33 b(where)591 1383 y(\003)659 1342 y Fs(\(3)p Fr(;i)p Fs(\))825 1383 y Fw(:=)27 b Fq(P)1032 1342 y Fs(\()p Fr(i)p Fs(\))1131 1213 y Fv(")1190 1383 y Fq(T)1261 1342 y Fs(\(5\))1377 1383 y Fw(+)22 b Fq(T)1546 1342 y Fs(\(4\))1656 1213 y Fv( )1735 1383 y Fw(\004)1800 1342 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1991 1383 y Fw(+)2089 1288 y Fv(X)2111 1501 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2250 1383 y Fw(\004)2315 1342 y Fs(\(1)p Fr(;l)q Fs(\))2451 1213 y Fv(!)2552 1383 y Fw(+)g Fq(T)2721 1342 y Fs(\(3\))2815 1383 y Fw(\004)2880 1342 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))3048 1213 y Fv(#)3123 1383 y Fq(P)3200 1342 y Fs(\()p Fr(i)p Fs(\))-71 1679 y Fw(By)33 b(assumption)f(\003)666 1642 y Fs(\(3)p Fr(;i)p Fs(\))836 1679 y Fw(has)h(only)f(one)h(eigen)m(v)-5 b(alue)32 b Fq(\013)1931 1694 y Fs(3)p Fr(;i)2047 1679 y Fw(so)h Fu(E)2220 1694 y Fs(3)2286 1679 y Fw(=)28 b Fq(\013)2452 1694 y Fs(3)p Fr(;i)2535 1679 y Fw(.)75 1799 y(No)m(w)34 b(for)e Fq(j)h Fu(6)p Fw(=)28 b Fq(i)33 b Fw(w)m(e)g(can)g(rewrite)g (\(24\))f(as)97 2068 y(\()p Fq(\013)197 2083 y Fs(1)p Fr(;i)302 2068 y Fu(\000)23 b Fq(\013)464 2083 y Fs(1)p Fr(;j)555 2068 y Fw(\))17 b Fq( )677 2017 y Fs(\()p Fr(j)t Fs(\))673 2092 y(2)851 2068 y Fw(=)83 b Fq(P)1087 2026 y Fs(\()p Fr(j)t Fs(\))1178 2068 y Fq(T)1249 2026 y Fs(\(5\))1343 2068 y Fq( )1406 2083 y Fs(0)1468 2068 y Fw(+)22 b Fq(P)1643 2026 y Fs(\()p Fr(j)t Fs(\))1734 2068 y Fq(T)1805 2026 y Fs(\(4\))1916 1897 y Fv( )1995 2068 y Fw(\004)2060 2026 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2251 2068 y Fw(+)2349 1973 y Fv(X)2370 2185 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2509 2068 y Fw(\004)2574 2026 y Fs(\(1)p Fr(;l)q Fs(\))2710 1897 y Fv(!)2805 2068 y Fq( )2868 2083 y Fs(0)1010 2342 y Fw(+)p Fq(P)1163 2301 y Fs(\()p Fr(j)t Fs(\))1254 2342 y Fq(T)1325 2301 y Fs(\(3\))1419 2342 y Fw(\004)1484 2301 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1653 2342 y Fq( )1716 2357 y Fs(0)1778 2342 y Fu(\000)h Fq(\013)1940 2357 y Fs(2)p Fr(;i)2023 2342 y Fw(\004)2088 2301 y Fs(\(1)p Fr(;j)t Fs(\))2234 2342 y Fq( )2297 2357 y Fs(0)2359 2342 y Fw(+)f Fq(P)2534 2301 y Fs(\()p Fr(j)t Fs(\))2625 2342 y Fq(T)2696 2301 y Fs(\(3\))2790 2342 y Fw(\004)2855 2301 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))3024 2342 y Fq( )3091 2291 y Fs(\()p Fr(i)p Fs(\))3087 2366 y(1)3196 2342 y Fw(+)g Fq(P)3371 2301 y Fs(\()p Fr(j)t Fs(\))3462 2342 y Fq(T)3533 2301 y Fs(\(4\))3627 2342 y Fq( )3694 2291 y Fs(\()p Fr(i)p Fs(\))3690 2366 y(1)-71 2523 y Fw(No)m(w)33 b(use)h(\(21\))e(and)g(de\014ne)42 2792 y(\004)107 2751 y Fs(\(2)p Fr(;j)t Fs(\))281 2792 y Fw(:=)27 b(\()p Fq(\013)511 2807 y Fs(1)p Fr(;i)616 2792 y Fu(\000)c Fq(\013)778 2807 y Fs(1)p Fr(;j)870 2792 y Fw(\))908 2743 y Fm(\000)p Fs(1)1018 2792 y Fq(P)1095 2751 y Fs(\()p Fr(j)t Fs(\))1203 2621 y Fv(")1261 2792 y Fq(T)1332 2751 y Fs(\(5\))1448 2792 y Fw(+)f Fq(T)1617 2751 y Fs(\(4\))1728 2621 y Fv( )1807 2792 y Fw(\004)1872 2751 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2063 2792 y Fw(+)2161 2697 y Fv(X)2182 2909 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2321 2792 y Fw(\004)2386 2751 y Fs(\(1)p Fr(;l)q Fs(\))2522 2621 y Fv(!)2623 2792 y Fw(+)g Fq(T)2792 2751 y Fs(\(3\))2886 2792 y Fw(\004)2951 2751 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))3142 2792 y Fu(\000)h Fq(\013)3304 2807 y Fs(2)p Fr(;i)3387 2792 y Fw(\004)3452 2751 y Fs(\(1)p Fr(;j)t Fs(\))3598 2621 y Fv(#)3673 2792 y Fq(P)3750 2751 y Fs(\()p Fr(i)p Fs(\))-71 3070 y Fw(to)32 b(obtain)1394 3191 y Fq( )1461 3140 y Fs(\()p Fr(j)t Fs(\))1457 3215 y(2)1580 3191 y Fw(=)c(\004)1749 3150 y Fs(\(2)p Fr(;j)t Fs(\))1895 3191 y Fq( )1958 3206 y Fs(0)2020 3191 y Fw(+)22 b(\004)2183 3150 y Fs(\(1)p Fr(;j)t Fs(\))2330 3191 y Fq( )2397 3140 y Fs(\()p Fr(i)p Fs(\))2393 3215 y(1)75 3349 y Fw(The)34 b(last)e(step)h(is)f(to)h(m)m(ultiply)d(\(23\))i(b)m (y)h(Q,)607 3530 y Fq(H)688 3545 y Fr(e)725 3530 y Fq( )792 3489 y Fm(?)788 3555 y Fs(3)878 3530 y Fw(=)28 b Fq(Q)1076 3449 y Fv(\000)1121 3530 y Fq(\013)1183 3545 y Fs(1)p Fr(;i)1266 3530 y Fq(A)1339 3489 y Fs(2)1339 3555 y Fr(e)1401 3530 y Fu(\000)23 b Fq(T)1572 3489 y Fs(\(3\))1666 3449 y Fv(\001)1728 3530 y Fq( )1791 3545 y Fs(2)1853 3530 y Fw(+)f Fq(Q)2045 3449 y Fv(\000)2091 3530 y Fq(\013)2153 3545 y Fs(2)p Fr(;i)2236 3530 y Fq(A)2309 3489 y Fs(2)2309 3555 y Fr(e)2370 3530 y Fu(\000)h Fq(T)2541 3489 y Fs(\(4\))2635 3449 y Fv(\001)2697 3530 y Fq( )2760 3545 y Fs(1)2822 3530 y Fu(\000)g Fq(QT)3070 3489 y Fs(\(5\))3164 3530 y Fq( )3227 3545 y Fs(0)3771 3530 y Fw(\(25\))-71 3711 y(W)-8 b(e)33 b(ha)m(v)m(e)73 3893 y Fq(Q)167 3812 y Fv(\000)212 3893 y Fq(\013)274 3908 y Fs(1)p Fr(;i)358 3893 y Fq(A)431 3851 y Fs(2)431 3917 y Fr(e)492 3893 y Fu(\000)23 b Fq(T)663 3851 y Fs(\(3\))757 3812 y Fv(\001)819 3893 y Fq( )882 3908 y Fs(2)1005 3893 y Fw(=)83 b Fq(Q)1258 3812 y Fv(\000)1304 3893 y Fq(\013)1366 3908 y Fs(1)p Fr(;i)1449 3893 y Fq(A)1522 3851 y Fs(2)1522 3917 y Fr(e)1583 3893 y Fu(\000)23 b Fq(T)1754 3851 y Fs(\(3\))1848 3812 y Fv(\001)1910 3893 y Fq( )1977 3851 y Fm(?)1973 3917 y Fs(2)2058 3893 y Fw(+)f Fq(\013)2218 3908 y Fs(1)p Fr(;i)2302 3893 y Fq(A)2375 3851 y Fs(2)2375 3917 y Fr(e)2414 3893 y Fq(Q )2558 3842 y Fm(k)2554 3917 y Fs(2)2620 3893 y Fu(\000)h Fq(QT)2868 3851 y Fs(\(3\))2979 3798 y Fv(X)3000 4010 y Fr(l)q Fm(6)p Fs(=)p Fr(i)3139 3893 y Fq( )3206 3842 y Fs(\()p Fr(l)q Fs(\))3202 3917 y(2)3309 3893 y Fu(\000)g Fq(QT)3557 3851 y Fs(\(3\))3651 3893 y Fq( )3718 3842 y Fs(\()p Fr(i)p Fs(\))3714 3917 y(2)1005 4167 y Fw(=)83 b Fq(Q)1258 4086 y Fv(\000)1304 4167 y Fq(\013)1366 4182 y Fs(1)p Fr(;i)1449 4167 y Fq(A)1522 4126 y Fs(2)1522 4192 y Fr(e)1583 4167 y Fu(\000)23 b Fq(T)1754 4126 y Fs(\(3\))1848 4086 y Fv(\001)1910 4167 y Fw(\004)1975 4126 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2144 4167 y Fq( )2207 4182 y Fs(0)2269 4167 y Fw(+)f Fq(Q)2461 4086 y Fv(\000)2507 4167 y Fq(\013)2569 4182 y Fs(1)p Fr(;i)2652 4167 y Fq(A)2725 4126 y Fs(2)2725 4192 y Fr(e)2786 4167 y Fu(\000)h Fq(T)2957 4126 y Fs(\(3\))3051 4086 y Fv(\001)3113 4167 y Fw(\004)3178 4126 y Fs(\(1)p Fr(;i)p Fs(\))3316 4167 y Fq( )3383 4116 y Fs(\()p Fr(i)p Fs(\))3379 4191 y(1)1164 4434 y Fu(\000)28 b Fq(QT)1417 4393 y Fs(\(3\))1528 4339 y Fv(X)1550 4552 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1689 4434 y Fw(\004)1754 4393 y Fs(\(2)p Fr(;l)q Fs(\))1889 4434 y Fq( )1952 4449 y Fs(0)2014 4434 y Fu(\000)23 b Fq(QT)2262 4393 y Fs(\(3\))2373 4339 y Fv(X)2395 4552 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2533 4434 y Fw(\004)2598 4393 y Fs(\(1)p Fr(;l)q Fs(\))2734 4434 y Fq( )2801 4383 y Fs(\()p Fr(i)p Fs(\))2797 4458 y(1)2906 4434 y Fu(\000)g Fq(QT)3154 4393 y Fs(\(3\))3248 4434 y Fq( )3315 4383 y Fs(\()p Fr(i)p Fs(\))3311 4458 y(2)1005 4782 y Fw(=)83 b Fq(Q)1258 4612 y Fv(")1316 4701 y(\000)1362 4782 y Fq(\013)1424 4797 y Fs(1)p Fr(;i)1507 4782 y Fq(A)1580 4741 y Fs(2)1580 4807 y Fr(e)1641 4782 y Fu(\000)23 b Fq(T)1812 4741 y Fs(\(3\))1906 4701 y Fv(\001)1968 4782 y Fw(\004)2033 4741 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2224 4782 y Fu(\000)2324 4687 y Fv(X)2346 4900 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2485 4782 y Fq(T)2556 4741 y Fs(\(3\))2650 4782 y Fw(\004)2715 4741 y Fs(\(2)p Fr(;l)q Fs(\))2850 4612 y Fv(#)2925 4782 y Fq( )2988 4797 y Fs(0)1164 5130 y Fw(+)28 b Fq(Q)1362 4960 y Fv(")1420 5049 y(\000)1465 5130 y Fq(\013)1527 5145 y Fs(1)p Fr(;i)1610 5130 y Fq(A)1683 5089 y Fs(2)1683 5155 y Fr(e)1745 5130 y Fu(\000)23 b Fq(T)1916 5089 y Fs(\(3\))2010 5049 y Fv(\001)2072 5130 y Fw(\004)2137 5089 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2328 5130 y Fu(\000)2428 5035 y Fv(X)2449 5248 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2588 5130 y Fq(T)2659 5089 y Fs(\(3\))2753 5130 y Fw(\004)2818 5089 y Fs(\(1)p Fr(;l)q Fs(\))2954 4960 y Fv(#)3029 5130 y Fq( )3096 5079 y Fs(\(1\))3092 5154 y(1)3212 5130 y Fu(\000)g Fq(QT)3460 5089 y Fs(\(3\))3554 5130 y Fq( )3621 5079 y Fs(\()p Fr(i)p Fs(\))3617 5154 y(2)3771 5130 y Fw(\(26\))1888 5470 y(13)p eop %%Page: 14 14 14 13 bop -71 123 a Fw(and)33 b(similarly)171 399 y Fq(Q)265 318 y Fv(\000)311 399 y Fq(\013)373 414 y Fs(2)p Fr(;i)456 399 y Fq(A)529 358 y Fs(2)529 424 y Fr(e)590 399 y Fu(\000)23 b Fq(T)761 358 y Fs(\(4\))855 318 y Fv(\001)917 399 y Fq( )980 414 y Fs(1)1048 399 y Fw(=)k Fq(Q)1245 229 y Fv(")1303 318 y(\000)1349 399 y Fq(\013)1411 414 y Fs(2)p Fr(;i)1494 399 y Fq(A)1567 358 y Fs(2)1567 424 y Fr(e)1629 399 y Fu(\000)c Fq(T)1800 358 y Fs(\(4\))1894 318 y Fv(\001)1956 399 y Fw(\004)2021 358 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2212 399 y Fu(\000)2311 304 y Fv(X)2333 516 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2472 399 y Fq(T)2543 358 y Fs(\(4\))2637 399 y Fw(\004)2702 358 y Fs(\(1)p Fr(;l)q Fs(\))2838 229 y Fv(#)2913 399 y Fq( )2976 414 y Fs(0)3038 399 y Fu(\000)f Fq(QT)3285 358 y Fs(\(4\))3379 399 y Fq( )3446 348 y Fs(\()p Fr(i)p Fs(\))3442 423 y(1)3771 399 y Fw(\(27\))-71 705 y(Insert)34 b(\(26\))e(and)g(\(27\))g(in)g(\(25\))g(and)h(m)m (ultiply)d(the)j(whole)f(equation)h(b)m(y)g(\()p Fq(H)2807 720 y Fr(e)2844 705 y Fw(\))2882 664 y Fm(\000)p Fs(1)2882 733 y Fm(?)3009 705 y Fw(to)f(obtain)1136 916 y Fq( )1203 874 y Fm(?)1199 940 y Fs(3)1290 916 y Fw(=)27 b(\004)1458 874 y Fs(\(3)p Fr(;)p Fm(?)p Fs(\))1627 916 y Fq( )1690 931 y Fs(0)1752 916 y Fw(+)22 b(\004)1915 874 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2084 916 y Fq( )2151 865 y Fs(\()p Fr(i)p Fs(\))2147 940 y(1)2256 916 y Fw(+)g(\004)2419 874 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2588 916 y Fq( )2655 865 y Fs(\()p Fr(i)p Fs(\))2651 940 y(2)-71 1109 y Fw(with)398 1303 y(\004)463 1262 y Fs(\(3)p Fr(;)p Fm(?)p Fs(\))715 1303 y Fw(:=)83 b(\()p Fq(H)1020 1318 y Fr(e)1056 1303 y Fw(\))1094 1262 y Fm(\000)p Fs(1)1094 1331 y Fm(?)1205 1223 y Fv(\002\000)1292 1303 y Fq(\013)1354 1318 y Fs(1)p Fr(;i)1437 1303 y Fw(\004)1502 1262 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))1693 1303 y Fw(+)22 b Fq(\013)1853 1318 y Fs(2)p Fr(;i)1937 1303 y Fw(\004)2002 1262 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2170 1223 y Fv(\001)2233 1303 y Fq(A)2306 1262 y Fs(2)2306 1328 y Fr(e)2367 1303 y Fu(\000)h Fq(QT)2615 1262 y Fs(\(5\))901 1593 y Fu(\000)54 b Fq(QT)1180 1552 y Fs(\(4\))1291 1423 y Fv( )1370 1593 y Fw(\004)1435 1552 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1626 1593 y Fw(+)1724 1498 y Fv(X)1746 1711 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1885 1593 y Fw(\004)1950 1552 y Fs(\(1)p Fr(;l)q Fs(\))2085 1423 y Fv(!)2186 1593 y Fu(\000)23 b Fq(QT)2434 1552 y Fs(\(5\))2545 1423 y Fv( )2624 1593 y Fw(\004)2689 1552 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))2880 1593 y Fw(+)2978 1498 y Fv(X)2999 1711 y Fr(l)q Fm(6)p Fs(=)p Fr(i)3138 1593 y Fw(\004)3203 1552 y Fs(\(2)p Fr(;l)q Fs(\))3339 1423 y Fv(!#)75 1884 y Fw(As)41 b(b)s(efore,)h(one)e(can)g(guess)i(the) e(solution)f(for)g(arbitrary)g Fq(n)p Fw(.)66 b(Let)40 b(us)h(summarize)d(h)m(yp)s(otheses)43 b(and)-71 2005 y(results:)-71 2203 y Fn(Prop)s(osition)36 b(2)48 b Fh(De\014ne)1454 2397 y Fw(\003)1522 2356 y Fs(\(1\))1699 2397 y Fw(:=)83 b Fq(P)14 b(T)2033 2356 y Fs(\(3\))2126 2397 y Fq(P)1382 2543 y Fw(\004)1447 2501 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1699 2543 y Fw(:=)83 b Fu(\000)p Fw(\()p Fq(H)2081 2558 y Fr(e)2118 2543 y Fw(\))2156 2501 y Fm(\000)p Fs(1)2156 2570 y Fm(?)2250 2543 y Fq(QT)2398 2501 y Fs(\(3\))-71 2737 y Fh(Supp)-5 b(ose)48 b(that)g Fw(\003)594 2700 y Fs(\(1\))737 2737 y Fh(has)g Fq(k)j Fh(distinct)d(eigenvalues)f Fq(\013)1975 2752 y Fs(1)p Fr(;)p Fs(1)2070 2737 y Fq(;)17 b(:)g(:)g(:)e(;)i(\013)2350 2752 y Fs(1)p Fr(;k)2496 2737 y Fh(with)48 b(eigensp)-5 b(ac)g(es)47 b Fq(B)3322 2752 y Fs(1)3361 2737 y Fq(;)17 b(:)g(:)g(:)f(;)h(B)3654 2752 y Fr(k)3696 2737 y Fh(.)86 b(L)-5 b(et)-71 2857 y Fq(P)6 2821 y Fs(\(1\))100 2857 y Fq(;)17 b(:)g(:)g(:)f(;)h(P)396 2821 y Fs(\()p Fr(k)r Fs(\))527 2857 y Fh(b)-5 b(e)35 b(the)g(asso)-5 b(ciate)g(d)33 b(pr)-5 b(oje)g(ction)35 b(op)-5 b(er)g(ators.)44 b(Given)34 b Fw(1)28 b Fu(\024)g Fq(i)g Fu(\024)g Fq(k)s Fh(,)35 b(set)449 3051 y Fw(\003)517 3010 y Fs(\(2)p Fr(;i)p Fs(\))737 3051 y Fw(:=)83 b Fq(P)1000 3010 y Fs(\()p Fr(i)p Fs(\))1099 2970 y Fv(\000)1145 3051 y Fq(T)1216 3010 y Fs(\(4\))1332 3051 y Fw(+)22 b Fq(T)1501 3010 y Fs(\(3\))1595 3051 y Fw(\004)1660 3010 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1829 2970 y Fv(\001)1892 3051 y Fq(P)1969 3010 y Fs(\()p Fr(i)p Fs(\))443 3281 y Fw(\004)508 3240 y Fs(\(1)p Fr(;j)t Fs(\))737 3281 y Fw(:=)83 b(\()p Fq(\013)1023 3296 y Fs(1)p Fr(;i)1129 3281 y Fu(\000)22 b Fq(\013)1290 3296 y Fs(1)p Fr(;j)1382 3281 y Fw(\))1420 3240 y Fm(\000)p Fs(1)1514 3281 y Fq(P)1591 3240 y Fs(\()p Fr(j)t Fs(\))1698 3200 y Fv(\000)1744 3281 y Fq(T)1815 3240 y Fs(\(4\))1931 3281 y Fw(+)g Fq(T)2100 3240 y Fs(\(3\))2194 3281 y Fw(\004)2259 3240 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2428 3200 y Fv(\001)2490 3281 y Fq(P)2567 3240 y Fs(\()p Fr(i)p Fs(\))421 3523 y Fw(\004)486 3482 y Fs(\(2)p Fr(;)p Fm(?)p Fs(\))737 3523 y Fw(:=)83 b(\()p Fq(H)1042 3538 y Fr(e)1079 3523 y Fw(\))1117 3482 y Fm(\000)p Fs(1)1117 3551 y Fm(?)1228 3353 y Fv(")1286 3523 y Fq(\013)1348 3538 y Fs(1)p Fr(;i)1431 3523 y Fw(\004)1496 3482 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1665 3523 y Fq(A)1738 3482 y Fs(2)1738 3548 y Fr(e)1800 3523 y Fu(\000)22 b Fq(QT)2047 3482 y Fs(\(4\))2164 3523 y Fu(\000)g Fq(QT)2411 3482 y Fs(\(3\))2522 3353 y Fv( )2601 3523 y Fw(\004)2666 3482 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))2857 3523 y Fw(+)2955 3429 y Fv(X)2977 3641 y Fr(l)q Fm(6)p Fs(=)p Fr(i)3116 3523 y Fw(\004)3181 3482 y Fs(\(1)p Fr(;l)q Fs(\))3316 3353 y Fv(!#)-71 3815 y Fh(and)34 b(then)h(r)-5 b(e)g(cursively)35 b(de\014ne)84 4093 y Fw(\003)152 4052 y Fs(\()p Fr(n;i)p Fs(\))381 4093 y Fw(:=)83 b Fq(P)644 4052 y Fs(\()p Fr(i)p Fs(\))743 3923 y Fv( )822 4093 y Fq(T)893 4052 y Fs(\()p Fr(n)p Fs(+2\))1106 4093 y Fw(+)1210 3969 y Fr(n)p Fm(\000)p Fs(1)1204 3998 y Fv(X)1215 4208 y Fr(s)p Fs(=1)1365 4093 y Fq(T)1436 4052 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))1715 4093 y Fw(\004)1780 4052 y Fs(\()p Fr(s;)p Fm(?)p Fs(\))1969 4093 y Fw(+)2072 3969 y Fr(n)p Fm(\000)p Fs(2)2067 3998 y Fv(X)2077 4208 y Fr(s)p Fs(=1)2227 3998 y Fv(X)2249 4211 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2388 4093 y Fq(T)2459 4052 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))2738 4093 y Fw(\004)2803 4052 y Fs(\()p Fr(s;l)q Fs(\))2936 3923 y Fv(!)3032 4093 y Fq(P)3109 4052 y Fs(\()p Fr(i)p Fs(\))-11 4444 y Fw(\004)54 4403 y Fs(\()p Fr(n)p Fm(\000)p Fs(1)p Fr(;j)t Fs(\))381 4444 y Fw(:=)g(\()p Fq(\013)667 4459 y Fs(1)p Fr(;i)772 4444 y Fu(\000)22 b Fq(\013)933 4459 y Fs(1)p Fr(;j)1025 4444 y Fw(\))1063 4403 y Fm(\000)p Fs(1)1157 4444 y Fq(P)1234 4403 y Fs(\()p Fr(j)t Fs(\))1342 4274 y Fv( )1420 4444 y Fq(T)1491 4403 y Fs(\()p Fr(n)p Fs(+2\))1705 4444 y Fw(+)1809 4319 y Fr(n)p Fm(\000)p Fs(1)1803 4349 y Fv(X)1814 4559 y Fr(s)p Fs(=1)1964 4444 y Fq(T)2035 4403 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))2314 4444 y Fw(\004)2379 4403 y Fs(\()p Fr(s;)p Fm(?)p Fs(\))567 4779 y Fw(+)702 4655 y Fr(n)p Fm(\000)p Fs(2)697 4684 y Fv(X)707 4894 y Fr(s)p Fs(=1)857 4684 y Fv(X)879 4897 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1018 4779 y Fq(T)1089 4738 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))1368 4779 y Fw(\004)1433 4738 y Fs(\()p Fr(s;l)q Fs(\))1588 4779 y Fu(\000)1693 4655 y Fr(n)p Fm(\000)p Fs(1)1688 4684 y Fv(X)1698 4894 y Fr(s)p Fs(=2)1848 4779 y Fq(\013)1910 4794 y Fr(s;i)1991 4779 y Fw(\004)2056 4738 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;j)t Fs(\))2297 4609 y Fv(!)2393 4779 y Fq(P)2470 4738 y Fs(\()p Fr(i)p Fs(\))56 5130 y Fw(\004)121 5089 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))381 5130 y Fw(:=)83 b(\()p Fq(H)686 5145 y Fr(e)722 5130 y Fw(\))760 5089 y Fm(\000)p Fs(1)760 5158 y Fm(?)871 4960 y Fv(")935 5005 y Fr(n)p Fm(\000)p Fs(1)929 5035 y Fv(X)940 5245 y Fr(s)p Fs(=1)1090 5130 y Fq(\013)1152 5145 y Fr(s;i)1232 5130 y Fw(\004)1297 5089 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;)p Fm(?)p Fs(\))1561 5130 y Fq(A)1634 5089 y Fs(2)1634 5155 y Fr(e)1696 5130 y Fu(\000)22 b Fq(QT)1943 5089 y Fs(\()p Fr(n)p Fs(+2\))2158 5130 y Fu(\000)2263 5005 y Fr(n)p Fm(\000)p Fs(1)2257 5035 y Fv(X)2268 5245 y Fr(s)p Fs(=1)2418 5130 y Fq(QT)2566 5089 y Fs(\()p Fr(s)p Fs(+2\))2764 4960 y Fv( )2843 5130 y Fw(\004)2908 5089 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;)p Fm(?)p Fs(\))3194 5130 y Fw(+)3292 5035 y Fv(X)3314 5248 y Fr(l)q Fm(6)p Fs(=)p Fr(i)3453 5130 y Fw(\004)3518 5089 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;l)q Fs(\))3748 4960 y Fv(!#)1888 5470 y Fw(14)p eop %%Page: 15 15 15 14 bop -71 123 a Fh(wher)-5 b(e)34 b Fq(\013)266 138 y Fr(s;i)382 123 y Fh(is)g(\(by)h(assumption\))f(the)h(unique)g (eigenvalue)f(of)g Fw(\003)2335 87 y Fs(\()p Fr(s;i)p Fs(\))2470 123 y Fh(.)75 244 y(L)-5 b(et)34 b Fu(E)295 259 y Fr(n)341 244 y Fq(;)17 b( )448 259 y Fr(n)528 244 y Fh(b)-5 b(e)33 b(the)g(R-S)f(c)-5 b(o)g(e\016cients.)43 b(Then)32 b Fu(E)1840 259 y Fs(1)1912 244 y Fh(has)h(to)g(b)-5 b(e)32 b(e)-5 b(qual)33 b(to)g(one)f(of)h(the)g(eigenvalues)e(of)i Fw(\003)3821 208 y Fs(\(1\))3915 244 y Fh(,)-71 364 y(let)i(us)g(say)g Fu(E)418 379 y Fs(1)485 364 y Fw(=)27 b Fq(\013)650 379 y Fs(1)p Fr(;i)733 364 y Fh(.)45 b(Conse)-5 b(quently,)34 b Fq( )1491 379 y Fs(0)1559 364 y Fu(2)28 b Fq(B)1727 379 y Fr(i)1790 364 y Fh(and)1244 560 y Fu(E)1297 575 y Fr(n)1426 560 y Fw(=)83 b Fq(\013)1647 575 y Fr(n;i)3771 560 y Fw(\(28\))1185 799 y Fq( )1252 758 y Fs(\()p Fr(j)t Fs(\))1248 823 y Fr(n)1426 799 y Fw(=)g(\004)1650 758 y Fs(\()p Fr(n;j)t Fs(\))1804 799 y Fq( )1867 814 y Fs(0)1929 799 y Fw(+)2033 674 y Fr(n)p Fm(\000)p Fs(1)2027 704 y Fv(X)2038 913 y Fr(s)p Fs(=1)2188 799 y Fw(\004)2253 758 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;j)t Fs(\))2494 799 y Fq( )2561 758 y Fs(\()p Fr(i)p Fs(\))2557 823 y Fr(s)3771 799 y Fw(\(29\))1217 1134 y Fq( )1284 1093 y Fm(?)1280 1158 y Fr(n)1426 1134 y Fw(=)g(\004)1650 1093 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))1827 1134 y Fq( )1890 1149 y Fs(0)1952 1134 y Fw(+)2055 1009 y Fr(n)p Fm(\000)p Fs(1)2050 1039 y Fv(X)2060 1248 y Fr(s)p Fs(=1)2210 1134 y Fw(\004)2275 1093 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;)p Fm(?)p Fs(\))2539 1134 y Fq( )2606 1093 y Fs(\()p Fr(i)p Fs(\))2602 1158 y Fr(s)3771 1134 y Fw(\(30\))1233 1443 y Fq( )1296 1458 y Fr(n)1426 1443 y Fw(=)g Fq( )1652 1402 y Fm(?)1648 1468 y Fr(n)1733 1443 y Fw(+)1831 1349 y Fv(X)1848 1561 y Fr(j)t Fm(6)p Fs(=)p Fr(i)1992 1443 y Fq( )2059 1402 y Fs(\()p Fr(j)t Fs(\))2055 1468 y Fr(n)2172 1443 y Fw(+)22 b Fq( )2337 1402 y Fs(\()p Fr(i)p Fs(\))2333 1468 y Fr(n)-71 1769 y Fh(The)34 b(ve)-5 b(ctors)35 b Fq( )523 1718 y Fs(\()p Fr(i)p Fs(\))519 1793 y(1)606 1769 y Fq(;)17 b(:)g(:)g(:)f(;)h( )892 1718 y Fs(\()p Fr(i)p Fs(\))888 1779 y Fr(n)1009 1769 y Fh(ar)-5 b(e)35 b(arbitr)-5 b(arily)35 b(chosen)f(fr)-5 b(om)34 b Fq(B)2260 1784 y Fr(i)2289 1769 y Fh(.)-71 1970 y(Pr)-5 b(o)g(of.)60 b Fw(Use)40 b(mathematical)35 b(induction.)60 b(Because)40 b(of)d(the)i(discussion) g(ab)s(o)m(v)m(e,)i(w)m(e)e(only)f(ha)m(v)m(e)i(to)e(pro)m(v)m(e)-71 2090 y(the)43 b(inductiv)m(e)f(step.)74 b(Th)m(us,)47 b(let)41 b(us)i(assume)g(that)f Fu(E)2006 2105 y Fr(m)2073 2090 y Fw(,)i Fq( )2211 2039 y Fs(\()p Fr(j)t Fs(\))2207 2115 y Fr(m)p Fm(\000)p Fs(1)2407 2090 y Fw(and)e Fq( )2673 2054 y Fm(?)2669 2115 y Fr(m)2779 2090 y Fw(are)g(giv)m(en)h(b)m(y)g (\(28\){\(30\),)g(for)-71 2231 y Fq(m)28 b Fw(=)g(2)p Fq(;)17 b(:)g(:)g(:)e(;)i(n)33 b Fw(.)43 b(Let)33 b(us)g(compute)g Fu(E)1323 2246 y Fr(n)p Fs(+1)1459 2231 y Fw(,)g Fq( )1586 2180 y Fs(\()p Fr(j)t Fs(\))1582 2241 y Fr(n)1710 2231 y Fw(and)g Fq( )1967 2195 y Fm(?)1963 2255 y Fr(n)p Fs(+1)2100 2231 y Fw(.)44 b(The)33 b(\()p Fq(n)22 b Fw(+)h(1\)-st)32 b(order)g(equation)h(is)1019 2510 y Fq(H)1100 2525 y Fr(e)1136 2510 y Fq( )1199 2525 y Fr(n)p Fs(+1)1359 2510 y Fw(+)1508 2385 y Fr(n)1457 2415 y Fv(X)1466 2624 y Fr(p)p Fs(=0)1617 2510 y Fq(T)1688 2468 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))1971 2510 y Fq( )2034 2525 y Fr(p)2101 2510 y Fw(=)2255 2385 y Fr(n)2205 2415 y Fv(X)2216 2624 y Fr(s)p Fs(=0)2365 2510 y Fu(E)2418 2525 y Fr(n)p Fs(+1)p Fm(\000)p Fr(s)2643 2510 y Fq(A)2716 2468 y Fs(2)2716 2534 y Fr(e)2755 2510 y Fq( )2818 2525 y Fr(s)3771 2510 y Fw(\(31\))-71 2800 y(W)-8 b(e)33 b(ha)m(v)m(e)59 2923 y Fr(n)8 2953 y Fv(X)17 3162 y Fr(p)p Fs(=0)169 3048 y Fq(T)240 3007 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))522 3048 y Fq( )585 3063 y Fr(p)708 3048 y Fw(=)83 b Fq(T)938 3007 y Fs(\()p Fr(n)p Fs(+3\))1129 3048 y Fq( )1192 3063 y Fs(0)1254 3048 y Fw(+)1403 2923 y Fr(n)1352 2953 y Fv(X)1362 3162 y Fr(p)p Fs(=1)1513 3048 y Fq(T)1584 3007 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))1866 3048 y Fq( )1933 3007 y Fm(?)1929 3072 y Fr(p)2014 3048 y Fw(+)2163 2923 y Fr(n)2112 2953 y Fv(X)2121 3162 y Fr(p)p Fs(=1)2273 3048 y Fq(T)2344 3007 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2643 2953 y Fv(X)2664 3165 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2803 3048 y Fq( )2870 3007 y Fs(\()p Fr(l)q Fs(\))2866 3072 y Fr(p)2973 3048 y Fw(+)3122 2923 y Fr(n)3071 2953 y Fv(X)3080 3162 y Fr(p)p Fs(=1)3232 3048 y Fq(T)3303 3007 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))3585 3048 y Fq( )3652 3007 y Fs(\()p Fr(i)p Fs(\))3648 3072 y Fr(p)708 3403 y Fw(=)g Fq(T)938 3362 y Fs(\()p Fr(n)p Fs(+3\))1129 3403 y Fq( )1192 3418 y Fs(0)1254 3403 y Fw(+)22 b Fq(T)1423 3362 y Fs(\()p Fr(n)p Fs(+2\))1615 3403 y Fw(\004)1680 3362 y Fs(\(1)p Fr(;)p Fm(?)p Fs(\))1849 3403 y Fq( )1912 3418 y Fs(0)1974 3403 y Fw(+)2123 3278 y Fr(n)2072 3308 y Fv(X)2081 3517 y Fr(p)p Fs(=2)2233 3403 y Fq(T)2304 3362 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2602 3232 y Fv( )2681 3403 y Fw(\004)2746 3362 y Fs(\()p Fr(p;)p Fm(?)p Fs(\))2915 3403 y Fq( )2978 3418 y Fs(0)3040 3403 y Fw(+)3147 3274 y Fr(p)p Fm(\000)p Fs(1)3138 3308 y Fv(X)3149 3517 y Fr(s)p Fs(=1)3299 3403 y Fw(\004)3364 3362 y Fs(\()p Fr(p)p Fm(\000)p Fr(s;)p Fm(?)p Fs(\))3620 3403 y Fq( )3687 3362 y Fs(\()p Fr(i)p Fs(\))3683 3427 y Fr(s)3770 3232 y Fv(!)867 3755 y Fw(+)987 3660 y Fv(X)1008 3872 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1147 3755 y Fq(T)1218 3714 y Fs(\()p Fr(n)p Fs(+2\))1410 3755 y Fw(\004)1475 3714 y Fs(\(1)p Fr(;l)q Fs(\))1611 3755 y Fq( )1674 3770 y Fs(0)1736 3755 y Fw(+)1839 3630 y Fr(n)p Fm(\000)p Fs(1)1834 3660 y Fv(X)1843 3869 y Fr(p)p Fs(=2)1994 3660 y Fv(X)2016 3872 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2155 3755 y Fq(T)2226 3714 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2525 3584 y Fv( )2604 3755 y Fw(\004)2669 3714 y Fs(\()p Fr(p;l)q Fs(\))2805 3755 y Fq( )2868 3770 y Fs(0)2930 3755 y Fw(+)3037 3626 y Fr(p)p Fm(\000)p Fs(1)3028 3660 y Fv(X)3038 3869 y Fr(s)p Fs(=1)3188 3755 y Fw(\004)3253 3714 y Fs(\()p Fr(p)p Fm(\000)p Fr(s;l)q Fs(\))3477 3755 y Fq( )3544 3714 y Fs(\()p Fr(i)p Fs(\))3540 3779 y Fr(s)3627 3584 y Fv(!)867 4091 y Fw(+)987 3997 y Fv(X)1008 4209 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1147 4091 y Fq(T)1218 4050 y Fs(\(3\))1312 4091 y Fq( )1379 4050 y Fs(\()p Fr(l)q Fs(\))1375 4116 y Fr(n)1482 4091 y Fw(+)1631 3967 y Fr(n)1580 3997 y Fv(X)1590 4206 y Fr(p)p Fs(=1)1741 4091 y Fq(T)1812 4050 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2094 4091 y Fq( )2161 4050 y Fs(\()p Fr(i)p Fs(\))2157 4116 y Fr(p)708 4442 y Fw(=)867 4272 y Fv( )946 4442 y Fq(T)1017 4401 y Fs(\()p Fr(n)p Fs(+3\))1230 4442 y Fw(+)1379 4318 y Fr(n)1328 4348 y Fv(X)1338 4557 y Fr(p)p Fs(=1)1489 4442 y Fq(T)1560 4401 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))1842 4442 y Fw(\004)1907 4401 y Fs(\()p Fr(p;)p Fm(?)p Fs(\))2098 4442 y Fw(+)2202 4318 y Fr(n)p Fm(\000)p Fs(1)2196 4348 y Fv(X)2206 4557 y Fr(p)p Fs(=1)2357 4348 y Fv(X)2378 4560 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2517 4442 y Fq(T)2588 4401 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2871 4442 y Fw(\004)2936 4401 y Fs(\()p Fr(p;l)q Fs(\))3072 4272 y Fv(!)3167 4442 y Fq( )3230 4457 y Fs(0)867 4793 y Fw(+)992 4669 y Fr(n)p Fm(\000)p Fs(1)987 4699 y Fv(X)997 4908 y Fr(s)p Fs(=1)1233 4669 y Fr(n)1182 4699 y Fv(X)1147 4908 y Fr(p)p Fs(=)p Fr(s)p Fs(+1)1377 4793 y Fq(T)1448 4752 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))1730 4793 y Fw(\004)1795 4752 y Fs(\()p Fr(p)p Fm(\000)p Fr(s;)p 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Fv(X)2326 666 y Fr(s)p Fs(=1)2476 427 y Fr(n)p Fm(\000)p Fs(1)p Fm(\000)p Fr(s)2514 457 y Fv(X)2510 666 y Fr(m)p Fs(=1)2713 457 y Fv(X)2735 669 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2873 552 y Fq(T)2944 511 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(s)p Fm(\000)p Fr(m)p Fs(\))3341 552 y Fw(\004)3406 511 y Fs(\()p Fr(m;l)q Fs(\))3569 552 y Fq( )3636 511 y Fs(\()p Fr(i)p Fs(\))3632 577 y Fr(s)867 889 y Fw(+)987 794 y Fv(X)1008 1006 y Fr(l)q Fm(6)p Fs(=)p Fr(i)1147 889 y Fq(T)1218 847 y Fs(\(3\))1312 889 y Fq( )1379 847 y Fs(\()p Fr(l)q Fs(\))1375 913 y Fr(n)1482 889 y Fw(+)1631 764 y Fr(n)1580 794 y Fv(X)1591 1003 y Fr(s)p Fs(=1)1741 889 y Fq(T)1812 847 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(s)p Fs(\))2091 889 y Fq( )2158 847 y Fs(\()p Fr(i)p Fs(\))2154 913 y Fr(s)-71 1228 y Fw(where)25 b(w)m(e)g(use)g(that)700 1153 y Fv(P)805 1180 y Fr(r)805 1257 y(p)p Fs(=1)952 1153 y Fv(P)1057 1180 y Fr(p)p Fm(\000)p Fs(1)1057 1257 y Fr(s)p Fs(=1)1203 1228 y Fq(F)1266 1243 y Fr(sp)1366 1228 y Fw(=)1470 1153 y Fv(P)1575 1180 y Fr(r)r Fm(\000)p Fs(1)1575 1257 y Fr(s)p Fs(=1)1720 1153 y Fv(P)1825 1180 y Fr(r)1825 1257 y(p)p Fs(=)p Fr(s)p Fs(+1)2059 1228 y Fq(F)2122 1243 y Fr(sp)2218 1228 y Fw(and)f(then)h(w)m(e)g(c)m(hange)g(index)f Fq(p)k Fu(!)f Fq(m)h Fw(=)f Fq(p)5 b Fu(\000)g Fq(s)p Fw(.)75 1364 y(Let)33 b(us)g(m)m(ultiply)e(\(31\))h(b)m(y)h Fq(P)1184 1328 y Fs(\()p Fr(i)p Fs(\))1267 1364 y Fw(.)43 b(Since)33 b Fq(P)1669 1328 y Fs(\()p Fr(i)p Fs(\))1751 1364 y Fq(H)1832 1379 y Fr(e)1897 1364 y Fw(=)27 b(0)32 b(and)h Fq(P)2348 1328 y Fs(\()p Fr(i)p Fs(\))2431 1364 y Fq(A)2504 1328 y Fs(2)2504 1389 y Fr(e)2571 1364 y Fw(=)27 b Fq(A)2747 1328 y Fs(2)2747 1389 y Fr(e)2787 1364 y Fq(P)2864 1328 y Fs(\()p Fr(i)p Fs(\))2974 1364 y Fw(=)h Fq(P)3155 1328 y Fs(\()p Fr(i)p Fs(\))3237 1364 y Fw(,)33 b(w)m(e)g(obtain)1007 1536 y Fr(n)956 1566 y Fv(X)966 1775 y Fr(p)p Fs(=0)1117 1660 y Fq(P)1194 1619 y Fs(\()p Fr(i)p Fs(\))1276 1660 y Fq(T)1347 1619 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))1630 1660 y Fq( )1693 1675 y Fr(p)1760 1660 y Fw(=)28 b Fu(E)1917 1675 y Fr(n)p Fs(+1)2054 1660 y Fq( )2117 1675 y Fs(0)2179 1660 y Fw(+)2327 1536 y Fr(n)2277 1566 y Fv(X)2287 1775 y Fr(s)p Fs(=1)2437 1660 y Fq(\013)2499 1675 y Fr(n)p Fs(+1)p Fm(\000)p Fr(s;i)2768 1660 y Fq( )2835 1619 y Fs(\()p Fr(i)p Fs(\))2831 1685 y Fr(s)3771 1660 y Fw(\(32\))-71 1975 y(The)34 b(left-hand)d(side)i(can)g(b)s(e)f(written)h(as)168 2136 y Fr(n)118 2165 y Fv(X)127 2375 y Fr(p)p Fs(=0)278 2260 y Fq(P)355 2219 y Fs(\()p Fr(i)p Fs(\))438 2260 y Fq(T)509 2219 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))791 2260 y Fq( )854 2275 y Fr(p)977 2260 y Fw(=)83 b Fq(P)1213 2219 y Fs(\()p Fr(i)p Fs(\))1312 2090 y Fv( )1391 2260 y Fq(T)1462 2219 y Fs(\()p Fr(n)p Fs(+3\))1676 2260 y Fw(+)1824 2136 y Fr(n)1774 2165 y Fv(X)1783 2375 y Fr(p)p Fs(=1)1934 2260 y Fq(T)2005 2219 y Fs(\()p Fr(n)p Fs(+3)p Fm(\000)p Fr(p)p Fs(\))2288 2260 y Fw(\004)2353 2219 y Fs(\()p Fr(p;)p Fm(?)p Fs(\))2544 2260 y Fw(+)2647 2136 y Fr(n)p 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Fr(n)p Fm(\000)p Fs(1)1994 645 y Fv(X)2004 854 y Fr(s)p Fs(=1)2138 739 y Fw(\()p Fq(\013)2238 754 y Fs(1)p Fr(;i)2343 739 y Fu(\000)g Fq(\013)2505 754 y Fs(1)p Fr(;j)2596 739 y Fw(\)\004)2699 698 y Fs(\()p Fr(n)p Fm(\000)p Fr(s;j)t Fs(\))2940 739 y Fq( )3007 698 y Fs(\()p Fr(i)p Fs(\))3003 764 y Fr(s)3113 739 y Fw(+)3263 615 y Fr(k)3211 645 y Fv(X)3226 857 y Fr(l)q Fs(=1)3371 739 y Fq(P)3448 698 y Fs(\()p Fr(j)t Fs(\))3539 739 y Fq(T)3610 698 y Fs(\(3\))3704 739 y Fq( )3771 698 y Fs(\()p Fr(l)q Fs(\))3767 764 y Fr(n)-71 1036 y Fw(As)33 b(pro)m(v)m(en)h(in)e(\(19\),)g(the)h(last)f (term)g(ab)s(o)m(v)m(e)h(is)f(equal)h(to)f Fq(\013)2136 1051 y Fs(1)p Fr(;j)2228 1036 y Fq( )2295 985 y Fs(\()p Fr(j)t Fs(\))2291 1046 y Fr(n)2386 1036 y Fw(.)43 b(Th)m(us)35 b(\(34\))c(leads)i(to)1263 1309 y Fq( )1330 1268 y Fs(\()p Fr(j)t Fs(\))1326 1333 y Fr(n)1449 1309 y Fw(=)27 b(\004)1617 1268 y Fs(\()p Fr(n;j)t Fs(\))1771 1309 y Fq( )1834 1324 y Fs(0)1896 1309 y Fw(+)2000 1184 y Fr(n)p Fm(\000)p 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2356 y Fq(A)837 2315 y Fs(2)837 2381 y Fr(e)877 2356 y Fq( )944 2315 y Fm(?)940 2381 y Fr(p)1086 2356 y Fw(=)1295 2232 y Fr(n)1245 2262 y Fv(X)1255 2471 y Fr(s)p Fs(=1)1405 2356 y Fq(\013)1467 2371 y Fr(n)p Fs(+1)p Fm(\000)p Fr(s;i)1736 2356 y Fq(A)1809 2315 y Fs(2)1809 2381 y Fr(e)1848 2356 y Fw(\004)1913 2315 y Fs(\()p Fr(s;)p Fm(?)p Fs(\))2079 2356 y Fq( )2142 2371 y Fs(0)2204 2356 y Fw(+)2353 2232 y Fr(n)2302 2262 y Fv(X)2312 2471 y Fr(p)p Fs(=2)2472 2228 y Fr(p)p Fm(\000)p Fs(1)2463 2262 y Fv(X)2474 2471 y Fr(s)p Fs(=1)2623 2356 y Fq(\013)2685 2371 y Fr(n)p Fs(+1)p Fm(\000)p Fr(p;i)2957 2356 y Fq(A)3030 2315 y Fs(2)3030 2381 y Fr(e)3069 2356 y Fw(\004)3134 2315 y Fs(\()p Fr(p)p Fm(\000)p Fr(s;)p Fm(?)p Fs(\))3391 2356 y Fq( )3458 2315 y Fs(\()p Fr(i)p Fs(\))3454 2381 y Fr(s)1086 2704 y Fw(=)1295 2580 y Fr(n)1245 2610 y Fv(X)1255 2819 y Fr(s)p Fs(=1)1405 2704 y Fq(\013)1467 2719 y Fr(n)p Fs(+1)p Fm(\000)p Fr(s;i)1736 2704 y Fq(A)1809 2663 y Fs(2)1809 2729 y Fr(e)1848 2704 y 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Fs(1)3700 4543 y Fm(?)3834 4515 y Fw(w)m(e)-71 4636 y(obtain)32 b(the)h(desired)g(result.)2872 b Fg(2)75 4827 y Fw(As)38 b(b)s(efore,)h(w)m(e)f(set)g Fq( )922 4776 y Fs(\()p Fr(i)p Fs(\))918 4837 y Fr(n)1041 4827 y Fw(=)e(0)h(for)f(all)f Fq(n)h Fw(=)g(1)p Fq(;)17 b Fw(2)p Fq(;)g(:)g(:)g(:)n Fw(.)58 b(Consequen)m(tly)-8 b(,)41 b Fq( )2815 4842 y Fr(n)2899 4827 y Fw(will)35 b(b)s(e)j(orthogonal)d(to)i Fq( )3905 4842 y Fs(0)-71 4947 y Fw(and)1312 5130 y Fq( )1375 5145 y Fr(n)1450 5130 y Fw(=)1554 4960 y Fv( )1633 5130 y Fw(\004)1698 5089 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))1896 5130 y Fw(+)1994 5035 y Fv(X)2016 5248 y Fr(l)q Fm(6)p Fs(=)p Fr(i)2155 5130 y Fw(\004)2220 5089 y Fs(\()p Fr(n;l)q Fs(\))2363 4960 y Fv(!)2459 5130 y Fq( )2522 5145 y Fs(0)1888 5470 y Fw(17)p eop %%Page: 18 18 18 17 bop 75 123 a Fw(The)34 b(follo)m(wing)c(expressions)k(will)c(b)s (e)j(useful)g(later:)472 407 y(\003)540 366 y Fs(\()p Fr(n;i)p Fs(\))685 407 y Fq( )748 422 y Fs(0)871 407 y Fw(=)83 b Fq(P)1107 366 y Fs(\()p Fr(i)p Fs(\))1189 407 y Fq(T)1260 366 y Fs(\()p Fr(n)p Fs(+2\))1452 407 y Fq( )1515 422 y Fs(0)1577 407 y Fw(+)1681 282 y Fr(n)p Fm(\000)p Fs(2)1675 312 y Fv(X)1686 521 y Fr(s)p Fs(=1)1836 407 y Fq(P)1913 366 y Fs(\()p Fr(i)p Fs(\))1995 407 y Fq(T)2066 366 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))2345 407 y Fq( )2408 422 y Fr(s)2468 407 y Fw(+)22 b Fq(P)2643 366 y Fs(\()p Fr(i)p Fs(\))2725 407 y Fq(T)2796 366 y Fs(\(3\))2890 407 y Fq( )2957 366 y Fm(?)2953 431 y Fr(n)p Fm(\000)p Fs(1)3771 407 y Fw(\(36\))662 742 y Fq( )729 701 y Fm(?)725 766 y Fr(n)871 742 y Fw(=)83 b(\()p Fq(H)1149 757 y Fr(e)1185 742 y Fw(\))1223 701 y Fm(\000)p Fs(1)1223 770 y Fm(?)1334 571 y Fv(")1398 617 y Fr(n)p Fm(\000)p Fs(1)1392 647 y Fv(X)1403 856 y Fr(s)p Fs(=1)1553 742 y Fu(E)1606 757 y Fr(s)1642 742 y Fq(A)1715 701 y Fs(2)1715 766 y Fr(e)1755 742 y Fq( )1822 701 y Fm(?)1818 766 y Fr(n)p Fm(\000)p Fr(s)1975 742 y Fu(\000)23 b Fq(QT)2223 701 y Fs(\()p Fr(n)p Fs(+2\))2415 742 y Fq( )2478 757 y Fs(0)2540 742 y Fu(\000)2645 617 y Fr(n)p Fm(\000)p Fs(1)2639 647 y Fv(X)2650 856 y Fr(s)p Fs(=1)2800 742 y Fq(QT)2948 701 y Fs(\()p Fr(s)p Fs(+2\))3130 742 y Fq( )3193 757 y Fr(n)p Fm(\000)p Fr(s)3327 571 y Fv(#)3771 742 y Fw(\(37\))587 1077 y Fq( )654 1026 y Fs(\()p Fr(j)t Fs(\))650 1101 y Fr(n)p Fm(\000)p Fs(1)871 1077 y Fw(=)83 b(\()p Fq(\013)1130 1092 y Fs(1)p Fr(;i)1235 1077 y Fu(\000)23 b Fq(\013)1397 1092 y Fs(1)p Fr(;j)1488 1077 y Fw(\))1526 1036 y Fm(\000)p Fs(1)1637 906 y Fv( )1716 1077 y Fq(P)1793 1036 y Fs(\()p Fr(j)t Fs(\))1884 1077 y Fq(T)1955 1036 y Fs(\()p Fr(n)p Fs(+2\))2146 1077 y Fq( )2209 1092 y Fs(0)2271 1077 y Fw(+)2375 952 y Fr(n)p Fm(\000)p Fs(2)2369 982 y Fv(X)2380 1191 y Fr(s)p Fs(=1)2530 1077 y Fq(P)2607 1036 y Fs(\()p Fr(j)t Fs(\))2698 1077 y Fq(T)2769 1036 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))3048 1077 y Fq( )3111 1092 y Fr(s)1030 1412 y Fw(+)43 b Fq(P)1226 1371 y Fs(\()p Fr(j)t Fs(\))1317 1412 y Fq(T)1388 1371 y Fs(\(3\))1482 1412 y Fq(P)1545 1427 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fs(1\))1991 1412 y Fq( )2058 1371 y Fm(?)2054 1436 y Fr(n)p Fm(\000)p Fs(1)2213 1412 y Fu(\000)2319 1287 y Fr(n)p Fm(\000)p Fs(1)2313 1317 y Fv(X)2324 1526 y Fr(s)p Fs(=2)2474 1412 y Fu(E)2527 1427 y Fr(s)2563 1412 y Fq( )2630 1361 y Fs(\()p Fr(j)t Fs(\))2626 1434 y Fr(n)p Fm(\000)p Fr(s)2761 1242 y Fv(!)3771 1412 y Fw(\(38\))75 1709 y(Next,)34 b(let)e(us)h(estimate)f(the)h(gro)m(wth)g(of)f(these)i(co)s(e\016cien)m (ts.)44 b(Since)33 b Fu(E)2694 1724 y Fr(n)2741 1709 y Fq( )2804 1724 y Fs(0)2871 1709 y Fw(=)28 b(\003)3043 1673 y Fs(\()p Fr(n;i)p Fs(\))3188 1709 y Fq( )3251 1724 y Fs(0)3291 1709 y Fw(,)238 1908 y Fu(jE)319 1923 y Fr(n)366 1908 y Fu(j)83 b Fw(=)637 1823 y Fv(\014)637 1883 y(\014)670 1827 y(\012)717 1908 y Fq( )780 1923 y Fs(0)820 1908 y Fq(;)17 b Fw(\003)932 1867 y Fs(\()p Fr(n;i)p Fs(\))1077 1908 y Fq( )1140 1923 y Fs(0)1180 1827 y Fv(\013)1227 1823 y(\014)1227 1883 y(\014)476 2201 y Fu(\024)637 2116 y Fv(\014)637 2176 y(\014)670 2120 y(\012)717 2201 y Fq( )780 2216 y Fs(0)820 2201 y Fq(;)g(P)941 2159 y Fs(\()p Fr(i)p Fs(\))1023 2201 y Fq(T)1094 2159 y Fs(\()p Fr(n)p Fs(+2\))1286 2201 y Fq( )1349 2216 y Fs(0)1389 2120 y Fv(\013)1436 2116 y(\014)1436 2176 y(\014)1491 2201 y Fw(+)1595 2076 y Fr(n)p Fm(\000)p Fs(2)1589 2106 y Fv(X)1600 2315 y Fr(s)p Fs(=1)1750 2116 y Fv(\014)1750 2176 y(\014)1783 2120 y(\012)1830 2201 y Fq( )1893 2216 y Fs(0)1933 2201 y Fq(;)g(P)2054 2159 y Fs(\()p Fr(i)p Fs(\))2136 2201 y Fq(T)2207 2159 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))2486 2201 y Fq( )2549 2216 y Fr(s)2586 2120 y Fv(\013)2633 2116 y(\014)2633 2176 y(\014)2689 2201 y Fw(+)2787 2116 y Fv(\014)2787 2176 y(\014)2820 2120 y(\012)2867 2201 y Fq( )2930 2216 y Fs(0)2970 2201 y Fq(;)g(P)3091 2159 y Fs(\()p Fr(i)p Fs(\))3173 2201 y Fq(T)3244 2159 y Fs(\(3\))3338 2201 y Fq( )3405 2159 y Fm(?)3401 2225 y Fr(n)p Fm(\000)p Fs(1)3539 2120 y Fv(\013)3586 2116 y(\014)3586 2176 y(\014)476 2536 y Fu(\024)637 2451 y Fv(\015)637 2511 y(\015)692 2536 y Fq(T)763 2494 y Fs(\()p Fr(n)p Fs(+2\))955 2536 y Fq(P)1018 2551 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1186 2451 y Fv(\015)1186 2511 y(\015)1263 2536 y Fw(+)1367 2411 y Fr(n)p Fm(\000)p Fs(2)1361 2441 y Fv(X)1372 2650 y Fr(s)p Fs(=1)1522 2451 y Fv(\014)1522 2511 y(\014)1555 2455 y(\012)1602 2536 y Fq(T)1673 2494 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))1952 2536 y Fq( )2015 2551 y Fs(0)2055 2536 y Fq(;)g( )2162 2551 y Fr(s)2199 2455 y Fv(\013)2246 2451 y(\014)2246 2511 y(\014)2302 2536 y Fw(+)2400 2451 y Fv(\014)2400 2511 y(\014)2433 2455 y(\012)2480 2536 y Fq(T)2551 2494 y Fs(\(3\))2645 2536 y Fq( )2708 2551 y Fs(0)2748 2536 y Fq(;)g( )2859 2494 y Fm(?)2855 2560 y Fr(n)p Fm(\000)p Fs(1)2992 2455 y Fv(\013)3039 2451 y(\014)3039 2511 y(\014)476 2871 y Fu(\024)637 2786 y Fv(\015)637 2846 y(\015)692 2871 y Fq(T)763 2829 y Fs(\()p Fr(n)p Fs(+2\))955 2871 y Fq(P)1018 2886 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1186 2786 y Fv(\015)1186 2846 y(\015)1263 2871 y Fw(+)1367 2746 y Fr(n)p Fm(\000)p Fs(2)1361 2776 y Fv(X)1372 2985 y Fr(s)p Fs(=1)1522 2786 y Fv(\015)1522 2846 y(\015)1577 2871 y Fq(T)1648 2829 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))1928 2871 y Fq(P)1991 2886 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2159 2786 y Fv(\015)2159 2846 y(\015)2230 2871 y Fu(k)p Fq( )2343 2886 y Fr(s)2380 2871 y Fu(k)22 b Fw(+)2550 2786 y Fv(\015)2550 2846 y(\015)2606 2871 y Fq(T)2677 2829 y Fs(\(3\))2771 2871 y Fq(P)2834 2886 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)3002 2786 y Fv(\015)3002 2846 y(\015)3074 2871 y Fu(k)p Fq( )3191 2829 y Fm(?)3187 2895 y Fr(n)p Fm(\000)p Fs(1)3324 2871 y Fu(k)477 3191 y Fw(=)687 3067 y Fr(n)637 3097 y Fv(X)647 3306 y Fr(s)p Fs(=2)797 3107 y Fv(\015)797 3167 y(\015)853 3191 y Fq(T)924 3150 y Fs(\()p Fr(s)p Fs(+2\))1105 3191 y Fq(P)1168 3207 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1336 3107 y Fv(\015)1336 3167 y(\015)1408 3191 y Fu(k)p Fq( )1521 3206 y Fr(n)p Fm(\000)p Fr(s)1656 3191 y Fu(k)g Fw(+)1826 3107 y Fv(\015)1826 3167 y(\015)1881 3191 y Fq(T)1952 3150 y Fs(\(3\))2046 3191 y Fq(P)2109 3207 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2277 3107 y Fv(\015)2277 3167 y(\015)2349 3191 y Fu(k)p Fq( )2466 3150 y Fm(?)2462 3216 y Fr(n)p Fm(\000)p Fs(1)2600 3191 y Fu(k)1121 b Fw(\(39\))-71 3489 y(This)33 b(calculation)d(follo)m(ws)h(from)h(\(36\),)g(the)h (self-adjoin)m(tness)f(of)g Fq(T)2455 3453 y Fs(\()p Fr(l)q Fs(\))2536 3489 y Fw(,)g(and)h(Lemma)e(1.)75 3609 y(It)i(is)f(easy)i(to)e(see)i(that)e Fu(k)p Fw(\()p Fq(H)1151 3624 y Fr(e)1188 3609 y Fw(\))1226 3568 y Fm(\000)p Fs(1)1226 3637 y Fm(?)1320 3609 y Fu(k)27 b Fw(=)h Fu(k)p Fq(A)1624 3624 y Fr(e)1661 3609 y Fu(k)f Fw(=)h(1.)43 b(Then)34 b(from)d(\(37\))h(w)m(e)i(ha)m(v)m(e)169 3816 y Fv(\015)169 3876 y(\015)225 3901 y Fq( )292 3860 y Fm(?)288 3925 y Fr(n)351 3816 y Fv(\015)351 3876 y(\015)489 3901 y Fu(\024)655 3776 y Fr(n)p Fm(\000)p Fs(1)650 3806 y Fv(X)660 4015 y Fr(s)p Fs(=1)810 3901 y Fu(jE)891 3916 y Fr(s)927 3901 y Fu(j)972 3816 y Fv(\015)972 3876 y(\015)1027 3901 y Fq( )1094 3860 y Fm(?)1090 3925 y Fr(n)p Fm(\000)p Fr(s)1225 3816 y Fv(\015)1225 3876 y(\015)1302 3901 y Fw(+)1400 3816 y Fv(\015)1400 3876 y(\015)1456 3901 y Fq(T)1527 3860 y Fs(\()p Fr(n)p Fs(+2\))1718 3901 y Fq(P)1781 3916 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1949 3816 y Fv(\015)1949 3876 y(\015)2021 3901 y Fu(k)p Fq( )2134 3916 y Fs(0)2174 3901 y Fu(k)22 b Fw(+)2349 3776 y Fr(n)p Fm(\000)p Fs(1)2344 3806 y Fv(X)2354 4015 y Fr(s)p Fs(=1)2504 3816 y Fv(\015)2504 3876 y(\015)2560 3901 y Fq(T)2631 3860 y Fs(\()p Fr(s)p Fs(+2\))2812 3901 y Fq(P)2875 3916 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fr(s)p Fs(\))3319 3816 y Fv(\015)3319 3876 y(\015)3391 3901 y Fu(k)o Fq( )3503 3916 y Fr(n)p Fm(\000)p Fr(s)3638 3901 y Fu(k)490 4236 y Fw(=)655 4111 y Fr(n)p Fm(\000)p Fs(1)650 4141 y Fv(X)660 4350 y Fr(s)p Fs(=1)810 4236 y Fu(jE)891 4251 y Fr(s)927 4236 y Fu(j)972 4151 y Fv(\015)972 4211 y(\015)1027 4236 y Fq( )1094 4195 y Fm(?)1090 4261 y Fr(n)p Fm(\000)p Fr(s)1225 4151 y Fv(\015)1225 4211 y(\015)1302 4236 y Fw(+)1451 4111 y Fr(n)1400 4141 y Fv(X)1411 4350 y Fr(s)p Fs(=1)1561 4151 y Fv(\015)1561 4211 y(\015)1616 4236 y Fq(T)1687 4195 y Fs(\()p Fr(s)p Fs(+2\))1869 4236 y Fq(P)1932 4251 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3\()p Fr(n)p Fm(\000)p Fr(s)p Fs(\))2375 4151 y Fv(\015)2375 4211 y(\015)2447 4236 y Fu(k)p Fq( )2560 4251 y Fr(n)p Fm(\000)p Fr(s)2695 4236 y Fu(k)1026 b Fw(\(40\))75 4522 y(Finally)30 b(let)i(us)h(consider)g(\(38\))339 4696 y Fv(\015)339 4756 y(\015)339 4816 y(\015)394 4811 y Fq( )461 4760 y Fs(\()p Fr(j)t Fs(\))457 4835 y Fr(n)p Fm(\000)p Fs(1)595 4696 y Fv(\015)595 4756 y(\015)595 4816 y(\015)733 4811 y Fu(\024)84 b(j)p Fq(\013)984 4826 y Fs(1)p Fr(;i)1089 4811 y Fu(\000)22 b Fq(\013)1250 4826 y Fs(1)p Fr(;j)1342 4811 y Fu(j)1370 4770 y Fm(\000)p Fs(1)1480 4640 y Fv( )1559 4726 y(\015)1559 4786 y(\015)1615 4811 y Fq(T)1686 4770 y Fs(\()p Fr(n)p Fs(+2\))1877 4811 y Fq(P)1940 4826 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2108 4726 y Fv(\015)2108 4786 y(\015)2180 4811 y Fu(k)p Fq( )2293 4826 y Fs(0)2333 4811 y Fu(k)g Fw(+)2508 4686 y Fr(n)p Fm(\000)p Fs(2)2503 4716 y Fv(X)2513 4925 y Fr(s)p Fs(=1)2663 4726 y Fv(\015)2663 4786 y(\015)2719 4811 y Fq(P)2796 4770 y Fs(\()p Fr(j)t Fs(\))2887 4811 y Fq(T)2958 4770 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))3237 4726 y Fv(\015)3237 4786 y(\015)3309 4811 y Fu(k)p Fq( )3422 4826 y Fr(s)3459 4811 y Fu(k)894 5146 y Fw(+)1013 5061 y Fv(\015)1013 5121 y(\015)1068 5146 y Fq(P)1145 5105 y Fs(\()p Fr(j)t Fs(\))1236 5146 y Fq(T)1307 5105 y Fs(\(3\))1401 5061 y Fv(\015)1401 5121 y(\015)1473 5061 y(\015)1473 5121 y(\015)1528 5146 y Fq( )1595 5105 y Fm(?)1591 5170 y Fr(n)p Fm(\000)p Fs(1)1729 5061 y Fv(\015)1729 5121 y(\015)1806 5146 y Fw(+)1910 5021 y Fr(n)p Fm(\000)p Fs(1)1904 5051 y Fv(X)1915 5260 y Fr(s)p Fs(=2)2065 5146 y Fu(jE)2146 5161 y Fr(s)2182 5146 y Fu(j)2227 5031 y Fv(\015)2227 5091 y(\015)2227 5151 y(\015)2281 5146 y Fq( )2348 5095 y Fs(\()p Fr(j)t Fs(\))2344 5168 y Fr(n)p Fm(\000)p Fr(s)2479 5031 y Fv(\015)2479 5091 y(\015)2479 5151 y(\015)2535 4975 y(!)1888 5470 y Fw(18)p eop %%Page: 19 19 19 18 bop -71 123 a Fw(Set)42 b Fq(D)j Fw(:=)e(min)540 138 y Fr(j)t Fm(6)p Fs(=)p Fr(i)672 123 y Fu(j)p Fq(\013)762 138 y Fs(1)p Fr(;i)873 123 y Fu(\000)29 b Fq(\013)1041 138 y Fs(1)p Fr(;j)1132 123 y Fu(j)1160 87 y Fm(\000)p Fs(1)1254 123 y Fw(.)70 b(Also,)43 b(let)d(us)i(notice)f(that)2408 39 y Fv(\015)2408 98 y(\015)2463 123 y Fq(P)2540 87 y Fs(\()p Fr(j)t Fs(\))2631 123 y Fq(T)2702 87 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))2982 39 y Fv(\015)2982 98 y(\015)3079 123 y Fw(=)3198 39 y Fv(\015)3198 98 y(\015)3253 123 y Fq(T)3324 87 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))3603 123 y Fq(P)3680 87 y Fs(\()p Fr(j)t Fs(\))3771 39 y Fv(\015)3771 98 y(\015)3869 123 y Fw(=)-71 168 y Fv(\015)-71 228 y(\015)-16 252 y Fq(T)55 216 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))335 252 y Fq(P)398 268 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)566 252 y Fq(P)643 216 y Fs(\()p Fr(j)t Fs(\))734 168 y Fv(\015)734 228 y(\015)817 252 y Fu(\024)922 168 y Fv(\015)922 228 y(\015)977 252 y Fq(T)1048 216 y Fs(\()p Fr(n)p Fs(+2)p Fm(\000)p Fr(s)p Fs(\))1327 252 y Fq(P)1390 268 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)1558 168 y Fv(\015)1558 228 y(\015)1614 252 y Fw(.)i(Th)m(us)91 424 y Fv(\015)91 484 y(\015)91 544 y(\015)146 539 y Fq( )213 488 y Fs(\()p Fr(j)t Fs(\))209 563 y Fr(n)p Fm(\000)p Fs(1)347 424 y Fv(\015)347 484 y(\015)347 544 y(\015)430 539 y Fu(\024)28 b Fq(D)641 414 y Fr(n)p Fm(\000)p Fs(1)635 444 y Fv(X)646 653 y Fr(s)p Fs(=2)796 539 y Fu(jE)877 554 y Fr(s)913 539 y Fu(j)958 424 y Fv(\015)958 484 y(\015)958 544 y(\015)1012 539 y Fq( )1079 488 y Fs(\()p Fr(j)t Fs(\))1075 560 y Fr(n)p Fm(\000)p Fr(s)1210 424 y Fv(\015)1210 484 y(\015)1210 544 y(\015)1288 539 y Fw(+)22 b Fq(D)1537 414 y Fr(n)1486 444 y Fv(X)1497 653 y Fr(s)p Fs(=2)1646 454 y Fv(\015)1646 514 y(\015)1702 539 y Fq(T)1773 498 y Fs(\()p Fr(s)p Fs(+2\))1954 539 y Fq(P)2017 554 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)2185 454 y Fv(\015)2185 514 y(\015)2257 539 y Fu(k)p Fq( )2370 554 y Fr(n)p Fm(\000)p Fr(s)2505 539 y Fu(k)g Fw(+)g Fq(D)2775 454 y Fv(\015)2775 514 y(\015)2831 539 y Fq(T)2902 498 y Fs(\(3\))2996 539 y Fq(P)3059 554 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)3227 454 y Fv(\015)3227 514 y(\015)3299 454 y(\015)3299 514 y(\015)3354 539 y Fq( )3421 498 y Fm(?)3417 563 y Fr(n)p Fm(\000)p Fs(1)3555 454 y Fv(\015)3555 514 y(\015)3771 539 y Fw(\(41\))75 806 y(These)37 b(inequalities)d(will)f(allo)m(w)g(us)j(to)f(obtain)f (upp)s(er)i(b)s(ounds)g(for)e(the)i(gro)m(wth)f(of)g(R-S)g(co)s (e\016cien)m(ts.)-71 927 y(T)-8 b(o)33 b(do)f(that,)h(w)m(e)g(need)h (some)e(tec)m(hnical)h(results.)-71 1116 y Fn(Lemma)k(3)49 b Fh(L)-5 b(et)35 b Fq(\024)28 b Fu(\025)g Fw(2)35 b Fh(b)-5 b(e)34 b(the)h(numb)-5 b(er)35 b(de\014ne)-5 b(d)34 b(in)g(L)-5 b(emma)34 b(2.)45 b(Then)45 1291 y(1.)j(F)-7 b(or)34 b(e)-5 b(ach)34 b Fq(a)28 b Fu(\025)g Fw(0)35 b Fh(ther)-5 b(e)35 b(is)f(a)h(c)-5 b(onstant)34 b Fq(\015)1713 1306 y Fr(a)1790 1291 y Fh(so)g(that,)h(for)g(al)5 b(l)34 b Fq(m)28 b Fu(\025)h Fw(0)p Fh(,)1239 1437 y Fr(m)1198 1467 y Fv(X)1214 1679 y Fr(l)q Fs(=0)1359 1421 y Fv(\024)1421 1494 y Fw(\(1)22 b(+)g Fq(a)h Fw(+)f Fq(m)g Fu(\000)h Fq(l)r Fw(\)!\(1)f(+)g Fq(a)g Fw(+)g Fq(l)r Fw(\)!)p 1421 1539 1157 4 v 1735 1630 a(\(1)g(+)g Fq(a)g Fw(+)g Fq(m)p Fw(\)!)2587 1421 y Fv(\025)2650 1416 y Fk(1)p 2650 1428 31 3 v 2650 1470 a(2)2722 1562 y Fu(\024)28 b Fq(\015)2878 1577 y Fr(a)45 1869 y Fh(2.)48 b(F)-7 b(or)34 b(al)5 b(l)34 b Fq(a)28 b Fu(\025)g(\000)p Fw(1)36 b Fh(ther)-5 b(e)34 b(is)h(a)g(c)-5 b(onstant)34 b Fq(\014)41 b Fh(so)34 b(that,)h(for)g(al)5 b(l)35 b Fq(m)28 b Fu(\025)g Fw(0)p Fh(,)1001 2015 y Fr(m)961 2045 y Fv(X)977 2257 y Fr(l)q Fs(=0)1121 2140 y Fq(\024)1177 2099 y Fm(\000)1242 2072 y Fk(5)p Fl(l)p 1242 2084 51 3 v 1252 2125 a Fk(2)1323 1999 y Fv(\024)1386 2072 y Fw(\(1)22 b(+)g Fq(a)g Fw(+)g(3)p Fq(m)g Fu(\000)h Fw(2)p Fq(l)r Fw(\)!\(1)f(+)g Fq(a)g Fw(+)g Fq(m)h Fu(\000)f Fq(l)r Fw(\)!)p 1386 2117 1461 4 v 1462 2208 a(\(1)g(+)g Fq(a)h Fw(+)f(3)p Fq(m)g Fu(\000)h Fw(3)p Fq(l)r Fw(\)!\(1)e(+)h Fq(a)h Fw(+)f Fq(m)p Fw(\)!)2857 1999 y Fv(\025)2919 1995 y Fk(1)p 2919 2007 31 3 v 2919 2048 a(2)2991 2140 y Fu(\024)29 b Fq(\014)45 2447 y Fh(3.)48 b(F)-7 b(or)34 b(e)-5 b(ach)34 b Fq(a)28 b Fu(\025)g Fw(0)35 b Fh(ther)-5 b(e)35 b(is)f(a)h(c)-5 b(onstant)34 b Fq(g)1709 2462 y Fr(a)1786 2447 y Fh(so)g(that,)h(for)g(al)5 b(l)35 b Fq(m)28 b Fu(\025)g Fw(0)p Fh(,)1140 2594 y Fr(m)1099 2623 y Fv(X)1115 2836 y Fr(l)q Fs(=1)1260 2718 y Fq(\024)1316 2677 y Fm(\000)1381 2650 y Fk(5)p Fl(l)p 1381 2662 51 3 v 1391 2703 a Fk(2)1462 2578 y Fv(\024)1524 2651 y Fw(\(1)22 b(+)g Fq(a)h Fw(+)f Fq(l)r Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(m)h Fu(\000)f Fq(l)r Fw(\)!)p 1524 2695 1157 4 v 1761 2786 a(\(1)g(+)g Fq(a)p Fw(\)!\()p Fq(a)g Fw(+)g Fq(m)p Fw(\)!)2690 2578 y Fv(\025)2753 2573 y Fk(1)p 2753 2585 31 3 v 2753 2626 a(2)2825 2718 y Fu(\024)28 b Fq(g)2977 2733 y Fr(a)-71 3026 y Fh(Pr)-5 b(o)g(of.)47 b Fw(Since)34 b Fq(\024)c Fu(\025)h Fw(2,)j(it)f(su\016ces)i(to)f(sho)m (w)h(the)f(assertions)h(for)e Fq(\024)d Fw(=)g(2.)47 b(Statemen)m(t)34 b(1)g(and)g(2)f(are)h(pro)m(v)m(en)-71 3146 y(in)e(Lemma)f(2)h(of)h([15)o(].)44 b(T)-8 b(o)32 b(sho)m(w)i(3,)e(notice)h(that)f(for)g(1)27 b Fu(\024)i Fq(l)h Fu(\024)e Fq(m)22 b Fu(\000)h Fw(1)32 b(w)m(e)i(ha)m(v)m(e)-61 3358 y(\(1)22 b(+)g Fq(a)g Fw(+)g Fq(l)r Fw(\)!\(1)g(+)g Fq(a)h Fw(+)f Fq(m)g Fu(\000)h Fq(l)r Fw(\)!)p -61 3403 1157 4 v 176 3494 a(\()p Fq(a)f Fw(+)g Fq(m)p Fw(\)!\(1)g(+)g Fq(a)p Fw(\)!)1133 3425 y(=)27 b(\(1)11 b(+)g Fq(a)g Fw(+)g Fq(l)r Fw(\))1656 3283 y Fv(Q)1750 3309 y Fr(m)p Fm(\000)p Fr(l)1750 3387 y(s)p Fs(=1)1893 3358 y Fw(\(1)22 b(+)g Fq(a)h Fw(+)f Fq(s)p Fw(\))p 1649 3403 713 4 v 1649 3431 a Fv(Q)1743 3457 y Fr(m)p Fm(\000)p Fs(1)1743 3535 y Fr(s)p Fs(=)p Fr(l)1900 3506 y Fw(\(1)g(+)g Fq(a)g Fw(+)g Fq(s)p Fw(\))2400 3425 y(=)27 b(\(1)11 b(+)g Fq(a)g Fw(+)g Fq(l)r Fw(\))2923 3301 y Fr(m)p Fm(\000)p Fr(l)2929 3331 y Fv(Y)2931 3540 y Fr(s)p Fs(=1)3089 3358 y Fw(1)22 b(+)g Fq(a)g Fw(+)g Fq(s)p 3089 3403 387 4 v 3098 3494 a(l)i Fw(+)e Fq(a)g Fw(+)g Fq(s)3513 3425 y Fu(\024)28 b Fw(1)11 b(+)g Fq(a)g Fw(+)g Fq(l)-71 3693 y Fw(Therefore)646 3754 y Fr(m)605 3784 y Fv(X)621 3996 y Fr(l)q Fs(=1)765 3879 y Fw(2)814 3838 y Fm(\000)879 3811 y Fk(5)p Fl(l)p 879 3823 51 3 v 889 3864 a Fk(2)960 3738 y Fv(\024)1023 3812 y Fw(\(1)21 b(+)h Fq(a)h Fw(+)f Fq(l)r Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(m)h Fu(\000)f Fq(l)r Fw(\)!)p 1023 3856 1157 4 v 1259 3947 a(\()p Fq(a)h Fw(+)f Fq(m)p Fw(\)!\(1)g(+)g Fq(a)p Fw(\)!)2189 3738 y Fv(\025)2251 3734 y Fk(1)p 2251 3746 31 3 v 2251 3787 a(2)2323 3879 y Fu(\024)2469 3754 y Fr(m)2429 3784 y Fv(X)2445 3996 y Fr(l)q Fs(=1)2589 3879 y Fw(2)2638 3838 y Fm(\000)2703 3811 y Fk(5)p Fl(l)p 2703 3823 51 3 v 2713 3864 a Fk(2)2767 3879 y Fw(\(1)g(+)g Fq(a)h Fw(+)f Fq(l)r Fw(\))3225 3811 y Fk(1)p 3225 3823 31 3 v 3225 3864 a(2)-71 4123 y Fw(where)34 b(the)f(righ)m(t-hand)f (side)g(con)m(v)m(erges)j(to)d(some)h(constan)m(t)g Fq(g)2295 4138 y Fr(a)2364 4123 y Fq(<)28 b Fu(1)p Fw(.)1275 b Fg(2)-71 4335 y Fn(Theorem)37 b(2)49 b Fh(De\014ne)42 b Fq(b)866 4350 y Fs(1)948 4335 y Fw(:=)g Fq(\015)1166 4224 y Fv(h)1213 4335 y Fq(k)s(g)1314 4350 y Fr(a)1378 4335 y Fw(+)22 b(\(2)g(+)g Fq(a)p Fw(\))1782 4271 y Fk(1)p 1782 4283 V 1782 4324 a(2)1826 4224 y Fv(i)1873 4335 y Fh(,)45 b Fq(b)1989 4350 y Fs(2)2071 4335 y Fw(:=)e(8)p Fq(D)2366 4224 y Fv(h)2413 4335 y Fq(b)2454 4350 y Fs(1)2494 4335 y Fq(\015)2545 4350 y Fr(a)2608 4335 y Fw(+)22 b Fq(\015)5 b Fw(\(2)22 b(+)g Fq(a)p Fw(\))3068 4271 y Fk(1)p 3068 4283 V 3068 4324 a(2)3135 4335 y Fw(+)g Fq(\015)5 b(k)s(g)3390 4350 y Fr(a)3431 4224 y Fv(i)3521 4335 y Fh(and)43 b Fq(b)3760 4350 y Fs(3)3842 4335 y Fw(:=)-71 4483 y Fq(b)-30 4498 y Fs(1)10 4483 y Fq(\015)61 4498 y Fr(a)124 4483 y Fw(+)22 b Fq(\015)5 b(\014)h Fw([1)22 b(+)g Fq(b)576 4498 y Fs(2)616 4483 y Fw(\()p Fq(k)j Fu(\000)e Fw(1\)])p Fh(.)44 b(Then)34 b(for)h(any)g Fq(b)28 b Fu(\025)g Fw(max)p Fu(f)p Fq(b)2062 4498 y Fs(1)2101 4483 y Fq(;)17 b(b)2186 4498 y Fs(2)2226 4483 y Fq(;)g(b)2311 4498 y Fs(3)2351 4483 y Fq(;)g Fw(1)p Fu(g)34 b Fh(and)g(for)h Fq(n)28 b Fw(=)f(1)p Fq(;)17 b Fw(2)p Fq(;)g(:)g(:)g(:)o Fh(,)1304 4689 y Fu(jE)1385 4704 y Fr(n)1431 4689 y Fu(j)83 b(\024)g Fq(b)1743 4704 y Fs(1)1783 4689 y Fq(\024)1839 4648 y Fs(3)p Fr(n)1921 4689 y Fq(b)1962 4648 y Fr(n)p Fm(\000)p Fs(2)2117 4689 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2513 4614 y Fk(1)p 2513 4626 V 2513 4667 a(2)3771 4689 y Fw(\(42\))1147 4818 y Fv(\015)1147 4878 y(\015)1147 4938 y(\015)1203 4933 y Fq( )1270 4882 y Fs(\()p Fr(l)q Fs(\))1266 4957 y Fr(n)p Fm(\000)p Fs(1)1403 4818 y Fv(\015)1403 4878 y(\015)1403 4938 y(\015)1542 4933 y Fu(\024)83 b Fq(b)1743 4948 y Fs(2)1783 4933 y Fq(\024)1839 4892 y Fs(3\()p Fr(n)p Fm(\000)p Fs(1\))2066 4933 y Fq(b)2107 4892 y Fr(n)p Fm(\000)p Fs(2)2262 4933 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2658 4857 y Fk(1)p 2658 4869 V 2658 4910 a(2)3771 4933 y Fw(\(43\))1222 5101 y Fv(\015)1222 5161 y(\015)1277 5186 y Fq( )1344 5145 y Fm(?)1340 5210 y Fr(n)1403 5101 y Fv(\015)1403 5161 y(\015)1542 5186 y Fu(\024)83 b Fq(b)1743 5201 y Fs(3)1783 5186 y Fq(\024)1839 5145 y Fs(3)p Fr(n)1921 5186 y Fq(b)1962 5145 y Fr(n)p Fm(\000)p Fs(2)2117 5186 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2682 5110 y Fk(1)p 2682 5122 V 2682 5163 a(2)3771 5186 y Fw(\(44\))1888 5470 y(19)p eop %%Page: 20 20 20 19 bop -71 123 a Fh(Pr)-5 b(o)g(of.)43 b Fw(Assume)34 b(the)f(estimates)f(are)g(true)h(for)f Fq(s)c Fw(=)g(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(n)22 b Fu(\000)h Fw(1.)43 b(This)33 b(implies)d(that)525 325 y Fu(k)p Fq( )638 340 y Fr(s)675 325 y Fu(k)e(\024)g Fw([1)22 b(+)g Fq(b)1095 340 y Fs(2)1135 325 y Fw(\()p Fq(k)j Fu(\000)e Fw(1\)])p Fq(\024)1519 283 y Fs(3)p Fr(s)1591 325 y Fq(b)1632 283 y Fr(s)p Fm(\000)p Fs(1)1759 325 y Fw([\(1)f(+)g Fq(a)h Fw(+)f Fq(s)p Fw(\)!])2313 256 y Fk(1)p 2313 268 31 3 v 2313 309 a(2)2385 325 y Fu(\024)28 b Fq(\024)2546 283 y Fs(3)p Fr(s)2618 325 y Fq(k)s(b)2713 283 y Fr(s)2751 325 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(s)p Fw(\)!])3304 256 y Fk(1)p 3304 268 V 3304 309 a(2)3771 325 y Fw(\(45\))-71 526 y(for)35 b Fq(s)e Fu(\024)h Fq(n)24 b Fu(\000)h Fw(2.)53 b(W)-8 b(e)36 b(will)d(use)k(the)f(second)h(inequalit)m(y)e(in)g(\(45\))g(to)g (pro)m(v)m(e)i(\(42\))f(and)f(\(43\),)h(and)g(the)g(\014rst)-71 646 y(one)d(to)f(pro)m(v)m(e)i(\(44\).)75 767 y(Let)f(us)g(start)g(b)m (y)g(sho)m(wing)g(\(42\).)43 b(Applying)32 b(Lemmas)f(2)i(and)f(3)h(w)m (e)g(obtain)26 933 y Fr(n)-25 963 y Fv(X)-14 1173 y Fr(s)p Fs(=2)136 973 y Fv(\015)136 1033 y(\015)191 1058 y Fq(T)262 1017 y Fs(\()p Fr(s)p Fs(+2\))444 1058 y Fq(P)507 1074 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)674 973 y Fv(\015)674 1033 y(\015)746 1058 y Fu(k)p Fq( )859 1073 y Fr(n)p Fm(\000)p Fr(s)994 1058 y Fu(k)83 b(\024)h Fq(\015)5 b(k)1465 933 y Fr(n)1414 963 y Fv(X)1425 1173 y Fr(s)p Fs(=2)1575 1058 y Fq(\024)1642 990 y Fl(s)p 1641 1002 V 1641 1043 a Fk(2)1702 918 y Fv(\024)1765 991 y Fw(\(1)21 b(+)h Fq(a)h Fw(+)f Fq(s)p Fw(\)!)p 1765 1035 490 4 v 1848 1126 a(\(1)f(+)h Fq(a)p Fw(\)!)2264 918 y Fv(\025)2326 913 y Fk(1)p 2326 925 31 3 v 2326 966 a(2)2387 1058 y Fq(\024)2443 1017 y Fs(3\()p Fr(n)p Fm(\000)p Fr(s)p Fs(\))2668 1058 y Fq(b)2709 1017 y Fr(n)p Fm(\000)p Fr(s)2844 1058 y Fw([\(1)g(+)g Fq(a)h Fw(+)f Fq(n)g Fu(\000)h Fq(s)p Fw(\)!])3578 990 y Fk(1)p 3578 1002 V 3578 1043 a(2)1127 1393 y Fu(\024)84 b Fq(\015)5 b(k)s(\024)1454 1352 y Fs(3)p Fr(n)1536 1393 y Fq(b)1577 1352 y Fr(n)p Fm(\000)p Fs(2)1715 1393 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2111 1325 y Fk(1)p 2111 1337 V 2111 1378 a(2)2223 1269 y Fr(n)2173 1299 y Fv(X)2183 1508 y Fr(s)p Fs(=2)2333 1393 y Fq(\024)2389 1352 y Fm(\000)2454 1325 y Fk(5)p Fl(s)p 2454 1337 60 3 v 2469 1378 a Fk(2)2544 1253 y Fv(\024)2607 1326 y Fw(\(1)g(+)g Fq(a)g Fw(+)g Fq(s)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)h Fu(\000)f Fq(s)p Fw(\)!)p 2607 1370 1159 4 v 2858 1462 a(\(1)g(+)g Fq(a)p Fw(\)!\()p Fq(a)h Fw(+)f Fq(n)p Fw(\)!)3775 1253 y Fv(\025)3838 1248 y Fk(1)p 3838 1260 31 3 v 3838 1301 a(2)1127 1699 y Fu(\024)84 b Fq(\015)5 b(k)s(g)1445 1714 y Fr(a)1486 1699 y Fq(\024)1542 1657 y Fs(3)p Fr(n)1624 1699 y Fq(b)1665 1657 y Fr(n)p Fm(\000)p Fs(2)1803 1699 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2199 1630 y Fk(1)p 2200 1642 V 2200 1683 a(2)-71 1900 y Fw(th)m(us)34 b(\(39\))e(yields)449 2114 y Fu(jE)530 2129 y Fr(n)576 2114 y Fu(j)83 b(\024)g Fq(\015)5 b(k)s(g)1004 2129 y Fr(a)1045 2114 y Fq(\024)1101 2073 y Fs(3)p Fr(n)1184 2114 y Fq(b)1225 2073 y Fr(n)p Fm(\000)p Fs(2)1363 2114 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])1759 2046 y Fk(1)p 1759 2058 V 1759 2099 a(2)1826 2114 y Fw(+)g Fq(\015)5 b(\024)2036 2073 y Fs(3\()p Fr(n)p Fm(\000)p Fs(1\))2263 2114 y Fq(b)2304 2129 y Fs(3)2344 2114 y Fq(b)2385 2073 y Fr(n)p Fm(\000)p Fs(3)2523 2114 y Fq(\024)2589 2046 y Fk(1)p 2589 2058 V 2589 2099 a(2)2634 2114 y Fw(\(2)21 b(+)h Fq(a)p Fw(\))2939 2046 y Fk(1)p 2940 2058 V 2940 2099 a(2)2984 2114 y Fw([\()p Fq(a)h Fw(+)f Fq(n)p Fw(\)!])3381 2046 y Fk(1)p 3381 2058 V 3381 2099 a(2)687 2320 y Fu(\024)83 b Fq(\015)5 b(k)s(g)1004 2335 y Fr(a)1045 2320 y Fq(\024)1101 2279 y Fs(3)p Fr(n)1184 2320 y Fq(b)1225 2279 y Fr(n)p Fm(\000)p Fs(2)1363 2320 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])1759 2252 y Fk(1)p 1759 2264 V 1759 2305 a(2)1826 2320 y Fw(+)g Fq(\015)5 b Fw(\(2)22 b(+)g Fq(a)p Fw(\))2286 2252 y Fk(1)p 2286 2264 V 2286 2305 a(2)2330 2320 y Fq(\024)2386 2279 y Fs(3)p Fr(n)2469 2320 y Fq(b)2510 2279 y Fr(n)p Fm(\000)p Fs(2)2647 2320 y Fw([\()p Fq(a)h Fw(+)f Fq(n)p Fw(\)!])3044 2252 y Fk(1)p 3044 2264 V 3044 2305 a(2)687 2526 y Fu(\024)83 b Fq(b)888 2541 y Fs(1)928 2526 y Fq(\024)984 2485 y Fs(3)p Fr(n)1066 2526 y Fq(b)1107 2485 y Fr(n)p Fm(\000)p Fs(2)1245 2526 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])1641 2457 y Fk(1)p 1642 2469 V 1642 2511 a(2)-71 2727 y Fw(whic)m(h)35 b(completes)g(the)g(pro)s(of)f(of)g(\(42\).)49 b(T)-8 b(o)34 b(pro)m(v)m(e)i(\(43\))e(w)m(e)i(start)f(from)e(\(41\))h(and)h (pro)s(ceed)g(in)f(the)h(same)-71 2847 y(fashion)24 2999 y Fv(\015)24 3059 y(\015)24 3119 y(\015)80 3114 y Fq( )147 3063 y Fs(\()p Fr(j)t Fs(\))143 3138 y Fr(n)p Fm(\000)p Fs(1)280 2999 y Fv(\015)280 3059 y(\015)280 3119 y(\015)418 3114 y Fu(\024)84 b Fq(D)s(\024)719 3073 y Fs(3)p Fr(n)801 3114 y Fq(b)842 3129 y Fs(1)882 3114 y Fq(b)923 3129 y Fs(2)963 3114 y Fq(b)1004 3073 y Fr(n)p Fm(\000)p Fs(3)1164 2989 y Fr(n)p Fm(\000)p Fs(1)1158 3019 y Fv(X)1169 3228 y Fr(s)p Fs(=2)1302 3114 y Fw([\()p Fq(a)22 b Fw(+)g Fq(s)p Fw(\)!\(1)g(+)g Fq(a)h Fw(+)f Fq(n)g Fu(\000)h Fq(s)p Fw(\)!])2356 3045 y Fk(1)p 2356 3057 V 2356 3099 a(2)2422 3114 y Fw(+)f Fq(D)s(\015)5 b(\024)2716 3073 y Fs(3)p Fr(n)2799 3114 y Fq(b)2840 3129 y Fs(3)2879 3114 y Fq(b)2920 3073 y Fr(n)p Fm(\000)p Fs(3)3058 3114 y Fw(\(2)22 b(+)g Fq(a)p Fw(\))3364 3045 y Fk(1)p 3364 3057 V 3364 3099 a(2)3409 3114 y Fw([\()p Fq(a)g Fw(+)g Fq(n)p Fw(\)!])3805 3045 y Fk(1)p 3805 3057 V 3805 3099 a(2)579 3449 y Fw(+)28 b Fq(D)s(\015)5 b(k)s(\024)933 3408 y Fs(3)p Fr(n)1015 3449 y Fq(b)1056 3408 y Fr(n)p Fm(\000)p Fs(2)1260 3324 y Fr(n)1210 3354 y Fv(X)1221 3564 y Fr(s)p Fs(=2)1370 3449 y Fq(\024)1426 3408 y Fm(\000)1491 3381 y Fk(5)p Fl(s)p 1491 3393 60 3 v 1506 3434 a Fk(2)1582 3308 y Fv(\024)1644 3382 y Fw(\(1)22 b(+)g Fq(a)g Fw(+)h Fq(s)p Fw(\)!\(1)e(+)h Fq(a)h Fw(+)f Fq(n)g Fu(\000)h Fq(s)p Fw(\)!)p 1644 3426 1159 4 v 2062 3517 a(\(1)f(+)g Fq(a)p Fw(\)!)2813 3308 y Fv(\025)2875 3304 y Fk(1)p 2875 3316 31 3 v 2875 3357 a(2)418 3784 y Fu(\024)84 b Fq(D)s(b)704 3799 y Fs(1)744 3784 y Fq(\024)800 3743 y Fs(3)p Fr(n)882 3784 y Fq(b)923 3743 y Fr(n)p Fm(\000)p Fs(2)1061 3784 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])1457 3716 y Fk(1)p 1457 3728 V 1457 3769 a(2)1528 3660 y Fr(n)p Fm(\000)p Fs(2)1523 3689 y Fv(X)1518 3899 y Fr(m)p Fs(=1)1688 3644 y Fv(\024)1750 3717 y Fw(\(1)g(+)g Fq(a)g Fw(+)g Fq(m)p Fw(\)!\()p Fq(a)h Fw(+)f Fq(n)g Fu(\000)h Fq(m)p Fw(\)!)p 1750 3761 1069 4 v 2118 3852 a(\()p Fq(a)f Fw(+)g Fq(n)p Fw(\)!)2828 3644 y Fv(\025)2891 3639 y Fk(1)p 2891 3651 31 3 v 2891 3692 a(2)579 4089 y Fw(+)28 b Fq(D)s(\015)5 b Fw(\(2)21 b(+)h Fq(a)p Fw(\))1128 4021 y Fk(1)p 1128 4033 V 1128 4074 a(2)1173 4089 y Fq(\024)1229 4048 y Fs(3)p Fr(n)1311 4089 y Fq(b)1352 4048 y Fr(n)p Fm(\000)p Fs(2)1490 4089 y Fw([\()p Fq(a)g Fw(+)g Fq(n)p Fw(\)!])1886 4021 y Fk(1)p 1887 4033 V 1887 4074 a(2)579 4372 y Fw(+)28 b Fq(D)s(\015)5 b(k)s(\024)933 4331 y Fs(3)p Fr(n)1015 4372 y Fq(b)1056 4331 y Fr(n)p Fm(\000)p Fs(2)1193 4372 y Fw([\()p Fq(a)23 b Fw(+)f Fq(n)p Fw(\)!])1590 4304 y Fk(1)p 1590 4316 V 1590 4357 a(2)1702 4248 y Fr(n)1651 4278 y Fv(X)1662 4487 y Fr(s)p Fs(=2)1812 4372 y Fq(\024)1868 4331 y Fm(\000)1933 4304 y Fk(5)p Fl(s)p 1933 4316 60 3 v 1947 4357 a Fk(2)2023 4232 y Fv(\024)2086 4305 y Fw(\(1)f(+)h Fq(a)h Fw(+)f Fq(s)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)g Fu(\000)h Fq(s)p Fw(\)!)p 2086 4349 1159 4 v 2337 4441 a(\(1)f(+)g Fq(a)p Fw(\)!\()p Fq(a)g Fw(+)g Fq(n)p Fw(\)!)3254 4232 y Fv(\025)3316 4227 y Fk(1)p 3316 4239 31 3 v 3316 4280 a(2)-71 4655 y Fw(where)31 b(w)m(e)f(ha)m(v)m(e)h(c)m(hanged)f(index)g Fq(s)d Fu(!)g Fq(m)i Fw(=)e Fq(s)15 b Fu(\000)g Fw(1)30 b(in)f(the)h(\014rst)f(term.) 42 b(F)-8 b(rom)28 b(this)h(and)g(statemen)m(ts)i(1)e(and)-71 4776 y(3)j(of)h(Lemma)e(3,)h(w)m(e)i(obtain)569 4884 y Fv(\015)569 4944 y(\015)569 5004 y(\015)624 4999 y Fq( )691 4948 y Fs(\()p Fr(j)t Fs(\))687 5023 y Fr(n)p Fm(\000)p Fs(1)825 4884 y Fv(\015)825 4944 y(\015)825 5004 y(\015)963 4999 y Fu(\024)84 b Fw(8)p Fq(D)1273 4888 y Fv(h)1320 4999 y Fq(b)1361 5014 y Fs(1)1401 4999 y Fq(\015)1452 5014 y Fr(a)1515 4999 y Fw(+)22 b Fq(\015)5 b Fw(\(2)22 b(+)g Fq(a)p Fw(\))1975 4930 y Fk(1)p 1975 4942 V 1975 4984 a(2)2042 4999 y Fw(+)g Fq(\015)5 b(k)s(g)2297 5014 y Fr(a)2338 4888 y Fv(i)2402 4999 y Fq(\024)2458 4957 y Fs(3\()p Fr(n)p Fm(\000)p Fs(1\))2685 4999 y Fq(b)2726 4957 y Fr(n)p Fm(\000)p Fs(2)2864 4999 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])3260 4930 y Fk(1)p 3261 4942 V 3261 4984 a(2)964 5221 y Fw(=)84 b Fq(b)1165 5236 y Fs(2)1205 5221 y Fq(\024)1261 5179 y Fs(3\()p Fr(n)p Fm(\000)p Fs(1\))1488 5221 y Fq(b)1529 5179 y Fr(n)p Fm(\000)p Fs(2)1667 5221 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2063 5152 y Fk(1)p 2063 5164 V 2063 5205 a(2)1888 5470 y Fw(20)p eop %%Page: 21 21 21 20 bop -71 123 a Fw(so)33 b(\(43\))f(is)g(done.)44 b(Consequen)m(tly)-8 b(,)35 b(\(45\))d(m)m(ust)g(b)s(e)h(v)-5 b(alid)31 b(for)h Fq(s)c Fw(=)f Fq(n)22 b Fu(\000)h Fw(1.)75 244 y(Finally)34 b(let)h(us)h(sho)m(w)h(\(44\).)52 b(Note)36 b(that)g(the)g(\014rst)g(term)f(of)h(\(40\))f(is)g(b)s(ounded)i(lik)m (e)e(the)h(\014rst)g(term)f(of)-71 364 y(\(41\).)43 b(Applying)32 b(statemen)m(t)h(2)f(of)g(Lemma)f(3,)i(it)e(follo)m(ws)h(that)-59 512 y Fv(\015)-59 571 y(\015)-4 596 y Fq( )63 555 y Fm(?)59 621 y Fr(n)122 512 y Fv(\015)122 571 y(\015)260 596 y Fu(\024)84 b Fq(b)462 611 y Fs(1)502 596 y Fq(b)543 611 y Fs(3)583 596 y Fq(\024)639 555 y Fs(3)p Fr(n)721 596 y Fq(b)762 555 y Fr(n)p Fm(\000)p Fs(3)900 596 y Fq(g)947 611 y Fr(a)988 596 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])1384 528 y Fk(1)p 1385 540 31 3 v 1385 581 a(2)421 842 y Fw(+)27 b Fq(\015)5 b Fw([1)22 b(+)g Fq(b)817 857 y Fs(2)857 842 y Fw(\()p Fq(k)j Fu(\000)e Fw(1\)])p Fq(\024)1241 801 y Fs(3)p Fr(n)1323 842 y Fq(b)1364 801 y Fr(n)p Fm(\000)p Fs(2)1502 842 y Fw([\(1)f(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2067 774 y Fk(1)p 2068 786 V 2068 827 a(2)2179 717 y Fr(n)2129 747 y Fv(X)2139 956 y Fr(s)p Fs(=1)2289 842 y Fq(\024)2345 801 y Fm(\000)2410 774 y Fk(5)p Fl(s)p 2410 786 60 3 v 2425 827 a Fk(2)2501 701 y Fv(\024)2563 774 y Fw(\(1)6 b(+)g Fq(a)g Fw(+)g(3)p Fq(n)g Fu(\000)g Fw(2)p Fq(s)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)g Fu(\000)g Fq(s)p Fw(\)!)p 2563 819 1237 4 v 2630 910 a(\(1)g(+)g Fq(a)g Fw(+)g(3)p Fq(n)g Fu(\000)g Fw(3)p Fq(s)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!)3810 701 y Fv(\025)3872 697 y Fk(1)p 3872 709 31 3 v 3872 750 a(2)260 1100 y Fu(\024)84 b Fq(b)462 1115 y Fs(1)502 1100 y Fq(g)549 1115 y Fr(a)590 1100 y Fq(\024)646 1059 y Fs(3)p Fr(n)729 1100 y Fq(b)770 1059 y Fr(n)p Fm(\000)p Fs(2)907 1100 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])1472 1032 y Fk(1)p 1473 1044 V 1473 1085 a(2)1539 1100 y Fw(+)g Fq(\015)5 b Fw([1)22 b(+)h Fq(b)1931 1115 y Fs(2)1970 1100 y Fw(\()p Fq(k)j Fu(\000)c Fw(1\)])p Fq(\014)6 b(\024)2415 1059 y Fs(3)p Fr(n)2497 1100 y Fq(b)2538 1059 y Fr(n)p Fm(\000)p Fs(2)2676 1100 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])3241 1032 y Fk(1)p 3241 1044 V 3241 1085 a(2)261 1377 y Fw(=)84 b Fq(b)462 1392 y Fs(3)502 1377 y Fq(\024)558 1335 y Fs(3)p Fr(n)640 1377 y Fq(b)681 1335 y Fr(n)p Fm(\000)p Fs(2)819 1377 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])1384 1308 y Fk(1)p 1384 1320 V 1384 1362 a(2)3870 1596 y Fg(2)-71 1823 y Fn(Corollary)36 b(1)1311 2057 y Fu(jE)1392 2072 y Fr(n)1439 2057 y Fu(j)82 b(\024)i Fq(\024)1766 2016 y Fs(3)p Fr(n)1848 2057 y Fq(b)1889 2016 y Fr(n)p Fm(\000)p Fs(1)2044 2057 y Fw([\()p Fq(a)22 b Fw(+)g Fq(n)p Fw(\)!])2440 1981 y Fk(1)p 2440 1993 V 2440 2034 a(2)1256 2293 y Fu(k)p Fq( )1369 2308 y Fr(n)1417 2293 y Fu(k)82 b(\024)i Fq(\024)1766 2252 y Fs(3)p Fr(n)1848 2293 y Fq(k)s(b)1943 2252 y Fr(n)2007 2293 y Fw([)q(\(1)21 b(+)h Fq(a)h Fw(+)f Fq(n)p Fw(\)!])2573 2218 y Fk(1)p 2573 2230 V 2573 2271 a(2)75 2521 y Fw(F)-8 b(or)29 b(the)g(case)h (where)h(degeneracy)f(is)f(partly)f(brok)m(en)j(only)d(up)i(to)e (second)j(order,)f(one)f(needs)i(to)e(de\014ne)-71 2641 y(certain)35 b(op)s(erators)g(\003)759 2605 y Fs(\()p Fr(n;i)873 2614 y Fk(1)907 2605 y Fr(;i)951 2614 y Fk(2)985 2605 y Fs(\))1017 2641 y Fw(,)h(\004)1145 2605 y Fs(\()p Fr(n)p Fm(\000)p Fs(2)p Fr(;i)1349 2614 y Fk(1)1383 2605 y Fr(;i)1427 2614 y Fk(2)1461 2605 y Fs(\))1493 2641 y Fw(,)g(\004)1621 2605 y Fs(\()p Fr(n;)p Fm(?)p Fs(\))1832 2641 y Fw(for)f Fq(n)d Fu(\025)h Fw(3,)i(in)f(addition)g(to)h(those)g (already)g(de\014ned)i(in)-71 2761 y(the)27 b(last)f(subsection.)43 b(No)m(w)27 b Fq( )1058 2776 y Fs(0)1124 2761 y Fw(w)m(ould)g(b)s(e)g (required)g(to)f(lie)f(in)h(a)g(certain)g(subspace)j Fq(B)3121 2776 y Fr(i)3145 2785 y Fk(1)3180 2776 y Fr(;)p Fs(1)3235 2785 y Fk(2)3301 2761 y Fu(2)f Fq(B)3469 2776 y Fr(i)3493 2785 y Fk(1)3560 2761 y Fu(2)g Fq(B)5 b Fw(,)28 b(and)-71 2882 y(one)i(w)m(ould)g(b)s(e)f(able)g(to)h(determine)f Fq( )1351 2897 y Fr(n)1428 2882 y Fw(mo)s(dule)f(an)h(arbitrary)g(comp) s(onen)m(t)h(in)f Fq(B)2994 2897 y Fr(i)3018 2906 y Fk(1)3052 2897 y Fr(;)p Fs(1)3107 2906 y Fk(2)3146 2882 y Fw(.)42 b(This)30 b(sc)m(heme)h(ma)m(y)-71 3002 y(b)s(e)39 b(extended)h(to)e (the)h(general)f(case.)62 b(But)39 b(the)f(complexit)m(y)g(of)g(the)h (set)g(of)f(equations)h(that)f(recursiv)m(ely)-71 3123 y(de\014nes)27 b(those)f(op)s(erators)f(rapidly)f(b)s(ecomes)h(wild.)40 b(F)-8 b(or)24 b(that)h(reason,)j(w)m(e)e(do)f(not)g(go)f(further.)42 b(W)-8 b(e)25 b(assume)-71 3243 y(instead)33 b(that,)f(in)g(general,) 1277 3477 y Fu(jE)1358 3492 y Fr(n)1404 3477 y Fu(j)82 b(\024)i Fq(\024)1731 3436 y Fs(3)p Fr(n)1813 3477 y Fq(b)1854 3436 y Fr(n)p Fs(+)p Fr(w)2026 3477 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2591 3401 y Fk(1)p 2591 3413 V 2591 3455 a(2)1221 3714 y Fu(k)p Fq( )1334 3729 y Fr(n)1382 3714 y Fu(k)82 b(\024)i Fq(\024)1731 3672 y Fs(3)p Fr(n)1813 3714 y Fq(b)1854 3672 y Fr(n)p Fs(+)p Fr(w)2026 3714 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2591 3638 y Fk(1)p 2591 3650 V 2591 3691 a(2)-71 3933 y Fw(for)32 b(some)h(p)s(ositiv)m(e)f(in)m(teger)g Fq(w)s Fw(,)g(whic)m(h)h(ma)m(y)f(dep)s(end)i(on)f(where)g(degeneracy)i (splits.)-71 4265 y Fx(4)161 b(The)54 b(t)l(w)l(o-side)d(error)i (function)-71 4484 y Fw(Our)25 b(upp)s(er)g(b)s(ounds)h(for)e Fu(jE)947 4499 y Fr(n)993 4484 y Fu(j)h Fw(and)g Fu(k)p Fq( )1341 4499 y Fr(n)1388 4484 y Fu(k)g Fw(will)d(allo)m(ws)i(us)h(to) g(estimate)f(the)h(error)g(made)f(in)g(the)i(Sc)m(hr\177)-49 b(odinger)-71 4605 y(equation)38 b(when)g(truncated)h(series)f(are)g (inserted)g(in)f(it.)58 b(W)-8 b(e)38 b(just)g(mimic)d(the)j(tec)m (hnique)h(dev)m(elop)s(ed)g(in)-71 4725 y([15],)32 b(whic)m(h)i(is)e (inspired)g(b)m(y)h([3].)75 4845 y(F)-8 b(or)32 b Fq(N)38 b Fu(\025)29 b Fw(1)j(de\014ne)725 5146 y Fq(E)797 5161 y Fr(N)892 5146 y Fw(=)27 b Fq(e)c Fw(+)1161 5021 y Fr(N)7 b Fm(\000)p Fs(1)1165 5051 y Fv(X)1171 5260 y Fr(n)p Fs(=1)1330 5146 y Ft(~)1394 5078 y Fl(n)p 1394 5090 39 3 v 1398 5131 a Fk(2)1447 5146 y Fu(E)1500 5161 y Fr(n)1862 5146 y Fw(\011)1938 5161 y Fr(N)2005 5146 y Fw(\()p Fq(x)p Fw(\))28 b(=)f Fq( )2330 5161 y Fs(0)2370 5146 y Fw(\()p Fq(x)p Fw(\))c(+)2622 5021 y Fr(N)7 b Fm(\000)p Fs(1)2626 5051 y Fv(X)2632 5260 y Fr(n)p Fs(=1)2792 5146 y Ft(~)2856 5078 y Fl(n)p 2855 5090 V 2859 5131 a Fk(2)2908 5146 y Fq( )2971 5161 y Fr(n)3018 5146 y Fw(\()p Fq(x)p Fw(\))1888 5470 y(21)p eop %%Page: 22 22 22 21 bop -71 123 a Fw(These)34 b(are)f(the)g(truncation)f(at)g(order)h Fq(N)43 b Fw(of)32 b(the)h(R-S)f(series.)44 b(Then)34 b(de\014ne)650 343 y Fq(\030)693 358 y Fr(N)760 343 y Fw(\()p Fq(x)p Fw(\))83 b(=)g Fq(A)1206 358 y Fr(e)1260 343 y Fw([)p Fq(H)1368 358 y Fs(0)1429 343 y Fw(+)22 b Fq(W)14 b Fw(\()p Ft(~)p Fw(;)j Fq(x)p Fw(\))22 b Fu(\000)h Fq(E)2056 358 y Fr(N)2123 343 y Fw(])17 b Fq(A)2240 358 y Fr(e)2277 343 y Fw(\011)2353 358 y Fr(N)2420 343 y Fw(\()p Fq(x)p Fw(\))974 605 y(=)1133 435 y Fv(")1191 605 y Fq(H)1272 620 y Fr(e)1331 605 y Fw(+)22 b Fq(A)1502 620 y Fr(e)1539 605 y Fq(W)14 b Fw(\()p Ft(~)p Fw(;)j Fq(x)p Fw(\))p Fq(A)1947 620 y Fr(e)2006 605 y Fu(\000)2106 480 y Fr(N)7 b Fm(\000)p Fs(1)2110 510 y Fv(X)2121 720 y Fr(j)t Fs(=1)2275 605 y Ft(~)2340 533 y Fl(j)p 2339 549 31 3 v 2339 590 a Fk(2)2384 605 y Fu(E)2437 620 y Fr(j)2473 605 y Fq(A)2546 564 y Fs(2)2546 630 y Fr(e)2585 435 y Fv(#)2660 480 y Fr(N)g Fm(\000)p Fs(1)2665 510 y Fv(X)2661 720 y Fr(m)p Fs(=0)2830 605 y Ft(~)2894 537 y Fl(m)p 2894 549 55 3 v 2906 590 a Fk(2)2962 605 y Fq( )3025 620 y Fr(m)3092 605 y Fw(\()p Fq(x)p Fw(\))548 b(\(46\))-71 926 y(W)-8 b(e)39 b(call)e Fq(\030)331 941 y Fr(N)398 926 y Fw(\()p Fq(x)p Fw(\))i(the)h(t)m(w)m(o-side)f(error)g(function)f (since)h(it)f(is)g(the)i(di\013erence)f(b)s(et)m(w)m(een)i(b)s(oth)e (sides)g(of)f(the)-71 1046 y(Sc)m(hr\177)-49 b(odinger)47 b(equation)g(when)i(exact)e(eigen)m(v)-5 b(alues)48 b(and)f (eigenfunctions)g(are)g(replaced)g(b)m(y)h(truncated)-71 1167 y(series.)69 b(It)41 b(can)g(b)s(e)g(p)s(ortra)m(y)m(ed)g(in)f(a)g (more)h(suitable)e(w)m(a)m(y)j(through)f(a)f(h)m(uge)i(n)m(um)m(b)s(er) f(of)f(cancellations.)-71 1287 y(This)33 b(calculation)d(is)i(outlined) f(in)h(the)h(app)s(endix.)44 b(The)33 b(result)g(is)380 1594 y Fq(\030)423 1609 y Fr(N)490 1594 y Fw(\()p Fq(x)p Fw(\))28 b(=)752 1470 y Fr(N)7 b Fm(\000)p Fs(1)757 1499 y Fv(X)763 1709 y Fr(n)p Fs(=0)922 1594 y Ft(~)986 1526 y Fl(n)p 986 1538 39 3 v 990 1579 a Fk(2)1039 1594 y Fq(A)1112 1609 y Fr(e)1149 1594 y Fq(W)1255 1553 y Fs([)p Fr(N)g Fm(\000)p Fr(n)p Fs(+1])1549 1594 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p Fq(A)1851 1609 y Fr(e)1887 1594 y Fq( )1950 1609 y Fr(n)1998 1594 y Fw(\()p Fq(x)p Fw(\))22 b Fu(\000)2251 1470 y Fs(2)p Fr(N)7 b Fm(\000)p Fs(2)2273 1499 y Fv(X)2265 1711 y Fr(n)p Fs(=)p Fr(N)2456 1594 y Ft(~)2520 1526 y Fl(n)p 2520 1538 V 2524 1579 a Fk(2)2681 1470 y Fr(N)g Fm(\000)p Fs(1)2686 1499 y Fv(X)2589 1711 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2944 1594 y Fu(E)2997 1609 y Fr(j)3033 1594 y Fq(A)3106 1553 y Fs(2)3106 1619 y Fr(e)3145 1594 y Fq( )3208 1609 y Fr(m)p Fm(\000)p Fr(j)3363 1594 y Fw(\()p Fq(x)p Fw(\))-71 1928 y(where)34 b Fq(W)317 1892 y Fs([)p Fr(j)t Fs(])392 1928 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))33 b(is)f(the)h(tail)d(of)i (the)h(T)-8 b(a)m(ylor)32 b(series)i(of)e Fq(V)21 b Fw(\()p Ft(~)p Fw(;)c Fq(x)p Fw(\))345 2243 y Fq(W)451 2202 y Fs([)p Fr(j)t Fs(])526 2243 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))27 b(=)h Fq(V)22 b Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))22 b Fu(\000)1371 2114 y Fr(j)1315 2148 y Fv(X)1331 2361 y Fr(l)q Fs(=2)1476 2243 y Ft(~)1540 2174 y Fl(l)p 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b(of)g(the)h(w)m(ork)g(is)f(in)m(v)m(olv)m(ed)h(in)e(the)i (outside)f(estimate,)h(whic)m(h)g(requires)-71 3705 y(con)m(trol)29 b(on)g(the)g(gro)m(wth)h(of)f(deriv)-5 b(ativ)m(es)29 b(of)g Fq(V)21 b Fw(\()p Fq(x)p Fw(\))30 b(far)e(a)m(w)m(a)m(y)j(from)d (the)h(minim)m(um)d(of)j Fq(V)21 b Fw(\()p Fq(x)p Fw(\).)43 b(That)30 b(is)e(done)-71 3826 y(b)m(y)34 b(resorting)e(to)g(the)h (Cauc)m(h)m(y)h(in)m(tegral)d(form)m(ula)g(and)i(h)m(yp)s(othesis)g (H2.)75 4017 y(Let)d Fq(\020)k Fw(=)28 b(\()p Fq(\020)509 4032 y Fs(1)548 4017 y Fq(;)17 b(:)g(:)g(:)f(;)h(\020)810 4032 y Fr(d)849 4017 y Fw(\))28 b Fu(2)g Ft(R)1075 3981 y Fr(d)1122 4017 y Fw(.)42 b(Let)29 b(\000)1423 4032 y Fr(i)1480 4017 y Fw(b)s(e)g(a)g(circle)f(of)g(radius)h Fq(\016)j Fw(in)d(the)g(complex)f(plane,)i(cen)m(tered)g(at)f Fq(\020)3890 4032 y Fr(i)3918 4017 y Fw(.)-71 4137 y(Then)38 b(the)g(Cauc)m(h)m(y)g(in)m(tegral)e(form)m(ula)f(applied)h(to)g Fq(V)22 b Fw(\()p Fq(x)p Fw(\),)38 b(whic)m(h)g(mak)m(es)f(sense)i(b)s 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Fs(2)1750 538 y Fw(0)82 b(otherwise.)-71 761 y Fn(Lemma)37 b(4)49 b Fh(Set)d Fq(R)k Fw(=)858 680 y Fu(p)p 941 680 699 4 v 81 x Fw(6)p Fq(N)32 b Fw(+)22 b(2)p Fq(a)g Fw(+)g Fq(d)g Fu(\000)h Fw(4)p Fh(.)79 b(Given)46 b Fq(m)j Fw(=)f(\()p Fq(m)2426 776 y Fs(1)2466 761 y Fq(;)17 b(:)g(:)g(:)f(;)h(m)2770 776 y Fr(d)2810 761 y Fw(\))47 b Fh(with)f Fu(j)p Fq(m)p Fu(j)i(\025)i Fw(2)p Fh(,)e(and)e Fq(n)j Fw(=)-71 881 y(0)p Fq(;)17 b(:)g(:)g(:)f(;)h(N)32 b Fu(\000)23 b Fw(1)p Fh(,)34 b(ther)-5 b(e)35 b(ar)-5 b(e)35 b(c)-5 b(ertain)34 b(c)-5 b(onstants)35 b Fq(F)48 b Fh(and)34 b Fq(G)h Fh(so)g(that)-32 1026 y Fv(\015)-32 1086 y(\015)-32 1145 y(\015)-32 1205 y(\015)33 1103 y Fq(\016)80 1067 y Fm(j)p Fr(m)p Fm(j)p 33 1147 153 4 v 53 1239 a Fq(m)p Fw(!)196 1170 y Fq(D)280 1129 y Fr(m)346 1170 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)607 1129 y Fr(m)669 1106 y Fi(0)712 1170 y Fw(\()o(1)22 b Fu(\000)h Fq(\037)981 1185 y Fr(R)1039 1170 y Fw(\))16 b Fq(P)1156 1186 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)p Fs(+1)1548 1026 y Fv(\015)1548 1086 y(\015)1548 1145 y(\015)1548 1205 y(\015)1631 1170 y Fu(\024)28 b Fq(F)14 b(G)1900 1101 y Fk(3)p Fl(n)p Fk(+2+)p Fl(a)p 1899 1114 227 3 v 1997 1155 a Fk(2)2149 1103 y Fw(\(3)p Fq(n)22 b Fw(+)g Fq(a)h Fw(+)f Fq(d)p Fw(\))2685 1038 y Fl(d)p Fi(\000)p Fk(1)p 2684 1051 111 3 v 2724 1092 a(2)p 2149 1147 660 4 v 2227 1208 a Fv(\020)2286 1319 y Fw(1)g Fu(\000)2487 1279 y Fr(\034)p 2467 1296 79 4 v 2467 1353 a(!)2511 1362 y Fk(0)2555 1208 y Fv(\021)2625 1197 y Fi(j)p Fl(m)p Fi(j)p 2625 1215 93 3 v 2656 1256 a Fk(2)2836 1030 y Fv(\024)2898 1103 y Fw(\(3)p Fq(n)6 b Fw(+)g Fu(j)p Fq(m)p Fu(j)g Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)g Fw(+)g Fq(a)p Fw(\)!)p 2898 1147 892 4 v 3153 1239 a(\(3)p Fq(n)22 b Fw(+)g Fq(a)p Fw(\)!)3799 1030 y Fv(\025)3862 1025 y Fk(1)p 3862 1037 31 3 v 3862 1078 a(2)-71 1550 y Fh(wher)-5 b(e)31 b Fu(j)p Fq(m)314 1514 y Fm(0)338 1550 y Fu(j)c Fw(=)h Fu(j)p Fq(m)p Fu(j)16 b(\000)g Fw(1)p Fh(,)31 b Fq(!)918 1565 y Fs(0)985 1550 y 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y(\015)111 2712 y(\015)176 2609 y Fq(\016)223 2573 y Fm(j)p Fr(m)p Fm(j)p 176 2654 V 196 2745 a Fq(m)p Fw(!)339 2677 y Fq(D)423 2636 y Fr(m)489 2677 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)750 2636 y Fr(m)812 2612 y Fi(0)855 2677 y Fw(\(1)21 b Fu(\000)i Fq(\037)1124 2692 y Fr(R)1182 2677 y Fw(\))16 b(\010)1306 2692 y Fr(j)1343 2677 y Fw(\()p Fq(x)p Fw(\))1474 2532 y Fv(\015)1474 2592 y(\015)1474 2652 y(\015)1474 2712 y(\015)1530 2550 y Fs(2)1653 2677 y Fw(=)1813 2541 y Fv(Z)1929 2532 y(\014)1929 2592 y(\014)1929 2652 y(\014)1929 2712 y(\014)1972 2609 y Fq(\016)2019 2573 y Fm(j)p Fr(m)p Fm(j)p 1972 2654 V 1993 2745 a Fq(m)p Fw(!)2135 2677 y Fq(D)2219 2636 y Fr(m)2285 2677 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\))2491 2532 y Fv(\014)2491 2592 y(\014)2491 2652 y(\014)2491 2712 y(\014)2523 2550 y Fs(2)2579 2677 y Fq(x)2634 2636 y Fs(2)p Fr(m)2731 2612 y Fi(0)2775 2677 y Fu(j)p Fw(\010)2873 2692 y Fr(j)2910 2677 y Fw(\()p Fq(x)p Fw(\))p Fu(j)3069 2628 y Fs(2)3125 2677 y Fw([1)22 b Fu(\000)h Fq(\037)3384 2692 y Fr(R)3442 2677 y Fw(\()p Fq(x)p Fw(\)])17 b Fq(d)3668 2636 y Fr(d)3708 2677 y Fq(x)1653 2940 y Fu(\024)83 b Fq(K)1903 2899 y Fs(2)1959 2805 y Fv(Z)2075 2940 y Fq(e)2120 2899 y Fs(4)p Fr(\034)8 b Fm(j)p Fr(x)p Fm(j)2274 2876 y Fk(2)2313 2940 y Fq(x)2368 2899 y Fs(2)p Fr(m)2465 2876 y Fi(0)2509 2940 y Fu(j)p Fw(\010)2607 2955 y Fr(j)2644 2940 y Fw(\()p Fq(x)p Fw(\))p Fu(j)2803 2892 y Fs(2)2859 2940 y Fw([1)22 b Fu(\000)g Fq(\037)3117 2955 y Fr(R)3175 2940 y Fw(\()p Fq(x)p Fw(\)])17 b Fq(d)3401 2899 y Fr(d)3441 2940 y Fq(x)-71 3193 y Fw(The)34 b(inequalit)m(y)d(comes)i(from)e(\(49\).)43 b(No)m(w)33 b(c)m(hange)h(v)-5 b(ariables)31 b Fq(x)2348 3208 y Fr(i)2404 3193 y Fu(!)d Fq(y)2580 3208 y Fr(i)2635 3193 y Fw(=)2739 3129 y Fu(p)p 2822 3129 89 4 v 64 x Fq(!)2883 3208 y Fr(i)2911 3193 y Fq(x)2966 3208 y Fr(i)3027 3193 y Fw(to)k(get)-66 3353 y Fv(\015)-66 3412 y(\015)-66 3472 y(\015)-66 3532 y(\015)0 3430 y Fq(\016)47 3393 y Fm(j)p Fr(m)p Fm(j)p 0 3474 153 4 v 20 3565 a Fq(m)p Fw(!)162 3497 y Fq(D)246 3456 y Fr(m)313 3497 y Fq(V)21 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)573 3456 y Fr(m)635 3432 y Fi(0)678 3497 y Fw(\(1)e Fu(\000)g Fq(\037)915 3512 y Fr(R)972 3497 y Fw(\))17 b(\010)1097 3512 y Fr(j)1134 3497 y Fw(\()p Fq(x)p Fw(\))1265 3353 y Fv(\015)1265 3412 y(\015)1265 3472 y(\015)1265 3532 y(\015)1321 3370 y Fs(2)1443 3497 y Fu(\024)67 b Fq(K)1677 3456 y Fs(2)1716 3327 y Fv( )1841 3372 y Fr(d)1795 3402 y Fv(Y)1802 3612 y Fr(i)p Fs(=1)1939 3497 y Fq(!)2004 3437 y Fm(\000)p Fr(m)2121 3413 y Fi(0)2121 3458 y Fl(i)2147 3437 y Fm(\000)2212 3409 y Fk(1)p 2212 3421 31 3 v 2212 3463 a(2)2000 3523 y Fr(i)2257 3327 y Fv(!)2335 3361 y(Z)2452 3497 y Fq(e)2497 3395 y Fd(P)2571 3465 y Fl(i)2610 3444 y Fs(4)2670 3417 y Fl(\034)p 2655 3429 66 3 v 2655 3470 a(!)2694 3486 y(i)2730 3444 y Fr(y)2767 3421 y Fk(2)2765 3466 y Fl(i)2806 3497 y Fq(y)2858 3456 y Fs(2)p Fr(m)2955 3432 y Fi(0)2998 3497 y Fu(j)p Fw(\010)3096 3512 y Fr(j)3133 3497 y Fw(\()p Fq(y)t Fw(\))p Fu(j)3287 3449 y Fs(2)3343 3497 y Fw([1)6 b Fu(\000)g Fq(\037)3569 3512 y Fr(R)3627 3497 y Fw(\()p Fq(y)t Fw(\)])15 b Fq(d)3848 3456 y Fr(d)3888 3497 y Fq(y)1443 3836 y Fu(\024)67 b Fq(K)1677 3795 y Fs(2)1733 3666 y Fv( )1857 3712 y Fr(d)1812 3741 y Fv(Y)1818 3951 y Fr(i)p Fs(=1)1956 3836 y Fq(!)2021 3776 y Fm(\000)p Fr(m)2138 3752 y Fi(0)2138 3797 y Fl(i)2164 3776 y Fm(\000)2229 3749 y Fk(1)p 2229 3761 31 3 v 2229 3802 a(2)2017 3862 y Fr(i)2273 3666 y Fv(!)2369 3701 y(Z)2485 3836 y Fq(e)2530 3784 y Fs(4)2595 3757 y Fl(\034)p 2575 3769 74 3 v 2575 3810 a(!)2614 3825 y Fk(0)2659 3784 y Fm(j)p Fr(y)r Fm(j)2736 3761 y Fk(2)2774 3836 y Fq(y)2826 3795 y Fs(2)p Fr(m)2923 3772 y Fi(0)2966 3836 y Fu(j)p Fw(\010)3064 3851 y Fr(j)3101 3836 y Fw(\()p Fq(y)t Fw(\))p Fu(j)3255 3788 y Fs(2)3311 3836 y Fw([1)6 b Fu(\000)g Fq(\037)3537 3851 y Fr(R)3595 3836 y Fw(\()p Fq(y)t Fw(\)])15 b Fq(d)3816 3795 y Fr(d)3856 3836 y Fq(y)1444 4126 y Fw(=)83 b Fq(F)1666 4141 y Fs(1)1723 4011 y Fv(\015)1723 4071 y(\015)1723 4131 y(\015)1778 4126 y Fq(e)1823 4074 y Fs(2)1888 4047 y Fl(\034)p 1868 4059 V 1868 4100 a(!)1907 4115 y Fk(0)1952 4074 y Fm(j)p Fr(y)r Fm(j)2029 4051 y Fk(2)2067 4126 y Fq(y)2119 4085 y Fr(m)2181 4061 y Fi(0)2224 4126 y Fw(\(1)6 b Fu(\000)g Fq(\037)2461 4141 y Fr(R)2518 4126 y Fw(\))16 b(\010)2642 4141 y Fr(j)2679 4126 y Fw(\()p Fq(y)t Fw(\))2807 4011 y Fv(\015)2807 4071 y(\015)2807 4131 y(\015)2862 4037 y Fs(2)3771 4126 y Fw(\(50\))-71 4352 y(where)36 b Fq(!)274 4367 y Fs(0)348 4352 y Fw(is)e(the)h(lo)m(w)m(est)g(eigenfrequency)i (of)d Fq(H)1768 4367 y Fs(0)1842 4352 y Fw(and)g Fq(F)2096 4367 y Fs(1)2170 4352 y Fw(is)g(de\014ned)j(in)c(the)i(ob)m(vious)g(w)m (a)m(y)-8 b(.)51 b(In)35 b(the)g(new)-71 4472 y(v)-5 b(ariables)1382 4661 y Fq(\037)1443 4676 y Fr(R)1500 4661 y Fw(\()p Fq(y)t Fw(\))27 b(=)1759 4491 y Fv(\()1880 4565 y Fw(1)83 b(if)27 b Fq(y)2149 4529 y Fs(2)2215 4565 y Fu(\024)h Fq(R)2395 4529 y Fs(2)1880 4756 y Fw(0)83 b(otherwise.)75 4922 y(Using)44 b(the)g(new)g(v)-5 b(ariables)42 b(in)h(\(5\),)j(w)m(e)f(see)g(that)e(\010)2105 4937 y Fr(j)2142 4922 y Fw(\()p Fq(y)t Fw(\))f(is)h(an)h(eigenfunction)f(of)g (the)h(normalized)-71 5043 y(harmonic)31 b(oscillator)f(op)s(erator) 1537 5202 y Fq(H)1626 5161 y Fm(0)1618 5227 y Fs(0)1685 5202 y Fw(=)e Fu(\000)1876 5135 y Fw(1)p 1876 5179 49 4 v 1876 5270 a(2)1935 5202 y(\001)2016 5217 y Fr(y)2080 5202 y Fw(+)2188 5135 y(1)p 2188 5179 V 2188 5270 a(2)2246 5202 y Fq(y)2298 5161 y Fs(2)1888 5470 y Fw(23)p eop %%Page: 24 24 24 23 bop -71 123 a Fw(with)32 b(energy)i Fq(e)508 138 y Fr(j)572 123 y Fw(=)28 b Fu(j)p Fq(j)6 b Fu(j)21 b Fw(+)h Fq(d=)p Fw(2.)43 b(F)-8 b(or)32 b Fq(d)27 b Fu(\025)h Fw(2)33 b(this)f(op)s(erator)g(is)g(equal)g(to)1105 398 y Fq(H)1194 357 y Fm(0)1186 423 y Fs(0)1253 398 y Fw(=)1366 331 y(1)p 1366 375 49 4 v 1366 466 a(2)1441 258 y Fv(\022)1515 398 y Fu(\000)1626 331 y Fq(@)1682 295 y Fs(2)p 1602 375 144 4 v 1602 466 a Fq(@)5 b(r)1705 438 y Fs(2)1777 398 y Fu(\000)1887 331 y Fq(d)22 b Fu(\000)g Fw(1)p 1887 375 222 4 v 1974 466 a Fq(r)2151 331 y(@)p 2128 375 104 4 v 2128 466 a(@)5 b(r)2264 398 y Fw(+)2372 331 y Fu(L)2441 295 y Fs(2)p 2372 375 109 4 v 2383 466 a Fq(r)2430 438 y Fs(2)2512 398 y Fw(+)22 b Fq(r)2657 357 y Fs(2)2696 258 y Fv(\023)-71 676 y Fw(in)38 b(spherical)h(co)s(ordinates,)h(where) g Fu(L)1373 639 y Fs(2)1451 676 y Fw(is)f(the)g(angular)f(momen)m(tum)f (op)s(erator)i(de\014ned)h(on)f Fq(S)3557 639 y Fr(d)p Fm(\000)p Fs(1)3687 676 y Fw(.)63 b(The)-71 796 y(eigen)m(v)-5 b(alues)33 b(no)m(w)g(read)g Fq(e)28 b Fw(=)f(2)p Fq(n)22 b Fw(+)g Fq(q)k Fw(+)c Fq(d=)p Fw(2)32 b(and)h(the)g(eigenfunctions)f (are)555 1129 y(\011)631 1144 y Fr(k)r(;q)r(;\027)786 1129 y Fw(\()p Fq(r)m(;)17 b(!)t Fw(\))27 b(=)1142 958 y Fv(")1429 1061 y Fw(2)p Fq(k)s Fw(!)p 1210 1106 567 4 v 1210 1203 a(\000)1288 1122 y Fv(\000)1334 1203 y Fq(k)e Fw(+)d Fq(q)k Fw(+)1685 1164 y Fr(d)p 1685 1180 37 4 v 1686 1238 a Fs(2)1731 1122 y Fv(\001)1787 958 y(#)1855 954 y Fk(1)p 1855 966 31 3 v 1855 1007 a(2)1916 1129 y Fq(r)1963 1088 y Fr(q)2001 1129 y Fq(L)2067 1068 y Fr(q)r Fs(+)2166 1041 y Fl(d)p 2166 1053 33 3 v 2167 1094 a Fk(2)2208 1068 y Fm(\000)p Fs(1)2067 1157 y Fr(k)2319 1048 y Fv(\000)2364 1129 y Fq(r)2411 1088 y Fs(2)2450 1048 y Fv(\001)2513 1129 y Fw(exp)2678 988 y Fv(\022)2752 1129 y Fu(\000)2839 1061 y Fq(r)2886 1025 y Fs(2)p 2839 1106 87 4 v 2858 1197 a Fw(2)2935 988 y Fv(\023)3025 1129 y Fq(Y)3082 1144 y Fr(q)r(;\027)3178 1129 y Fw(\()p Fq(!)t Fw(\))-71 1433 y(Here)41 b Fq(Y)224 1448 y Fr(q)r(;\027)320 1433 y Fw(\()p Fq(!)t Fw(\))e(are)h(the)g(normalized)d(eigenfunctions)j (of)f Fu(L)2182 1397 y Fs(2)2221 1433 y Fw(,)j(with)d(quan)m(tum)h(n)m (um)m(b)s(ers)h Fq(q)t(;)17 b(\027)6 b Fw(.)65 b(F)-8 b(or)39 b(eac)m(h)-71 1553 y Fq(q)46 b Fw(=)c(0)p Fq(;)17 b Fw(1)p Fq(;)g(:)g(:)g(:)40 b Fw(there)i(are)f Fq(\027)954 1568 y Fr(q)1033 1553 y Fw(v)-5 b(alues)41 b(of)g Fq(\027)6 b Fw(.)69 b(Although)41 b(the)g(explicit)f(form)m(ula)f(for)i Fq(\027)3143 1568 y Fr(q)3222 1553 y Fw(is)g(rather)g(clumsy)-8 b(,)-71 1674 y(there)34 b(is)f(a)g(simple)f(b)s(ound)h(for)g(it,)f (namely)h Fq(\027)1629 1689 y Fr(q)1696 1674 y Fu(\024)c Fq(C)1872 1689 y Fr(d)1912 1674 y Fq(e)1957 1637 y Fr(\013)2002 1649 y Fl(d)2039 1637 y Fr(q)2077 1674 y Fw(.)46 b(This)33 b(b)s(ound)g(su\016ces)j(for)c(the)i(purp)s(ose)g(of)f(our)-71 1803 y(pro)s(of.)47 b Fq(L)292 1756 y Fr(\014)292 1831 y(k)339 1803 y Fw(\()p Fq(x)p Fw(\))34 b(denotes)h(the)g(Laguerre)f(p)s (olynomial.)43 b(By)35 b(Lemma)d(6.2)i(of)f([3],)h(this)g(p)s (olynomial)c(satis\014es)-71 1847 y Fv(\014)-71 1907 y(\014)-71 1966 y(\014)-38 1961 y Fq(L)28 1901 y Fr(q)r Fs(+)127 1874 y Fl(d)p 127 1886 33 3 v 128 1927 a Fk(2)169 1901 y Fm(\000)p Fs(1)28 1972 y Fr(n)264 1961 y Fw(\()p Fq(x)p Fw(\))395 1847 y Fv(\014)395 1907 y(\014)395 1966 y(\014)461 1961 y Fu(\024)581 1922 y Fr(x)621 1899 y Fl(k)p 581 1938 78 4 v 591 1996 a Fr(k)r Fs(!)704 1961 y Fw(for)35 b(all)e Fq(x)g(>)g Fw(4)p Fq(k)27 b Fw(+)d(2)p Fq(q)j Fw(+)d Fq(d)p Fw(.)52 b(Finally)-8 b(,)33 b(b)m(y)k(equating)e (the)h(expressions)h(for)e(the)h(energy)-71 2109 y(w)m(e)e(obtain)d Fu(j)p Fq(j)6 b Fu(j)27 b Fw(=)h(2)p Fq(k)d Fw(+)d Fq(q)t Fw(.)75 2229 y(No)m(w)34 b(\010)368 2244 y Fr(j)405 2229 y Fw(\()p Fq(y)t Fw(\))d(is)h(certain)g(linear)f(com)m(bination)g(of)h (\011)2001 2244 y Fr(k)r(;q)r(;\027)2156 2229 y Fw(\()p Fq(r)m(;)17 b(!)t Fw(\),)1234 2463 y(\010)1304 2478 y Fr(j)1341 2463 y Fw(\()p Fq(y)t Fw(\))26 b(=)1713 2368 y Fv(X)1700 2573 y Fr(k)r(;q)r(;\027)t Fs(:)1640 2646 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)1988 2463 y Fq(c)2030 2478 y Fr(k)r(;q)r(;\027)2184 2463 y Fw(\011)2260 2478 y Fr(k)r(;q)r(;\027)2415 2463 y Fw(\()p Fq(r)m(;)17 b(!)t Fw(\))-71 2884 y(with)148 2810 y Fv(P)269 2884 y Fu(j)p Fq(c)339 2899 y Fr(k)r(;q)r(;\027)494 2884 y Fu(j)522 2836 y Fs(2)589 2884 y Fw(=)27 b(1.)42 b(Inserting)29 b(this)g(linear)e(con)m(bination)h(in)m(to)g(\(50\))g(and)h(using)f (the)i(H\177)-49 b(older)28 b(inequalit)m(y)-8 b(,)-71 3005 y(it)32 b(follo)m(ws)f(that)43 3131 y Fv(\015)43 3190 y(\015)43 3250 y(\015)43 3310 y(\015)108 3208 y Fq(\016)155 3172 y Fm(j)p Fr(m)p Fm(j)p 108 3252 153 4 v 128 3343 a Fq(m)p Fw(!)271 3275 y Fq(D)355 3234 y Fr(m)421 3275 y Fq(V)21 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)681 3234 y Fr(m)743 3210 y Fi(0)786 3275 y Fw(\(1)e Fu(\000)g Fq(\037)1023 3290 y Fr(R)1080 3275 y Fw(\))17 b(\010)1205 3290 y Fr(j)1242 3275 y Fw(\()p Fq(x)p Fw(\))1373 3131 y Fv(\015)1373 3190 y(\015)1373 3250 y(\015)1373 3310 y(\015)1429 3148 y Fs(2)598 3534 y Fu(\024)84 b Fq(F)822 3549 y Fs(1)992 3439 y Fv(X)979 3645 y Fr(k)r(;q)r(;\027)t Fs(:)919 3718 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)1266 3419 y Fv(\015)1266 3479 y(\015)1266 3539 y(\015)1322 3534 y Fq(e)1367 3482 y Fs(2)1432 3455 y Fl(\034)p 1412 3467 74 3 v 1412 3508 a(!)1451 3523 y Fk(0)1496 3482 y Fr(y)1533 3459 y Fk(2)1572 3534 y Fq(y)1624 3493 y Fr(m)1686 3469 y Fi(0)1728 3534 y Fw(\(1)6 b Fu(\000)g Fq(\037)1965 3549 y Fr(R)2022 3534 y Fw(\))17 b(\011)2153 3549 y Fr(k)r(;q)r(;m)2331 3534 y Fw(\()p Fq(y)t Fw(\))2459 3419 y Fv(\015)2459 3479 y(\015)2459 3539 y(\015)2513 3446 y Fs(2)598 3936 y Fu(\024)84 b Fq(F)822 3951 y Fs(1)992 3841 y Fv(X)979 4046 y Fr(k)r(;q)r(;\027)t Fs(:)919 4120 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)1393 3868 y Fw(2)p Fq(k)s Fw(!)p Fq(A)1596 3883 y Fr(d)p Fm(\000)p Fs(1)p 1276 3913 567 4 v 1276 4010 a Fw(\000)1354 3929 y Fv(\000)1400 4010 y Fq(k)25 b Fw(+)d Fq(q)k Fw(+)1751 3971 y Fr(d)p 1751 3987 37 4 v 1752 4044 a Fs(2)1797 3929 y Fv(\001)1869 3800 y(Z)1969 3826 y Fm(1)1925 4026 y Fr(R)2060 3936 y Fq(e)2105 3882 y Fm(\000)2160 3808 y Fd(\020)2202 3882 y Fs(1)p Fm(\000)2307 3855 y Fk(4)p Fl(\034)p 2303 3867 74 3 v 2303 3908 a(!)2342 3923 y Fk(0)2386 3808 y Fd(\021)2428 3882 y Fr(r)2462 3858 y Fk(2)2501 3936 y Fq(r)2548 3895 y Fs(2\()p Fm(j)p Fr(m)p Fm(j\000)p Fs(1+)p Fr(q)r Fs(\))2939 3821 y Fv(\014)2939 3881 y(\014)2939 3941 y(\014)2972 3936 y Fq(L)3038 3875 y Fr(q)r Fs(+)3137 3848 y Fl(d)p 3137 3860 33 3 v 3138 3901 a Fk(2)3179 3875 y Fm(\000)p Fs(1)3038 3963 y Fr(k)3290 3855 y Fv(\000)3336 3936 y Fq(r)3383 3895 y Fs(2)3422 3855 y Fv(\001)3468 3821 y(\014)3468 3881 y(\014)3468 3941 y(\014)3501 3840 y Fs(2)3557 3936 y Fq(r)3604 3895 y Fr(d)p Fm(\000)p Fs(1)3734 3936 y Fq(dr)-71 4353 y Fw(where)33 b Fq(A)283 4368 y Fr(d)p Fm(\000)p Fs(1)446 4353 y Fw(is)e(the)h(area)g(of)f(the)h(\()p Fq(d)20 b Fu(\000)h Fw(1\))32 b(dimensional)d(unit)i(sphere.)45 b(W)-8 b(e)32 b(also)f(ha)m(v)m(e)i(used)g(that)e Fq(y)3677 4317 y Fs(2)p Fm(j)p Fr(m)3794 4294 y Fi(0)3816 4317 y Fm(j)3867 4353 y Fu(\024)-71 4474 y Fq(r)-24 4438 y Fs(2)p Fm(j)p Fr(m)93 4414 y Fi(0)115 4438 y Fm(j)167 4474 y Fw(=)c Fq(r)317 4438 y Fs(2\()p Fm(j)p Fr(m)p Fm(j\000)p Fs(1\))603 4474 y Fw(.)44 b(Since)32 b Fq(R)d Fu(\025)1136 4389 y Fv(p)p 1236 4389 321 4 v 85 x Fw(2)p Fu(j)p Fq(j)6 b Fu(j)21 b Fw(+)h Fq(d)p Fw(,)32 b(Lemma)g(6.2)g(of)g ([3])g(applies)g(so)235 4630 y Fv(\015)235 4690 y(\015)235 4750 y(\015)235 4810 y(\015)300 4707 y Fq(\016)347 4671 y Fm(j)p Fr(m)p Fm(j)p 300 4752 153 4 v 320 4843 a Fq(m)p Fw(!)463 4775 y Fq(D)547 4734 y Fr(m)613 4775 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)874 4734 y Fr(m)936 4710 y Fi(0)979 4775 y Fw(\(1)e Fu(\000)g Fq(\037)1216 4790 y Fr(R)1273 4775 y Fw(\))16 b(\010)1397 4790 y Fr(j)1434 4775 y Fw(\()p Fq(x)p Fw(\))1565 4630 y Fv(\015)1565 4690 y(\015)1565 4750 y(\015)1565 4810 y(\015)1621 4648 y Fs(2)909 5040 y Fu(\024)83 b Fq(F)1132 5055 y Fs(1)1302 4946 y Fv(X)1289 5151 y Fr(k)r(;q)r(;\027)t Fs(:)1230 5224 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)1784 4973 y Fw(2)p Fq(A)1906 4988 y Fr(d)p Fm(\000)p Fs(1)p 1587 5018 648 4 v 1587 5115 a Fq(k)s Fw(!\000)1746 5034 y Fv(\000)1791 5115 y Fq(k)25 b Fw(+)e Fq(q)i Fw(+)2142 5076 y Fr(d)p 2142 5092 37 4 v 2143 5149 a Fs(2)2189 5034 y Fv(\001)2261 4905 y(Z)2361 4931 y Fm(1)2316 5130 y Fr(R)2452 5040 y Fq(e)2497 4987 y Fm(\000)2552 4913 y Fd(\020)2594 4987 y Fs(1)p Fm(\000)2698 4959 y Fk(4)p Fl(\034)p 2694 4971 74 3 v 2694 5013 a(!)2733 5028 y Fk(0)2778 4913 y Fd(\021)2820 4987 y Fr(r)2854 4963 y Fk(2)2892 5040 y Fq(r)2939 4999 y Fs(2)p Fm(j)p Fr(m)p Fm(j)p Fs(+2)p Fr(q)r Fs(+4)p Fr(k)r Fs(+)p Fr(d)p Fm(\000)p Fs(3\))3542 5040 y Fq(dr)1888 5470 y Fw(24)p eop %%Page: 25 25 25 24 bop 909 183 a Fw(=)84 b Fq(F)1132 198 y Fs(1)1302 88 y Fv(X)1289 293 y Fr(k)r(;q)r(;\027)t Fs(:)1230 367 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)1784 115 y Fw(2)p Fq(A)1906 130 y Fr(d)p Fm(\000)p Fs(1)p 1587 160 648 4 v 1587 257 a Fq(k)s Fw(!\000)1746 176 y Fv(\000)1791 257 y Fq(k)25 b Fw(+)e Fq(q)i Fw(+)2142 218 y Fr(d)p 2142 234 37 4 v 2143 292 a Fs(2)2189 176 y Fv(\001)2254 109 y Fw(\000)2332 28 y Fv(\000)2378 109 y Fu(j)p Fq(m)p Fu(j)c Fw(+)h Fq(q)k Fw(+)c(2)p Fq(k)j Fw(+)3038 70 y Fr(d)p 3038 86 V 3039 143 a Fs(2)3107 109 y Fu(\000)d Fw(1)3255 28 y Fv(\001)p 2254 160 1047 4 v 2291 322 a Fw(2)2357 211 y Fv(\020)2416 322 y Fw(1)g Fu(\000)2598 283 y Fs(4)p Fr(\034)p 2596 299 79 4 v 2596 356 a(!)2640 365 y Fk(0)2685 211 y Fv(\021)2744 234 y Fm(j)p Fr(m)p Fm(j)p Fs(+)p Fr(q)r Fs(+2)p Fr(k)r Fs(+)3129 206 y Fl(d)p 3128 218 33 3 v 3129 260 a Fk(2)3170 234 y Fm(\000)p Fs(1)909 605 y Fw(=)84 b Fq(F)1132 620 y Fs(1)1172 605 y Fq(A)1245 620 y Fr(d)p Fm(\000)p Fs(1)1385 531 y Fw(\000)1463 451 y Fv(\000)1509 531 y Fu(j)p Fq(m)p Fu(j)21 b Fw(+)h Fu(j)p Fq(j)6 b Fu(j)22 b Fw(+)2001 492 y Fr(d)p 2001 508 37 4 v 2002 566 a Fs(2)2069 531 y Fu(\000)h Fw(1)2218 451 y Fv(\001)p 1385 582 878 4 v 1416 633 a(\020)1475 744 y Fw(1)f Fu(\000)1658 705 y Fs(4)p Fr(\034)p 1656 721 79 4 v 1656 778 a(!)1700 787 y Fk(0)1744 633 y Fv(\021)1804 656 y Fm(j)p Fr(m)p Fm(j)p Fs(+)p Fm(j)p Fr(j)t Fm(j)p Fs(+)2099 629 y Fl(d)p 2097 641 33 3 v 2098 682 a Fk(2)2139 656 y Fm(\000)p Fs(1)2404 510 y Fv(X)2420 716 y Fr(k)r(;q)r Fs(:)2331 789 y(2)p Fr(k)r Fs(+)p Fr(q)r Fs(=)p Fm(j)p Fr(j)t Fm(j)2969 538 y Fq(\027)3017 553 y Fr(q)p 2689 582 648 4 v 2689 679 a Fq(k)s 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1682 y Fr(k)r Fs(=0)3018 1497 y Fq(e)3063 1461 y Fm(\000)p Fs(2)p Fr(\013)3198 1473 y Fl(d)3234 1461 y Fr(k)p 2795 1542 704 4 v 2795 1639 a Fq(k)s Fw(!\000)2954 1558 y Fv(\000)3000 1639 y Fu(j)p Fq(j)6 b Fu(j)21 b(\000)i Fq(k)i Fw(+)3407 1600 y Fr(d)p 3407 1616 37 4 v 3408 1673 a Fs(2)3453 1558 y Fv(\001)3771 1565 y Fw(\(51\))-71 1984 y(F)-8 b(or)28 b Fu(j)p Fq(j)6 b Fu(j)27 b(\025)h Fw(1,)h Fu(j)p Fq(j)6 b Fu(j)14 b(\000)g Fq(k)j Fw(+)d Fq(d=)p Fw(2)27 b Fu(\025)h Fw(1)14 b(+)g Fq(d=)p Fw(2)27 b Fu(\025)h Fw(2)g(for)g(all)f(0)g Fu(\024)h Fq(k)j Fu(\024)2241 1873 y Fe(r)2309 1936 y Fm(j)p Fr(j)t Fm(j)p 2309 1961 72 4 v 2328 2018 a Fs(2)2391 1873 y Fe(z)2449 1984 y Fw(.)42 b(Since)29 b(\000\()p Fq(x)p Fw(\))g(is)f(increasing)g(for)g Fq(x)g Fu(\025)g Fw(2,)-71 2126 y(w)m(e)34 b(ha)m(v)m(e)488 2223 y Fe(J)538 2182 y Fi(j)p Fl(j)s Fi(j)p 538 2200 67 3 v 556 2241 a Fk(2)614 2223 y Fe(K)499 2270 y Fv(X)507 2482 y Fr(k)r Fs(=0)904 2297 y Fq(e)949 2261 y Fm(\000)p Fs(2)p Fr(\013)1084 2273 y 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Fq(m)p Fw(!)680 3585 y Fq(D)764 3544 y Fr(m)830 3585 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)1091 3544 y Fr(m)1153 3520 y Fi(0)1196 3585 y Fw(\(1)e Fu(\000)g Fq(\037)1433 3600 y Fr(R)1490 3585 y Fw(\))16 b(\010)1614 3600 y Fr(j)1651 3585 y Fw(\()p Fq(x)p Fw(\))1782 3440 y Fv(\015)1782 3500 y(\015)1782 3560 y(\015)1782 3620 y(\015)1838 3458 y Fs(2)1905 3585 y Fu(\024)28 b Fq(F)2073 3600 y Fs(2)2113 3585 y Fw(2)2162 3544 y Fm(j)p Fr(j)t Fm(j)2237 3585 y Fq(e)2282 3544 y Fr(\013)2327 3556 y Fl(d)2364 3544 y Fm(j)p Fr(j)t Fm(j)2492 3511 y Fw(\000)2570 3430 y Fv(\000)2615 3511 y Fu(j)p Fq(m)p Fu(j)22 b Fw(+)g Fu(j)p Fq(j)6 b Fu(j)21 b Fw(+)3107 3472 y Fr(d)p 3107 3488 37 4 v 3108 3545 a Fs(2)3176 3511 y Fu(\000)h Fw(1)3324 3430 y Fv(\001)p 2450 3562 963 4 v 2450 3724 a Fu(j)p Fq(j)6 b Fu(j)p Fw(!)2596 3613 y Fv(\020)2654 3724 y Fw(1)22 b Fu(\000)2837 3685 y Fs(4)p Fr(\034)p 2835 3701 79 4 v 2835 3758 a(!)2879 3767 y Fk(0)2923 3613 y Fv(\021)2983 3636 y Fm(j)p Fr(m)p Fm(j)p Fs(+)p 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Fv(\015)3802 4503 y(\015)3802 4562 y(\015)3802 4622 y(\015)3858 4460 y Fs(2)-71 4905 y Fw(Therefore)97 5031 y Fv(\015)97 5091 y(\015)97 5151 y(\015)97 5211 y(\015)163 5108 y Fq(\016)210 5072 y Fm(j)p Fr(m)p Fm(j)p 163 5153 V 183 5244 a Fq(m)p Fw(!)325 5176 y Fq(D)409 5135 y Fr(m)476 5176 y Fq(V)k Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)736 5135 y Fr(m)798 5111 y Fi(0)841 5176 y Fw(\(1)22 b Fu(\000)h Fq(\037)1111 5191 y Fr(R)1168 5176 y Fw(\))17 b Fq(P)1286 5191 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)p Fs(+1)1677 5031 y Fv(\015)1677 5091 y(\015)1677 5151 y(\015)1677 5211 y(\015)1732 5049 y Fs(2)1888 5470 y Fw(25)p eop %%Page: 26 26 26 25 bop 1243 183 a Fu(\024)84 b Fq(F)1467 198 y Fs(2)1549 115 y Fw(2)1598 79 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1863 115 y Fq(e)1908 79 y Fr(\013)1953 91 y Fl(d)1990 79 y Fs(\(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1\))p 1516 160 826 4 v 1516 211 a Fv(\020)1576 322 y Fw(1)22 b Fu(\000)1758 283 y Fs(4)p Fr(\034)p 1756 299 79 4 v 1756 356 a(!)1800 365 y Fk(0)1845 211 y Fv(\021)1904 234 y Fs(3)p Fr(n)p Fs(+)p Fm(j)p Fr(m)p Fm(j)p Fs(+)p Fr(a)p Fs(+)2296 206 y Fl(d)p 2296 218 33 3 v 2297 260 a Fk(2)2490 88 y Fv(X)2369 304 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)2782 109 y Fw(\000)2860 28 y Fv(\000)2906 109 y Fu(j)p Fq(m)p Fu(j)g Fw(+)g Fu(j)p Fq(j)6 b Fu(j)21 b Fw(+)3398 70 y Fr(d)p 3398 86 37 4 v 3399 143 a Fs(2)3466 109 y Fu(\000)i Fw(1)3615 28 y Fv(\001)p 2782 160 878 4 v 3157 251 a Fu(j)p Fq(j)6 b Fu(j)p Fw(!)1243 573 y Fu(\024)84 b Fq(F)1467 588 y Fs(2)1549 506 y Fw(2)1598 470 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1863 506 y Fq(e)1908 470 y Fr(\013)1953 482 y Fl(d)1990 470 y Fs(\(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1\))p 1516 550 826 4 v 1516 602 a Fv(\020)1576 712 y Fw(1)22 b Fu(\000)1758 673 y Fs(4)p Fr(\034)p 1756 689 79 4 v 1756 747 a(!)1800 756 y Fk(0)1845 602 y Fv(\021)1904 624 y Fs(3)p Fr(n)p Fs(+)p Fm(j)p Fr(m)p Fm(j)p Fs(+)p Fr(a)p Fs(+)2296 597 y Fl(d)p 2296 609 33 3 v 2297 650 a Fk(2)2490 479 y Fv(X)2369 694 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)2782 506 y Fw(\()p Fu(j)p Fq(m)p Fu(j)g Fw(+)g Fu(j)p Fq(j)6 b Fu(j)22 b Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)f Fu(\000)i Fw(1\)!)p 2782 550 985 4 v 3210 642 a Fu(j)p Fq(j)6 b Fu(j)p Fw(!)-71 947 y(where)32 b(w)m(e)f(use)g(that)f(0)d Fq(<)h Fw(\(1)17 b Fu(\000)h Fw(4)p Fq(\034)11 b(=!)1316 962 y Fs(0)1355 947 y Fw(\))27 b Fq(<)h Fw(1.)42 b(The)31 b(terms)g(under)g(the)f (summation)e(sign)i(are)g(increasing)g(in)-71 1067 y Fu(j)p Fq(j)6 b Fu(j)p Fw(.)43 b(Also,)1136 1270 y Fv(X)1015 1486 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1368 1365 y Fw(1)84 b(=)1661 1240 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1719 1270 y Fv(X)1730 1479 y Fr(s)p Fs(=0)1905 1365 y Fw(#)p Fu(f)p Fq(j)33 b Fw(:)28 b Fu(j)p Fq(j)6 b Fu(j)27 b Fw(=)h Fq(s)p Fu(g)1501 1723 y Fw(=)1661 1599 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1719 1628 y Fv(X)1730 1838 y Fr(s)p Fs(=0)1948 1656 y Fw(\()p Fq(s)22 b Fw(+)g Fq(d)g Fu(\000)g Fw(1\)!)p 1948 1700 491 4 v 1994 1791 a Fq(s)p Fw(!\()p Fq(d)g Fu(\000)h Fw(1\)!)1500 2058 y Fu(\024)1661 1934 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1719 1964 y Fv(X)1730 2173 y Fr(s)p Fs(=0)1948 1991 y Fw(\()p Fq(s)f Fw(+)g Fq(d)g Fu(\000)g Fw(1\))2411 1955 y Fr(d)p Fm(\000)p Fs(1)p 1948 2035 594 4 v 2083 2127 a Fw(\()p Fq(d)g Fu(\000)g Fw(1\)!)1500 2366 y Fu(\024)1671 2298 y Fw(\(3)p Fq(n)g Fw(+)g Fq(a)g Fw(+)g Fq(d)p Fw(\))2196 2262 y Fr(d)p Fm(\000)p Fs(1)p 1671 2343 656 4 v 1836 2434 a Fw(\()p Fq(d)g Fu(\000)h Fw(1\)!)2336 2366 y(\(3)p Fq(n)f Fw(+)g Fq(a)h Fw(+)f(2\))-71 2629 y(Finally)-8 b(,)30 b(\(3)p Fq(n)22 b Fw(+)g(2)g(+)g Fq(a)p Fw(\))p Fq(=)p Fw(\(3)p Fq(n)g Fw(+)g(1)g(+)g Fq(a)p Fw(\))28 b Fu(\024)g Fw(2.)43 b(Th)m(us,)-24 2774 y Fv(\015)-24 2834 y(\015)-24 2894 y(\015)-24 2954 y(\015)42 2851 y Fq(\016)89 2815 y Fm(j)p Fr(m)p Fm(j)p 42 2896 153 4 v 62 2987 a Fq(m)p Fw(!)204 2919 y Fq(D)288 2878 y Fr(m)355 2919 y Fq(V)21 b Fw(\()p Fq(\020)8 b Fw(\))p Fq(x)615 2878 y Fr(m)677 2854 y Fi(0)720 2919 y Fw(\(1)22 b Fu(\000)h Fq(\037)990 2934 y Fr(R)1047 2919 y Fw(\))17 b Fq(P)1165 2934 y Fm(j)p Fr(j)t Fm(j\024)p Fr(a)p Fs(+3)p Fr(n)p Fs(+1)1556 2774 y Fv(\015)1556 2834 y(\015)1556 2894 y(\015)1556 2954 y(\015)1611 2792 y Fs(2)1122 3200 y Fu(\024)84 b Fq(F)1346 3215 y Fs(2)1428 3133 y Fw(2)1477 3097 y Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+2)1742 3133 y Fq(e)1787 3097 y Fr(\013)1832 3109 y Fl(d)1869 3097 y Fs(\(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1\))p 1395 3177 826 4 v 1395 3229 a Fv(\020)1455 3339 y Fw(1)22 b Fu(\000)1638 3300 y Fs(4)p Fr(\034)p 1635 3316 79 4 v 1635 3374 a(!)1679 3383 y Fk(0)1724 3229 y Fv(\021)1783 3251 y Fs(3)p Fr(n)p Fs(+)p Fm(j)p Fr(m)p Fm(j)p Fs(+)p Fr(a)p Fs(+)2175 3224 y Fl(d)p 2175 3236 33 3 v 2176 3277 a Fk(2)2231 3200 y Fw(\(3)p Fq(n)g Fw(+)g Fq(a)h Fw(+)f Fq(d)p Fw(\))2757 3159 y Fr(d)p Fm(\000)p Fs(1)2897 3133 y Fw(\(3)p Fq(n)g Fw(+)g Fu(j)p Fq(m)p Fu(j)g Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)f Fw(+)h Fq(a)p Fw(\))q(!)p 2897 3177 991 4 v 3202 3268 a(\(3)p Fq(n)g Fw(+)g Fq(a)p Fw(\)!)-71 3568 y(Finally)-8 b(,)30 b(de\014ne)k Fq(F)46 b Fw(and)33 b Fq(G)f Fw(in)g(the)h(ob)m (vious)g(w)m(a)m(y)-8 b(.)2082 b Fg(2)-71 3782 y Fn(Theorem)37 b(3)49 b Fh(Given)35 b Fq(a)p Fh(,)g(ther)-5 b(e)34 b(ar)-5 b(e)35 b(p)-5 b(ositive)34 b(c)-5 b(onstants)35 b Fq(A)g Fh(,)f Fq(B)40 b Fh(and)35 b Fq(N)2652 3797 y Fs(0)2726 3782 y Fh(so)g(that)1042 3992 y Fv(\015)1042 4052 y(\015)1097 4077 y Fq(A)1170 4036 y Fm(\000)p Fs(1)1170 4101 y Fr(e)1264 4077 y Fq(\030)1307 4092 y Fr(N)1374 4077 y Fw(\()p Fq(x)p Fw(\))1505 3992 y Fv(\015)1505 4052 y(\015)1589 4077 y Fu(\024)1725 3952 y Fs(2)p Fr(N)1702 3982 y Fv(X)1694 4194 y Fr(n)p Fs(=)p Fr(N)1871 4077 y Fq(AB)2023 4036 y Fr(N)2091 4077 y Ft(~)2155 4008 y Fl(N)p 2154 4020 54 3 v 2166 4062 a Fk(2)2222 4077 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2787 4008 y Fk(1)p 2788 4020 31 3 v 2788 4062 a(2)-71 4368 y Fh(whenever)34 b Fq(N)k Fu(\025)28 b Fq(N)653 4383 y Fs(0)728 4368 y Fh(and)34 b Ft(~)27 b Fu(\024)i Fw(1)p Fh(.)-71 4582 y(Pr)-5 b(o)g(of.)64 b Fw(Recall)37 b(that)j(w)m(e)g(assume)g(that)f Fq(\016)44 b Fu(\024)39 b Fw(1.)64 b(W)-8 b(e)40 b(already)f(kno)m(w)h(that)g Fq(b)f Fu(\025)h Fw(1.)64 b(F)-8 b(rom)38 b(the)i(pro)s(of)e(of)-71 4702 y(Lemma)31 b(2,)i(w)m(e)g(also)f(kno)m(w)i(that)e Fu(k)p Fq(x)1296 4717 y Fr(\013)1346 4702 y Fq(A)1419 4717 y Fr(e)1456 4702 y Fu(k)27 b(\024)h Fq(\015)1689 4717 y Fs(1)1728 4702 y Fw(.)75 4823 y(F)-8 b(rom)32 b(\(47\))f(it)h(follo)m(ws)f(that)13 5038 y Fv(\015)13 5098 y(\015)69 5123 y Fq(A)142 5081 y Fm(\000)p Fs(1)142 5147 y Fr(e)236 5123 y Fq(\030)279 5138 y Fr(N)346 5123 y Fw(\()p Fq(x)p Fw(\))477 5038 y Fv(\015)477 5098 y(\015)616 5123 y Fu(\024)83 b Ft(~)840 5054 y Fl(N)p 840 5066 54 3 v 852 5108 a Fk(2)924 4998 y Fr(N)7 b Fm(\000)p Fs(1)929 5028 y Fv(X)934 5237 y Fr(n)p Fs(=0)1226 5028 y Fv(X)1094 5244 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1518 4978 y Fv(\015)1518 5038 y(\015)1518 5098 y(\015)1518 5157 y(\015)1584 5055 y Fq(D)1668 5019 y Fr(m)1734 5055 y Fq(V)38 b Fw(\()p Fq(\020)1910 5070 y Fr(n)1957 5055 y Fw(\))p 1584 5100 411 4 v 1733 5191 a Fq(m)p Fw(!)2004 5123 y(\(1)22 b Fu(\000)h Fq(\037)2274 5138 y Fr(R)2332 5123 y Fw(\()p Fq(x)p Fw(\)\))p Fq(x)2556 5081 y Fr(m)2623 5123 y Fq(A)2696 5138 y Fr(e)2733 5123 y Fq( )2796 5138 y Fr(n)2843 5123 y Fw(\()p Fq(x)p Fw(\))2974 4978 y Fv(\015)2974 5038 y(\015)2974 5098 y(\015)2974 5157 y(\015)1888 5470 y Fw(26)p eop %%Page: 27 27 27 26 bop 776 204 a Fw(+)28 b Ft(~)944 135 y Fl(N)p 944 147 54 3 v 955 189 a Fk(2)1028 79 y Fr(N)7 b Fm(\000)p Fs(1)1033 109 y Fv(X)1038 318 y Fr(n)p Fs(=0)1330 109 y Fv(X)1198 325 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1622 59 y Fv(\015)1622 119 y(\015)1622 179 y(\015)1622 238 y(\015)1687 136 y Fq(D)1771 100 y Fr(m)1837 136 y Fq(V)39 b Fw(\()p Fq(\020)2014 151 y Fr(n)2060 136 y Fw(\))p 1687 181 411 4 v 1836 272 a Fq(m)p Fw(!)2108 204 y Fq(\037)2169 219 y Fr(R)2227 204 y Fw(\()p Fq(x)p Fw(\))p Fq(x)2413 162 y Fr(m)2480 204 y Fq(A)2553 219 y Fr(e)2590 204 y Fq( )2653 219 y Fr(n)2701 204 y Fw(\()p Fq(x)p Fw(\))2832 59 y Fv(\015)2832 119 y(\015)2832 179 y(\015)2832 238 y(\015)776 564 y Fw(+)885 440 y Fs(2)p Fr(N)7 b Fm(\000)p Fs(2)908 470 y Fv(X)899 681 y Fr(n)p Fs(=)p Fr(N)1091 564 y Ft(~)1155 496 y Fl(n)p 1154 508 39 3 v 1158 549 a Fk(2)1311 440 y Fr(N)g Fm(\000)p Fs(1)1315 470 y Fv(X)1223 682 y Fr(l)q Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)1568 564 y Fu(k)o(E)1670 579 y Fr(l)1696 564 y Fq(A)1769 579 y Fr(e)1806 564 y Fq( )1869 579 y Fr(n)p Fm(\000)p Fr(l)1993 564 y Fw(\()p Fq(x)p Fw(\))p Fu(k)616 912 y(\024)83 b Ft(~)840 844 y Fl(N)p 840 856 54 3 v 852 897 a Fk(2)924 787 y Fr(N)7 b Fm(\000)p Fs(1)929 817 y Fv(X)934 1027 y Fr(n)p Fs(=0)1226 817 y Fv(X)1094 1033 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1518 768 y Fv(\015)1518 827 y(\015)1518 887 y(\015)1518 947 y(\015)1584 845 y Fq(D)1668 808 y Fr(m)1734 845 y Fq(V)38 b Fw(\()p Fq(\020)1910 860 y Fr(n)1957 845 y Fw(\))p 1584 889 411 4 v 1733 980 a Fq(m)p Fw(!)2004 912 y(\(1)22 b Fu(\000)h Fq(\037)2274 927 y Fr(R)2332 912 y Fw(\()p Fq(x)p Fw(\)\))p Fq(x)2556 871 y Fr(m)2618 847 y Fi(0)2645 912 y Fq(P)2708 927 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)3100 768 y Fv(\015)3100 827 y(\015)3100 887 y(\015)3100 947 y(\015)3171 912 y Fu(k)p Fq(x)3276 927 y Fr(\013)3326 912 y Fq(A)3399 927 y Fr(e)3436 912 y Fu(k)17 b(k)o Fq( )3615 927 y Fr(n)3663 912 y Fw(\()p Fq(x)p Fw(\))p Fu(k)776 1273 y Fw(+)28 b Ft(~)944 1205 y Fl(N)p 944 1217 54 3 v 955 1258 a Fk(2)1028 1148 y Fr(N)7 b Fm(\000)p Fs(1)1033 1178 y Fv(X)1038 1387 y Fr(n)p Fs(=0)1330 1178 y Fv(X)1198 1394 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1622 1128 y Fv(\015)1622 1188 y(\015)1622 1248 y(\015)1622 1308 y(\015)1687 1205 y Fq(D)1771 1169 y Fr(m)1837 1205 y Fq(V)39 b Fw(\()p Fq(\020)2014 1220 y Fr(n)2060 1205 y Fw(\))p 1687 1250 411 4 v 1836 1341 a Fq(m)p Fw(!)2108 1273 y Fq(\037)2169 1288 y Fr(R)2227 1273 y Fw(\()p Fq(x)p Fw(\))p Fq(x)2413 1232 y Fr(m)2475 1208 y Fi(0)2503 1273 y Fq(P)2566 1288 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)2957 1128 y Fv(\015)2957 1188 y(\015)2957 1248 y(\015)2957 1308 y(\015)3029 1273 y Fu(k)o Fq(x)3133 1288 y Fr(\013)3183 1273 y Fq(A)3256 1288 y Fr(e)3293 1273 y Fu(k)17 b(k)p Fq( )3473 1288 y Fr(n)3520 1273 y Fw(\()p Fq(x)p Fw(\))p Fu(k)776 1634 y Fw(+)885 1509 y Fs(2)p Fr(N)7 b Fm(\000)p Fs(2)908 1539 y Fv(X)899 1750 y Fr(n)p Fs(=)p Fr(N)1091 1634 y Ft(~)1155 1565 y Fl(n)p 1154 1577 39 3 v 1158 1619 a Fk(2)1311 1509 y Fr(N)g Fm(\000)p Fs(1)1315 1539 y Fv(X)1223 1751 y Fr(l)q Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)1568 1634 y Fu(j)o(E)1648 1649 y Fr(l)1674 1634 y Fu(j)16 b(k)p Fq( )1831 1649 y Fr(n)p Fm(\000)p Fr(l)1955 1634 y Fw(\()p Fq(x)p Fw(\))p Fu(k)-71 1938 y Fw(Let)31 b(us)g(estimates)g(eac)m(h)g (term)g(individually)-8 b(.)39 b(Applying)30 b(Lemma)f(4)i(and)g(the)g (estimates)f(for)g Fu(k)p Fq(x)3547 1953 y Fr(\013)3597 1938 y Fq(A)3670 1953 y Fr(e)3707 1938 y Fu(k)h Fw(and)-71 2058 y Fu(k)p Fq( )42 2073 y Fr(n)89 2058 y Fu(k)p Fw(,)i(w)m(e)g (obtain)282 2368 y(1)331 2331 y Fs(st)423 2368 y Fw(term)83 b Fu(\024)g Ft(~)931 2299 y Fl(N)p 931 2311 54 3 v 942 2352 a Fk(2)1015 2243 y Fr(N)7 b Fm(\000)p Fs(1)1020 2273 y Fv(X)1025 2482 y Fr(n)p Fs(=0)1317 2273 y Fv(X)1185 2489 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fs(+2)p Fm(\000)p Fr(n)1609 2368 y Fq(\016)1656 2326 y Fm(j)p Fr(m)p Fm(j)1762 2368 y Fq(F)14 b(G)1926 2298 y Fk(3)p Fl(n)p Fk(+)p Fl(a)p Fk(+2)p 1925 2311 227 3 v 2023 2352 a(2)2182 2227 y Fv(\022)2255 2368 y Fw(1)22 b Fu(\000)2436 2300 y Fw(4)p Fq(\034)p 2436 2345 103 4 v 2437 2436 a(!)2498 2451 y Fs(0)2548 2227 y Fv(\023)2621 2249 y Fm(\000)2686 2216 y Fi(j)p Fl(m)p Fi(j)p 2686 2234 93 3 v 2717 2275 a Fk(2)2809 2368 y Fw(\(3)p Fq(n)g Fw(+)g Fq(a)h Fw(+)f Fq(d)p Fw(\))3344 2291 y Fl(d)p Fi(\000)p Fk(1)p 3344 2304 111 3 v 3384 2345 a(2)867 2726 y Fu(\002)989 2586 y Fv(\024)1051 2659 y Fw(\(3)p Fq(n)g Fw(+)g Fu(j)p Fq(m)p Fu(j)g Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)g Fw(+)g Fq(a)p Fw(\)!)p 1051 2703 991 4 v 1356 2794 a(\(3)p Fq(n)g Fw(+)g Fq(a)p Fw(\)!)2052 2586 y Fv(\025)2115 2581 y Fk(1)p 2115 2593 31 3 v 2115 2634 a(2)2176 2726 y Fq(\015)2227 2741 y Fs(1)2266 2726 y Fq(\024)2322 2685 y Fs(3)p Fr(n)2404 2726 y Fq(b)2445 2685 y Fr(n)p Fs(+)p Fr(w)2600 2726 y Fw([\(1)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])3165 2658 y Fk(1)p 3166 2670 V 3166 2711 a(2)707 3058 y Fu(\024)83 b Fq(F)14 b(\015)995 3073 y Fs(1)1034 3058 y Fq(\016)1081 3017 y Fm(\000)p Fs(\()p Fr(N)7 b Fs(+2\))1348 3058 y Fq(G)1435 2989 y Fk(3)p Fl(N)e Fk(+)p Fl(a)p Fk(+)p Fl(d)p Fk(+1)p 1435 3002 320 3 v 1579 3043 a(2)1785 2917 y Fv(\022)1859 3058 y Fw(1)22 b Fu(\000)2039 2990 y Fw(4)p Fq(\034)p 2039 3035 103 4 v 2040 3126 a(!)2101 3141 y Fs(0)2151 2917 y Fv(\023)2225 2940 y Fm(\000)2290 2912 y Fl(N)5 b Fk(+2)p 2290 2925 131 3 v 2340 2966 a(2)2451 3058 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)h Fw(+)f Fq(d)f Fu(\000)i Fw(3\))3187 2989 y Fl(d)p Fi(\000)p Fk(1)p 3187 3002 111 3 v 3227 3043 a(2)867 3365 y Fu(\002)989 3241 y Fr(N)7 b Fm(\000)p Fs(1)993 3271 y Fv(X)999 3480 y Fr(n)p Fs(=0)1159 3365 y Fq(\024)1215 3324 y Fs(3)p Fr(n)1314 3225 y Fv(\024)1376 3298 y Fw(\(2)p Fq(n)22 b Fw(+)g Fq(N)33 b Fw(+)22 b Fe(J)p Fq(d=)p Fw(2)p Fe(K)g Fw(+)g Fq(a)g Fw(+)g(2\)!\()p Fq(n)g Fw(+)g Fq(a)g Fw(+)g(1\)!)p 1376 3343 1610 4 v 1990 3434 a(\(3)p Fq(n)g Fw(+)g Fq(a)p Fw(\)!)2995 3225 y Fv(\025)3058 3220 y Fk(1)p 3058 3232 31 3 v 3058 3273 a(2)3251 3271 y Fv(X)3119 3487 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)7 b Fs(+2)p Fm(\000)p Fr(n)3543 3365 y Fw(1)-71 3694 y(Note)33 b(that)f(\(2)p Fq(n)22 b Fw(+)g Fq(N)33 b Fw(+)22 b Fe(J)p Fq(d=)p Fw(2)p Fe(K)g Fw(+)g Fq(a)g Fw(+)g(2\)!)27 b Fu(\024)i Fw(\(2)p Fq(n)22 b Fw(+)g Fq(N)32 b Fw(+)22 b Fq(a)h Fw(+)f(2\)!\(2)p Fq(n)g Fw(+)g Fq(N)32 b Fw(+)22 b Fe(J)p Fq(d=)p Fw(2)p Fe(K)g Fw(+)g Fq(a)g Fw(+)g(2\))3456 3658 y Fc(J)p Fr(d=)p Fs(2)p Fc(K)3624 3694 y Fw(.)43 b(Then)171 3926 y(1)220 3890 y Fs(st)311 3926 y Fw(term)83 b Fu(\024)g Fq(A)828 3941 y Fs(1)868 3926 y Fq(B)947 3885 y Fr(N)942 3951 y Fs(1)1014 3926 y Ft(~)1078 3858 y Fl(N)p 1078 3870 54 3 v 1090 3911 a Fk(2)1146 3926 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)g Fw(+)g Fq(d)g Fu(\000)h Fw(3\))1882 3857 y Fl(d)p Fi(\000)p Fk(1)p 1882 3870 111 3 v 1922 3911 a(2)2006 3926 y Fw(\()p Fq(N)33 b Fw(+)22 b Fq(d)g Fw(+)g(1\))2511 3885 y Fr(d)p Fm(\000)p Fs(1)2641 3926 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)h Fw(+)f Fe(J)p Fq(d=)p Fw(2)p Fe(K)p Fw(\))3385 3852 y Fb(J)p Fl(d=)p Fk(2)p Fb(K)p 3384 3870 147 3 v 3442 3911 a Fk(2)755 4212 y Fu(\002)28 b Fw([\(2)22 b(+)g Fq(a)h Fw(+)f Fq(N)10 b Fw(\)!])1456 4143 y Fk(1)p 1456 4155 31 3 v 1456 4197 a(2)1518 4087 y Fr(N)d Fm(\000)p Fs(1)1522 4117 y Fv(X)1528 4326 y Fr(n)p Fs(=0)1687 4212 y Fq(\024)1743 4171 y Fs(3)p Fr(n)1842 4071 y Fv(\024)1905 4144 y Fw(\(2)22 b(+)g Fq(a)g Fw(+)g Fq(N)33 b Fw(+)22 b(2)p Fq(n)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!)p 1905 4189 1261 4 v 2079 4280 a(\()p Fq(a)g Fw(+)g(3)p Fq(n)p Fw(\)!\(2)g(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!)3175 4071 y Fv(\025)3238 4066 y Fk(1)p 3238 4078 31 3 v 3238 4120 a(2)595 4529 y Fu(\024)83 b Fq(A)828 4544 y Fs(1)868 4529 y Fq(B)947 4488 y Fr(N)942 4553 y Fs(1)1014 4529 y Fq(\024)1070 4488 y Fs(3)p Fr(N)1173 4529 y Ft(~)1237 4460 y Fl(N)p 1237 4472 54 3 v 1248 4514 a Fk(2)1304 4529 y Fw(\(3)p Fq(N)33 b Fw(+)22 b Fq(a)g Fw(+)g Fq(d)g Fu(\000)h Fw(3\))2041 4460 y Fl(d)p Fi(\000)p Fk(1)p 2040 4473 111 3 v 2080 4514 a(2)2165 4529 y Fw(\()p Fq(N)32 b Fw(+)22 b Fq(d)g Fw(+)g(1\))2669 4488 y Fr(d)p Fm(\000)p Fs(1)2800 4529 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)g Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)p Fw(\))3543 4454 y Fb(J)p Fl(d=)p Fk(2)p Fb(K)p 3543 4473 147 3 v 3601 4514 a Fk(2)755 4835 y Fu(\002)28 b Fw([\(2)22 b(+)g Fq(a)h Fw(+)f Fq(N)10 b Fw(\)!])1456 4767 y Fk(1)p 1456 4779 31 3 v 1456 4820 a(2)1542 4835 y Fw(max)1518 4898 y Fs(1)p Fm(\024)p Fr(l)q Fm(\024)p Fr(N)1764 4695 y Fv(\024)1827 4768 y Fw(\(3)p Fq(N)32 b Fu(\000)23 b Fw(3)p Fq(l)h Fw(+)e Fq(a)g Fw(+)g(1\)\(3)p Fq(N)32 b Fu(\000)23 b Fw(3)p Fq(l)h Fw(+)e Fq(a)g Fw(+)g(2\))p 1827 4812 1511 4 v 2253 4903 a(\()p Fq(N)33 b Fu(\000)22 b Fq(l)j Fw(+)d Fq(a)g Fw(+)g(2\))3347 4695 y Fv(\025)3410 4690 y Fk(1)p 3410 4702 31 3 v 3410 4743 a(2)755 5143 y Fu(\002)918 5018 y Fr(N)877 5048 y Fv(X)893 5260 y Fr(l)q Fs(=1)1038 5143 y Fq(\024)1094 5102 y Fm(\000)1159 5075 y Fk(5)p Fl(l)p 1159 5087 51 3 v 1169 5128 a Fk(2)1240 5002 y Fv(\024)1302 5075 y Fw(\(2)g(+)g Fq(a)h Fw(+)f(3)p Fq(N)32 b Fw(+)22 b(2)p Fq(l)r Fw(\)!\(2)g(+)g Fq(a)g Fw(+)g Fq(N)33 b Fu(\000)23 b Fq(l)r Fw(\)!)p 1302 5120 1466 4 v 1378 5211 a(\(2)f(+)g Fq(a)g Fw(+)g(3)p Fq(N)33 b Fu(\000)22 b Fw(3)p Fq(l)r Fw(\)!\(2)g(+)g Fq(a)h Fw(+)f Fq(N)10 b Fw(\)!)2778 5002 y Fv(\025)2840 4998 y Fk(1)p 2840 5010 31 3 v 2840 5051 a(2)1888 5470 y Fw(27)p eop %%Page: 28 28 28 27 bop -71 123 a Fw(where)32 b(w)m(e)g(use)f(that)726 49 y Fv(P)832 152 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)7 b Fs(+2)p Fm(\000)p Fr(n)1260 123 y Fw(1)27 b Fu(\024)i Fw([\()p Fq(d)17 b Fu(\000)i Fw(1\)!])1812 87 y Fm(\000)p Fs(1)1906 123 y Fw(\()p Fq(N)29 b Fw(+)18 b Fq(d)f Fw(+)h(1\))2394 87 y Fr(d)p Fm(\000)p Fs(1)2525 123 y Fw(,)31 b(and)f(then)i(w)m(e)f (collect)f(constan)m(ts)i(in)-71 244 y Fq(A)2 259 y Fs(1)78 244 y Fw(and)37 b Fq(B)346 259 y Fs(1)385 244 y Fw(.)55 b(A)37 b(c)m(hange)g(of)f(index)h(is)f(p)s(erformed)g(in)g(the)h (summation)d(ab)s(o)m(v)m(e.)56 b(No)m(w)37 b(w)m(e)h(need)f(to)f (apply)-71 364 y(Lemma)31 b(3)i(to)f(obtain)140 562 y(1)189 526 y Fs(st)281 562 y Fw(term)83 b Fu(\024)g Fq(\014)6 b(A)859 577 y Fs(1)898 562 y Fq(B)977 521 y Fr(N)972 586 y Fs(1)1045 562 y Fq(\024)1101 521 y Fs(3)p Fr(N)1203 562 y Ft(~)1267 493 y Fl(N)p 1267 505 54 3 v 1279 547 a Fk(2)1335 562 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)g Fw(+)g Fq(d)g Fu(\000)h Fw(3\))2071 493 y Fl(d)p Fi(\000)p Fk(1)p 2071 506 111 3 v 2111 547 a(2)2195 562 y Fw(\()p Fq(N)33 b Fw(+)22 b Fq(d)g Fw(+)g(1\))2700 521 y Fr(d)p Fm(\000)p Fs(1)2830 562 y Fw(\(3)p Fq(N)32 b Fw(+)22 b Fq(a)h Fw(+)f Fe(J)p Fq(d=)p Fw(2)p Fe(K)p Fw(\))3574 487 y Fb(J)p Fl(d=)p Fk(2)p Fb(K)p 3573 505 147 3 v 3631 547 a Fk(2)725 744 y Fu(\002)28 b Fw([3\(3)p Fq(N)k Fw(+)22 b Fq(a)h Fw(+)f(2\)])1497 676 y Fk(1)p 1497 688 31 3 v 1497 729 a(2)1541 744 y Fw([\(2)g(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!])2136 676 y Fk(1)p 2137 688 V 2137 729 a(2)-71 941 y Fw(Finally)30 b(de\014ne)k Fq(N)621 956 y Fs(1)693 941 y Fw(through)e(the)h(inequalit)m(y)197 1158 y(\(3)p Fq(N)f Fw(+)22 b Fq(a)g Fw(+)g Fe(J)p Fq(d=)p Fw(2)p Fe(K)p Fw(\))940 1084 y Fb(J)p Fl(d=)p Fk(2)p Fb(K)p 940 1102 147 3 v 998 1143 a Fk(2)1100 1158 y Fw(\(3)p Fq(N)1265 1173 y Fs(1)1327 1158 y Fw(+)g Fq(a)g Fw(+)g Fq(d)g Fu(\000)h Fw(3\))1866 1089 y Fl(d)p Fi(\000)p Fk(1)p 1865 1102 111 3 v 1905 1143 a(2)1990 1158 y Fw(\()p Fq(N)2106 1173 y Fs(1)2168 1158 y Fw(+)f Fq(d)f Fw(+)h(1\))2523 1117 y Fr(d)p Fm(\000)p Fs(1)2654 1158 y Fw([3\(3)p Fq(N)2895 1173 y Fs(1)2956 1158 y Fw(+)g Fq(a)g Fw(+)g(2\)])3349 1090 y Fk(1)p 3349 1102 31 3 v 3349 1143 a(2)3421 1158 y Fu(\024)29 b Fq(\024)3583 1117 y Fr(N)3639 1126 y Fk(1)-71 1356 y Fw(Then)1075 1476 y(1)1124 1440 y Fs(st)1216 1476 y Fw(term)e Fu(\024)h Fq(\014)6 b(A)1683 1491 y Fs(1)1722 1476 y Fq(B)1801 1435 y Fr(N)1796 1501 y Fs(1)1869 1476 y Fq(\024)1925 1435 y Fs(4)p Fr(N)2027 1476 y Ft(~)2091 1408 y Fl(N)p 2091 1420 54 3 v 2103 1461 a Fk(2)2159 1476 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!])2754 1408 y Fk(1)p 2755 1420 31 3 v 2755 1461 a(2)-71 1641 y Fw(whenev)m(er)35 b Fq(N)j Fu(\025)29 b Fq(N)657 1656 y Fs(1)696 1641 y Fw(.)75 1832 y(Inequalit)m(y)k(\(48\))f(yields)243 2029 y Fq(\016)290 1992 y Fm(j)p Fr(m)p Fm(j)p 243 2073 153 4 v 263 2164 a Fq(m)p Fw(!)422 2096 y Fu(j)p Fq(D)534 2055 y Fr(m)600 2096 y Fq(V)22 b Fw(\()p Fq(\020)8 b Fw(\()p Fq(x)p Fw(\)\))p Fu(j)26 b(\024)j Fq(K)23 b Fw(exp)1369 1955 y Fv(\022)1452 2029 y Fw(2)p Fq(\034)11 b(d)p 1452 2073 V 1477 2164 a(!)1542 2130 y Fs(2)1538 2189 y(0)1615 2096 y Fq(R)1690 2055 y Fs(2)1730 1955 y Fv(\023)1831 2096 y Fw(=)27 b Fq(K)d Fw(exp)2207 1955 y Fv(\024)2269 2029 y Fw(2)p Fq(\034)11 b(d)p 2269 2073 V 2294 2164 a(!)2359 2130 y Fs(2)2355 2189 y(0)2432 2096 y Fw(\(2)p Fq(a)22 b Fw(+)g Fq(d)g Fu(\000)h Fw(4\))2950 1955 y Fv(\025)3019 2096 y Fw(exp)3184 1955 y Fv(\022)3267 2029 y Fw(12)p Fq(\034)11 b(d)p 3267 2073 202 4 v 3316 2164 a(!)3381 2130 y Fs(2)3377 2189 y(0)3479 2096 y Fq(N)3567 1955 y Fv(\023)-71 2351 y Fw(on)33 b(the)g(supp)s(ort)g(of)f Fq(\037)766 2366 y Fr(R)823 2351 y Fw(\()p Fq(x)p Fw(\).)44 b(Th)m(us,)34 b(the)f(second)h(term)e(satis\014es)-1 2606 y(2)48 2570 y Fs(nd)163 2606 y Fw(term)82 b Fu(\024)i Ft(~)671 2537 y Fl(N)p 671 2549 54 3 v 682 2591 a Fk(2)738 2606 y Fq(\015)789 2621 y Fs(1)828 2606 y Fq(\016)875 2565 y Fm(\000)p Fs(\()p Fr(N)7 b Fs(+2\))1142 2606 y Fq(K)24 b Fw(exp)1415 2465 y Fv(\024)1477 2538 y Fw(2)p Fq(\034)11 b(d)p 1477 2583 153 4 v 1502 2674 a(!)1567 2640 y Fs(2)1563 2698 y(0)1640 2606 y Fw(\(2)p Fq(a)22 b Fw(+)g Fq(d)g Fu(\000)h Fw(4\))2158 2465 y Fv(\025)2227 2606 y Fw(exp)2392 2465 y Fv(\022)2476 2538 y Fw(12)p Fq(\034)11 b(d)p 2476 2583 202 4 v 2525 2674 a(!)2590 2640 y Fs(2)2586 2698 y(0)2687 2606 y Fq(N)2775 2465 y Fv(\023)607 2913 y Fu(\002)729 2789 y Fr(N)c Fm(\000)p Fs(1)733 2819 y Fv(X)739 3028 y Fr(n)p Fs(=0)1030 2819 y Fv(X)898 3035 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1323 2799 y Fv(\015)1323 2859 y(\015)1323 2918 y(\015)1378 2913 y Fq(x)1433 2872 y Fr(m)1495 2849 y Fi(0)1522 2913 y Fq(P)1585 2929 y Fm(j)p Fr(j)t Fm(j\024)p Fs(3)p Fr(n)p Fs(+)p Fr(a)p Fs(+1)1976 2799 y Fv(\015)1976 2859 y(\015)1976 2918 y(\015)2048 2913 y Fu(k)p Fq( )2161 2928 y Fr(n)2209 2913 y Fw(\()p Fq(x)p Fw(\))p Fu(k)446 3239 y(\024)84 b Ft(~)671 3171 y Fl(N)p 671 3183 54 3 v 682 3224 a Fk(2)738 3239 y Fq(\015)789 3254 y Fs(1)828 3239 y Fq(\016)875 3198 y Fm(\000)p Fs(\()p Fr(N)7 b Fs(+2\))1142 3239 y Fq(K)24 b Fw(exp)1415 3098 y Fv(\024)1477 3171 y Fw(2)p Fq(\034)11 b(d)p 1477 3216 153 4 v 1502 3307 a(!)1567 3273 y Fs(2)1563 3332 y(0)1640 3239 y Fw(\(2)p Fq(a)22 b Fw(+)g Fq(d)g Fu(\000)h Fw(4\))2158 3098 y Fv(\025)2227 3239 y Fw(exp)2392 3098 y Fv(\022)2476 3171 y Fw(12)p Fq(\034)11 b(d)p 2476 3216 202 4 v 2525 3307 a(!)2590 3273 y Fs(2)2586 3332 y(0)2687 3239 y Fq(N)2775 3098 y Fv(\023)607 3547 y Fu(\002)729 3422 y Fr(N)c Fm(\000)p Fs(1)733 3452 y Fv(X)739 3661 y Fr(n)p Fs(=0)1030 3452 y Fv(X)898 3668 y Fm(j)p Fr(m)p Fm(j)p Fs(=)p Fr(N)g Fm(\000)p Fr(n)p Fs(+2)1323 3547 y Fq(\024)1389 3472 y Fi(j)p Fl(m)p Fi(j\000)p Fk(1)p 1389 3490 171 3 v 1459 3531 a(2)1590 3406 y Fv(\024)1653 3479 y Fw(\()p Fq(a)22 b Fw(+)g Fu(j)p Fq(m)p Fu(j)g Fw(+)g(3)p Fq(n)p Fw(\)!)p 1653 3524 643 4 v 1699 3615 a(\(1)f(+)h Fq(a)h Fw(+)f(3)p Fq(n)p Fw(\)!)2305 3406 y Fv(\025)2367 3401 y Fk(1)p 2367 3413 31 3 v 2367 3454 a(2)2428 3547 y Fq(\024)2484 3505 y Fs(3)p Fr(n)2567 3547 y Fq(b)2608 3505 y Fr(n)p Fs(+)p Fr(w)2763 3547 y Fw([\(1)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])3328 3478 y Fk(1)p 3328 3490 V 3328 3531 a(2)446 3907 y Fu(\024)84 b Fq(A)680 3922 y Fs(2)719 3907 y Fq(B)798 3866 y Fr(N)793 3932 y Fs(2)866 3907 y Ft(~)930 3839 y Fl(N)p 930 3851 54 3 v 941 3892 a Fk(2)997 3907 y Fw(\()p Fq(N)33 b Fw(+)22 b Fq(d)g Fw(+)g(1\))1502 3866 y Fr(d)p Fm(\000)p Fs(1)1649 3783 y Fr(N)7 b Fm(\000)p Fs(1)1653 3813 y Fv(X)1659 4022 y Fr(n)p Fs(=0)1818 3907 y Fq(\024)1874 3866 y Fs(3)p Fr(n)1973 3767 y Fv(\024)2036 3840 y Fw(\(2)22 b(+)g Fq(a)g Fw(+)g Fq(N)33 b Fw(+)22 b(2)p Fq(n)p Fw(\)!\(1)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!)p 2036 3885 1261 4 v 2391 3976 a(\(1)g(+)g Fq(a)g Fw(+)g(3)p Fq(n)p Fw(\)!)3306 3767 y Fv(\025)3369 3762 y Fk(1)p 3369 3774 31 3 v 3369 3815 a(2)446 4215 y Fu(\024)84 b Fq(A)680 4230 y Fs(2)719 4215 y Fq(B)798 4174 y Fr(N)793 4239 y Fs(2)866 4215 y Fq(\024)922 4174 y Fs(3)p Fr(N)1024 4215 y Ft(~)1088 4146 y Fl(N)p 1088 4158 54 3 v 1100 4200 a Fk(2)1156 4215 y Fw(\()p Fq(N)32 b Fw(+)22 b Fq(d)g Fw(+)g(1\))1660 4174 y Fr(d)p Fm(\000)p Fs(1)1791 4215 y Fw([\(2)g(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!])2386 4146 y Fk(1)p 2387 4158 31 3 v 2387 4200 a(2)607 4500 y Fu(\002)69 b Fw(max)729 4563 y Fs(1)p Fm(\024)p Fr(l)q Fm(\024)p Fr(N)975 4360 y Fv(\024)1038 4433 y Fw(2)22 b(+)g Fq(a)g Fw(+)g(3)p Fq(N)32 b Fu(\000)23 b Fw(3)p Fq(l)p 1038 4477 680 4 v 1086 4568 a Fw(2)f(+)g Fq(a)h Fw(+)f Fq(N)32 b Fu(\000)23 b Fq(l)1727 4360 y Fv(\025)1790 4355 y Fk(1)p 1790 4367 31 3 v 1790 4408 a(2)1891 4375 y Fr(N)1851 4405 y Fv(X)1867 4618 y Fr(l)q Fs(=1)2011 4500 y Fq(\024)2067 4459 y Fm(\000)2132 4432 y Fk(5)p Fl(l)p 2132 4444 51 3 v 2142 4485 a Fk(2)2213 4360 y Fv(\024)2276 4433 y Fw(\(2)f(+)g Fq(a)g Fw(+)g(3)p Fq(N)33 b Fw(+)22 b(2)p Fq(l)r Fw(\)!\(2)g(+)g Fq(a)g Fw(+)g Fq(N)33 b Fu(\000)22 b Fq(l)r Fw(\)!)p 2276 4477 1466 4 v 2352 4568 a(\(2)g(+)g Fq(a)g Fw(+)g(3)p Fq(N)32 b Fu(\000)23 b Fw(3)p Fq(l)r Fw(\)!\(2)f(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!)3751 4360 y Fv(\025)3814 4355 y Fk(1)p 3814 4367 31 3 v 3814 4408 a(2)446 4810 y Fu(\024)84 b Fw(3)666 4742 y Fk(1)p 666 4754 V 666 4795 a(2)710 4810 y Fq(\014)6 b(A)844 4825 y Fs(2)883 4810 y Fq(B)962 4769 y Fr(N)957 4835 y Fs(2)1030 4810 y Fq(\024)1086 4769 y Fs(3)p Fr(N)1188 4810 y Ft(~)1252 4742 y Fl(N)p 1252 4754 54 3 v 1264 4795 a Fk(2)1320 4810 y Fw(\()p Fq(N)32 b Fw(+)22 b Fq(d)g Fw(+)g(1\))1824 4769 y Fr(d)p Fm(\000)p Fs(1)1955 4810 y Fw([\(2)f(+)h Fq(a)h Fw(+)f Fq(N)10 b Fw(\)!])2550 4742 y Fk(1)p 2550 4754 31 3 v 2550 4795 a(2)-71 5008 y Fw(No)m(w)33 b(de\014ne)h Fq(N)511 5023 y Fs(2)583 5008 y Fw(to)e(sati\014es)h(\()p Fq(N)1139 5023 y Fs(2)1201 5008 y Fw(+)22 b Fq(d)g Fw(+)g(1\))1557 4972 y Fr(d)p Fm(\000)p Fs(1)1715 5008 y Fu(\024)28 b Fq(\024)1876 4972 y Fr(N)1932 4981 y Fk(2)1971 5008 y Fw(.)43 b(Then)34 b(for)e Fq(N)38 b Fu(\025)28 b Fq(N)2744 5023 y Fs(2)1012 5221 y Fw(2)1061 5184 y Fs(nd)1176 5221 y Fw(term)f Fu(\024)h Fw(3)1568 5152 y Fk(1)p 1568 5164 V 1568 5205 a(2)1612 5221 y Fq(\014)6 b(A)1746 5236 y Fs(2)1785 5221 y Fq(B)1864 5179 y Fr(N)1859 5245 y Fs(2)1932 5221 y Fq(\024)1988 5179 y Fs(4)p Fr(N)2090 5221 y Ft(~)2154 5152 y Fl(N)p 2154 5164 54 3 v 2166 5205 a Fk(2)2222 5221 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(N)10 b Fw(\)!])2817 5152 y Fk(1)p 2818 5164 31 3 v 2818 5205 a(2)1888 5470 y Fw(28)p eop %%Page: 29 29 29 28 bop 75 123 a Fw(F)-8 b(or)32 b(the)h(third)f(term)g(w)m(e)i(only) e(need)h(to)g(use)g(Lemma)e(1.)44 b(The)33 b(result)g(is)1004 401 y(3)1053 364 y Fs(rd)1156 401 y Fw(term)27 b Fu(\024)1521 276 y Fs(2)p Fr(N)1498 306 y Fv(X)1489 517 y Fr(n)p Fs(=)p Fr(N)1667 401 y Fq(\015)1718 416 y Fr(a)1759 401 y Fq(\024)1815 359 y Fs(3)p Fr(n)1897 401 y Fq(b)1938 359 y Fr(n)p Fs(+2)p Fr(w)2128 401 y Ft(~)2192 332 y Fl(N)p 2192 344 54 3 v 2204 385 a Fk(2)2260 401 y Fw([\(1)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2825 332 y Fk(1)p 2826 344 31 3 v 2826 385 a(2)75 701 y Fw(T)-8 b(o)38 b(complete)f(the)i(pro)s(of)e (de\014ne)i Fq(N)1440 716 y Fs(0)1516 701 y Fw(=)d(max)p Fu(f)p Fq(N)1938 716 y Fs(1)1977 701 y Fq(;)17 b(N)2099 716 y Fs(2)2139 701 y Fu(g)p Fw(,)39 b Fq(A)d Fw(=)h(max)o Fu(f)p Fq(\014)6 b(A)2842 716 y Fs(1)2881 701 y Fq(;)17 b Fw(3)2984 637 y Fk(1)p 2984 649 V 2984 691 a(2)3028 701 y Fq(\014)6 b(A)3162 716 y Fs(2)3201 701 y Fq(;)17 b(\015)3296 716 y Fr(a)3337 701 y Fq(b)3378 665 y Fs(2)p Fr(w)3471 701 y Fu(g)37 b Fw(and)h Fq(B)k Fw(=)-71 822 y(max)p Fu(f)p Fq(\024)217 785 y Fs(4)256 822 y Fq(B)330 837 y Fs(1)370 822 y Fq(;)17 b(\024)470 785 y Fs(3)509 822 y Fq(B)583 837 y Fs(2)622 822 y Fq(;)g(\024)722 785 y Fs(3)762 822 y Fq(b)p Fu(g)p Fw(.)2990 b Fg(2)-71 1150 y Fx(5)161 b(Optimal)55 b(truncation)-71 1369 y Fw(In)33 b(this)f(section)h(w)m(e)g(pro)m(v)m(e)h(that)e(exact)h(eigen)m(v)-5 b(alues)33 b(and)f(eigenfunctions)g(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\))27 b(:=)h Fu(\000)3347 1330 y Fs(1)p 3347 1346 36 4 v 3347 1403 a(2)3393 1369 y Fw(\001)3474 1384 y Fr(x)3540 1369 y Fw(+)22 b Fq(V)f Fw(\()p Ft(~)p Fq(;)c(x)p Fw(\))-71 1489 y(can)30 b(b)s(e)h(appro)m(ximated)e(b)m(y)i(truncated)g (R-S)e(series,)j(up)e(to)g(an)g(exp)s(onen)m(tially)f(small)f(error.)42 b(T)-8 b(o)30 b(that)g(end,)-71 1610 y(w)m(e)k(will)c(exploit)i(our)g (estimate)g(of)g(the)h(norm)f Fq(A)1751 1574 y Fm(\000)p Fs(1)1751 1634 y Fr(e)1845 1610 y Fq(\030)1888 1625 y Fr(N)1955 1610 y Fw(\()p Fq(x)p Fw(\).)44 b(W)-8 b(e)33 b(will)d(also)i(need)i(a)e(couple)h(of)f(results.)44 b(The)-71 1730 y(\014rst)31 b(is)f(a)g(lo)m(w)m(er)g(b)s(ound)g(for)g (the)h(distance)f(b)s(et)m(w)m(een)j(p)s(erturb)s(ed)d(eigen)m(v)-5 b(alues)31 b(that)f(degenerate)h(at)f Ft(~)d Fw(=)h(0.)-71 1851 y(The)38 b(second)g(is)e(a)h(\\rev)m(erse")h(de\014nition)e(of)g (asymptoticness,)j(whic)m(h)e(is)g(a)f(generalization)e(of)j(Lemma)e(4) -71 1971 y(of)d([15].)75 2162 y(Let)41 b(us)f(consider)g(t)m(w)m(o)h (distinct)e(eigen)m(v)-5 b(alues)40 b(of)f Fq(H)8 b Fw(\()p Ft(~)p Fw(\),)41 b Fq(E)6 b Fw(\()p Ft(~)p Fw(\))40 b(and)g Fq(E)2769 2126 y Fm(0)2792 2162 y Fw(\()p Ft(~)p Fw(\),)i(whic)m(h)e (con)m(v)m(erge)i(to)d(the)-71 2283 y(same)g(eigen)m(v)-5 b(alue)39 b(of)g Fq(H)852 2298 y Fs(0)931 2283 y Fw(as)h Ft(~)f Fw(go)s(es)g(to)g(0.)64 b(Also,)41 b(let)e(us)h(assume)g(that)f (their)g(asymptotic)f(series)i(ha)m(v)m(e)-71 2403 y(only)32 b(a)g(\014nite)h(n)m(um)m(b)s(er)g(of)f(common)f(R-S)h(co)s(e\016cien)m (ts.)45 b(That)33 b(is,)540 2603 y Fq(E)6 b Fw(\()p Ft(~)p Fw(\))83 b Fu(\030)g Fq(e)23 b Fw(+)f Fu(E)1210 2618 y Fs(1)1249 2603 y Ft(~)1313 2535 y Fk(1)p 1312 2547 31 3 v 1312 2588 a(2)1379 2603 y Fw(+)g Fq(:)17 b(:)g(:)22 b Fw(+)g Fu(E)1765 2618 y Fr(M)7 b Fm(\000)p Fs(1)1934 2603 y Ft(~)1998 2534 y Fl(M)e Fi(\000)p Fk(1)p 1997 2547 142 3 v 2053 2588 a(2)2175 2603 y Fw(+)22 b Fu(E)2326 2618 y Fr(M)2405 2603 y Ft(~)2469 2535 y Fl(M)p 2469 2547 64 3 v 2485 2588 a Fk(2)2568 2603 y Fw(+)g Fu(E)2719 2618 y Fr(M)7 b Fs(+1)2888 2603 y Ft(~)2952 2534 y Fl(M)e Fk(+1)p 2952 2547 141 3 v 3007 2588 a(2)3128 2603 y Fw(+)22 b Fq(:)17 b(:)g(:)517 2765 y(E)595 2724 y Fm(0)618 2765 y Fw(\()p Ft(~)p Fw(\))83 b Fu(\030)g Fq(e)23 b Fw(+)f Fu(E)1210 2780 y Fs(1)1249 2765 y Ft(~)1313 2696 y Fk(1)p 1312 2708 31 3 v 1312 2750 a(2)1379 2765 y Fw(+)g Fq(:)17 b(:)g(:)22 b Fw(+)g Fu(E)1765 2780 y Fr(M)7 b Fm(\000)p Fs(1)1934 2765 y Ft(~)1998 2696 y Fl(M)e Fi(\000)p Fk(1)p 1997 2709 142 3 v 2053 2750 a(2)2175 2765 y Fw(+)22 b Fu(E)2335 2724 y Fm(0)2326 2789 y Fr(M)2405 2765 y Ft(~)2469 2696 y Fl(M)p 2469 2708 64 3 v 2485 2750 a Fk(2)2568 2765 y Fw(+)g Fu(E)2728 2724 y Fm(0)2719 2789 y Fr(M)7 b Fs(+1)2888 2765 y Ft(~)2952 2696 y Fl(M)e Fk(+1)p 2952 2709 141 3 v 3007 2750 a(2)3128 2765 y Fw(+)22 b Fq(:)17 b(:)g(:)-71 2955 y Fw(with)32 b Fu(E)204 2970 y Fr(M)311 2955 y Fu(6)p Fw(=)27 b Fu(E)476 2919 y Fm(0)467 2980 y Fr(M)546 2955 y Fw(.)43 b(Then,)658 3161 y Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1065 3120 y Fm(0)1089 3161 y Fw(\()p Ft(~)p Fw(\))27 b Fu(\030)h Fw(\()p Fu(E)1442 3176 y Fr(M)1543 3161 y Fu(\000)23 b(E)1705 3120 y Fm(0)1696 3186 y Fr(M)1774 3161 y Fw(\))17 b Ft(~)1893 3093 y Fl(M)p 1893 3105 64 3 v 1909 3146 a Fk(2)1992 3161 y Fw(+)2090 3080 y Fv(\000)2136 3161 y Fu(E)2189 3176 y Fr(M)7 b Fs(+1)2380 3161 y Fu(\000)22 b(E)2541 3120 y Fm(0)2532 3186 y Fr(M)7 b Fs(+1)2701 3080 y Fv(\001)2763 3161 y Ft(~)2827 3092 y Fl(M)e Fk(+1)p 2827 3105 141 3 v 2882 3146 a(2)3004 3161 y Fw(+)22 b Fq(:)17 b(:)g(:)-71 3360 y Fw(so)25 b(w)m(e)g(exp)s(ect)h(that)e(the)h (di\013erence)g(b)s(et)m(w)m(een)h(these)g(exact)f(eigen)m(v)-5 b(alues)24 b(b)s(e)h(b)s(ounded)g(b)s(elo)m(w)f(b)m(y)h Fq(O)3627 3279 y Fv(\000)3673 3360 y Ft(~)3727 3324 y Fr(M)s(=)p Fs(2)3872 3279 y Fv(\001)3918 3360 y Fw(.)-71 3480 y(Since)33 b(the)g(series)g(ab)s(o)m(v)m(e)g(is)g(asymptotic,)e (there)j(are)e Fq(C)1998 3495 y Fr(M)2105 3480 y Fq(>)27 b Fw(0)33 b(and)f Ft(~)2533 3495 y Fs(1)2573 3480 y Fw(\()p Fq(M)10 b Fw(\))28 b Fq(>)f Fw(0)33 b(so)g(that)1003 3583 y Fv(\014)1003 3643 y(\014)1003 3703 y(\014)1037 3698 y Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1444 3656 y Fm(0)1468 3698 y Fw(\()p Ft(~)p Fw(\))f Fu(\000)i Fw(\()p Fu(E)1810 3713 y Fr(M)1911 3698 y Fu(\000)f(E)2072 3656 y Fm(0)2063 3722 y Fr(M)2142 3698 y Fw(\))16 b Ft(~)2260 3629 y Fl(M)p 2260 3641 64 3 v 2277 3683 a Fk(2)2338 3583 y Fv(\014)2338 3643 y(\014)2338 3703 y(\014)2398 3698 y Fu(\024)29 b Fq(C)2574 3713 y Fr(M)2652 3698 y Ft(~)2716 3628 y Fl(M)5 b Fk(+1)p 2716 3641 141 3 v 2771 3683 a(2)-71 3915 y Fw(whenev)m(er)35 b Ft(~)28 b Fu(\024)g Ft(~)598 3930 y Fs(1)637 3915 y Fw(\()p Fq(M)10 b Fw(\).)44 b(Then)1019 4121 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1454 4080 y Fm(0)1478 4121 y Fw(\()p Ft(~)p Fw(\))p Fu(j)27 b(\025)h(jE)1849 4136 y Fr(M)1949 4121 y Fu(\000)23 b(E)2111 4080 y Fm(0)2102 4146 y Fr(M)2181 4121 y Fu(j)16 b Ft(~)2289 4053 y Fl(M)p 2289 4065 64 3 v 2305 4106 a Fk(2)2388 4121 y Fu(\000)23 b Fq(C)2558 4136 y Fr(M)2637 4121 y Ft(~)2701 4052 y Fl(M)5 b Fk(+1)p 2700 4065 141 3 v 2755 4106 a(2)-71 4311 y Fw(Set)33 b Ft(~)151 4326 y Fs(2)190 4311 y Fw(\()p Fq(M)10 b Fw(\))29 b(=)e Fu(jE)583 4326 y Fr(M)684 4311 y Fu(\000)22 b(E)845 4275 y Fm(0)836 4336 y Fr(M)915 4311 y Fu(j)16 b Fq(=)p Fw(2)p Fq(C)1127 4326 y Fr(M)1205 4311 y Fw(.)44 b(Then)34 b(for)e Ft(~)27 b Fu(\024)h Ft(~)1920 4326 y Fs(2)1960 4311 y Fw(\()p Fq(M)10 b Fw(\),)1345 4546 y Fq(C)1415 4561 y Fr(M)1494 4546 y Ft(~)1558 4477 y Fl(M)5 b Fk(+1)p 1558 4490 V 1613 4531 a(2)1740 4546 y Fu(\024)1855 4478 y Fw(1)p 1855 4523 49 4 v 1855 4614 a(2)1931 4546 y Fu(j)o(E)2011 4561 y Fr(M)2112 4546 y Fu(\000)23 b(E)2274 4505 y Fm(0)2265 4570 y Fr(M)2343 4546 y Fu(j)16 b Ft(~)2451 4478 y Fl(M)p 2451 4490 64 3 v 2468 4531 a Fk(2)-71 4767 y Fw(Th)m(us)34 b(for)e Ft(~)c Fu(\024)g Fw(min)o Fu(f)p Ft(~)779 4782 y Fs(1)818 4767 y Fw(\()p Fq(M)10 b Fw(\))p Fq(;)17 b Ft(~)1096 4782 y Fs(2)1135 4767 y Fw(\()p Fq(M)10 b Fw(\))p Fu(g)33 b Fw(w)m(e)h(ha)m(v)m(e)1221 5001 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1656 4960 y Fm(0)1680 5001 y Fw(\()p Ft(~)p Fw(\))p Fu(j)27 b(\025)1980 4934 y Fw(1)p 1980 4978 49 4 v 1980 5070 a(2)2055 5001 y Fu(jE)2136 5016 y Fr(M)2236 5001 y Fu(\000)c(E)2398 4960 y Fm(0)2389 5026 y Fr(M)2468 5001 y Fu(j)16 b Ft(~)2576 4933 y Fl(M)p 2576 4945 64 3 v 2592 4986 a Fk(2)-71 5221 y Fw(Let)33 b(us)g(denote)g Fu(E)596 5236 y Fr(M)697 5221 y Fu(\000)23 b(E)859 5184 y Fm(0)850 5246 y Fr(M)961 5221 y Fw(as)33 b(\001)p Fu(E)1215 5236 y Fr(M)1293 5221 y Fw(.)44 b(Therefore,)34 b(so)e(far)h(w)m(e)g(kno)m(w)h(that)1888 5470 y(29)p eop %%Page: 30 30 30 29 bop -71 123 a Fn(Lemma)37 b(5)49 b Fh(L)-5 b(et)44 b Fq(E)6 b Fw(\()p Ft(~)p Fw(\))43 b Fh(and)g Fq(E)1135 87 y Fm(0)1159 123 y Fw(\()p Ft(~)p Fw(\))g Fh(b)-5 b(e)43 b(distinct)h(eigenvalues)e(of)i Fq(H)8 b Fw(\()p Ft(~)p Fw(\))p Fh(,)44 b(which)f(de)-5 b(gener)g(ate)43 b(at)h Ft(~)f Fw(=)h(0)p Fh(.)-71 244 y(Then)34 b(either)45 447 y(1.)48 b Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)608 411 y Fm(0)632 447 y Fw(\()p Ft(~)p Fw(\))p Fu(j)27 b(\024)h Fq(O)1016 337 y Fv(\020)1075 447 y Ft(~)1139 383 y Fl(N)p 1139 395 54 3 v 1150 436 a Fk(2)1207 337 y Fv(\021)1301 447 y Fh(for)35 b(al)5 b(l)34 b(non-ne)-5 b(gative)33 b(inte)-5 b(gers)35 b Fq(N)10 b Fh(,)35 b(or)45 678 y(2.)48 b(Ther)-5 b(e)34 b(exist)h Fq(M)46 b Fh(and)34 b Ft(~)1063 693 y Fs(1)1130 678 y Fw(=)27 b Ft(~)1287 693 y Fs(1)1327 678 y Fw(\()p Fq(M)10 b Fw(\))35 b Fh(such)g(that)1429 942 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1864 901 y Fm(0)1887 942 y Fw(\()p Ft(~)p Fw(\))p Fu(j)28 b(\025)2188 875 y Fw(1)p 2188 919 49 4 v 2188 1011 a(2)2263 942 y Fu(j)p Fw(\001)p Fu(E)2425 957 y Fr(M)2504 942 y Fu(j)16 b Ft(~)2612 874 y Fl(M)p 2612 886 64 3 v 2628 927 a Fk(2)173 1188 y Fh(whenever)34 b Ft(~)27 b Fu(\024)h Ft(~)838 1203 y Fs(1)878 1188 y Fh(.)75 1416 y Fw(Ob)m(viously)j(this)f (lemma)e(is)i(v)-5 b(alid)29 b(for)h(sev)m(eral)h(eigen)m(v)-5 b(alues)31 b(of)f Fq(H)8 b Fw(\()p Ft(~)p Fw(\))29 b(con)m(v)m(erging)i (to)f(the)h(same)g(eigen-)-71 1537 y(v)-5 b(alue)30 b(of)g Fq(H)369 1552 y Fs(0)408 1537 y Fw(.)42 b(As)31 b(a)f(shorthand,)i(w)m (e)f(will)d(sa)m(y)j(that)f Fq(E)6 b Fw(\()p Ft(~)p Fw(\))30 b(is)g Fh(quasi-de)-5 b(gener)g(ate)28 b Fw(if)i(the)g(condition)f(1)h (in)f(the)-71 1657 y(lemma)h(ab)s(o)m(v)m(e)k(o)s(ccurs.)-71 1956 y Fn(Lemma)j(6)49 b Fh(Supp)-5 b(ose)808 1881 y Fv(P)913 1985 y Fr(n)p Fs(=0)1067 1956 y Fq(f)1115 1971 y Fr(n)1162 1956 y Fq(\014)1223 1920 y Fr(n)1310 1956 y Fh(is)40 b(asymptotic)g(to)h Fq(f)11 b Fw(\()p Fq(\014)6 b Fw(\))40 b Fh(in)g(the)g(sense)g(that)h(given)f Fq(N)48 b Fu(\025)39 b Fq(N)3617 1971 y Fs(0)3695 1956 y Fu(\025)f Fq(M)10 b Fh(,)-71 2076 y(ther)-5 b(e)35 b(exists)f Fq(C)508 2091 y Fr(N)610 2076 y Fh(and)h Fq(\014)6 b Fw(\()p Fq(N)k Fw(\))35 b Fh(such)f(that)h(for)g(al)5 b(l)35 b Fq(\014)e Fu(\024)28 b Fq(\014)6 b Fw(\()p Fq(N)k Fw(\))1255 2215 y Fv(\014)1255 2275 y(\014)1255 2334 y(\014)1255 2394 y(\014)1255 2454 y(\014)1304 2389 y Fq(f)h Fw(\()p Fq(\014)6 b Fw(\))38 b Fu(\000)1654 2265 y Fr(N)7 b Fm(\000)p Fs(1)1659 2295 y Fv(X)1665 2504 y Fr(n)p Fs(=0)1841 2389 y Fq(f)1889 2404 y Fr(n)1953 2389 y Fq(\014)2014 2348 y Fr(n)2077 2215 y Fv(\014)2077 2275 y(\014)2077 2334 y(\014)2077 2394 y(\014)2077 2454 y(\014)2173 2389 y Fq(<)62 b(C)2381 2404 y Fr(N)2465 2389 y Fq(\014)2526 2348 y Fr(N)2592 2389 y Fq(:)-71 2696 y Fh(Then)38 b(given)f Fq(\017)e(>)e Fw(0)p Fh(,)39 b(ther)-5 b(e)39 b(exists)e Fq(\014)6 b Fw(\()p Fq(\017)p Fw(\))34 b Fq(>)g Fw(0)p Fh(,)39 b(such)f(that)h(for)f(e)-5 b(ach)38 b Fq(\014)h Fu(\024)c Fq(\014)6 b Fw(\()p Fq(\017)p Fw(\))p Fh(,)39 b(ther)-5 b(e)38 b(is)g(an)g Fq(N)10 b Fw(\()p Fq(\014)c Fw(\))34 b Fu(\025)g Fq(N)3905 2711 y Fs(0)-71 2817 y Fh(\(mayb)-5 b(e)34 b(e)-5 b(qual)35 b(to)g Fu(1)p Fh(\),)f(so)h(that)1318 2955 y Fv(\014)1318 3015 y(\014)1318 3075 y(\014)1318 3134 y(\014)1318 3194 y(\014)1368 3129 y Fq(f)11 b Fw(\()p Fq(\014)6 b Fw(\))38 b Fu(\000)1718 3005 y Fr(N)7 b Fm(\000)p Fs(1)1723 3035 y Fv(X)1729 3244 y Fr(n)p Fs(=0)1905 3129 y Fq(f)1953 3144 y Fr(n)2017 3129 y Fq(\014)2078 3088 y Fr(n)2141 2955 y Fv(\014)2141 3015 y(\014)2141 3075 y(\014)2141 3134 y(\014)2141 3194 y(\014)2237 3129 y Fu(\024)63 b Fq(\017\014)2477 3088 y Fr(M)3771 3129 y Fw(\(52\))-71 3437 y Fh(whenever)34 b Fq(N)432 3452 y Fs(0)499 3437 y Fu(\024)28 b Fq(N)38 b(<)28 b(N)10 b Fw(\()p Fq(\014)c Fw(\))p Fh(.)-71 3689 y(Pr)-5 b(o)g(of.)45 b Fw(Fix)32 b Fq(\017)d(>)f Fw(0.)45 b(De\014ne)34 b Fq(\014)1057 3704 y Fs(1)1096 3689 y Fw(\()p Fq(N)1212 3704 y Fs(0)1252 3689 y Fw(\))28 b(=)h(\()p Fq(\017)17 b(C)1594 3648 y Fm(\000)p Fs(1)1587 3716 y Fr(N)1643 3725 y Fk(0)1688 3689 y Fw(\))1818 3610 y Fk(1)p 1736 3622 194 3 v 1736 3663 a Fl(N)1784 3678 y Fk(0)1819 3663 y Fi(\000)p Fl(M)1944 3689 y Fw(.)45 b(Then)34 b(for)f Fq(N)39 b(>)29 b(N)2721 3704 y Fs(0)2760 3689 y Fw(,)k(recursiv)m(ely)h (c)m(ho)s(ose)g(p)s(ositiv)m(e)-71 3836 y(n)m(um)m(b)s(ers)g Fq(\014)378 3851 y Fs(1)417 3836 y Fw(\()p Fq(N)10 b Fw(\))33 b(that)f(satisfy)h Fq(\014)1185 3851 y Fs(1)1224 3836 y Fw(\()p Fq(N)10 b Fw(\))29 b Fq(<)e Fw(min)o Fu(f)p Fw(\()p Fq(\017)17 b(C)1904 3795 y Fm(\000)p Fs(1)1897 3863 y Fr(N)1998 3836 y Fw(\))2113 3765 y Fk(1)p 2046 3777 165 3 v 2046 3819 a Fl(N)5 b Fi(\000)p Fl(M)2224 3836 y Fq(;)34 b(\014)2340 3851 y Fs(1)2379 3836 y Fw(\()p Fq(N)f Fu(\000)22 b Fw(1\))p Fu(g)p Fw(.)43 b(Then)479 3977 y Fv(\014)479 4036 y(\014)479 4096 y(\014)479 4156 y(\014)479 4216 y(\014)529 4151 y Fq(f)11 b Fw(\()p Fq(\014)6 b Fw(\))38 b Fu(\000)879 4027 y Fr(N)7 b Fm(\000)p Fs(1)884 4056 y Fv(X)889 4266 y Fr(n)p Fs(=0)1066 4151 y Fq(f)1114 4166 y Fr(n)1177 4151 y Fq(\014)1238 4110 y Fr(n)1302 3977 y Fv(\014)1302 4036 y(\014)1302 4096 y(\014)1302 4156 y(\014)1302 4216 y(\014)1362 4151 y Fu(\024)1468 4070 y Fv(\000)1513 4151 y Fq(C)1583 4166 y Fr(N)1650 4151 y Fq(\014)1711 4110 y Fr(N)g Fm(\000)p Fr(M)1908 4070 y Fv(\001)1970 4151 y Fq(\014)2031 4110 y Fr(M)2137 4151 y Fu(\024)2243 4070 y Fv(\000)2288 4151 y Fq(C)2358 4166 y Fr(N)2425 4151 y Fq(\014)2480 4166 y Fs(1)2520 4151 y Fw(\()p Fq(N)j Fw(\))2684 4110 y Fr(N)d Fm(\000)p Fr(M)2881 4070 y Fv(\001)2943 4151 y Fq(\014)3004 4110 y Fr(M)3111 4151 y Fu(\024)28 b Fq(\017\014)3316 4110 y Fr(M)-71 4458 y Fw(whenev)m(er)35 b Fq(\014)f(<)27 b(\014)604 4473 y Fs(1)644 4458 y Fw(\()p Fq(N)10 b Fw(\).)75 4579 y(De\014ne)34 b Fq(\014)6 b Fw(\()p Fq(\017)p Fw(\))27 b(=)h Fq(\014)740 4594 y Fs(1)779 4579 y Fw(\()p Fq(N)895 4594 y Fs(0)935 4579 y Fw(\),)k(and)h(de\014ne)894 4886 y Fq(N)10 b Fw(\()p Fq(\014)c Fw(\))60 b(=)1315 4715 y Fv(\()1437 4801 y Fq(N)33 b Fw(+)22 b(1)82 b(if)g Fq(\014)1972 4816 y Fs(1)2012 4801 y Fw(\()p Fq(N)32 b Fw(+)22 b(1\))28 b Fq(<)f(\014)34 b Fu(\024)28 b Fq(\014)2725 4816 y Fs(1)2764 4801 y Fw(\()p Fq(N)10 b Fw(\))1516 4969 y Fu(1)161 b Fw(if)149 b Fq(\014)33 b(<)28 b(\014)2231 4984 y Fs(1)2270 4969 y Fw(\()p Fq(N)10 b Fw(\))33 b(for)f(all)f Fq(N)5 b(:)-71 5193 y Fw(Then)34 b(\(52\))e(holds)g(whenev)m(er)j Fq(N)1151 5208 y Fs(0)1218 5193 y Fu(\024)29 b Fq(N)38 b Fu(\024)28 b Fq(N)10 b Fw(\()p Fq(\014)c Fw(\).)2073 b Fg(2)1888 5470 y Fw(30)p eop %%Page: 31 31 31 30 bop 75 123 a Fw(Let)43 b Fu(f)p Fq(e)355 138 y Fr(\014)403 123 y Fu(g)f Fw(b)s(e)h(an)g(arrangemen)m(t)g(in)f (increasing)g(order)h(of)f(the)h(eigen)m(v)-5 b(alues)43 b(of)f Fq(H)3232 138 y Fs(0)3272 123 y Fw(,)j(coun)m(ting)d(m)m(ul-)-71 244 y(tiplicities.)61 b(Theorem)40 b(1.1)f(of)h([14)o(])g(states)g (that)g(giv)m(en)f(a)h(non-negativ)m(e)f(in)m(teger)h Fq(\013)q Fw(,)h(w)m(e)f(can)g(c)m(ho)s(ose)h Ft(~)3906 259 y Fs(0)-71 364 y Fw(so)e(that)f(for)g(eac)m(h)h Ft(~)e Fu(\024)i Ft(~)913 379 y Fs(0)990 364 y Fw(there)g(are)g(at)f(least)g Fq(\013)27 b Fw(+)e(1)39 b(eigen)m(v)-5 b(alues)38 b(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\),)39 b(coun)m(ting)f(m)m(ultiplicities.) -71 484 y(F)-8 b(urthermore,)32 b(eac)m(h)i(one)f(of)f(them)g(con)m(v)m (erges)j(to)d(one)h(of)f(the)h(\014rst)g Fq(\013)23 b Fw(+)f(1)32 b(eigen)m(v)-5 b(alues)33 b(of)f Fq(H)3456 499 y Fs(0)3495 484 y Fw(.)75 605 y(In)27 b(the)f(follo)m(wing)d(prop)s (osition,)j(w)m(e)h(study)g(the)f(b)s(eha)m(vior)g(of)f(truncations)h (of)g(the)g(R-S)f(series)i(of)e Fq(E)3738 620 y Fr(\013)3788 605 y Fw(\()p Ft(~)p Fw(\),)-71 725 y(the)33 b Fq(\013)q Fw(-th)f(eigen)m(v)-5 b(alue)32 b(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\).)43 b(Without)32 b(loss)g(w)m(e)i(ma)m(y)e(assume)h(that)f Fq(e)2709 740 y Fr(\013)p Fs(+1)2877 725 y Fq(>)c(e)3026 740 y Fr(\013)3075 725 y Fw(.)-71 944 y Fn(Prop)s(osition)36 b(3)48 b Fh(L)-5 b(et)41 b Fq(E)6 b Fw(\()p Ft(~)p Fw(\))36 b(=)h Fq(E)1242 959 y Fr(\013)1291 944 y Fw(\()p Ft(~)p Fw(\))j Fh(b)-5 b(e)39 b(a)h(non-quasi-de)-5 b(gener)g(ate)38 b(eigenvalue)g(of)i Fq(H)8 b Fw(\()p Ft(~)p Fw(\))p Fh(,)40 b(which)f(c)-5 b(on-)-71 1064 y(ver)g(ges)42 b(to)h Fq(e)f Fw(=)f Fq(e)601 1079 y Fr(\013)651 1064 y Fh(.)68 b(L)-5 b(et)43 b Fq(E)997 1079 y Fr(N)1064 1064 y Fw(\()p Ft(~)p Fw(\))f Fh(b)-5 b(e)42 b(the)h(asso)-5 b(ciate)g(d)41 b(R-S)i(series,)g(trunc)-5 b(ate)g(d)43 b(at)g(or)-5 b(der)42 b Fq(N)10 b Fh(.)68 b(L)-5 b(et)43 b Fq(N)3778 1079 y Fs(0)3860 1064 y Fh(b)-5 b(e)-71 1184 y(as)42 b(de\014ne)-5 b(d)41 b(in)h(The)-5 b(or)g(em)41 b(3.)66 b(Then)42 b(ther)-5 b(e)42 b(exists)g Ft(~)1922 1199 y Fr(e)1999 1184 y Fq(>)f Fw(0)h Fh(and)g(so)g(that)g(for)g(e)-5 b(ach)42 b Ft(~)f Fu(\024)g Ft(~)3399 1199 y Fr(e)3478 1184 y Fh(ther)-5 b(e)42 b(is)g(an)-71 1305 y Fq(N)7 1320 y Fr(e)44 1305 y Fw(\()p Ft(~)p Fw(\))28 b Fu(\025)g Fq(N)385 1320 y Fs(0)459 1305 y Fh(such)35 b(that)992 1609 y Fu(j)p Fq(E)1092 1624 y Fr(N)1159 1609 y Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)h Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fu(j)27 b(\024)1810 1484 y Fs(2)p Fr(N)1787 1514 y Fv(X)1779 1725 y Fr(n)p Fs(=)p Fr(N)1956 1609 y Fq(AB)2108 1567 y Fr(n)2156 1609 y Ft(~)2220 1540 y Fl(n)p 2219 1552 39 3 v 2223 1593 a Fk(2)2272 1609 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2837 1540 y Fk(1)p 2837 1552 31 3 v 2837 1593 a(2)-71 1909 y Fh(for)35 b(al)5 b(l)34 b Ft(~)28 b Fu(\024)g Ft(~)465 1924 y Fr(e)537 1909 y Fh(and)34 b Fq(N)804 1924 y Fs(0)871 1909 y Fu(\024)28 b Fq(N)38 b Fu(\024)29 b Fq(N)1276 1924 y Fr(e)1313 1909 y Fw(\()p Ft(~)p Fw(\))p Fh(.)-71 2127 y(Pr)-5 b(o)g(of.)52 b Fw(If)36 b(no)f(other)h(eigen)m(v)-5 b(alue)35 b(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\))35 b(con)m(v)m(erges)j(to)d Fq(e)p Fw(,)i(then)f(the)g(pro)s(of)f(reduces)i(to)f(the)g(one)g(giv)m (en)-71 2248 y(in)31 b(Prop)s(osition)e(3)j(of)e([15].)43 b(Otherwise,)33 b(let)d Fq(E)1682 2212 y Fm(0)1706 2248 y Fw(\()p Ft(~)p Fw(\))h(b)s(e)g(another)h(eigen)m(v)-5 b(alue)31 b(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\))30 b(con)m(v)m(erging)i (to)f Fq(e)h Fw(as)-71 2379 y Ft(~)37 b Fu(&)h Fw(0.)60 b(By)39 b(Lemma)e(5,)j(there)f(are)f Fq(M)49 b Fw(and)39 b Ft(~)1739 2394 y Fs(1)1816 2379 y Fw(so)g(that)f Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))21 b Fu(\000)i Fq(E)2594 2343 y Fm(0)2617 2379 y Fw(\()p Ft(~)p Fw(\))p Fu(j)37 b(\025)2937 2340 y Fs(1)p 2937 2356 36 4 v 2937 2413 a(2)2999 2379 y Fu(j)p Fw(\001)p Fu(E)3161 2394 y Fr(M)3240 2379 y Fu(j)16 b Ft(~)3348 2315 y Fl(M)p 3348 2327 64 3 v 3364 2368 a Fk(2)3463 2379 y Fw(for)38 b Ft(~)g Fu(\024)g Ft(~)3879 2394 y Fs(1)3918 2379 y Fw(.)-71 2499 y(Without)28 b(loss)g(w)m(e)i(ma)m(y)e(assume)h(that)g Fq(N)1465 2514 y Fs(0)1532 2499 y Fu(\025)f Fq(M)10 b Fw(.)43 b(T)-8 b(o)29 b(simplify)d(the)j(pro)s(of,)f(w)m(e)i(furthermore)e(assume)h (that)-71 2620 y(no)k(other)f(eigen)m(v)-5 b(alue)32 b(of)h Fq(H)8 b Fw(\()p Ft(~)p Fw(\))31 b(con)m(v)m(erges)k(to)d Fq(e)p Fw(.)75 2740 y(No)m(w)i(set)f Fq(N)528 2755 y Fs(1)567 2740 y Fw(\()p Ft(~)p Fw(\))g(as)f(the)h(largest)f Fq(N)39 b Fu(\025)28 b Fq(N)1632 2755 y Fs(0)1704 2740 y Fw(suc)m(h)34 b(that)1058 2924 y Fs(2)p Fr(N)1149 2933 y Fk(1)1183 2924 y Fs(\()p Fj(~)p Fs(\))1095 2962 y Fv(X)1026 3177 y Fr(n)p Fs(=)p Fr(N)1180 3186 y Fk(1)1215 3177 y Fs(\()p Fj(~)p Fs(\))1324 3056 y Fq(AB)1476 3015 y Fr(n)1524 3056 y Ft(~)1588 2987 y Fl(n)p Fi(\000)p Fl(M)p 1587 3000 150 3 v 1647 3041 a Fk(2)1751 3056 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2316 2988 y Fk(1)p 2317 3000 31 3 v 2317 3041 a(2)2389 3056 y Fu(\024)2504 2989 y Fw(1)p 2504 3033 49 4 v 2504 3125 a(4)2579 3056 y Fu(j)p Fw(\001)p Fu(E)2741 3071 y Fr(M)2820 3056 y Fu(j)-71 3372 y Fw(Then,)34 b(from)d(Theorem)i(3)f(it)g(follo)m(ws)f (that)366 3671 y Fu(k)p Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)855 3686 y Fr(N)923 3671 y Fw(\()p Ft(~)p Fw(\)])16 b Fq(A)1169 3686 y Fr(e)1206 3671 y Fw(\011)1282 3686 y Fr(N)1349 3671 y Fw(\()p Ft(~)p Fw(;)h Fq(x)p Fw(\))p Fu(k)27 b(\024)1792 3546 y Fs(2)p Fr(N)1769 3576 y Fv(X)1761 3787 y Fr(n)p Fs(=)p Fr(N)1938 3671 y Fq(AB)2090 3630 y Fr(n)2137 3671 y Ft(~)2201 3602 y Fl(n)p 2201 3614 39 3 v 2205 3656 a Fk(2)2253 3671 y Fw([\(2)22 b(+)g Fq(a)h Fw(+)f Fq(n)p Fw(\)!])2819 3602 y Fk(1)p 2819 3614 31 3 v 2819 3656 a(2)2891 3671 y Fu(\024)3006 3603 y Fw(1)p 3006 3648 49 4 v 3006 3739 a(4)3082 3671 y Fu(j)o Fw(\001)p Fu(E)3243 3686 y Fr(M)3322 3671 y Fu(j)17 b Ft(~)3431 3602 y Fl(M)p 3430 3614 64 3 v 3447 3656 a Fk(2)3771 3671 y Fw(\(53\))-71 3971 y(whenev)m(er)42 b Ft(~)d Fu(\024)h Ft(~)628 3986 y Fs(0)706 3971 y Fw(\(assumed)h(to)e (b)s(e)g(less)h(or)f(equal)g(than)h(1\))f(and)g Fq(N)2620 3986 y Fs(0)2699 3971 y Fu(\024)h Fq(N)49 b Fu(\024)40 b Fq(N)3138 3986 y Fs(1)3178 3971 y Fw(\()p Ft(~)p Fw(\).)63 b(By)40 b(the)g(same)-71 4092 y(argumen)m(t)32 b(used)i(in)e(the)h(pro) s(of)f(of)g(Prop)s(osition)f(3)h(of)g([15],)g(the)h(inequalit)m(y)f(ab) s(o)m(v)m(e)h(implies)d(that)592 4396 y(dist)16 b Fu(f)p Fq(E)888 4411 y Fr(N)955 4396 y Fw(\()p Ft(~)p Fw(\))p Fq(;)h(\033)t Fw(\()p Fq(H)8 b Fw(\))p Fu(g)27 b(\024)1566 4271 y Fs(2)p Fr(N)1543 4301 y Fv(X)1535 4512 y Fr(n)p Fs(=)p Fr(N)1712 4396 y Fq(AB)1864 4354 y Fr(n)1912 4396 y Ft(~)1976 4327 y Fl(n)p 1975 4339 39 3 v 1979 4381 a Fk(2)2028 4396 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2593 4327 y Fk(1)p 2593 4339 31 3 v 2593 4381 a(2)2666 4396 y Fu(\024)2781 4328 y Fw(1)p 2781 4373 49 4 v 2781 4464 a(4)2856 4396 y Fu(j)p Fw(\001)p Fu(E)3018 4411 y Fr(M)3097 4396 y Fu(j)16 b Ft(~)3205 4327 y Fl(M)p 3205 4339 64 3 v 3221 4381 a Fk(2)3771 4396 y Fw(\(54\))-71 4696 y(for)33 b Ft(~)c Fu(\024)h Ft(~)323 4711 y Fs(0)396 4696 y Fw(and)k Fq(N)665 4711 y Fs(0)734 4696 y Fu(\024)c Fq(N)40 b Fu(\024)30 b Fq(N)1144 4711 y Fs(1)1183 4696 y Fw(\()p Ft(~)p Fw(\).)47 b(Let)33 b(\001)h(b)s(e)g(minim)m(um)c (distance)k(b)s(et)m(w)m(een)i(distinct)d(eigen)m(v)-5 b(alues)33 b(of)-71 4817 y Fq(H)10 4832 y Fs(0)49 4817 y Fw(,)25 b(among)c(the)h(\014rst)h Fq(\013)r Fw(+)q(1)f(of)g(them.)39 b(Since)23 b Fq(E)1666 4832 y Fr(\014)1713 4817 y Fw(\()p Ft(~)p Fw(\))k Fu(!)h Fq(e)2043 4832 y Fr(\014)2090 4817 y Fw(,)c(w)m(e)g(can)e(set)h Ft(~)2639 4832 y Fs(\001)2730 4817 y Fq(>)k Fw(0)22 b(so)h(that)f(for)f(0)28 b Fu(\024)g Fq(\014)33 b Fu(\024)28 b Fq(\013)r Fw(+)q(1,)-71 4937 y Fu(j)p Fq(E)29 4952 y Fr(\014)76 4937 y Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)h Fq(e)373 4952 y Fr(\014)420 4937 y Fu(j)k(\024)590 4898 y Fs(1)p 590 4914 36 4 v 590 4971 a(4)636 4937 y Fw(\001)h(if)f Ft(~)g Fu(\024)i Ft(~)1071 4952 y Fs(\001)1133 4937 y Fw(.)42 b(That)29 b(implies)c(that,)k(for)f Ft(~)f Fu(\024)i Ft(~)2389 4952 y Fs(\001)2479 4937 y Fw(and)g Fq(E)2743 4901 y Fm(00)2785 4937 y Fw(\()p Ft(~)p Fw(\))e Fu(2)h Fq(\033)t Fw(\()p Fq(H)8 b Fw(\))13 b Fu(n)g(f)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fq(;)17 b(E)3716 4901 y Fm(0)3738 4937 y Fw(\()p Ft(~)p Fw(\))p Fu(g)p Fw(,)1441 5117 y Fv(\014)1441 5177 y(\014)1474 5202 y Fq(E)1552 5161 y Fs(#)1615 5202 y Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)h Fq(E)1945 5161 y Fm(00)1987 5202 y Fw(\()p Ft(~)p Fw(\))2117 5117 y Fv(\014)2117 5177 y(\014)2178 5202 y Fu(\025)2293 5135 y Fw(1)p 2293 5179 49 4 v 2293 5270 a(2)2352 5202 y(\001)1888 5470 y(31)p eop %%Page: 32 32 32 31 bop -71 131 a Fw(where)43 b Fq(E)298 95 y Fs(#)403 131 y Fw(denotes)g(either)f Fq(E)48 b Fw(or)41 b Fq(E)1377 95 y Fm(0)1400 131 y Fw(.)72 b(No)m(w)42 b(set)h Ft(~)1946 146 y Fs(2)2028 131 y Fw(=)h(\(\001)p Fq(=)17 b Fu(j)o Fw(\001)p Fu(E)2494 146 y Fr(M)2573 131 y Fu(j)p Fw(\))2665 56 y Fk(2)p 2648 68 64 3 v 2648 109 a Fl(M)2726 131 y Fw(.)71 b(Then)43 b(for)e Ft(~)j Fu(\024)g Ft(~)3519 146 y Fs(2)3600 131 y Fw(w)m(e)e(ha)m(v)m(e)-61 223 y Fs(1)p -61 240 36 4 v -61 297 a(2)-16 263 y Fw(\001)28 b Fu(\025)208 223 y Fs(1)p 208 240 V 208 297 a(2)270 263 y Fu(j)p Fw(\001)p Fu(E)432 278 y Fr(M)511 263 y Fu(j)16 b Ft(~)619 199 y Fl(M)p 619 211 64 3 v 635 252 a Fk(2)696 263 y Fw(.)44 b(As)33 b(a)f(consequence,)1242 532 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1677 491 y Fm(00)1720 532 y Fw(\()p Ft(~)p Fw(\))p Fu(j)82 b(\025)2131 465 y Fw(1)p 2131 509 49 4 v 2131 601 a(2)2206 532 y Fu(j)p Fw(\001)p Fu(E)2368 547 y Fr(M)2447 532 y Fu(j)16 b Ft(~)2555 464 y Fl(M)p 2555 476 64 3 v 2571 517 a Fk(2)1261 765 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1696 724 y Fm(0)1720 765 y Fw(\()p Ft(~)p Fw(\))p Fu(j)82 b(\025)2131 698 y Fw(1)p 2131 742 49 4 v 2131 834 a(2)2206 765 y Fu(j)p Fw(\001)p Fu(E)2368 780 y Fr(M)2447 765 y Fu(j)16 b Ft(~)2555 697 y Fl(M)p 2555 709 64 3 v 2571 750 a Fk(2)-71 1006 y Fw(whic)m(h)33 b(ultimately)d(implies)g(that)1090 1261 y(dist)16 b Fu(f)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fq(;)17 b(\033)t Fw(\()p Fq(H)8 b Fw(\))21 b Fu(n)h Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fu(g)27 b(\025)2283 1194 y Fw(1)p 2283 1238 49 4 v 2283 1329 a(2)2358 1261 y Fu(j)p Fw(\001)p Fu(E)2520 1276 y Fr(M)2599 1261 y Fu(j)16 b Ft(~)2707 1193 y Fl(M)p 2707 1205 64 3 v 2723 1246 a Fk(2)3771 1261 y Fw(\(55\))-71 1508 y(whenev)m(er)35 b Ft(~)28 b Fu(\024)g Fw(min)o Fu(f)p Ft(~)811 1523 y Fs(0)850 1508 y Fq(;)17 b Ft(~)948 1523 y Fs(1)987 1508 y Fq(;)g Ft(~)1085 1523 y Fs(\001)1147 1508 y Fq(;)g Ft(~)1245 1523 y Fs(2)1284 1508 y Fu(g)75 1628 y Fw(Since)40 b Fq(E)409 1643 y Fr(N)476 1628 y Fw(\()p Ft(~)p Fw(\))f(is)g(asymptotic) g(to)g Fq(E)6 b Fw(\()p Ft(~)p Fw(\),)41 b(w)m(e)f(ma)m(y)f(apply)h (Lemma)d(6.)64 b(Then)41 b(there)f(is)f Ft(~)3476 1643 y Fs(3)3554 1628 y Fq(>)g Fw(0)g(suc)m(h)-71 1749 y(that)32 b(for)h(eac)m(h)g Ft(~)28 b Fu(\024)g Ft(~)750 1764 y Fs(3)821 1749 y Fw(w)m(e)34 b(can)f(\014x)g Fq(N)1360 1764 y Fs(2)1400 1749 y Fw(\()p Ft(~)p Fw(\))27 b Fu(\025)h Fq(N)1740 1764 y Fs(0)1812 1749 y Fw(so)33 b(that)1288 2009 y Fu(j)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1717 2024 y Fr(N)1785 2009 y Fw(\()p Ft(~)p Fw(\))p Fu(j)27 b(\024)2085 1941 y Fw(1)p 2085 1986 49 4 v 2085 2077 a(4)2160 2009 y Fu(j)p Fw(\001)p Fu(E)2322 2024 y Fr(M)2401 2009 y Fu(j)16 b Ft(~)2509 1940 y Fl(M)p 2509 1952 64 3 v 2525 1994 a Fk(2)3771 2009 y Fw(\(56\))-71 2255 y(for)32 b Fq(N)156 2270 y Fs(0)223 2255 y Fu(\024)d Fq(N)38 b Fu(\024)28 b Fq(N)628 2270 y Fs(2)667 2255 y Fw(\()p Ft(~)p Fw(\).)75 2376 y(No)m(w)34 b(\(55\),)e(\(56\))g(and)g (the)h(second)h(inequalit)m(y)e(of)g(\(54\))g(implies)e(that)1139 2591 y(dist)17 b Fu(f)o Fq(E)1435 2606 y Fr(N)1503 2591 y Fw(\()p Ft(~)p Fw(\))p Fq(;)g(\033)t Fw(\()p Fq(H)8 b Fw(\))p Fu(g)26 b Fw(=)i Fu(j)o Fq(E)6 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)h Fq(E)2510 2606 y 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Fw(\))32 b(is)g(quasi-degenerate,)i(then)f(Prop)s(osition)e(3)h(is)g(v) -5 b(alid)31 b(in)h(the)h(follo)m(wing)d(w)m(eak)m(er)k(v)m(ersion:)-71 3895 y Fn(Prop)s(osition)i(4)48 b Fh(L)-5 b(et)40 b Fq(E)6 b Fw(\()p Ft(~)p Fw(\))34 b(=)h Fq(E)1237 3910 y Fr(\013)1287 3895 y Fw(\()p Ft(~)p Fw(\))j Fh(b)-5 b(e)38 b(an)h(eigenvalue)e(of)i Fq(H)8 b Fw(\()p Ft(~)p Fw(\))p Fh(,)39 b(which)e(c)-5 b(onver)g(ges)38 b(to)h Fq(e)c Fw(=)g Fq(e)3676 3910 y Fr(\013)3725 3895 y Fh(.)57 b(L)-5 b(et)-71 4015 y Fq(E)1 4030 y Fr(N)69 4015 y Fw(\()p Ft(~)p Fw(\))34 b Fh(b)-5 b(e)34 b(the)g(asso)-5 b(ciate)g(d)33 b(R-S)h(series,)g (trunc)-5 b(ate)g(d)34 b(at)h(or)-5 b(der)34 b Fq(N)10 b Fh(.)45 b(A)n(lso)33 b(let)i Fq(E)2863 3979 y Fs(#)2926 4015 y Fw(\()p Ft(~)p Fw(\))f Fh(b)-5 b(e)34 b(any)g(eigenvalue)f(of) -71 4135 y Fq(H)8 b Fw(\()p Ft(~)p Fw(\))30 b Fh(that)h(satis\014es)f (the)h(c)-5 b(ondition)29 b(1)i(of)f(L)-5 b(emma)30 b(5)g(\(including)g Fq(E)6 b Fw(\()p Ft(~)p Fw(\))30 b Fh(itself.\))g(Then)g(ther)-5 b(e)31 b(exists)f Ft(~)3728 4150 y Fr(e)3793 4135 y Fq(>)d Fw(0)-71 4256 y Fh(so)35 b(that)g(for)g(e)-5 b(ach)34 b Ft(~)27 b Fu(\024)h Ft(~)868 4271 y Fr(e)940 4256 y Fh(ther)-5 b(e)35 b(is)f(an)h Fq(N)1505 4271 y Fr(e)1542 4256 y Fw(\()p Ft(~)p Fw(\))27 b Fu(\025)h Fq(N)1882 4271 y Fs(0)1957 4256 y Fh(such)35 b(that)955 4479 y Fv(\014)955 4539 y(\014)988 4564 y Fq(E)1060 4579 y Fr(N)1128 4564 y Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1457 4523 y Fs(#)1520 4564 y Fw(\()p Ft(~)p Fw(\))1650 4479 y Fv(\014)1650 4539 y(\014)1711 4564 y Fu(\024)1847 4440 y Fs(2)p Fr(N)1824 4469 y Fv(X)1816 4681 y Fr(n)p Fs(=)p Fr(N)1993 4564 y Fq(AB)2145 4523 y Fr(n)2193 4564 y Ft(~)2257 4496 y Fl(n)p 2256 4508 39 3 v 2260 4549 a Fk(2)2309 4564 y Fw([\(2)g(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])2874 4496 y Fk(1)p 2874 4508 31 3 v 2874 4549 a(2)-71 4877 y Fh(for)35 b(al)5 b(l)34 b Ft(~)28 b Fu(\024)g Ft(~)465 4892 y Fr(e)502 4877 y Fh(,)34 b Fq(N)644 4892 y Fs(0)712 4877 y Fu(\024)28 b Fq(N)38 b Fu(\024)28 b Fq(N)1116 4892 y Fr(e)1153 4877 y Fw(\()p Ft(~)p Fw(\))p Fh(,)35 b(and)f Fq(E)1615 4841 y Fs(#)1678 4877 y Fw(\()p Ft(~)p Fw(\))p Fh(.)75 5100 y Fw(In)i(the)f(follo)m(wing)e(theorem)i(w)m(e)h (assume)g(the)f(h)m(yp)s(otheses)j(of)c(Prop)s(osition)g(3.)51 b(An)35 b(analogous)f(result)-71 5221 y(follo)m(ws)d(from)h(the)h(h)m (yp)s(otheses)i(Prop)s(osition)c(4.)1888 5470 y(32)p eop %%Page: 33 33 33 32 bop -71 123 a Fn(Theorem)37 b(4)49 b Fh(Assume)41 b(the)h(hyp)-5 b(otheses)40 b(of)h(Pr)-5 b(op)g(osition)41 b(3.)64 b(Then)40 b(for)i(e)-5 b(ach)40 b Fw(0)g Fq(<)f(g)k(<)d(B)3459 87 y Fm(\000)p Fs(2)3553 123 y Fh(,)j(ther)-5 b(e)41 b(is)-71 244 y Ft(~)-17 259 y Fr(g)51 244 y Fq(>)27 b Fw(0)35 b Fh(such)g(that)g(for)g(e)-5 b(ach)34 b Ft(~)27 b Fu(\024)h Ft(~)1274 259 y Fr(g)1349 244 y Fh(ther)-5 b(e)35 b(exists)g Fq(N)10 b Fw(\()p Ft(~)p Fw(\))35 b Fh(such)f(that)1214 408 y Fv(\014)1214 468 y(\014)1247 493 y Fq(E)1319 508 y Fr(N)7 b Fs(\()p Fj(~)p Fs(\))1480 493 y Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)6 b Fw(\()p Ft(~)p Fw(\))1939 408 y Fv(\014)1939 468 y(\014)2000 493 y Fu(\024)28 b Fw(\003)17 b(exp)2355 352 y Fv(\022)2428 493 y Fu(\000)2515 425 y Fw(\000)p 2515 470 61 4 v 2519 561 a Ft(~)2587 352 y Fv(\023)-71 734 y Fh(for)35 b(some)f Fw(\003)27 b Fq(>)h Fw(0)34 b Fh(and)h Fw(\000)27 b Fq(>)h Fw(0)35 b Fh(indep)-5 b(endent)33 b(of)i Ft(~)p Fh(.)-71 926 y(Pr)-5 b(o)g(of.)42 b Fw(W)-8 b(e)31 b(follo)m(w)d(the)i(pro)s(of) f(of)h(Theorem)g(2)g(of)f([15].)43 b(So)29 b(w)m(e)i(only)f(sk)m(etc)m (h)i(the)f(argumen)m(t,)f(p)s(oin)m(ting)e(out)-71 1047 y(the)33 b(small)d(c)m(hanges)k(needed)g(here.)75 1167 y(Fix)e(0)c Fq(<)f(g)k(<)d(B)691 1131 y Fm(\000)p Fs(2)785 1167 y Fw(.)44 b(Then)33 b(w)m(e)h(ha)m(v)m(e)g Fq(B)1558 1131 y Fs(2)1597 1167 y Fq(g)d Fw(=)d(exp)q(\()p Fu(\000)p Fq(!)t Fw(\))k(for)g(some)h Fq(!)e(>)c Fw(0.)43 b(Consider)33 b(the)g(function)867 1413 y Fq(f)11 b Fw(\()p Ft(~)p Fw(\))27 b(:=)h Fq(Ag)20 b Fw(exp)1519 1273 y Fv(\022)1593 1413 y Fu(\000)1680 1346 y Fq(!)t Fw(\(2)h(+)h Fq(a)p Fw(\))p 1680 1390 361 4 v 1836 1482 a(4)2051 1273 y Fv(\023)2140 1413 y Ft(~)2194 1372 y Fm(\000)2259 1344 y Fk(4+)p Fl(a)p Fk(+)p Fl(M)p 2259 1357 221 3 v 2354 1398 a Fk(2)2510 1413 y Fw(exp)2676 1303 y Fv(\020)2735 1413 y Fu(\000)2822 1346 y Fq(!)t(g)p 2822 1390 115 4 v 2828 1482 a Fw(4)p Ft(~)2947 1303 y Fv(\021)-71 1654 y Fw(and)33 b(set)1277 1807 y Ft(~)1331 1822 y Fs(4)1398 1807 y Fw(=)28 b(inf)1636 1666 y Fv(\032)1711 1807 y Ft(~)g Fw(:)f Fq(f)11 b Fw(\()p Ft(~)p Fw(\))27 b Fu(\025)2179 1739 y Fw(1)p 2179 1784 49 4 v 2179 1875 a(4)2254 1807 y Fu(j)p Fw(\001)p Fu(E)2416 1822 y Fr(M)2494 1807 y Fu(j)2522 1666 y Fv(\033)-71 2019 y Fw(then)33 b(set)754 2114 y(^)752 2140 y Ft(~)806 2155 y Fr(g)873 2140 y Fw(=)28 b(sup)1140 2029 y Fv(n)1207 2140 y Ft(~)f Fw(:)h Ft(~)f Fu(\024)i Fw(min)n Fu(f)p Ft(~)1796 2155 y Fr(e)1833 2140 y Fq(;)17 b Ft(~)1931 2155 y Fs(4)1970 2140 y Fu(g)33 b Fw(and)2227 2029 y Fe(r)2296 2072 y Fq(g)p 2295 2117 54 4 v 2295 2208 a Ft(~)2358 2029 y Fe(z)2444 2140 y Fu(\025)28 b Fw(2)22 b(+)g Fq(a)h Fw(+)f(2)p Fq(N)3017 2155 y Fs(0)3056 2029 y Fv(o)-71 2352 y Fw(No)m(w)43 b(for)f Ft(~)i Fu(\024)543 2326 y Fw(^)540 2352 y Ft(~)594 2367 y Fr(g)676 2352 y Fw(de\014ne)g Fq(N)10 b Fw(\()p Ft(~)p Fw(\))43 b(b)m(y)g(2)29 b(+)f Fq(a)h Fw(+)g(2)p Fq(N)10 b Fw(\()p Ft(~)p Fw(\))45 b(=)f Fe(J)2224 2308 y Fr(g)p 2223 2329 39 4 v 2223 2386 a Fj(~)2271 2352 y Fe(K)p Fw(.)73 b(So)43 b(de\014ned,)j Fq(N)10 b Fw(\()p Ft(~)p Fw(\))45 b Fu(\025)g Fq(N)3395 2367 y Fs(0)3434 2352 y Fw(.)73 b(Making)42 b(a)-71 2472 y(computation)31 b(similar)e(to)k(one)g(in)e(the)i(pro)s(of)f(of)g (Theorem)h(2)f(of)g([15],)h(w)m(e)g(can)g(sho)m(w)h(that)618 2634 y Fs(2)p Fr(N)7 b Fs(\()p Fj(~)p Fs(\))642 2671 y Fv(X)587 2887 y Fr(n)p Fs(=)p Fr(N)g Fs(\()p Fj(~)p Fs(\))857 2766 y Fq(AB)1009 2725 y Fr(n)1057 2766 y Ft(~)1121 2698 y Fl(n)p 1120 2710 39 3 v 1124 2751 a Fk(2)1189 2766 y Fw([)q(\(2)21 b(+)h Fq(a)h Fw(+)f Fq(n)p Fw(\)!])1755 2691 y Fk(1)p 1755 2703 31 3 v 1755 2744 a(2)1883 2766 y Fu(\024)83 b Fq(Ag)t(e)2212 2725 y Fm(\000)2277 2698 y Fl(!)p 2277 2710 41 3 v 2282 2751 a Fk(4)2327 2725 y Fs(\(1+)p Fr(a)p Fs(\))2514 2766 y Ft(~)2568 2725 y Fm(\000)2633 2697 y Fk(4+)p Fl(a)p 2633 2710 111 3 v 2673 2751 a Fk(2)2774 2766 y Fw(exp)2940 2656 y Fv(\020)2999 2766 y Fu(\000)3086 2699 y Fq(!)t(g)p 3086 2743 115 4 v 3092 2834 a Fw(4)p Ft(~)3211 2656 y Fv(\021)3771 2766 y Fw(\(57\))1883 3079 y Fu(\024)2053 3011 y Fw(1)p 2053 3056 49 4 v 2053 3147 a(4)2128 3079 y Fu(j)p Fw(\001)p Fu(E)2290 3094 y Fr(M)2369 3079 y Fu(j)16 b Ft(~)2477 3010 y Fl(M)p 2477 3022 64 3 v 2493 3064 a Fk(2)-71 3299 y Fw(whic)m(h)33 b(means)g(that)f Fq(N)10 b Fw(\()p Ft(~)p Fw(\))28 b Fu(\024)g Fq(N)1147 3314 y Fr(e)1184 3299 y Fw(\()p Ft(~)p Fw(\).)43 b(Th)m(us,)35 b(w)m(e)e(can)g(apply)f(Prop)s (osition)f(3.)43 b(Finally)30 b(de\014ne)1008 3515 y Ft(~)1062 3530 y Fr(g)1130 3515 y Fw(=)d(max)1432 3404 y Fv(n)1498 3515 y Ft(~)g Fu(\024)1687 3489 y Fw(^)1685 3515 y Ft(~)1739 3530 y Fr(g)1806 3515 y Fw(:)h Ft(~)1915 3474 y Fm(\000)1980 3446 y Fk(4+)p Fl(a)p 1980 3459 111 3 v 2020 3500 a Fk(2)2121 3515 y Fw(exp)2287 3404 y Fv(\020)2346 3515 y Fu(\000)2433 3447 y Fq(!)t(g)p 2433 3492 115 4 v 2439 3583 a Fw(8)p Ft(~)2558 3404 y Fv(\021)2645 3515 y Fu(\024)h Fw(1)2800 3404 y Fv(o)-71 3735 y Fw(Then,)34 b(for)e(eac)m(h)h Ft(~)28 b Fu(\024)g Ft(~)820 3750 y Fr(g)860 3735 y Fw(,)33 b(Prop)s(osition)e(3)h(along)f(with)h(\(57\))g (implies)e(the)j(assertion.)723 b Fg(2)75 3926 y Fw(Similar)23 b(results)j(can)h(b)s(e)f(pro)m(v)m(en)h(for)f(eigen)m(v)m(ectors,)j (but)d(only)g(for)f(non-quasi-degenerate)i(eigen)m(v)-5 b(alues)-71 4046 y(of)32 b Fq(H)8 b Fw(\()p Ft(~)p Fw(\).)43 b(Recall)31 b(that)h(\011)h(is)f(an)g(eigen)m(v)m(ector)i(of)e Fq(H)1875 4061 y Fr(e)1912 4046 y Fw(\()p Ft(~)p Fw(\))g(if)f(and)i (only)f(if)2670 4021 y(~)2657 4046 y(\011)g(is)g(an)h(eigen)m(v)m (ector)g(of)f Fq(H)8 b Fw(\()p Ft(~)p Fw(\).)-71 4239 y Fn(Prop)s(osition)36 b(5)48 b Fh(L)-5 b(et)41 b Fq(E)6 b Fw(\()p Ft(~)p Fw(\))39 b Fh(b)-5 b(e)40 b(a)f(non-quasi-de)-5 b(gener)g(ate)38 b(eigenvalue)h(of)g Fq(H)8 b Fw(\()p Ft(~)p Fw(\))p Fh(,)40 b(with)g(eigensp)-5 b(ac)g(e)38 b Fq(B)3855 4254 y Fr(E)3915 4239 y Fh(.)-71 4359 y(L)-5 b(et)42 b Fq(P)167 4374 y Fr(E)267 4359 y Fh(b)-5 b(e)41 b(the)h(\(ortho)-5 b(gonal\))40 b(pr)-5 b(oje)g(ctor)41 b(onto)g Fq(B)1842 4374 y Fr(E)1901 4359 y Fh(.)64 b(L)-5 b(et)2183 4334 y Fw(~)2170 4359 y(\011)2246 4374 y Fr(N)2313 4359 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))41 b Fh(b)-5 b(e)41 b(the)g Fq(N)2965 4323 y Fa(th)3071 4359 y Fh(trunc)-5 b(ation)42 b(of)f(the)g(R-)-71 4480 y(S)k(series)g(\(6\).)75 b(L)-5 b(et)45 b Ft(~)782 4495 y Fr(e)864 4480 y Fh(and)f Fq(N)1141 4495 y Fr(e)1178 4480 y Fw(\()p Ft(~)p Fw(\))h Fh(b)-5 b(e)45 b(de\014ne)-5 b(d)44 b(as)h(in)g(Pr)-5 b(op)g(osition)44 b(3.)75 b(Then)45 b(for)g(e)-5 b(ach)44 b Ft(~)i Fu(\024)i Ft(~)3709 4495 y Fr(e)3790 4480 y Fh(and)-71 4600 y Fq(N)7 4615 y Fs(0)74 4600 y Fu(\024)29 b Fq(N)38 b Fu(\024)28 b Fq(N)479 4615 y Fr(e)516 4600 y Fw(\()p Ft(~)p Fw(\))p Fh(,)360 4702 y Fv(\015)360 4762 y(\015)360 4822 y(\015)360 4881 y(\015)360 4941 y(\015)360 5001 y(\015)494 4814 y Fw(~)480 4839 y(\011)556 4854 y Fr(N)623 4839 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p 425 4883 483 4 v 425 4895 a Fv(\015)425 4955 y(\015)425 5015 y(\015)494 4985 y Fw(~)480 5010 y(\011)556 5025 y Fr(N)623 5010 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))852 4895 y Fv(\015)852 4955 y(\015)852 5015 y(\015)940 4906 y Fu(\000)1105 4839 y Fq(P)1168 4854 y Fr(E)1241 4814 y Fw(~)1227 4839 y(\011)1303 4854 y Fr(N)1370 4839 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))p 1049 4883 606 4 v 1049 4895 a Fv(\015)1049 4955 y(\015)1049 5015 y(\015)1105 5010 y Fq(P)1168 5025 y Fr(E)1241 4985 y Fw(~)1227 5010 y(\011)1303 5025 y Fr(N)1370 5010 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))1599 4895 y Fv(\015)1599 4955 y(\015)1599 5015 y(\015)1664 4702 y(\015)1664 4762 y(\015)1664 4822 y(\015)1664 4881 y(\015)1664 4941 y(\015)1664 5001 y(\015)1747 4906 y Fu(\024)29 b Fw(8)17 b Fu(j)o Fw(\001)p Fu(E)2080 4921 y Fr(M)2158 4906 y Fu(j)2186 4858 y Fm(\000)p Fs(1)2328 4782 y(2)p Fr(N)2305 4812 y Fv(X)2297 5023 y Fr(n)p Fs(=)p Fr(N)2474 4906 y Fq(AB)2626 4865 y Fr(n)2674 4906 y Ft(~)2728 4865 y Fr(n)p Fm(\000)p Fr(M)2904 4906 y Fw([\(2)22 b(+)g Fq(a)h Fw(+)f Fq(n)p Fw(\)!])3470 4838 y Fk(1)p 3470 4850 31 3 v 3470 4891 a(2)-71 5221 y Fh(for)35 b(some)f Fq(M)k Fu(\024)28 b Fq(N)649 5236 y Fs(0)689 5221 y Fh(.)1888 5470 y Fw(33)p eop %%Page: 34 34 34 33 bop -71 123 a Fh(Pr)-5 b(o)g(of.)58 b Fw(As)38 b(b)s(efore,)g(w)m(e)h(only)e(need)h(to)f(mak)m(e)h(a)f(couple)g(of)g (c)m(hanges)i(to)e(the)h(pro)s(of)f(of)f(Prop)s(osition)g(3)h(of)-71 244 y([15].)43 b(So)33 b(details)e(of)h(the)h(follo)m(wing)d(argumen)m (t)j(can)f(b)s(e)h(found)g(there.)75 364 y(W)-8 b(e)33 b(ha)m(v)m(e)463 450 y Fv(\015)463 510 y(\015)463 569 y(\015)532 539 y Fw(~)519 564 y(\011)595 579 y Fr(N)662 564 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))891 450 y Fv(\015)891 510 y(\015)891 569 y(\015)946 476 y Fm(\000)p Fs(1)1070 539 y Fw(~)1057 564 y(\011)1133 579 y Fr(N)1200 564 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))27 b(=)h Fq(w)1630 579 y Fr(N)1714 450 y Fv(\015)1714 510 y(\015)1714 569 y(\015)1769 564 y Fq(P)1832 579 y Fr(E)1905 539 y Fw(~)1891 564 y(\011)1967 579 y Fr(N)2035 564 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))2264 450 y Fv(\015)2264 510 y(\015)2264 569 y(\015)2319 476 y Fm(\000)p Fs(1)2430 564 y Fq(P)2493 579 y Fr(E)2566 539 y Fw(~)2552 564 y(\011)2628 579 y Fr(N)2695 564 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))22 b(+)g(\012)3114 579 y Fr(N)3182 564 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))-71 777 y(where)34 b(\012)281 792 y Fr(N)349 777 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))32 b(is)g(orthogonal)f(to)h Fq(B)1389 792 y Fr(E)1449 777 y Fw(,)g(and)h Fu(j)p Fq(w)1796 792 y Fr(N)1863 777 y Fu(j)1891 741 y Fs(2)1952 777 y Fw(+)22 b Fu(k)p Fw(\012)2170 792 y Fr(N)2237 777 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p Fu(k)2516 741 y Fs(2)2583 777 y Fw(=)28 b(1.)43 b(Then)34 b(w)m(e)f(can)g(pro)m(v)m(e)h(that)401 871 y Fv(\015)401 931 y(\015)401 991 y(\015)401 1051 y(\015)456 901 y(\015)456 961 y(\015)456 1021 y(\015)525 991 y Fw(~)511 1016 y(\011)587 1031 y Fr(N)654 1016 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))883 901 y Fv(\015)883 961 y(\015)883 1021 y(\015)939 928 y Fm(\000)p Fs(1)1063 991 y Fw(~)1050 1016 y(\011)1126 1031 y Fr(N)1193 1016 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))22 b Fu(\000)1543 901 y Fv(\015)1543 961 y(\015)1543 1021 y(\015)1599 1016 y Fq(P)1662 1031 y Fr(E)1735 991 y Fw(~)1721 1016 y(\011)1797 1031 y Fr(N)1864 1016 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))2093 901 y Fv(\015)2093 961 y(\015)2093 1021 y(\015)2148 928 y Fm(\000)p Fs(1)2259 1016 y Fq(P)2322 1031 y Fr(E)2395 991 y Fw(~)2382 1016 y(\011)2458 1031 y Fr(N)2525 1016 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))2754 871 y Fv(\015)2754 931 y(\015)2754 991 y(\015)2754 1051 y(\015)2837 1016 y Fu(\024)28 b Fw(2)17 b Fu(k)o Fw(\012)3127 1031 y Fr(N)3195 1016 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))p Fu(k)297 b Fw(\(58\))-71 1254 y(On)33 b(the)g(other)f(hand,)h(\(53\))f(implies)e (that)441 1384 y Fv(\015)441 1444 y(\015)441 1504 y(\015)441 1563 y(\015)496 1528 y Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)935 1543 y Fr(N)1003 1528 y Fw(\()p Ft(~)p Fw(\)])1176 1414 y Fv(\015)1176 1474 y(\015)1176 1533 y(\015)1245 1503 y Fw(~)1232 1528 y(\011)1308 1543 y Fr(N)1375 1528 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))1604 1414 y Fv(\015)1604 1474 y(\015)1604 1533 y(\015)1659 1440 y Fm(\000)p Fs(1)1783 1503 y Fw(~)1770 1528 y(\011)1846 1543 y Fr(N)1913 1528 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))2142 1384 y Fv(\015)2142 1444 y(\015)2142 1504 y(\015)2142 1563 y(\015)2225 1528 y Fu(\024)2361 1404 y Fs(2)p Fr(N)2338 1434 y Fv(X)2330 1645 y Fr(n)p Fs(=)p Fr(N)2507 1528 y Fq(AB)2659 1487 y Fr(n)2707 1528 y Ft(~)2771 1460 y Fl(n)p 2770 1472 39 3 v 2774 1513 a Fk(2)2823 1528 y Fw([\(2)22 b(+)g Fq(a)g Fw(+)g Fq(n)p Fw(\)!])3388 1460 y Fk(1)p 3388 1472 31 3 v 3388 1513 a(2)-71 1794 y Fw(from)31 b(whic)m(h,)j(after)e(some)g(algebra,)g(Prop)s(osition)63 b(3)33 b(implies)d(that)713 2062 y Fu(k)p Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1202 2077 y Fr(N)1270 2062 y Fw(\()p Ft(~)p Fw(\)])16 b(\012)1513 2077 y Fr(N)1581 2062 y Fw(\()p Ft(~)p Fw(;)h Fq(x)p Fw(\))p Fu(k)27 b(\024)h Fw(2)2089 1938 y Fs(2)p Fr(N)2066 1967 y Fv(X)2058 2179 y Fr(n)p Fs(=)p Fr(N)2235 2062 y Fq(AB)2387 2021 y Fr(n)2434 2062 y Ft(~)2498 1994 y Fl(n)p 2498 2006 39 3 v 2502 2047 a Fk(2)2550 2062 y Fw([\(2)22 b(+)g Fq(a)h Fw(+)f Fq(n)p Fw(\)!])3116 1994 y Fk(1)p 3116 2006 31 3 v 3116 2047 a(2)3771 2062 y Fw(\(59\))-71 2333 y(for)32 b Ft(~)c Fu(\024)g Ft(~)319 2348 y Fr(e)388 2333 y Fw(and)33 b Fq(N)656 2348 y Fs(0)723 2333 y Fu(\024)28 b Fq(N)38 b Fu(\024)28 b Fq(N)1127 2348 y Fr(e)1164 2333 y Fw(\()p Ft(~)p Fw(\).)75 2453 y(Recall)38 b(that)h Fq(E)666 2468 y Fr(N)733 2453 y Fw(\()p Ft(~)p Fw(\))g Fu(62)g Fq(\033)t Fw(\()p Fq(H)8 b Fw(\).)62 b(F)-8 b(rom)38 b(the)h(fact)g(that)g([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))21 b Fu(\000)i Fq(E)2613 2468 y Fr(N)2680 2453 y Fw(\()p Ft(~)p Fw(\)])17 b(\012)2924 2468 y Fr(N)2992 2453 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))38 b(is)h(orthogonal)e(to)-71 2574 y Fq(B)3 2589 y Fr(E)63 2574 y Fw(,)32 b(it)g(follo)m(ws)f(that) 622 2755 y Fu(k)p Fw(\012)742 2770 y Fr(N)810 2755 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p Fu(k)27 b(\024)1221 2670 y Fv(\015)1221 2730 y(\015)1276 2755 y Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)1715 2770 y Fr(N)1783 2755 y Fw(\()p Ft(~)p Fw(\)])1940 2706 y Fm(\000)p Fs(1)1940 2784 y Fm(?)2034 2670 y Fv(\015)2034 2730 y(\015)2106 2755 y Fu(k)p Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))21 b Fu(\000)i Fq(E)2595 2770 y Fr(N)2662 2755 y Fw(\()p Ft(~)p Fw(\)])17 b(\012)2906 2770 y Fr(N)2973 2755 y Fw(\()p Ft(~)p Fw(;)g Fq(x)p Fw(\))p Fu(k)519 b Fw(\(60\))-71 2936 y(where)32 b([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))21 b Fu(\000)i Fq(E)648 2951 y Fr(N)716 2936 y Fw(\()p Ft(~)p Fw(\)])872 2965 y Fm(?)962 2936 y Fw(is)30 b(the)h(restriction)f(of)g([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))22 b Fu(\000)g Fq(E)2237 2951 y Fr(N)2305 2936 y Fw(\()p Ft(~)p Fw(\)])30 b(to)g(the)h(subspace)i(orthogonal)c (to)h Fq(B)3858 2951 y Fr(E)3918 2936 y Fw(.)-71 3057 y(F)-8 b(or)31 b(simplicit)m(y)-8 b(,)28 b(let)j(us)h(assume)g(that)f Fq(E)1453 3020 y Fm(0)1476 3057 y Fw(\()p Ft(~)p Fw(\))g(con)m(v)m (erges)j(to)d(the)g(same)g(eigen)m(v)-5 b(alue)31 b(of)g Fq(H)3256 3072 y Fs(0)3326 3057 y Fw(as)h Fq(E)6 b Fw(\()p Ft(~)p Fw(\).)43 b(Then)-71 3177 y(\(55\))32 b(and)h(the)g(sp)s(ectral) f(theorem)g(imply)f(that)439 3289 y Fv(\015)439 3348 y(\015)495 3373 y Fw([)p Fq(H)8 b Fw(\()p Ft(~)p Fw(\))21 b Fu(\000)i Fq(E)934 3388 y Fr(N)1001 3373 y Fw(\()p Ft(~)p Fw(\)])1158 3325 y Fm(\000)p Fs(1)1158 3402 y Fm(?)1252 3289 y Fv(\015)1252 3348 y(\015)1335 3373 y Fu(\024)28 b Fw(\(dist)17 b Fu(f)p Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fq(;)17 b(\033)t Fw(\()p Fq(H)8 b Fw(\))21 b Fu(n)h Fq(E)6 b Fw(\()p Ft(~)p Fw(\))p Fu(g)o Fw(\))2567 3325 y Fm(\000)p Fs(1)2689 3373 y Fu(\024)28 b Fw(2)17 b Fu(j)o Fw(\001)p Fu(E)3021 3388 y Fr(M)3100 3373 y Fu(j)3128 3325 y Fm(\000)p Fs(1)3239 3373 y Ft(~)3293 3332 y Fm(\000)3358 3305 y Fl(M)p 3357 3317 64 3 v 3374 3358 a Fk(2)3771 3373 y Fw(\(61\))75 3555 y(No)m(w)34 b(the)f(assertion)f(follo)m(ws)f(from)h(\(58\){\(61\).)2022 b Fg(2)-71 3746 y Fh(R)-5 b(emarks.)65 b Fw(The)40 b(assumption)g(of)f (non-quasi-degeneracy)i(of)e Fq(E)6 b Fw(\()p Ft(~)p Fw(\))40 b(is)f(critical,)h(as)g(one)g(can)g(see)h(in)e(the)-71 3866 y(argumen)m(t)32 b(that)h(leads)f(\(61\).)75 4058 y(The)e(optimal)c(truncation)j(for)f(eigen)m(v)m(ectors)j(follo)m(ws)c (from)h(Prop)s(osition)f(5)i(as)g(Theorem)g(4)g(do)s(es)g(from)-71 4178 y(Prop)s(osition)i(3:)-71 4362 y Fn(Theorem)37 b(5)49 b Fh(F)-7 b(or)39 b Fw(0)d Fu(\024)h Fq(g)i Fu(\024)e Fq(B)1174 4326 y Fm(\000)p Fs(2)1308 4362 y Fh(ther)-5 b(e)40 b(is)f Fw(\003)1733 4326 y Fm(0)1756 4362 y Fh(,)i Fw(\000)1888 4326 y Fm(0)1951 4362 y Fh(and)e Ft(~)2199 4326 y Fm(0)2199 4387 y Fr(g)2275 4362 y Fq(>)d Fw(0)j Fh(such)h(that)g(for)f(e)-5 b(ach)39 b Ft(~)d Fu(\024)h Ft(~)3548 4377 y Fr(g)3628 4362 y Fh(ther)-5 b(e)39 b(is)-71 4482 y Fq(N)17 4446 y Fm(0)41 4482 y Fw(\()p Ft(~)p Fw(\))c Fh(so)f(that)797 4516 y Fv(\015)797 4575 y(\015)797 4635 y(\015)797 4695 y(\015)797 4755 y(\015)797 4814 y(\015)931 4624 y Fw(~)917 4649 y(\011)993 4664 y Fr(N)7 b Fs(\()p Fj(~)p Fs(\))1154 4649 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p 862 4697 576 4 v 862 4709 a Fv(\015)862 4769 y(\015)862 4828 y(\015)931 4798 y Fw(~)917 4823 y(\011)993 4839 y Fr(N)7 b Fs(\()p Fj(~)p Fs(\))1154 4823 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))1383 4709 y Fv(\015)1383 4769 y(\015)1383 4828 y(\015)1470 4720 y Fu(\000)1635 4649 y Fq(P)1698 4664 y Fr(E)1771 4624 y Fw(~)1757 4649 y(\011)1833 4664 y Fr(N)7 b Fs(\()p Fj(~)p Fs(\))1994 4649 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))p 1579 4697 699 4 v 1579 4709 a Fv(\015)1579 4769 y(\015)1579 4828 y(\015)1635 4823 y Fq(P)1698 4838 y Fr(E)1771 4798 y Fw(~)1757 4823 y(\011)1833 4839 y Fr(N)7 b Fs(\()p Fj(~)p Fs(\))1994 4823 y Fw(\()p Ft(~)p Fw(;)17 b Fq(x)p Fw(\))2223 4709 y Fv(\015)2223 4769 y(\015)2223 4828 y(\015)2288 4516 y(\015)2288 4575 y(\015)2288 4635 y(\015)2288 4695 y(\015)2288 4755 y(\015)2288 4814 y(\015)2371 4720 y Fu(\024)28 b Fw(\003)2544 4679 y Fm(0)2583 4720 y Fw(exp)2749 4579 y Fv(\022)2822 4720 y Fu(\000)2909 4652 y Fw(\000)2970 4616 y Fm(0)p 2909 4697 85 4 v 2925 4788 a Ft(~)3004 4579 y Fv(\023)-71 5100 y Fh(A)-5 b(cknow)5 b(le)-5 b(dgements.)43 b(The)33 b(author)i(would)e(like)h(to)g(thank)g(Ge)-5 b(or)g(ge)34 b(Hage)-5 b(dorn)34 b(for)g(his)f(supp)-5 b(ort)35 b(and)e(valu-)-71 5221 y(able)h(c)-5 b(omments.)1888 5470 y Fw(34)p eop %%Page: 35 35 35 34 bop -71 123 a Fx(6)161 b(App)t(endix)-71 342 y Fw(Here)47 b(w)m(e)g(simplify)c(the)j(form)m(ula)e(\(46\))h(b)m(y)i (using)e(the)i(iden)m(tit)m(y)e(\(10\).)83 b(This)46 b(calculation)d(is)j(formally)-71 463 y(iden)m(tical)29 b(the)j(one)f(done)h(in)e([15)o(],)i(with)f(some)f(notational)f(c)m (hange.)44 b(W)-8 b(e)31 b(repro)s(duce)h(it)e(here)i(for)e(a)h(sak)m (e)h(of)-71 583 y(con)m(v)m(enience.)547 857 y Fq(\030)590 872 y Fr(N)740 857 y Fw(=)899 686 y Fv(")957 857 y Fq(H)1038 872 y Fr(e)1097 857 y Fw(+)22 b Fq(A)1268 872 y Fr(e)1305 857 y Fq(W)14 b(A)1484 872 y Fr(e)1542 857 y Fu(\000)1642 732 y Fr(N)7 b Fm(\000)p Fs(1)1647 762 y Fv(X)1657 972 y Fr(j)t Fs(=1)1812 857 y Ft(~)1877 785 y Fl(j)p 1876 800 31 3 v 1876 841 a Fk(2)1920 857 y Fu(E)1973 872 y Fr(j)2009 857 y Fq(A)2082 815 y Fs(2)2082 881 y Fr(e)2122 686 y Fv(#)2197 732 y Fr(N)g Fm(\000)p Fs(1)2201 762 y Fv(X)2197 971 y Fr(m)p Fs(=0)2367 857 y Ft(~)2431 788 y Fl(m)p 2430 800 55 3 v 2442 841 a Fk(2)2499 857 y Fq( )2562 872 y Fr(m)740 1208 y Fw(=)899 1083 y Fr(N)g Fm(\000)p Fs(1)903 1113 y Fv(X)899 1322 y Fr(m)p Fs(=0)1068 1208 y Ft(~)1132 1139 y Fl(m)p 1132 1151 V 1144 1193 a Fk(2)1201 1208 y Fq(H)1282 1223 y Fr(e)1319 1208 y Fq( )1382 1223 y Fr(m)1471 1208 y Fw(+)1569 1083 y Fr(N)g Fm(\000)p Fs(1)1573 1113 y Fv(X)1569 1322 y Fr(m)p Fs(=0)1739 1208 y Ft(~)1803 1139 y Fl(m)p 1802 1151 V 1814 1193 a Fk(2)1871 1208 y Fq(A)1944 1223 y Fr(e)1981 1208 y Fq(W)14 b(A)2160 1223 y Fr(e)2197 1208 y Fq( )2260 1223 y Fr(m)2349 1208 y Fu(\000)2448 1083 y Fr(N)7 b Fm(\000)p Fs(1)2453 1113 y Fv(X)2464 1323 y Fr(j)t Fs(=1)2618 1083 y Fr(N)g Fm(\000)p Fs(1)2623 1113 y Fv(X)2619 1322 y Fr(m)p Fs(=0)2788 1208 y Ft(~)2852 1136 y Fl(j)s Fk(+)p Fl(m)p 2852 1152 130 3 v 2901 1193 a Fk(2)2995 1208 y Fu(E)3048 1223 y Fr(j)3085 1208 y Fq(A)3158 1167 y Fs(2)3158 1232 y Fr(e)3197 1208 y Fq( )3260 1223 y Fr(m)-71 1540 y Fw(W)-8 b(e)39 b(use)g Fq(A)350 1555 y Fr(e)387 1540 y Fq(W)14 b(A)566 1555 y Fr(e)639 1540 y Fw(=)752 1465 y Fv(P)858 1492 y Fr(N)7 b Fs(+2)858 1569 y Fr(j)t Fs(=3)1032 1540 y Ft(~)1096 1472 y Fl(j)s Fi(\000)p Fk(2)p 1095 1488 107 3 v 1134 1529 a(2)1216 1540 y Fq(T)1287 1504 y Fs(\()p Fr(j)t Fs(\))1404 1540 y Fw(+)26 b Fq(A)1579 1555 y Fr(e)1616 1540 y Fq(W)1722 1504 y Fs([)p Fr(N)7 b Fs(+2])1918 1540 y Fq(A)1991 1555 y Fr(e)2066 1540 y Fw(and)39 b(c)m(hange)g(the)f (index)g(b)m(y)h Fq(j)k Fu(!)37 b Fq(j)32 b Fu(\000)26 b Fw(2.)60 b(Using)-71 1660 y Fq(H)10 1675 y Fr(e)47 1660 y Fq( )110 1675 y Fs(0)177 1660 y Fw(=)28 b(0,)k(w)m(e)i(then)f (obtain)592 1953 y Fq(\030)635 1968 y Fr(N)785 1953 y Fw(=)944 1829 y Fr(N)7 b Fm(\000)p Fs(1)949 1858 y Fv(X)944 2068 y Fr(m)p Fs(=1)1114 1953 y Ft(~)1178 1885 y Fl(m)p 1178 1897 55 3 v 1190 1938 a Fk(2)1246 1953 y Fq(H)1327 1968 y Fr(e)1364 1953 y Fq( )1427 1968 y Fr(m)1516 1953 y Fw(+)1614 1829 y Fr(N)g Fm(\000)p Fs(1)1619 1858 y Fv(X)1615 2068 y Fr(m)p Fs(=0)1824 1829 y Fr(N)1784 1858 y Fv(X)1795 2068 y Fr(j)t Fs(=1)1945 1953 y Ft(~)2009 1881 y Fl(m)p Fk(+)p Fl(j)p 2008 1897 130 3 v 2058 1938 a Fk(2)2152 1953 y Fq(T)2223 1912 y Fs(\()p Fr(j)t Fs(+2\))2404 1953 y Fq( )2467 1968 y Fr(m)944 2304 y Fw(+)1064 2180 y Fr(N)g Fm(\000)p Fs(1)1069 2210 y Fv(X)1065 2419 y Fr(m)p Fs(=0)1234 2304 y Ft(~)1298 2236 y Fl(m)p 1298 2248 55 3 v 1310 2289 a Fk(2)1366 2304 y Fq(A)1439 2319 y Fr(e)1477 2304 y Fq(W)1583 2263 y Fs([)p Fr(N)g Fs(+2])1779 2304 y Fq(A)1852 2319 y Fr(e)1889 2304 y Fq( )1952 2319 y Fr(m)2041 2304 y Fu(\000)2141 2180 y Fr(N)g Fm(\000)p Fs(1)2145 2210 y Fv(X)2141 2419 y Fr(m)p Fs(=0)2310 2180 y Fr(N)g Fm(\000)p Fs(1)2315 2210 y Fv(X)2326 2420 y Fr(j)t Fs(=1)2480 2304 y Ft(~)2544 2232 y Fl(j)s Fk(+)p Fl(m)p 2544 2248 130 3 v 2594 2289 a Fk(2)2688 2304 y Fu(E)2741 2319 y Fr(j)2777 2304 y Fq(A)2850 2263 y Fs(2)2850 2329 y Fr(e)2889 2304 y Fq( )2952 2319 y Fr(m)785 2656 y Fw(=)944 2531 y Fr(N)g Fm(\000)p Fs(1)949 2561 y Fv(X)954 2770 y Fr(n)p Fs(=1)1114 2656 y Ft(~)1178 2587 y Fl(n)p 1178 2599 39 3 v 1182 2641 a Fk(2)1230 2656 y Fq(H)1311 2671 y Fr(e)1348 2656 y Fq( )1411 2671 y Fr(n)1480 2656 y Fw(+)1578 2531 y Fr(N)g Fm(\000)p Fs(1)1583 2561 y Fv(X)1589 2770 y Fr(n)p Fs(=1)1748 2656 y Ft(~)1812 2587 y Fl(n)p 1812 2599 V 1816 2641 a Fk(2)1932 2531 y Fr(n)1881 2561 y Fv(X)1892 2771 y Fr(j)t Fs(=1)2042 2656 y Fq(T)2113 2615 y Fs(\()p Fr(j)t Fs(+2\))2294 2656 y Fq( )2357 2671 y Fr(n)p Fm(\000)p Fr(j)944 3007 y Fw(+)1064 2882 y Fs(2)p Fr(N)g Fm(\000)p Fs(1)1087 2912 y Fv(X)1078 3124 y Fr(n)p Fs(=)p Fr(N)1269 3007 y Ft(~)1333 2939 y Fl(n)p 1333 2951 V 1337 2992 a Fk(2)1540 2882 y Fr(N)1499 2912 y Fv(X)1402 3124 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)1757 3007 y Fq(T)1828 2966 y Fs(\()p Fr(j)t Fs(+2\))2009 3007 y Fq( )2072 3022 y Fr(n)p Fm(\000)p Fr(j)2229 3007 y Fw(+)2327 2882 y Fr(N)g Fm(\000)p Fs(1)2332 2912 y Fv(X)2327 3122 y Fr(m)p Fs(=0)2497 3007 y Ft(~)2561 2939 y Fl(m)p 2561 2951 55 3 v 2573 2992 a Fk(2)2629 3007 y Fq(A)2702 3022 y Fr(e)2739 3007 y Fq(W)2845 2966 y Fs([)p Fr(N)g Fs(+2])3042 3007 y Fq(A)3115 3022 y Fr(e)3152 3007 y Fq( )3215 3022 y Fr(m)944 3360 y Fu(\000)1066 3235 y Fr(N)g Fm(\000)p Fs(1)1071 3265 y Fv(X)1076 3474 y Fr(n)p Fs(=1)1236 3360 y Ft(~)1300 3291 y Fl(n)p 1300 3303 39 3 v 1304 3345 a Fk(2)1419 3235 y Fr(n)1369 3265 y Fv(X)1379 3475 y Fr(j)t Fs(=1)1529 3360 y Fu(E)1582 3375 y Fr(j)1618 3360 y Fq(A)1691 3319 y Fs(2)1691 3384 y Fr(e)1731 3360 y Fq( )1794 3375 y Fr(n)p Fm(\000)p Fr(j)1951 3360 y Fu(\000)2050 3235 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)2072 3265 y Fv(X)2064 3476 y Fr(n)p Fs(=)p Fr(N)2255 3360 y Ft(~)2319 3291 y Fl(n)p 2319 3303 V 2323 3345 a Fk(2)2481 3235 y Fr(N)g Fm(\000)p Fs(1)2485 3265 y Fv(X)2388 3476 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2743 3360 y Fu(E)2796 3375 y Fr(j)2832 3360 y Fq(A)2905 3319 y Fs(2)2905 3384 y Fr(e)2945 3360 y Fq( )3008 3375 y Fr(n)p Fm(\000)p Fr(j)-71 3668 y Fw(The)43 b(\014rst,)h(second)f(and) f(\014fth)g(terms)f(of)g(last)g(equation)h(cancel)f(b)s(ecause)i(of)e (\(10\).)70 b(In)42 b(the)g(third)f(term)-71 3788 y(de\014ne)34 b Fq(m)28 b Fw(=)f Fq(n)c Fu(\000)f Fq(j)39 b Fw(and)33 b(then)g Fq(p)27 b Fw(=)h Fq(n)22 b Fu(\000)h Fq(N)10 b Fw(.)44 b(This)33 b(yields)617 4081 y Fq(\030)660 4096 y Fr(N)810 4081 y Fw(=)969 3956 y Fs(2)p Fr(N)7 b Fm(\000)p Fs(1)991 3986 y Fv(X)982 4197 y Fr(n)p Fs(=)p Fr(N)1174 4081 y Ft(~)1238 4012 y Fl(n)p 1238 4024 V 1242 4066 a Fk(2)1369 3956 y Fr(N)g Fm(\000)p Fs(1)1374 3986 y Fv(X)1307 4197 y Fr(m)p Fs(=)p Fr(n)p Fm(\000)p Fr(N)1601 4081 y Fq(T)1672 4040 y Fs(\()p Fr(n)p Fm(\000)p Fr(m)p Fs(+2\))1981 4081 y Fq( )2044 4096 y Fr(m)969 4427 y Fw(+)1089 4303 y Fr(N)g Fm(\000)p Fs(1)1093 4333 y Fv(X)1089 4542 y Fr(m)p Fs(=0)1259 4427 y Ft(~)1323 4359 y Fl(m)p 1322 4371 55 3 v 1334 4412 a Fk(2)1391 4427 y Fq(A)1464 4442 y Fr(e)1501 4427 y Fq(W)1607 4386 y Fs([)p Fr(N)g Fs(+2])1803 4427 y Fq(A)1876 4442 y Fr(e)1913 4427 y Fq( )1976 4442 y Fr(m)2065 4427 y Fu(\000)2165 4303 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)2187 4333 y Fv(X)2179 4544 y Fr(n)p Fs(=)p Fr(N)2370 4427 y Ft(~)2434 4359 y Fl(n)p 2434 4371 39 3 v 2438 4412 a Fk(2)2596 4303 y Fr(N)g Fm(\000)p Fs(1)2600 4333 y Fv(X)2503 4544 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2858 4427 y Fu(E)2911 4442 y Fr(j)2947 4427 y Fq(A)3020 4386 y Fs(2)3020 4452 y Fr(e)3060 4427 y Fq( )3123 4442 y Fr(n)p Fm(\000)p Fr(j)810 4780 y Fw(=)969 4656 y Fr(N)g Fm(\000)p Fs(1)973 4686 y Fv(X)982 4895 y Fr(p)p Fs(=0)1138 4656 y Fr(N)g Fm(\000)p Fs(1)1143 4686 y Fv(X)1139 4895 y Fr(m)p Fs(=)p Fr(p)1308 4780 y Ft(~)1372 4708 y Fl(p)p Fk(+)p Fl(N)p 1372 4724 133 3 v 1423 4765 a Fk(2)1518 4780 y Fq(T)1589 4739 y Fs(\()p Fr(p)p Fs(+)p Fr(N)g Fm(\000)p Fr(m)p Fs(+2\))2009 4780 y Fq( )2072 4795 y Fr(m)969 5131 y Fw(+)1089 5006 y Fr(N)g Fm(\000)p Fs(1)1093 5036 y Fv(X)1089 5245 y Fr(m)p Fs(=0)1259 5131 y Ft(~)1323 5062 y Fl(m)p 1322 5074 55 3 v 1334 5116 a Fk(2)1391 5131 y Fq(A)1464 5146 y Fr(e)1501 5131 y Fq(W)1607 5090 y Fs([)p Fr(N)g Fs(+2])1803 5131 y Fq(A)1876 5146 y Fr(e)1913 5131 y Fq( )1976 5146 y Fr(m)2065 5131 y Fu(\000)2165 5006 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)2187 5036 y Fv(X)2179 5248 y Fr(n)p Fs(=)p Fr(N)2370 5131 y Ft(~)2434 5062 y Fl(n)p 2434 5074 39 3 v 2438 5116 a Fk(2)2596 5006 y Fr(N)g Fm(\000)p Fs(1)2600 5036 y Fv(X)2503 5248 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2858 5131 y Fu(E)2911 5146 y Fr(j)2947 5131 y Fq(A)3020 5090 y Fs(2)3020 5155 y Fr(e)3060 5131 y Fq( )3123 5146 y Fr(n)p Fm(\000)p Fr(j)1888 5470 y Fw(35)p eop %%Page: 36 36 36 35 bop 810 204 a Fw(=)969 79 y Fr(N)7 b Fm(\000)p Fs(1)973 109 y Fv(X)969 318 y Fr(m)p Fs(=0)1138 204 y Ft(~)1202 135 y Fl(m)p 1202 147 55 3 v 1214 189 a Fk(2)1328 79 y Fr(m)1287 109 y Fv(X)1297 318 y Fr(p)p Fs(=0)1448 204 y Ft(~)1512 132 y Fl(p)p Fk(+)p Fl(N)e Fi(\000)p Fl(m)p 1512 147 235 3 v 1614 189 a Fk(2)1760 204 y Fq(T)1831 162 y Fs(\()p Fr(p)p Fs(+)p Fr(N)i Fm(\000)p Fr(m)p Fs(+2\))2251 204 y Fq( )2314 219 y Fr(m)969 554 y Fw(+)1089 430 y Fr(N)g Fm(\000)p Fs(1)1093 459 y Fv(X)1089 669 y Fr(m)p Fs(=0)1259 554 y Ft(~)1323 486 y Fl(m)p 1322 498 55 3 v 1334 539 a Fk(2)1391 554 y Fq(A)1464 569 y Fr(e)1501 554 y Fq(W)1607 513 y Fs([)p Fr(N)g Fs(+2])1803 554 y Fq(A)1876 569 y Fr(e)1913 554 y Fq( )1976 569 y Fr(m)2065 554 y Fu(\000)2165 430 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)2187 459 y Fv(X)2179 671 y Fr(n)p Fs(=)p Fr(N)2370 554 y Ft(~)2434 486 y Fl(n)p 2434 498 39 3 v 2438 539 a Fk(2)2596 430 y Fr(N)g Fm(\000)p Fs(1)2600 459 y Fv(X)2503 671 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2858 554 y Fu(E)2911 569 y Fr(j)2947 554 y Fq(A)3020 513 y Fs(2)3020 579 y Fr(e)3060 554 y Fq( )3123 569 y Fr(n)p Fm(\000)p Fr(j)810 907 y Fw(=)969 782 y Fr(N)g Fm(\000)p Fs(1)973 812 y Fv(X)969 1021 y Fr(m)p Fs(=0)1138 907 y Ft(~)1202 839 y Fl(m)p 1202 851 55 3 v 1214 892 a Fk(2)1287 737 y Fv(")1346 782 y Fr(m)p Fs(+2)1350 812 y Fv(X)1365 1022 y Fr(i)p Fs(=2)1515 907 y Ft(~)1579 838 y Fl(i)p Fk(+)p Fl(N)e Fi(\000)p Fl(m)p Fi(\000)p Fk(2)p 1579 851 304 3 v 1715 892 a(2)1896 907 y Fq(T)1967 866 y Fs(\()p Fr(i)p Fs(+)p Fr(N)i Fm(\000)p Fr(m)p Fs(\))2307 907 y Fw(+)22 b Fq(A)2478 922 y Fr(e)2515 907 y Fq(W)2621 866 y Fs([)p Fr(N)7 b Fs(+2])2817 907 y Fq(A)2890 922 y Fr(e)2927 737 y Fv(#)3002 907 y Fq( )3065 922 y Fr(m)969 1245 y Fu(\000)1090 1121 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)1113 1151 y Fv(X)1104 1362 y Fr(n)p Fs(=)p Fr(N)1296 1245 y Ft(~)1360 1177 y Fl(n)p 1359 1189 39 3 v 1363 1230 a Fk(2)1521 1121 y Fr(N)g Fm(\000)p Fs(1)1526 1151 y Fv(X)1428 1362 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)1783 1245 y Fu(E)1836 1260 y Fr(j)1872 1245 y Fq(A)1945 1204 y Fs(2)1945 1270 y Fr(e)1985 1245 y Fq( )2048 1260 y Fr(n)p Fm(\000)p Fr(j)-71 1593 y Fw(Finally)-8 b(,)30 b(note)j(that)f Ft(~)772 1526 y Fl(j)s Fi(\000)p Fk(2)p 772 1542 107 3 v 810 1583 a(2)893 1593 y Fq(T)964 1557 y Fs(\()p Fr(j)t Fs(\))1077 1593 y Fw(+)22 b Fq(A)1248 1608 y Fr(e)1285 1593 y Fq(W)1391 1557 y Fs([)p Fr(j)t Fs(+1])1557 1593 y Fq(A)1630 1608 y Fr(e)1694 1593 y Fw(=)28 b Fq(A)1871 1608 y Fr(e)1908 1593 y Fq(W)2014 1557 y Fs([)p Fr(j)t Fs(])2089 1593 y Fq(A)2162 1608 y Fr(e)2199 1593 y Fw(.)44 b(It)32 b(then)i(follo)m(ws)d(that)664 1901 y Fq(\030)707 1916 y Fr(N)801 1901 y Fw(=)905 1776 y Fr(N)7 b Fm(\000)p Fs(1)910 1806 y Fv(X)905 2015 y Fr(m)p Fs(=0)1075 1901 y Ft(~)1139 1832 y Fl(m)p 1139 1844 55 3 v 1151 1885 a Fk(2)1207 1901 y Fq(A)1280 1916 y Fr(e)1317 1901 y Fq(W)1423 1859 y Fs([)p Fr(N)g Fm(\000)p Fr(m)p Fs(+1])1737 1901 y Fq(A)1810 1916 y Fr(e)1847 1901 y Fq( )1910 1916 y Fr(m)1999 1901 y Fu(\000)2098 1776 y Fs(2)p Fr(N)g Fm(\000)p Fs(2)2121 1806 y Fv(X)2112 2017 y Fr(n)p Fs(=)p Fr(N)2304 1901 y Ft(~)2368 1832 y Fl(n)p 2367 1844 39 3 v 2371 1885 a Fk(2)2529 1776 y Fr(N)g Fm(\000)p Fs(1)2534 1806 y Fv(X)2436 2017 y Fr(j)t Fs(=)p Fr(n)p Fm(\000)p Fr(N)g Fs(+1)2791 1901 y Fu(E)2844 1916 y Fr(j)2880 1901 y Fq(A)2953 1859 y Fs(2)2953 1925 y Fr(e)2993 1901 y Fq( )3056 1916 y Fr(m)p Fm(\000)p Fr(j)-71 2298 y Fx(References)-22 2517 y Fw([1])49 b(Com)m(b)s(es)33 b(J)f(M,)h(Duclos)f(P)h(and)g (Seiler)e(R)i(1983)e Fh(J.)k(F)-7 b(unct.)34 b(A)n(nal.)e Fn(52)h Fw(257{301)-22 2721 y([2])49 b(Da)m(vies)32 b(E)h(B)g(1982)e Fh(Commun.)j(Math.)h(Phys.)e Fn(85)f Fw(471{479)-22 2924 y([3])49 b(Hagedorn)32 b(G)g(A)h(and)g(Jo)m(y)m(e)h(A)e(2000)g Fh(A)n(nn.)i(Henri)h(Poinc)-5 b(ar)n(\023)-47 b(e.)31 b Fn(1)h Fw(837{883)-22 3128 y([4])49 b(Hel\013er)32 b(B)h(and)f(Sj\177)-49 b(ostrand)33 b(J)g(1984)e Fh(Comm.)j(PDE)e Fn(9)h Fw(337{408)-22 3331 y([5])49 b(Hel\013er)28 b(B)h(1988)g Fh(Semiclassic)-5 b(al)29 b(A)n(nalysis)i(for)g(the)h(Schr\177)-50 b(odinger)30 b(Op)-5 b(er)g(ator)31 b(and)h(Applic)-5 b(ations)30 b(\(L)-5 b(e)g(c-)130 3451 y(tur)g(e)35 b(Notes)g(in)g (Mathematics)f(1336\))e Fw(\(Berlin:)42 b(Springer\))-22 3655 y([6])49 b(Ho)m(wland)32 b(J)h(S)f(1997)g Fh(J.)j(Phys.)g(A:)g (Math.)g(Gen.)d Fn(30)h Fw(2069{2076)-22 3858 y([7])49 b(Hunzik)m(er)33 b(W)g(and)g(Pillet)d(C)j(A)g(1983)e Fh(Commun.)j(Math.)h(Phys.)e Fn(90)f Fw(219{233)-22 4062 y([8])49 b(Hunzik)m(er)33 b(W)g(1988)f Fh(Helv.)i(Phys.)h(A)-5 b(cta)33 b Fn(61)g Fw(257{304)-22 4265 y([9])49 b(Kato)31 b(T)i(1966)f Fh(Perturb)-5 b(ation)35 b(The)-5 b(ory)35 b(for)f(Line)-5 b(ar)35 b(Op)-5 b(er)g(ators)32 b Fw(\(Berlin:)42 b(Springer\))-71 4468 y([10])49 b(Messiah)33 b(A)g(1962)e Fh(Quantum)k(Me)-5 b(chanics)32 b Fw(v)m(ol)g(I)s(I)g(\(New)i(Y)-8 b(ork:)43 b(Wiley\))-71 4672 y([11])49 b(Reed)35 b(M)g(and)g(Simon)f(B) h(1978)e Fh(Metho)-5 b(ds)37 b(of)g(Mo)-5 b(dern)37 b(Mathematic)-5 b(al)36 b(Physics)f Fw(v)m(ol)f(I)s(I)s(I)h(\(New)h(Y)-8 b(ork:)130 4792 y(Academic\))-71 4996 y([12])49 b(Reed)35 b(M)h(and)f(Simon)e(B)j(1978)e Fh(Metho)-5 b(ds)37 b(of)g(Mo)-5 b(dern)36 b(Mathematic)-5 b(al)37 b(Physics)e Fw(v)m(ol)f(IV)i(\(New)g (Y)-8 b(ork:)130 5116 y(Academic\))1888 5470 y(36)p eop %%Page: 37 37 37 36 bop -71 123 a Fw([13])49 b(Simon)31 b(B)h(1983)g Fh(A)n(nn.)i(Inst.)h(Henri)f(Poinc)-5 b(ar)n(\023)-47 b(e)33 b Fw(A)g Fn(38)g Fw(295{307)-71 327 y([14])49 b(Simon)31 b(B)h(1984)g Fh(A)n(nn.)i(of)h(Math.)g(\(2\))d Fn(120)h Fw(89{118)-71 530 y([15])49 b(T)-8 b(oloza)31 b(J)i(H)g(2001)e Fh(J.)k(Phys.)g(A:)g(Math.)g(Gen.)e Fn(34)f Fw(1203{1218)1888 5470 y(37)p eop 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