Content-Type: multipart/mixed; boundary="-------------9911260657900" This is a multi-part message in MIME format. ---------------9911260657900 Content-Type: text/plain; name="99-445.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-445.keywords" Semiclassical methods, quantum dynamics, exponential asymptotics ---------------9911260657900 Content-Type: application/postscript; name="hagjoy5.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hagjoy5.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: hagjoy5.dvi %%Pages: 43 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips hagjoy5 -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.11.26:1344 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 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Fo(de\014ned)34 b(for)e Fk(m)c Fo(=)g(1)p Fk(;)33 b Fo(2)p Fk(;)g(:)17 b(:)g(:)f(;)33 b(d)p Fo(.)247 230 y Fj(A)327 245 y Fh(m)393 230 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))955 189 y Fg(\003)1054 230 y Fo(=)1269 162 y(1)p 1200 206 187 4 v 1200 224 a Fj(p)p 1283 224 104 4 v 85 x Fo(2\026)-50 b Fk(h)1430 84 y Fd(")1543 122 y Fh(d)1501 147 y Fd(X)1495 328 y Fh(n)p Fi(=1)p 1661 152 80 4 v 1661 230 a Fk(B)1740 245 y Fh(n)12 b(m)1878 230 y Fo(\()p Fk(x)1971 245 y Fh(n)2040 230 y Fj(\000)23 b Fk(a)2191 245 y Fh(n)2238 230 y Fo(\))55 b Fj(\000)39 b Fk(i)2578 122 y Fh(d)2536 147 y Fd(X)2529 328 y Fh(n)p Fi(=1)p 2696 152 74 4 v 2696 230 a Fk(A)2769 245 y Fh(n)11 b(m)2906 230 y Fo(\()p Fj(\000)p Fk(i)p Fo(\026)-50 b Fk(h)3120 162 y(@)p 3120 206 160 4 v 3120 298 a(@)5 b(x)3231 313 y Fh(n)3312 230 y Fj(\000)22 b Fk(\021)3459 245 y Fh(n)3506 230 y Fo(\))3561 84 y Fd(#)3626 230 y Fk(:)0 502 y Fo(The)34 b(corresp)s(onding)e(lo)m (w)m(ering)g(op)s(erators)g Fj(A)1723 517 y Fh(m)1789 502 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))32 b(are)g(their)g(formal)f(adjoin)m(ts.)146 622 y(These)k(op)s(erators)d(satisfy)h(comm)m(utation)d(relations)h (that)i(lead)f(to)g(the)h(prop)s(erties)f(of)g(the)0 743 y Fk(\036)58 758 y Fh(j)94 743 y Fo(\()p Fk(A;)i(B)5 b(;)33 b Fo(\026)-50 b Fk(h;)33 b(a;)h(\021)t(;)f(x)p Fo(\))g(that)f(w)m(e)i(list)d(b)s(elo)m(w.)43 b(The)34 b(raising)d(op)s(erators)h Fj(A)2729 758 y Fh(m)2795 743 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))3357 707 y Fg(\003)3428 743 y Fo(for)0 863 y Fk(m)42 b Fo(=)g(1)p Fk(;)33 b Fo(2)p Fk(;)g(:)17 b(:)g(:)f(;)33 b(d)40 b Fo(comm)m(ute)h(with)f(one)h(another,)i(and)e (the)g(lo)m(w)m(ering)f(op)s(erators)h Fj(A)3273 878 y Fh(m)3339 863 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))0 983 y(comm)m(ute)32 b(with)g(one)h(another.)44 b(Ho)m(w)m(ev)m(er,)218 1187 y Fj(A)298 1202 y Fh(m)364 1187 y 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Fk(h)1711 1715 y Fg(\000)p Fh(d=)p Fi(4)1894 1758 y Fo(\(det\()p Fk(A)p Fo(\)\))2254 1717 y Fg(\000)p Fi(1)p Fh(=)p Fi(2)898 1966 y Fj(\002)114 b Fo(exp)1255 1870 y Fd(n)1327 1966 y Fj(\000)17 b(h)g Fo(\()p Fk(x)22 b Fj(\000)g Fk(a)p Fo(\))p Fk(;)34 b(B)22 b(A)2010 1925 y Fg(\000)p Fi(1)2121 1966 y Fo(\()p Fk(x)g Fj(\000)h Fk(a)p Fo(\))17 b Fj(i)p Fk(=)p Fo(\(2\026)-50 b Fk(h)o Fo(\))39 b(+)f Fk(i)17 b Fj(h)g Fk(\021)t(;)32 b Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\))17 b Fj(i)p Fk(=)p Fo(\026)-50 b Fk(h)3560 1870 y Fd(o)3632 1966 y Fk(:)0 2155 y Fo(Then,)34 b(for)e(an)m(y)h(non-zero)g(m)m(ulti-index) d Fk(j)6 b Fo(,)33 b(w)m(e)g(de\014ne)244 2383 y Fk(\036)302 2398 y Fh(j)338 2383 y Fo(\()p Fk(A;)g(B)5 b(;)34 b Fo(\026)-50 b Fk(h;)33 b(a;)h(\021)t(;)f Fj(\001)17 b Fo(\))82 b(=)1376 2315 y(1)p 1322 2360 157 4 v 1322 2377 a Fj(p)p 1405 2377 74 4 v 75 x Fk(j)6 b Fo(!)1537 2383 y(\()17 b Fj(A)1672 2398 y Fi(1)1711 2383 y Fo(\()p Fk(A;)g(B)5 b(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a;)g(\021)t 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y(tra)5 b(jectory)33 b(selected)h(b)m(y)f(the)g (initial)c(conditions.)146 2215 y(In)37 b(the)f(rest)h(of)f(the)g(pap)s (er,)i(whenev)m(er)g(w)m(e)g(write)d Fk(\036)2118 2230 y Fh(j)2155 2215 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\),)36 b(w)m(e)h(tacitly)e(as-)0 2335 y(sume)h(that)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\),)35 b(and)h Fk(S)6 b Fo(\()p Fk(t)p Fo(\))35 b(are)g(solutions)f(to)h (\(2.3\))g(with)g(initial)c(conditions)k(satis-)0 2455 y(fying)d(\(2.1\).)0 2788 y Fl(3)161 b(The)53 b(Main)i(Results)0 3066 y Fo(In)32 b(this)g(section,)g(w)m(e)i(list)c(our)i(results)g (concerning)h(the)f(propagation)f(of)g(semiclassical)f(w)m(a)m(v)m(e)k (pac)m(k)m(ets.)0 3187 y(The)26 b(\014rst)f(is)g(the)g(construction)g 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Fe(and)f(satis\014es)0 4051 y Fj(j)p Fk(V)21 b Fo(\()p Fk(z)t Fo(\))p Fj(j)45 b(\024)g Fk(M)f Fo(exp)17 b(\()p Fk(\034)11 b Fj(j)p Fk(z)t Fj(j)925 4015 y Fi(2)965 4051 y Fo(\))35 b Fe(for)g Fk(z)d Fj(2)c Fk(S)1425 4066 y Fh(\016)1498 4051 y Fe(and)34 b(some)g(p)-5 b(ositive)34 b(c)-5 b(onstants)35 b Fk(M)45 b Fe(and)35 b Fk(\034)11 b Fe(.)146 4171 y(Fix)36 b Fk(T)14 b Fe(,)35 b(cho)-5 b(ose)35 b(a)h(classic)-5 b(al)35 b(orbit)g Fk(a)p Fo(\()p Fk(t)p Fo(\))i Fe(for)e Fo(0)30 b Fj(\024)g Fk(t)g Fj(\024)g Fk(T)14 b Fe(,)36 b(and)f(c)-5 b(onsider)35 b(an)g(arbitr)-5 b(ary)36 b(normalize)-5 b(d)0 4291 y(c)g(oher)g(ent)34 b(state)h(of)g(the)g(form)837 4511 y Fk( )t Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))28 b(=)1384 4428 y Fd(X)1358 4617 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)1563 4511 y Fk(c)1605 4526 y Fh(j)1641 4511 y Fo(\(0\))17 b Fk(\036)1841 4526 y Fh(j)1877 4511 y Fo(\()p Fk(A)p Fo(\(0\))p Fk(;)g(B)5 b Fo(\(0\))p Fk(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a)p Fo(\(0\))p Fk(;)g(\021)t Fo(\(0\))p Fk(;)g(x)p Fo(\))p Fk(:)0 4824 y Fe(Ther)-5 b(e)37 b(exists)h(a)g(numb)-5 b(er)37 b Fk(G)d(>)f Fo(0)p Fe(,)38 b(such)g(that)h(for)e(e)-5 b(ach)38 b(choic)-5 b(e)37 b(of)g(the)h(p)-5 b(ar)g(ameter)38 b Fk(g)f Fj(2)c Fo(\(0)p Fk(;)g(G)p Fo(\))p Fe(,)39 b(ther)-5 b(e)0 4945 y(exists)35 b(an)f(exact)h(solution)f(to)h(the)g(Schr\177)-50 b(odinger)34 b(e)-5 b(quation,)1328 5225 y Fk(i)17 b Fo(\026)-50 b Fk(h)1460 5158 y(@)5 b Fo(\011)p 1460 5202 133 4 v 1480 5293 a Fk(@)g(t)1665 5225 y Fo(=)62 b Fj(\000)1907 5158 y Fo(\026)-50 b Fk(h)1962 5115 y Fi(2)p 1907 5202 95 4 v 1930 5293 a Fo(2)2028 5225 y(\001)17 b(\011)57 b(+)39 b Fk(V)f Fo(\011)p Fk(;)1926 5506 y Fo(6)p eop %%Page: 7 7 7 6 bop 0 -43 a Fe(with)35 b Fo(\011\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)o Fo(\))28 b(=)f Fk( )t Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))p Fe(,)35 b(that)g(agr)-5 b(e)g(es)34 b(with)h(the)g(appr)-5 b(oximate)33 b(solution)493 150 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))63 b(=)f Fk(e)1115 109 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1551 67 y Fd(X)1379 256 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(g)r(=)p Fi(\026)-35 b Fh(h)o Fg(\000)p Fi(3)1876 150 y Fk(c)1918 165 y Fh(j)1955 150 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(\036)2240 165 y Fh(j)2275 150 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fk(;)0 450 y Fe(up)26 b(to)h(an)f(err)-5 b(or)26 b(whose)f Fk(L)948 414 y Fi(2)988 450 y Fo(\()p Fk(I)-20 b(R)1124 408 y Fh(d)1165 450 y Fo(\))26 b Fe(norm)g(is)g(b)-5 b(ounde)g(d)25 b(by)i Fk(C)c Fo(exp)35 b Fj(f)16 b(\000)p Fk(\015)2528 465 y Fh(g)2568 450 y Fk(=)p Fo(\026)-50 b Fk(h)17 b Fj(g)o Fe(,)28 b(with)e Fk(\015)3050 465 y Fh(g)3118 450 y Fk(>)h Fo(0)p Fe(.)42 b(F)-7 b(urthermor)i(e,)0 570 y(the)35 b(c)-5 b(omplex)34 b(c)-5 b(o)g(e\016cients)33 b Fk(c)1069 585 y Fh(j)1106 570 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))35 b Fe(ar)-5 b(e)34 b(determine)-5 b(d)34 b(by)h(an)g(explicit)f(pr)-5 b(o)g(c)g(e)g(dur)g(e.)146 768 y Fo(The)36 b(second)g(result)e(sho)m(ws)i(that)e(the)h(appro)m (ximate)f(w)m(a)m(v)m(e)i(function)e(of)g(Theorem)h(3.1)f(is)g(concen-) 0 889 y(trated)c(within)f(an)h(arbitrarily)d(small)g(distance)k(of)e (the)h(classical)f(path)h(up)g(to)g(an)f(exp)s(onen)m(tially)g(small)0 1009 y(error)j(if)g Fk(g)k Fo(is)c(c)m(hosen)i(su\016cen)m(tly)g (small.)0 1248 y Fa(Theorem)j(3.2)49 b Fe(Supp)-5 b(ose)40 b(that)h(the)f(hyp)-5 b(otheses)40 b(of)g(The)-5 b(or)g(em)39 b(3.1)h(ar)-5 b(e)40 b(satis\014e)-5 b(d)40 b(and)g(that)h Fk(b)d(>)g Fo(0)i Fe(is)0 1368 y(given.)71 b(F)-7 b(or)44 b(su\016ciently)g(smal)5 b(l)43 b(values)g(of)h(the)g(p)-5 b(ar)g(ameter)44 b Fk(g)k(>)c Fo(0)p Fe(,)i(the)e(wave)g(p)-5 b(acket)43 b Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))44 b Fe(is)0 1488 y(lo)-5 b(c)g(alize)g(d)32 b(within)h(a)h(distanc)-5 b(e)32 b Fk(b)i Fe(of)g Fk(a)p Fo(\()p Fk(t)p Fo(\))p Fe(,)g(up)f(to)h(an)f(err)-5 b(or)34 b Fo(exp)g Fj(f)16 b(\000)p Fo(\000)2567 1503 y Fh(g)2608 1488 y Fk(=)p Fo(\026)-50 b Fk(h)16 b Fj(g)p Fe(,)34 b(with)f Fo(\000)3113 1503 y Fh(g)3181 1488 y Fk(>)28 b Fo(0)p Fe(,)33 b(in)g(the)h(sense)0 1609 y(that)875 1644 y Fd( )957 1673 y(Z)1003 1862 y Fg(j)p Fh(x)p Fg(\000)p Fh(a)p Fi(\()p Fh(t)p Fi(\))p Fg(j)p Fh(>b)1396 1790 y Fj(j)p Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)o Fo(\))p Fj(j)1827 1749 y Fi(2)1883 1790 y Fk(dx)2006 1644 y Fd(!)2071 1667 y Fi(1)p Fh(=)p Fi(2)2244 1790 y Fj(\024)63 b Fo(exp)34 b Fj(f)17 b(\000)p Fo(\000)2771 1805 y Fh(g)2811 1790 y Fk(=)p Fo(\026)-50 b Fk(h)17 b Fj(g)f Fk(:)146 2116 y Fo(Next,)33 b(w)m(e)g(turn)f(to)f(the)i(v)-5 b(alidit)m(y)29 b(of)j(the)g(appro)m(ximation)d(and)j(its)f(lo)s (calization)d(prop)s(erties)k(on)g(the)0 2236 y(Ehrenfest)i(time)d (scale,)i(i.e.)43 b(when)34 b Fk(T)46 b Fo(is)32 b(allo)m(w)m(ed)g(to)g (increase)h(with)f(\026)-50 b Fk(h)33 b Fo(as)g(ln)o(\(1)p Fk(=)p Fo(\026)-50 b Fk(h)p Fo(\).)0 2475 y Fa(Theorem)37 b(3.3)49 b Fe(Supp)-5 b(ose)38 b(the)h(assumptions)e(of)i(The)-5 b(or)g(em)37 b(3.1)h(ar)-5 b(e)38 b(satis\014e)-5 b(d)38 b(exc)-5 b(ept)38 b(that)h(the)g(upp)-5 b(er)0 2595 y(b)g(ound)33 b(on)g Fk(V)55 b Fe(is)34 b(r)-5 b(eplac)g(e)g(d)32 b(by)i Fj(j)p Fk(V)21 b Fo(\()p Fk(z)t Fo(\))p Fj(j)45 b(\024)g Fk(M)f Fo(exp)17 b(\()p Fk(\034)11 b Fj(j)p Fk(z)t Fj(j)p Fo(\))34 b Fe(for)g Fk(z)e Fj(2)c Fk(S)2511 2610 y Fh(\016)2583 2595 y Fe(and)k(some)h(p)-5 b(ositive)33 b(c)-5 b(onstants)33 b Fk(M)0 2715 y Fe(and)e Fk(\034)11 b Fe(.)44 b(F)-7 b(urther,)31 b(assume)g(the)h(existenc)-5 b(e)30 b(of)h(a)g(c)-5 b(onstant)31 b Fk(N)38 b(>)28 b Fo(0)j Fe(and)g(a)g(p)-5 b(ositive)31 b(Lyapunov)g(exp)-5 b(onent)0 2836 y Fk(\025)35 b Fe(so)f(that)h Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)44 b(\024)h Fk(N)27 b Fo(exp)q(\()p Fk(\025t)p Fo(\))p Fe(,)34 b(for)h(al)5 b(l)34 b Fk(t)28 b Fj(\025)g Fo(0)p Fe(.)45 b(Then,)33 b(for)i(su\016ciently)f(smal)5 b(l)34 b Fk(T)3198 2800 y Fg(0)3249 2836 y Fk(>)27 b Fo(0)p Fe(,)35 b(ther)-5 b(e)34 b(exist)0 2956 y(c)-5 b(onstants)28 b Fk(C)499 2920 y Fg(0)550 2956 y Fk(>)f Fo(0)p Fe(,)p Fk(\015)788 2920 y Fg(0)839 2956 y Fk(>)h Fo(0)p Fe(,)h Fk(\033)j Fj(2)c Fo(\(0)p Fk(;)33 b Fo(1\))p Fe(,)c(and)f Fk(\033)1767 2920 y Fg(0)1818 2956 y Fj(2)g Fo(\(0)p Fk(;)33 b Fo(1\))p Fe(,)c(and)f(an)g(exact)g(solution)g(to)h(the)f (Schr\177)-50 b(odinger)0 3077 y(e)-5 b(quation)35 b(that)g(agr)-5 b(e)g(es)34 b(with)h(the)g(appr)-5 b(oximation)479 3270 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))63 b(=)f Fk(e)1101 3229 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1551 3187 y Fd(X)1365 3393 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(=)p Fi(\026)-35 b Fh(h)1703 3367 y Ff(\033)1741 3353 y Fb(0)1767 3393 y Fg(\000)p Fi(3)1890 3270 y Fk(c)1932 3285 y Fh(j)1969 3270 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(\036)2254 3285 y Fh(j)2289 3270 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fk(;)0 3574 y Fe(up)35 b(to)g(an)g(err)-5 b(or)34 b(whose)g(norm)h(is)f(b)-5 b(ounde)g(d)34 b(by)h Fk(C)1865 3538 y Fg(0)1905 3574 y Fo(exp)f Fj(f)17 b(\000)p Fk(\015)2287 3538 y Fg(0)2311 3574 y Fk(=)p Fo(\026)-50 b Fk(h)2415 3531 y Fh(\033)2478 3574 y Fj(g)p Fe(,)35 b(whenever)e Fo(0)28 b Fj(\024)g Fk(t)g Fj(\024)g Fk(T)3438 3538 y Fg(0)3477 3574 y Fo(ln\(1)p Fk(=)p Fo(\026)-50 b Fk(h)o Fo(\))p Fe(.)0 3695 y(Mor)-5 b(e)g(over,)28 b(if)e Fk(\034)37 b Fe(c)-5 b(an)26 b(b)-5 b(e)26 b(taken)g(arbitr)-5 b(arily)26 b(smal)5 b(l,)27 b(we)f(c)-5 b(an)26 b(chose)f Fk(T)2520 3658 y Fg(0)2571 3695 y Fo(=)2705 3655 y Fi(1)p 2684 3671 77 4 v 2684 3729 a(6)p Fh(\025)2771 3695 y Fo(\(1)s Fj(\000)s Fk(\017)p Fo(\))i Fe(wher)-5 b(e)25 b Fk(\017)i Fe(is)f(arbitr)-5 b(arily)0 3815 y(smal)5 b(l.)0 4013 y Fa(Remark:)97 b Fo(The)35 b(semiclassical)d(appro)m(ximation)f(of)j (observ)-5 b(ables)34 b(in)f(the)i(Heisen)m(b)s(erg)f(picture)g(holds)0 4133 y(for)k(an)m(y)h Fk(T)416 4097 y Fg(0)476 4133 y Fk(<)e Fo(2)p Fk(=)p Fo(\(3)p Fk(\025)p Fo(\),)i(when)g Fk(\034)49 b(<<)37 b Fo(1,)i(as)g(sho)m(wn)g(recen)m(tly)g(in)f([3].)60 b(That)38 b(time)f(in)m(terv)-5 b(al)37 b(is)h(longer)0 4254 y(than)e(those)h(for)e(whic)m(h)i(a)f(lo)s(calized)d(coheren)m(t) 38 b(state)e(can)g(appro)m(ximate)g(the)g(ev)m(olution)f(of)h(an)g (initial)0 4374 y(coheren)m(t)i(state,)g(whic)m(h)f(is)f(c)m (haracterized)h(b)m(y)g Fk(T)1860 4338 y Fg(0)1917 4374 y Fk(<)e Fo(1)p Fk(=)p Fo(\(2)p Fk(\025)p Fo(\).)54 b(See)37 b([2])f(for)g(a)g(study)i(of)e(related)g(issues)0 4495 y(on)c(quan)m(tized)i(h)m(yp)s(erb)s(olic)e(maps)g(on)h(the)g(torus.)0 4693 y Fa(Theorem)k(3.4)49 b Fe(Supp)-5 b(ose)34 b(the)h(hyp)-5 b(otheses)35 b(of)g(The)-5 b(or)g(em)33 b(3.3)i(ar)-5 b(e)35 b(satis\014e)-5 b(d)34 b(and)h(that)g Fk(b)28 b(>)g Fo(0)35 b Fe(is)g(given.)0 4813 y(Then,)g(for)g(su\016ciently)g (smal)5 b(l)35 b Fk(T)1264 4777 y Fg(0)1316 4813 y Fk(>)28 b Fo(0)p Fe(,)35 b(ther)-5 b(e)36 b(exist)f Fo(\000)2066 4777 y Fg(0)2118 4813 y Fk(>)28 b Fo(0)p Fe(,)35 b Fk(\033)e Fj(2)c Fo(\(0)p Fk(;)k Fo(1\))p Fe(,)i(and)g Fk(\033)3067 4777 y Fg(0)3119 4813 y Fj(2)29 b Fo(\(0)p Fk(;)k Fo(1\))p Fe(,)i(such)h(that)0 4934 y(the)f(appr)-5 b(oximation)33 b(of)i(The)-5 b(or)g(em)34 b(3.3)g(satis\014es)860 5061 y Fd( )942 5090 y(Z)988 5279 y Fg(j)p Fh(x)p Fg(\000)p Fh(a)p Fi(\()p Fh(t)p Fi(\))p Fg(j)p Fh(>b)1381 5207 y Fj(j)p Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)o Fo(\))p Fj(j)1812 5166 y Fi(2)1868 5207 y Fk(dx)1991 5061 y Fd(!)2056 5084 y Fi(1)p Fh(=)p Fi(2)2229 5207 y Fj(\024)63 b Fo(exp)34 b Fj(f)17 b(\000)p Fo(\000)2756 5166 y Fg(0)2779 5207 y Fk(=)p Fo(\026)-50 b Fk(h)2883 5164 y Fh(\033)2947 5207 y Fj(g)16 b Fk(;)1926 5506 y Fo(7)p eop %%Page: 8 8 8 7 bop 0 -43 a Fe(whenever)34 b Fo(0)27 b Fj(\024)h Fk(t)g Fj(\024)g Fk(T)845 -79 y Fg(0)885 -43 y Fo(ln)o(\(1)p Fk(=)p Fo(\026)-50 b Fk(h)p Fo(\))p Fe(.)0 77 y(Mor)-5 b(e)g(over,)28 b(if)e Fk(\034)37 b Fe(c)-5 b(an)26 b(b)-5 b(e)26 b(taken)g(arbitr)-5 b(arily)26 b(smal)5 b(l,)27 b(we)f(c)-5 b(an)26 b(chose)f Fk(T)2520 41 y Fg(0)2571 77 y Fo(=)2705 38 y Fi(1)p 2684 54 77 4 v 2684 112 a(6)p Fh(\025)2771 77 y Fo(\(1)s Fj(\000)s Fk(\017)p Fo(\))i Fe(wher)-5 b(e)25 b Fk(\017)i Fe(is)f(arbitr)-5 b(arily)0 198 y(smal)5 b(l.)146 384 y Fo(W)-8 b(e)31 b(also)f(explore)h(the)g(v) -5 b(alidit)m(y)29 b(of)h(the)i(appro)m(ximation)c(in)i(a)h(scattering) f(framew)m(ork)h(and)g(its)f(con-)0 504 y(sequences)40 b(on)d(the)g(corresp)s(onding)f(semiclassical)f(appro)m(ximation)f(of)i (the)h(scattering)g(matrix)e Fk(S)6 b Fo(\(\026)-50 b Fk(h)p Fo(\).)0 625 y(This)33 b(requires)g(assumptions)g(on)f(the)h (deca)m(y)h(of)e(the)h(p)s(oten)m(tial)e(and)i(its)f(deriv)-5 b(ativ)m(es)32 b(at)g(in\014nit)m(y)-8 b(.)146 865 y(F)g(or)32 b(scattering)g(theory)-8 b(,)34 b(w)m(e)f(assume)g Fk(V)54 b Fo(satis\014es)34 b(the)f(follo)m(wing)d(deca)m(y)k(h)m(yp)s (othesis.)0 1106 y Fa(D:)98 b Fe(Ther)-5 b(e)37 b(exist)g Fk(\014)i(>)33 b Fo(1)p Fe(,)38 b Fk(v)1094 1121 y Fi(0)1167 1106 y Fk(>)32 b Fo(0)p Fe(,)39 b(and)e Fk(v)1632 1121 y Fi(1)1704 1106 y Fk(>)c Fo(0)p Fe(,)38 b(such)g(that)g(for)g(al)5 b(l)37 b Fk(x)c Fj(2)h Fk(I)-20 b(R)2944 1064 y Fh(d)3022 1106 y Fe(and)38 b(al)5 b(l)37 b(multi-indic)-5 b(es)0 1226 y Fk(m)28 b Fj(2)g Fa(N)295 1184 y Fh(d)335 1226 y Fe(,)244 1474 y Fk(D)328 1433 y Fh(m)394 1474 y Fk(V)22 b Fo(\()p Fk(x)p Fo(\))63 b Fj(\024)817 1406 y Fk(v)864 1421 y Fi(0)920 1406 y Fk(v)971 1355 y Fg(j)p Fh(m)p Fg(j)967 1428 y Fi(1)1093 1406 y Fk(m)p Fo(!)p 817 1450 389 4 v 843 1542 a Fj(h)p Fk(x)p Fj(i)976 1513 y Fh(\014)s Fi(+)p Fg(j)p Fh(m)p Fg(j)1215 1474 y Fk(;)2457 b Fo(\(3.1\))0 1717 y Fe(wher)-5 b(e)34 b Fj(h)p Fk(x)p Fj(i)44 b Fo(=)573 1635 y Fj(p)p 656 1635 264 4 v 82 x Fo(1)22 b(+)g Fk(x)880 1688 y Fi(2)919 1717 y Fe(.)146 1957 y Fo(Theorem)31 b(1.2)f(of)g([7)o(])h(sho)m(ws)h(that)e(under)h(the)g(h)m(yp)s(othesis) g(D,)f(the)h(solution)d(of)i(the)h(classical)e(equa-)0 2078 y(tions)j(\(2.3\))g(satis\014es)h(the)g(follo)m(wing)d(asymptotic) i(estimates:)146 2198 y(F)-8 b(or)38 b(an)m(y)h Fk(a)568 2213 y Fg(\000)665 2198 y Fj(2)f Fk(I)-19 b(R)867 2156 y Fh(d)908 2198 y Fo(,)40 b(0)d Fj(6)p Fo(=)g Fk(\021)1222 2213 y Fg(\000)1319 2198 y Fj(2)h Fk(I)-19 b(R)1522 2156 y Fh(d)1601 2198 y Fo(suc)m(h)39 b(that)f(\()p Fk(a)2132 2213 y Fg(\000)2192 2198 y Fk(;)33 b(\021)2300 2213 y Fg(\000)2359 2198 y Fo(\))k Fj(2)h Fk(I)-19 b(R)2637 2156 y Fi(2)p Fh(d)2713 2198 y Fj(nE)9 b Fo(,)39 b(where)h Fj(E)46 b(\022)38 b(f)p Fo(\()p Fk(a)3532 2213 y Fg(\000)3591 2198 y Fk(;)33 b(\021)3699 2213 y Fg(\000)3758 2198 y Fo(\))38 b Fj(2)0 2318 y Fk(I)-19 b(R)99 2277 y Fi(2)p Fh(d)208 2318 y Fo(:)33 b Fk(\021)316 2333 y Fg(\000)408 2318 y Fj(6)p Fo(=)g(0)p Fj(g)i Fo(is)g(closed)h(and)g(of)f(Leb)s (esgue)i(measure)f(zero)g(in)f Fk(I)-19 b(R)2583 2277 y Fh(d)2624 2318 y Fo(,)36 b(there)h(exists)f(\()p Fk(a)3301 2333 y Fi(+)3360 2318 y Fk(;)e(\021)3469 2333 y Fi(+)3528 2318 y Fo(\))f Fj(2)g Fk(I)-19 b(R)3797 2277 y Fi(2)p Fh(d)3873 2318 y Fo(,)0 2439 y Fk(\021)48 2454 y Fi(+)135 2439 y Fj(6)p Fo(=)27 b(0,)33 b(and)f Fk(S)596 2454 y Fi(+)683 2439 y Fj(2)c Fk(I)-19 b(R)34 b Fo(suc)m(h)g(that)453 2611 y(lim)410 2665 y Fh(t)p Fg(!\0061)665 2611 y Fj(j)p Fk(a)p Fo(\()p Fk(t)p Fo(\))22 b Fj(\000)g Fk(a)1027 2626 y Fg(\006)1109 2611 y Fj(\000)g Fk(\021)1256 2626 y Fg(\006)1316 2611 y Fk(t)p Fj(j)60 b Fo(=)g(0)p Fk(;)453 2822 y Fo(lim)410 2876 y Fh(t)p Fg(!\0061)665 2822 y Fj(j)p Fk(\021)t Fo(\()p Fk(t)p Fo(\))21 b Fj(\000)i Fk(\021)1025 2837 y Fg(\006)1084 2822 y Fj(j)60 b Fo(=)g(0)p Fk(;)453 3042 y Fo(lim)410 3097 y Fh(t)p Fg(!\0001)665 3042 y Fj(j)p Fk(S)6 b Fo(\()p Fk(t)p Fo(\))21 b Fj(\000)i Fk(t\021)1078 3001 y Fi(2)1074 3067 y Fg(\000)1133 3042 y Fk(=)p Fo(2)p Fj(j)60 b Fo(=)g(0)p Fk(;)453 3262 y Fo(lim)410 3317 y Fh(t)p Fg(!)p Fi(+)p Fg(1)665 3262 y Fj(j)p Fk(S)6 b Fo(\()p Fk(t)p Fo(\))21 b Fj(\000)i Fk(S)1051 3277 y Fi(+)1132 3262 y Fj(\000)g Fk(t\021)1319 3221 y Fi(2)1315 3287 y(+)1374 3262 y Fk(=)p Fo(2)p Fj(j)60 b Fo(=)g(0)p Fk(:)1927 b Fo(\(3.2\))0 3467 y(Moreo)m(v)m(er,)41 b(for)c(an)m(y)i Fk(d)26 b Fj(\002)g Fk(d)38 b Fo(matrices)f(\()p Fk(A)1582 3482 y Fg(\000)1641 3467 y Fk(;)c(B)1775 3482 y Fg(\000)1834 3467 y Fo(\))38 b(satisfying)f(condition)g(\(2.1\),)i (there)f(exist)h(matrices)0 3587 y(\()p Fk(A)111 3602 y Fi(+)170 3587 y Fk(;)33 b(B)304 3602 y Fi(+)364 3587 y Fo(\))27 b Fj(2)h Fk(M)617 3602 y Fh(d)658 3587 y Fo(\()p Fk(I)-51 b(C)7 b Fo(\))803 3551 y Fi(2)875 3587 y Fo(satisfying)32 b(\(2.1\),)g(suc)m(h)i(that)453 3759 y(lim)410 3813 y Fh(t)p Fg(!\0061)665 3759 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))22 b Fj(\000)g Fk(A)1093 3774 y Fg(\006)1175 3759 y Fj(\000)g Fk(iB)1381 3774 y Fg(\006)1441 3759 y Fk(t)p Fj(k)60 b Fo(=)g(0)p Fk(;)453 3970 y Fo(lim)410 4024 y Fh(t)p Fg(!\0061)665 3970 y Fj(k)p Fk(B)5 b Fo(\()p Fk(t)p Fo(\))22 b Fj(\000)g Fk(B)1100 3985 y Fg(\006)1160 3970 y Fj(k)60 b Fo(=)g(0)p Fk(:)2217 b Fo(\(3.3\))146 4253 y(Our)39 b(assumption)e(D)h(implies)e(that)i Fk(V)60 b Fo(is)38 b(short)g(range.)61 b(It)38 b(follo)m(ws)g(that)g(if)f Fk(H)3169 4268 y Fi(0)3208 4253 y Fo(\(\026)-50 b Fk(h)p Fo(\))76 b(=)f Fj(\000)3670 4185 y Fo(\026)-50 b Fk(h)3726 4142 y Fi(2)p 3670 4229 95 4 v 3693 4321 a Fo(2)3792 4253 y(\001,)0 4398 y(then)33 b(the)g(w)m(a)m(v)m(e)i(op)s(erators)d (de\014ned)i(b)m(y)244 4570 y(\012)314 4529 y Fg(\007)373 4570 y Fo(\(\026)-50 b Fk(h)p Fo(\))61 b(=)f Fk(s)22 b Fj(\000)69 b Fo(lim)868 4624 y Fh(s)p Fg(!\0061)1130 4570 y Fo(e)1174 4529 y Fh(iH)5 b Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(s=)p Fi(\026)g Fh(h)1486 4570 y Fo(e)1529 4529 y Fg(\000)p Fh(iH)1666 4538 y Fc(0)1700 4529 y Fi(\(\026)g Fh(h)p Fi(\))p Fh(s=)p Fi(\026)g Fh(h)3699 4570 y Fo(\(3.4\))0 4774 y(exist)37 b(and)f(ha)m(v)m(e)i(iden)m(tical)c(ranges)j(equal)f (to)g(the)h(absolutely)e(con)m(tin)m(uous)i(subspace)i(of)c Fk(H)8 b Fo(\(\026)-50 b Fk(h)p Fo(\).)54 b(As)37 b(a)0 4894 y(result,)c(the)g(scattering)f(matrix)244 5066 y Fk(S)6 b Fo(\(\026)-50 b Fk(h)p Fo(\))60 b(=)g(\012)707 5025 y Fg(\000)766 5066 y Fo(\(\026)-50 b Fk(h)p Fo(\))897 5025 y Fg(\003)953 5066 y Fo(\012)1023 5025 y Fi(+)1083 5066 y Fo(\(\026)g Fk(h)p Fo(\))2485 b(\(3.5\))0 5238 y(is)32 b(unitary)-8 b(.)1926 5506 y(8)p eop %%Page: 9 9 9 8 bop 0 -43 a Fa(Theorem)37 b(3.5)49 b Fe(Supp)-5 b(ose)57 b Fk(d)68 b Fj(\025)i Fo(3)57 b Fe(and)f(assume)h(hyp)-5 b(othesis)57 b(D.)f(L)-5 b(et)58 b Fo(\()p Fk(a)2934 -28 y Fg(\000)2993 -43 y Fk(;)33 b(\021)3101 -28 y Fg(\000)3161 -43 y Fo(\))69 b Fj(2)g Fk(I)-20 b(R)3501 -85 y Fi(2)p Fh(d)3577 -43 y Fj(nE)66 b Fe(and)0 77 y Fo(\()p Fk(A)111 92 y Fg(\000)170 77 y Fk(;)33 b(B)304 92 y Fg(\000)364 77 y Fo(\))41 b Fj(2)h Fk(M)645 92 y Fh(d)686 77 y Fo(\()p Fk(I)-52 b(C)7 b Fo(\))830 41 y Fi(2)912 77 y Fe(satisfy)43 b(c)-5 b(ondition)41 b(\(2.1\).)67 b(L)-5 b(et)43 b Fk(c)2182 92 y Fh(j)2218 77 y Fo(\()p Fj(\0001)p Fo(\))f Fj(2)g Fk(I)-52 b(C)7 b Fe(,)45 b(for)d Fk(j)48 b Fj(2)42 b Fa(N)3211 35 y Fh(d)3251 77 y Fe(,)i(with)f Fj(j)p Fk(j)6 b Fj(j)41 b(\024)h Fk(J)9 b Fe(,)0 198 y(such)35 b(that)421 131 y Fd(P)509 218 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)717 198 y Fj(j)p Fk(c)787 213 y Fh(j)823 198 y Fo(\()p Fj(\0001)p Fo(\))p Fj(j)1104 161 y Fi(2)1171 198 y Fo(=)28 b(1)p Fe(.)45 b(Then,)34 b(ther)-5 b(e)34 b(exist)h Fo(\()p Fk(a)2241 213 y Fi(+)2300 198 y Fk(;)f(\021)2409 213 y Fi(+)2468 198 y Fo(\))27 b Fj(2)i Fk(I)-20 b(R)2726 156 y Fi(2)p Fh(d)2802 198 y Fe(,)35 b Fo(\()p Fk(A)2978 213 y Fi(+)3037 198 y Fk(;)e(B)3171 213 y Fi(+)3230 198 y Fo(\))28 b Fj(2)g Fk(M)3484 213 y Fh(d)3525 198 y Fo(\()p Fk(I)-52 b(C)7 b Fo(\))3669 161 y Fi(2)3743 198 y Fe(sat-)0 335 y(isfying)36 b(\(2.1\),)f Fk(S)648 350 y Fi(+)737 335 y Fj(2)c Fk(I)-20 b(R)37 b Fe(and)f(explicit)g(c)-5 b(o)g(e\016cients)35 b Fk(c)2037 350 y Fh(j)2073 335 y Fo(\(+)p Fj(1)p Fk(;)17 b Fo(\026)-50 b Fk(h)o Fo(\))31 b Fj(2)f Fk(I)-52 b(C)7 b Fe(,)36 b(for)g(al)5 b(l)36 b Fk(j)g Fj(2)31 b Fa(N)3242 292 y Fh(d)3283 335 y Fe(,)36 b(with)g Fj(j)p Fk(j)6 b Fj(j)30 b(\024)3825 309 y Fo(~)3801 335 y Fk(J)3855 350 y Fi(\026)-35 b Fh(h)0 455 y Fe(with)243 430 y Fo(~)219 455 y Fk(J)273 470 y Fi(\026)g Fh(h)360 455 y Fo(=)42 b Fk(J)37 b Fo(+)27 b(3)p Fk(g)t(=)p Fo(\026)-50 b Fk(h)27 b Fj(\000)h Fo(3)43 b Fe(such)f(that)h(for)f(some)g Fk(\015)47 b(>)42 b Fo(0)p Fe(,)i Fk(C)k(>)42 b Fo(0)p Fe(,)i Fk(g)h(>)d Fo(0)g Fe(\(dep)-5 b(ending)41 b(on)h(the)h(the)0 575 y(classic)-5 b(al)34 b(data\),)g(the)h(states)g(de\014ne)-5 b(d)34 b(by)244 730 y Fo(\010)314 745 y Fg(\000)373 730 y Fo(\()p Fk(A)484 745 y Fg(\000)543 730 y Fk(;)17 b(B)661 745 y Fg(\000)720 730 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)914 745 y Fg(\000)973 730 y Fk(;)g(\021)1065 745 y Fg(\000)1124 730 y Fk(;)g(x)p Fo(\))83 b(=)1529 647 y Fd(X)1503 835 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)1691 730 y Fk(c)1733 745 y Fh(j)1769 730 y Fo(\()p Fj(\0001)p Fo(\))p Fk(\036)2080 745 y Fh(j)2117 730 y Fo(\()p Fk(A)2228 745 y Fg(\000)2287 730 y Fk(;)17 b(B)2405 745 y Fg(\000)2463 730 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)2657 745 y Fg(\000)2716 730 y Fk(;)g(\021)2808 745 y Fg(\000)2867 730 y Fk(;)g(x)p Fo(\))244 1026 y(\010)314 1041 y Fi(+)373 1026 y Fo(\()p Fk(A)484 1041 y Fi(+)543 1026 y Fk(;)g(B)661 1041 y Fi(+)720 1026 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)914 1041 y Fi(+)973 1026 y Fk(;)g(\021)1065 1041 y Fi(+)1124 1026 y Fk(;)g(x)p Fo(\))83 b(=)g Fe(e)1548 985 y Fh(iS)1615 994 y Fc(+)1665 985 y Fh(=)p Fi(\026)-35 b Fh(h)1804 943 y Fd(X)1762 1143 y Fg(j)p Fh(j)t Fg(j\024)1905 1126 y Fi(~)1890 1143 y Fh(J)1929 1155 y Fc(\026)j Ff(h)1983 1026 y Fk(c)2025 1041 y Fh(j)2061 1026 y Fo(\(+)p Fj(1)p Fk(;)17 b Fo(\026)-50 b Fk(h)o Fo(\))p Fk(\036)2469 1041 y Fh(j)2506 1026 y Fo(\()p Fk(A)2617 1041 y Fi(+)2676 1026 y Fk(;)17 b(B)2794 1041 y Fi(+)2853 1026 y Fk(;)g Fo(\026)-50 b Fk(h)o(;)17 b(a)3046 1041 y Fi(+)3105 1026 y Fk(;)g(\021)3197 1041 y Fi(+)3256 1026 y Fk(;)g(x)p Fo(\))306 b(\(3.6\))0 1279 y Fe(satisfy)331 1442 y Fj(k)p Fk(S)6 b Fo(\(\026)-50 b Fk(h)p Fo(\))17 b(\010)665 1457 y Fg(\000)724 1442 y Fo(\()p Fk(A)835 1457 y Fg(\000)894 1442 y Fk(;)g(B)1012 1457 y Fg(\000)1071 1442 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)1265 1457 y Fg(\000)1324 1442 y Fk(;)g(\021)1416 1457 y Fg(\000)1475 1442 y Fk(;)g Fj(\001)p Fo(\))38 b Fj(\000)h Fo(\010)1809 1457 y Fi(+)1869 1442 y Fo(\()p Fk(A)1980 1457 y Fi(+)2039 1442 y Fk(;)17 b(B)2157 1457 y Fi(+)2215 1442 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)2409 1457 y Fi(+)2468 1442 y Fk(;)g(\021)2560 1457 y Fi(+)2619 1442 y Fk(;)g Fj(\001)p Fo(\))p Fj(k)2779 1467 y Fh(L)2827 1448 y Fc(2)2861 1467 y Fi(\()p Fh(I)-15 b(R)2957 1442 y Ff(d)2994 1467 y Fi(\))3088 1442 y Fj(\024)63 b Fk(C)23 b Fe(e)3366 1401 y Fg(\000)p Fh(\015)t(=)p Fi(\026)-35 b Fh(h)3541 1442 y Fk(;)0 1605 y Fe(if)35 b Fo(\026)-50 b Fk(h)35 b Fe(is)f(smal)5 b(l)34 b(enough.)146 1768 y Fo(Finally)-8 b(,)29 b(w)m(e)j(address)g(the)f(question)g(of)g(the)g (generalization)d(of)j(the)g(initial)c(coheren)m(t)32 b(state,)g(whose)0 1889 y(ev)m(olution)k(can)i(b)s(e)f(con)m(trolled)g (up)g(to)g(exp)s(onen)m(tial)g(accuracy)h(in)f(the)g(di\013eren)m(t)h (settings)f(considered)0 2009 y(ab)s(o)m(v)m(e.)146 2129 y(F)-8 b(or)32 b(\()p Fk(a;)17 b(\021)t Fo(\))27 b Fj(2)h Fk(I)-19 b(R)764 2088 y Fi(2)p Fh(d)840 2129 y Fo(,)33 b(w)m(e)g(de\014ne)h(\003)1393 2144 y Fh(h)1438 2129 y Fo(\()p Fk(a;)17 b(\021)t Fo(\))32 b(to)g(b)s(e)h(the)g(op)s(erator) 885 2303 y(\(\003)991 2318 y Fi(\026)-35 b Fh(h)1035 2303 y Fo(\()p Fk(a;)17 b(\021)t Fo(\))p Fk(f)11 b Fo(\)\()p Fk(x)p Fo(\))44 b(=)g(\026)-50 b Fk(h)1705 2260 y Fg(\000)p Fh(d=)p Fi(2)1887 2303 y Fo(e)1931 2261 y Fh(i)p Fg(h)p Fh(\021)r(;)12 b Fi(\()p Fh(x)p Fg(\000)p Fh(a)p Fi(\))p Fg(i)p Fh(=)p Fi(\026)-35 b Fh(h)2363 2303 y Fk(f)11 b Fo(\(\()p Fk(x)22 b Fj(\000)g Fk(a)p Fo(\))p Fk(=)2812 2213 y Fj(p)p 2895 2213 56 4 v 90 x Fo(\026)-50 b Fk(h)p Fo(\))p Fk(:)0 2466 y Fo(W)-8 b(e)33 b(de\014ne)h(a)e(dense)i(set)f Fj(C)39 b Fo(in)32 b Fk(L)1220 2429 y Fi(2)1260 2466 y Fo(\()p Fk(I)-19 b(R)1397 2424 y Fh(d)1437 2466 y Fo(\),)33 b(that)f(is)g(con)m(tained)h(in)f(the)h(set)g Fj(S)40 b Fo(of)32 b(Sc)m(h)m(w)m(artz)j(functions,)e(b)m(y)244 2707 y Fj(C)89 b Fo(=)544 2533 y Fd(8)544 2608 y(<)544 2757 y(:)618 2707 y Fk(f)11 b Fo(\()p Fk(x)p Fo(\))28 b(=)939 2624 y Fd(X)983 2806 y Fh(j)1092 2707 y Fk(c)1134 2722 y Fh(j)1187 2707 y Fk(\036)1245 2722 y Fh(j)1282 2707 y Fo(\()p Fk(I)-67 b(I)7 b(;)33 b(I)-67 b(I)8 b(;)33 b Fo(1)p Fk(;)g Fo(0)p Fk(;)g Fo(0)p Fk(;)h(x)p Fo(\))44 b Fj(2)h(S)7 b Fk(;)587 3039 y Fo(suc)m(h)34 b(that)49 b Fj(9)17 b Fk(K)34 b(>)28 b Fo(0)k(with)1674 2956 y Fd(X)1648 3144 y Fg(j)p Fh(j)t Fg(j)p Fh(>J)1836 3039 y Fj(j)p Fk(c)1906 3054 y Fh(j)1942 3039 y Fj(j)1970 2998 y Fi(2)2037 3039 y Fj(\024)c Fo(e)2185 2998 y Fg(\000)p Fh(K)5 b(J)2353 3039 y Fk(;)50 b Fo(for)32 b(large)f Fk(J)2880 2865 y Fd(9)2880 2940 y(=)2880 3089 y(;)2971 3039 y Fk(:)701 b Fo(\(3.7\))0 3334 y Fa(Remark)97 b Fo(It)38 b(is)g(easy)h(to)f(c)m(hec)m(k)i(that)e(the)h(inequalit)m(y)e (in)g(\(3.7\))h(is)f(equiv)-5 b(alen)m(t)38 b(to)g(the)h(requiremen)m (t)0 3454 y(that)32 b(the)h(co)s(e\016cien)m(ts)h(of)e Fk(f)43 b Fo(satisfy)1667 3575 y Fj(j)p Fk(c)1737 3590 y Fh(j)1774 3575 y Fj(j)h(\024)g Fo(e)2011 3533 y Fg(\000)p Fh(K)5 b Fg(j)p Fh(j)t Fg(j)2205 3575 y Fk(;)0 3724 y Fo(for)32 b(large)f Fj(j)p Fk(j)6 b Fj(j)p Fo(.)43 b(Another)33 b(equiv)-5 b(alen)m(t)33 b(de\014nition)e(of)h Fj(C)39 b Fo(is)1553 3887 y Fj(C)51 b Fo(=)44 b Fj([)1842 3902 y Fh(t>)p Fi(0)1979 3887 y Fo(e)2022 3846 y Fg(\000)p Fh(tH)2160 3858 y Ff(ho)2252 3887 y Fj(S)7 b Fk(;)0 4050 y Fo(where)30 b Fk(H)359 4065 y Fh(ho)482 4050 y Fo(=)45 b Fj(\000)17 b Fo(\001)p Fk(=)p Fo(2)31 b(+)h Fk(x)1070 4014 y Fi(2)1110 4050 y Fk(=)p Fo(2)c(is)h(the)g(harmonic)f(oscillator) e(Hamiltonian.)39 b(The)30 b(set)g Fj(C)35 b Fo(is)29 b(sometimes)0 4171 y(called)i(the)i(set)h(of)e(analytic)f(v)m(ectors)j ([17])e(for)g(the)h(harmonic)f(oscillator)d(Hamiltonian.)0 4409 y Fa(Theorem)37 b(3.6)49 b Fe(A)n(l)5 b(l)35 b(the)-5 b(or)g(ems)34 b(ab)-5 b(ove)34 b(r)-5 b(emain)34 b(true)h(if)g(the)g (initial)f(c)-5 b(ondition)34 b(has)h(the)f(form)1373 4572 y Fk( )t Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))27 b(=)h(\(\003)2000 4587 y Fi(\026)-35 b Fh(h)2044 4572 y Fo(\()p Fk(a;)17 b(\021)t Fo(\))p Fk(')p Fo(\)\()p Fk(x)p Fo(\))p Fk(;)0 4735 y Fe(wher)-5 b(e)34 b Fk(')28 b Fj(2)g(C)6 b Fe(.)146 5016 y Fo(Theorem)36 b(3.1)g(is)f(pro)m(v)m(ed)i(in)e(Section)h(6.)53 b(Theorem)36 b(3.2)g(is)f(pro)m(v)m(ed)i(in)e(Section)h(7.)53 b(Theorems)36 b(3.3)0 5137 y(and)c(3.4)f(are)h(pro)m(v)m(ed)i(in)d(Section)g(8.)43 b(Theorem)33 b(3.5)e(is)g(pro)m(v)m(ed)j(in)d(Section)h(9.)42 b(Theorem)33 b(3.6)e(is)g(pro)m(v)m(ed)0 5257 y(in)h(Section)g(10.)1926 5506 y(9)p eop %%Page: 10 10 10 9 bop 0 -43 a Fl(4)161 b(An)53 b(Alternativ)l(e)h(Semiclassical)i (Expansion)0 243 y Fo(In)27 b(this)f(section)h(w)m(e)h(deriv)m(e)f(an)f (expansion)h(in)f(p)s(o)m(w)m(ers)i(of)e(\026)-50 b Fk(h)2200 201 y Fi(1)p Fh(=)p Fi(2)2310 243 y Fo(.)42 b(In)27 b(later)e(sections) j(w)m(e)f(p)s(erform)f(optimal)0 364 y(truncation)32 b(of)g(this)g(expansion)h(to)g(obtain)e(exp)s(onen)m(tially)h(accurate) h(appro)m(ximations.)146 602 y(W)-8 b(e)33 b(wish)g(appro)m(ximately)e (to)h(solv)m(e)h(the)g(equation)244 854 y Fk(i)17 b Fo(\026)-50 b Fk(h)375 787 y(@)5 b( )p 375 831 124 4 v 391 922 a(@)g(t)570 854 y Fo(=)60 b Fj(\000)810 787 y Fo(\026)-50 b Fk(h)865 744 y Fi(2)p 810 831 95 4 v 833 922 a Fo(2)931 854 y(\001)17 b Fk( )42 b Fo(+)d Fk(V)22 b Fo(\()p Fk(x)p Fo(\))17 b Fk( )t(;)2129 b Fo(\(4.1\))0 1071 y(with)32 b(initial)d(conditions)j (of)g(the)h(form)244 1262 y Fk( )t Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))60 b(=)855 1179 y Fd(X)830 1367 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)1034 1262 y Fk(c)1076 1277 y Fi(0)p Fh(;j)1168 1262 y Fo(\(0\))17 b Fk(\036)1368 1277 y Fh(j)1403 1262 y Fo(\()p Fk(A)p Fo(\(0\))p Fk(;)g(B)5 b Fo(\(0\))p Fk(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a)p Fo(\(0\))p Fk(;)g(\021)t Fo(\(0\))p Fk(;)g(x)p Fo(\))p Fk(;)1109 b Fo(\(4.2\))0 1563 y(where)307 1480 y Fd(X)282 1668 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)486 1563 y Fj(j)17 b Fk(c)573 1578 y Fi(0)p Fh(;j)664 1563 y Fo(\(0\))g Fj(j)833 1515 y Fi(2)932 1563 y Fo(=)61 b(1.)146 1771 y(W)-8 b(e)25 b(can)g(write)f(the)h(exact)h(solution)d (to)h(this)g(equation)h(in)e(the)i(basis)g(of)f(semiclassical)e(w)m(a)m (v)m(e)27 b(pac)m(k)m(ets,)244 1962 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))60 b(=)g Fk(e)861 1921 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1142 1879 y Fd(X)1185 2061 y Fh(j)1295 1962 y Fk(c)1337 1977 y Fh(j)1373 1962 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(\036)1658 1977 y Fh(j)1694 1962 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fk(:)873 b Fo(\(4.3\))0 2231 y(Note)33 b(that)f(the)g(sum)h(is)f(o)m(v)m(er)h(m)m(ulti-indices)d Fk(j)6 b Fo(.)43 b(The)33 b(in\014nite)f(v)m(ector)h Fk(c)f Fo(whose)i(en)m(tries)f(are)f(the)h(co)s(e\016-)0 2351 y(cien)m(ts)g Fk(c)316 2366 y Fh(j)385 2351 y Fo(satis\014es)244 2543 y Fk(i)17 b Fo(\026)-50 b Fk(h)29 b Fo(_)-40 b Fk(c)61 b Fo(=)f Fk(K)7 b Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(c;)2709 b Fo(\(4.4\))0 2734 y(where)34 b Fk(K)7 b Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))32 b(is)g(an)h(in\014nite)f(self{adjoin)m(t)f(matrix.)146 2854 y(The)39 b(matrix)e Fk(K)7 b Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))38 b(has)g(an)g(asymptotic)f(expansion)i(in) e(p)s(o)m(w)m(ers)i(of)f(\026)-50 b Fk(h)2918 2812 y Fi(1)p Fh(=)p Fi(2)3028 2854 y Fo(.)59 b(The)39 b(cubic)f(term)g(in)0 2975 y(the)f(expansion)g(of)e Fk(V)22 b Fo(\()p Fk(x)p Fo(\))36 b(around)h Fk(x)d Fo(=)g Fk(a)p Fo(\()p Fk(t)p Fo(\))j(giv)m(es)f(the)h(leading)e(non-zero)h(term)g(of)g(order)g(\026) -50 b Fk(h)3540 2932 y Fi(3)p Fh(=)p Fi(2)3650 2975 y Fo(.)55 b(The)0 3095 y(quartic)32 b(term)g(in)g(the)h(expansion)g(of)f Fk(V)22 b Fo(\()p Fk(x)p Fo(\))33 b(giv)m(es)g(the)g(term)f(of)g(order) h(\026)-50 b Fk(h)2716 3052 y Fi(4)p Fh(=)p Fi(2)2826 3095 y Fo(,)32 b(etc.)44 b(Th)m(us,)35 b(w)m(e)e(can)g(write)244 3344 y Fk(K)7 b Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))60 b Fj(\030)771 3236 y Fg(1)746 3261 y Fd(X)742 3446 y Fh(k)r Fi(=3)904 3344 y Fo(\026)-50 b Fk(h)959 3301 y Fh(k)r(=)p Fi(2)1089 3344 y Fk(K)1172 3359 y Fh(k)1214 3344 y Fo(\()p Fk(t)p Fo(\))p Fk(;)2347 b Fo(\(4.5\))0 3603 y(with)244 3824 y Fk(K)327 3839 y Fh(k)370 3824 y Fo(\()p Fk(t)p Fo(\))60 b(=)714 3741 y Fd(X)677 3929 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(k)915 3757 y Fo(\()p Fk(D)1037 3720 y Fh(m)1103 3757 y Fk(V)22 b Fo(\)\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 915 3801 543 4 v 1130 3892 a Fk(m)p Fo(!)1500 3824 y Fk(X)8 b Fo(\()p Fk(t)p Fo(\))1700 3783 y Fh(m)1767 3824 y Fk(;)1905 b Fo(\(4.6\))0 4126 y(where)39 b Fk(X)8 b Fo(\()p Fk(t)p Fo(\))487 4090 y Fh(m)591 4126 y Fo(is)38 b(the)g(in\014nite)f(matrix)f(that)i (represen)m(ts)i(\026)-50 b Fk(h)2262 4083 y Fg(\000j)p Fh(m)p Fg(j)p Fh(=)p Fi(2)2510 4126 y Fo(\()p Fk(x)26 b Fj(\000)g Fk(a)p Fo(\))2821 4090 y Fh(m)2888 4126 y Fo(.)59 b(Explicit)37 b(form)m(ulas)f([10])0 4246 y(sho)m(w)e(that)e (en)m(tries)h(of)f Fk(X)8 b Fo(\()p Fk(t)p Fo(\))1076 4210 y Fh(m)1175 4246 y Fo(and)33 b Fk(K)1448 4261 y Fh(k)1491 4246 y Fo(\()p Fk(t)p Fo(\))f(do)h(not)f(dep)s(end)i(on)e (\026)-50 b Fk(h)p Fo(.)146 4367 y(W)-8 b(e)33 b(formally)d(expand)k (the)f(v)m(ector)g Fk(c)g Fo(as)244 4558 y Fk(c)p Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))83 b(=)g Fk(c)780 4573 y Fi(0)819 4558 y Fo(\()p Fk(t)p Fo(\))39 b(+)g(\026)-50 b Fk(h)1139 4515 y Fi(1)p Fh(=)p Fi(2)1265 4558 y Fk(c)1307 4573 y Fi(1)1347 4558 y Fo(\()p Fk(t)p Fo(\))39 b(+)f(\026)-50 b Fk(h)1666 4515 y Fi(2)p Fh(=)p Fi(2)1793 4558 y Fk(c)1835 4573 y Fi(2)1874 4558 y Fo(\()p Fk(t)p Fo(\))39 b(+)55 b Fk(:)17 b(:)g(:)579 4745 y Fo(=)738 4662 y Fd(X)778 4847 y Fh(k)891 4745 y Fo(\026)-50 b Fk(h)946 4702 y Fh(k)r(=)p Fi(2)1076 4745 y Fk(c)1118 4760 y Fh(k)1160 4745 y Fo(\()p Fk(t)p Fo(\))p Fk(:)2401 b Fo(\(4.7\))0 5016 y(W)-8 b(e)37 b(denote)g(the)g Fk(j)708 4980 y Fi(th)816 5016 y Fo(en)m(try)h(of)e Fk(c)1232 5031 y Fh(k)1274 5016 y Fo(\()p Fk(t)p Fo(\))h(b)m(y)g Fk(c)1603 5031 y Fh(k)r(;j)1698 5016 y Fo(\()p Fk(t)p Fo(\).)56 b(Note)37 b(that)f Fk(k)k Fo(is)c(a)g(non-negativ)m(e)h(in)m(teger,)g(and)g Fk(j)42 b Fo(is)0 5137 y(a)33 b(m)m(ulti-index.)42 b(W)-8 b(e)34 b(substitute)g(the)f(t)m(w)m(o)h(expansions)g(\(4.5\))f(and)g (\(4.7\))f(in)m(to)h(\(4.4\))f(and)h(divide)g(b)m(y)h(\026)-50 b Fk(h)p Fo(.)0 5257 y(W)-8 b(e)33 b(then)g(equate)h(terms)e(of)g(the)h (same)g(orders)g(on)f(the)h(t)m(w)m(o)h(sides)f(of)f(the)h(resulting)f (equation.)1901 5506 y(10)p eop %%Page: 11 11 11 10 bop 0 -43 a Fa(Order)37 b Fo(0)p Fa(.)98 b Fo(The)33 b(zeroth)h(order)e(terms)h(simply)e(require)244 160 y Fk(i)f Fo(_)-40 b Fk(c)336 175 y Fi(0)435 160 y Fo(=)60 b(0)p Fk(:)3052 b Fo(\(4.8\))0 364 y(F)-8 b(rom)31 b(\(4.2\),)h(the)h (solution)e(is)h(ob)m(viously)h Fk(c)1625 379 y Fi(0)p Fh(;j)1716 364 y Fo(\()p Fk(t)p Fo(\))60 b(=)g Fk(c)2065 379 y Fi(0)p Fh(;j)2157 364 y Fo(\(0\).)43 b(W)-8 b(e)33 b(note)g(that)f Fk(c)2990 379 y Fi(0)p Fh(;j)3081 364 y Fo(\()p Fk(t)p Fo(\))c(=)g(0)k(if)f Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)9 b Fo(.)0 602 y Fa(Order)37 b Fo(1)p Fa(.)98 b Fo(The)33 b(\014rst)h(order)e(terms)h(require)244 806 y Fk(i)d Fo(_)-40 b Fk(c)336 821 y Fi(1)435 806 y Fo(=)60 b Fk(K)654 821 y Fi(3)694 806 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)864 821 y Fi(0)903 806 y Fo(\()p Fk(t)p Fo(\))p Fk(:)2658 b Fo(\(4.9\))0 1009 y(W)-8 b(e)42 b(solv)m(e)g(this)g (b)m(y)g(in)m(tegrating.)69 b(Because)44 b(of)d(\(4.2\),)j Fk(c)2147 1024 y Fi(1)2186 1009 y Fo(\(0\))59 b(=)h(0.)71 b(F)-8 b(rom)40 b(the)i(form)f(of)g Fk(c)3497 1024 y Fi(0)3536 1009 y Fo(\()p Fk(t)p Fo(\),)j(only)0 1129 y(\014nitely)36 b(man)m(y)g(of)f(the)i(en)m(tries)g(of)f Fk(c)1360 1144 y Fi(1)1399 1129 y Fo(\()p Fk(t)p Fo(\))g(are)h (non-zero,)g(and)f Fk(c)2375 1144 y Fi(1)p Fh(;j)2517 1129 y Fo(=)51 b(0)36 b(whenev)m(er)i Fj(j)p Fk(j)6 b Fj(j)34 b Fk(>)f(J)h Fo(+)25 b(3.)54 b(In)36 b Fk(d)0 1315 y Fo(space)e(dimensions,)d Fk(c)829 1330 y Fi(1)869 1315 y Fo(\()p Fk(t)p Fo(\))h(has)h(at)g(most)1545 1169 y Fd( )1652 1254 y Fk(J)e Fo(+)22 b(3)g(+)g Fk(d)1828 1374 y(d)2096 1169 y Fd(!)2195 1315 y Fo(non-zero)32 b(en)m(tries.)0 1610 y Fa(Order)37 b Fo(2)p Fa(.)98 b Fo(The)33 b(second)h(order)f(terms)g(require)244 1814 y Fk(i)d Fo(_)-40 b Fk(c)336 1829 y Fi(2)435 1814 y Fo(=)60 b Fk(K)654 1829 y Fi(4)694 1814 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)864 1829 y Fi(0)903 1814 y Fo(\()p Fk(t)p Fo(\))39 b(+)f Fk(K)1250 1829 y Fi(3)1290 1814 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)1460 1829 y Fi(1)1499 1814 y Fo(\()p Fk(t)p Fo(\))p Fk(:)2014 b Fo(\(4.10\))0 2017 y(Again,)36 b(w)m(e)h(can)g(solv)m(e)f(this)g(b)m(y)h(in)m(tegrating)d(with)i Fk(c)1988 2032 y Fi(2)2027 2017 y Fo(\(0\))50 b(=)g(0.)k(The)37 b(only)f(en)m(tries)h(of)e Fk(c)3352 2032 y Fi(2)3392 2017 y Fo(\()p Fk(t)p Fo(\))h(that)g(can)0 2200 y(b)s(e)f(non-zero)h (are)f Fk(c)741 2215 y Fi(2)p Fh(;j)832 2200 y Fo(\()p Fk(t)p Fo(\))h(with)f Fj(j)p Fk(j)6 b Fj(j)31 b(\024)i Fk(J)g Fo(+)24 b(6.)51 b(In)36 b Fk(d)f Fo(dimensions,)g(there)h(are)f (at)g(most)3283 2053 y Fd( )3390 2139 y Fk(J)c Fo(+)22 b(6)g(+)g Fk(d)3566 2259 y(d)3834 2053 y Fd(!)0 2374 y Fo(non-zero)33 b(en)m(tries.)0 2612 y Fa(Order)k Fk(n)p Fa(.)98 b Fo(In)33 b(general,)f(the)h Fk(n)1224 2576 y Fi(th)1328 2612 y Fo(order)g(terms)f(require)244 2882 y Fk(i)e Fo(_)-40 b Fk(c)336 2897 y Fh(n)443 2882 y Fo(=)579 2774 y Fh(n)p Fg(\000)p Fi(1)585 2799 y Fd(X)581 2983 y Fh(k)r Fi(=0)745 2882 y Fk(K)828 2897 y Fh(n)p Fi(+2)p Fg(\000)p Fh(k)1059 2882 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)1229 2897 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y(a)f(normalized)f(w)m(a)m(v) m(e)j(function.)75 b(The)45 b(expansion)e(deriv)m(ed)i(ab)s(o)m(v)m(e)f (do)s(es)g(not)f(generate)h(normalized)0 4126 y(w)m(a)m(v)m(epac)m(k)m (ets.)146 4364 y(T)-8 b(o)37 b(pro)m(v)m(e)h(that)f(this)f(expansion)i (is)e(asymptotic,)h(w)m(e)h(apply)f(Lemma)e(2.8)h(of)h([10)o(].)57 b(T)-8 b(o)36 b(c)m(hec)m(k)k(the)0 4484 y(h)m(yp)s(otheses)34 b(of)e(that)f(lemma,)f(w)m(e)j(do)f(the)g(expansion)g(ab)s(o)m(v)m(e)h (through)f(order)g(\()p Fk(l)23 b Fj(\000)e Fo(1\))31 b(to)h(obtain)e Fk(c)3722 4499 y Fi(0)3762 4484 y Fo(\()p Fk(t)p Fo(\),)0 4605 y Fk(c)42 4620 y Fi(1)81 4605 y Fo(\()p Fk(t)p Fo(\),)38 b(.)16 b(.)g(.)g(,)38 b Fk(c)493 4620 y Fh(l)q Fg(\000)p Fi(1)609 4605 y Fo(\()p Fk(t)p Fo(\).)54 b(W)-8 b(e)37 b(substitute)g(these)g(in)m(to)f(\(4.7\))f (with)h(the)h(sum)f(cut)g(o\013)g(after)g Fk(k)h Fo(=)d Fk(l)26 b Fj(\000)g Fo(1.)54 b(W)-8 b(e)0 4725 y(then)33 b(use)h(the)f(result)f(in)g(\(4.3\))g(and)h(compute)244 4995 y Fk(\030)287 5010 y Fh(l)312 4995 y Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)q Fo(\))60 b(=)g Fk(i)17 b Fo(\026)-50 b Fk(h)949 4927 y(@)5 b( )p 949 4971 124 4 v 965 5063 a(@)g(t)1083 4995 y Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))55 b(+)1587 4927 y(\026)-50 b Fk(h)1643 4884 y Fi(2)p 1587 4971 95 4 v 1610 5063 a Fo(2)1709 4995 y(\001)17 b Fk( )t Fo(\()p Fk(x;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))38 b Fj(\000)h Fk(V)22 b Fo(\()p Fk(x)p Fo(\))17 b Fk( )t Fo(\()p Fk(x;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))711 b(\(4.12\))1901 5506 y(11)p eop %%Page: 12 12 12 11 bop 0 -43 a Fo(Because)34 b(the)f Fk(c)583 -28 y Fh(k)626 -43 y Fo(\()p Fk(t)p Fo(\))f(solv)m(e)h(\(4.8\),)g(\(4.9\),) f(\(4.10\),)g(etc.,)h(there)g(are)g(man)m(y)f(cancellations.)42 b(W)-8 b(e)33 b(obtain)244 234 y Fk(\030)287 249 y Fh(l)312 234 y Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)q Fo(\))83 b(=)f Fk(e)908 193 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1181 126 y(l)q Fg(\000)p Fi(1)1177 151 y Fd(X)1173 336 y Fh(k)r Fi(=0)1335 234 y Fo(\026)-50 b Fk(h)1390 191 y Fh(k)r(=)p Fi(2)1520 234 y Fk(W)1626 183 y Fi(\()p Fh(l)q Fi(+1)p Fg(\000)p Fh(k)r Fi(\))1612 263 y Fh(a)p Fi(\()p Fh(t)p Fi(\))1890 234 y Fo(\()p Fk(x)p Fo(\))2119 151 y Fd(X)2054 359 y Fg(j)p Fh(j)t Fg(j\024)2192 349 y Fd(e)2182 359 y Fh(J)5 b Fi(\()p Fh(l)q Fi(\))2336 234 y Fk(c)2378 249 y Fh(k)r(;j)2472 234 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\036)2658 249 y Fh(j)2694 234 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fk(:)3651 492 y Fo(\(4.13\))0 707 y(Here,)282 675 y Fd(e)257 707 y Fk(J)10 b Fo(\()p Fk(l)r Fo(\))27 b(=)h Fk(J)j Fo(+)22 b(3)p Fk(l)i Fj(\000)f Fo(3,)32 b(and)h(for)f(eac)m(h)i Fk(q)t Fo(,)e Fk(W)1823 657 y Fi(\()p Fh(q)r Fi(\))1809 736 y Fh(a)p Fi(\()p Fh(t)p Fi(\))1931 707 y Fo(\()p Fk(x)p Fo(\))h(denotes)h(the)f(T)-8 b(a)m(ylor)32 b(series)h(error)244 985 y Fk(W)350 934 y Fi(\()p Fh(q)r Fi(\))336 1014 y Fh(a)p Fi(\()p Fh(t)p Fi(\))458 985 y Fo(\()p Fk(x)p Fo(\))83 b(=)g Fk(V)22 b Fo(\()p Fk(x)p Fo(\))39 b Fj(\000)1231 902 y Fd(X)1196 1090 y Fg(j)p Fh(m)p Fg(j\024)p Fh(q)1430 918 y Fk(D)1514 881 y Fh(m)1580 918 y Fk(V)21 b Fo(\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 1430 962 468 4 v 1607 1053 a Fk(m)p Fo(!)1923 985 y(\()p Fk(x)i Fj(\000)f Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2338 944 y Fh(m)672 1332 y Fo(=)912 1249 y Fd(X)831 1437 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(q)r Fi(+1)1155 1264 y Fk(D)1239 1228 y Fh(m)1305 1264 y Fk(V)g Fo(\()p Fk(\020)1465 1279 y Fh(m)1531 1264 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\))p 1155 1309 714 4 v 1455 1400 a Fk(m)p Fo(!)1895 1332 y(\()p Fk(x)22 b Fj(\000)h Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2310 1291 y Fh(m)2377 1332 y Fk(;)1247 b Fo(\(4.14\))0 1627 y(for)32 b(some)g Fk(\020)436 1642 y Fh(m)503 1627 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\))44 b(=)g Fk(a)p Fo(\()p Fk(t)p Fo(\))23 b(+)f Fk(\022)1332 1642 y Fh(m;x;a)p Fi(\()p Fh(t)p Fi(\))1596 1627 y Fo(\()p Fk(x)g Fj(\000)h Fk(a)p Fo(\()p Fk(t)p Fo(\)\),)33 b(with)f Fk(\022)2338 1642 y Fh(m;x;a)p Fi(\()p Fh(t)p Fi(\))2629 1627 y Fj(2)c Fo(\(0)p Fk(;)33 b Fo(1\).)146 1768 y(If)47 b Fk(V)69 b Fo(is)46 b Fk(C)573 1732 y Fh(l)q Fi(+2)736 1768 y Fo(on)h(some)g(neigh)m(b)s(orho)s(o)s(d)f(of)g Fj(f)p Fk(a)p Fo(\()p Fk(t)p Fo(\))69 b(:)g Fk(t)53 b Fj(2)f Fo([0)p Fk(;)33 b(T)14 b Fo(])j Fj(g)p Fo(,)50 b(then)e(eac)m(h)g Fk(W)3435 1717 y Fi(\()p Fh(q)r Fi(\))3421 1797 y Fh(a)p Fi(\()p Fh(t)p Fi(\))3543 1768 y Fo(\()p Fk(x)p Fo(\))f(that)0 1888 y(o)s(ccurs)34 b(in)e(\(4.13\))g(is)h(b)s(ounded)h(on)f(a)g(sligh) m(tly)e(smaller)g(neigh)m(b)s(orho)s(o)s(d)h(of)g Fj(f)p Fk(a)p Fo(\()p Fk(t)p Fo(\))46 b(:)f Fk(t)29 b Fj(2)f Fo([0)p Fk(;)34 b(T)14 b Fo(])j Fj(g)p Fo(.)44 b(Since)0 2009 y Fj(k)16 b Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\()p Fk(t)p Fo(\)\))482 1972 y Fh(m)548 2009 y Fk(\036)606 2024 y Fh(j)643 2009 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))g Fj(k)39 b Fo(has)j(order)f(\026)-50 b Fk(h)2355 1966 y Fg(j)p Fh(m)p Fg(j)p Fh(=)p Fi(2)2531 2009 y Fo(,)44 b(it)c(follo)m(ws)f(that)i Fj(k)17 b Fk(\030)5 b Fo(\()p Fj(\001)p Fk(;)17 b(t;)g Fo(\026)-50 b Fk(h)n Fo(\))17 b Fj(k)41 b Fo(has)0 2129 y(order)35 b(\026)-50 b Fk(h)312 2086 y Fh(l)q Fi(+2)428 2129 y Fo(.)49 b(Applying)34 b(Lemma)f(2.8)h(of)g([10)o(],)i(w)m(e)f(learn)f(that)g(the)h Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))34 b(solv)m(es)i(the)f(Sc)m(hr\177)-49 b(odinger)0 2249 y(equation)32 b(up)h(to)f(an)h(error)f(whose)i(norm)e(is)g(b)s(ounded)h (b)m(y)h Fk(C)2277 2264 y Fh(l)2319 2249 y Fo(\026)-50 b Fk(h)2374 2206 y Fh(l)q(=)p Fi(2)2471 2249 y Fo(,)32 b(when)i(\026)-50 b Fk(h)33 b Fo(is)f(su\016cien)m(tly)h(small.)146 2370 y(Note)39 b(that)f(the)i(argumen)m(t)e(ab)s(o)m(v)m(e)h(requires)h (the)f(insertion)e(of)i(cuto\013s)g(to)f(handle)g(the)h(Gaussian)0 2490 y(tails)30 b(or)i(some)f(other)h(assumption,)g(suc)m(h)h(as)f Fk(V)49 b Fj(2)28 b Fk(C)1985 2454 y Fh(l)q Fi(+2)2101 2490 y Fo(\()p Fk(I)-19 b(R)2238 2448 y Fh(d)2279 2490 y Fo(\))31 b(with)h Fj(j)17 b Fk(D)2699 2454 y Fh(m)2764 2490 y Fk(V)22 b Fo(\()p Fk(x)p Fo(\))17 b Fj(j)44 b(\024)g Fk(M)3278 2505 y Fh(m)3378 2490 y Fo(exp)r(\()p Fk(\034)11 b(x)3674 2454 y Fi(2)3714 2490 y Fo(\))31 b(for)0 2610 y Fj(j)p Fk(m)p Fj(j)c(\024)i Fk(l)24 b Fo(+)e(2.)0 2943 y Fl(5)161 b(Estimates)53 b(of)h(the)f(Expansion)g(Co)t(e\016cien)l(ts) 0 3221 y Fo(In)33 b(this)f(section)h(w)m(e)g(study)h(the)f(b)s(eha)m (vior)g(of)f Fk(c)1770 3236 y Fh(k)1812 3221 y Fo(\()p Fk(t)p Fo(\).)146 3460 y(The)40 b(\014rst)e(step)h(is)f(to)g(get)g(a)g (go)s(o)s(d)f(estimate)h(of)g(the)g(op)s(erator)g(norm)f(of)h(the)h(b)s (ounded)g(op)s(erator)0 3580 y(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\))304 3544 y Fh(m)387 3580 y Fk(P)450 3595 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)624 3580 y Fo(,)32 b(where)i Fk(P)1028 3595 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)1234 3580 y Fo(denotes)g(the)f(pro)5 b(jection)32 b(on)m(to)h(the)g(span)g(of)f (the)h Fk(\036)3171 3595 y Fh(j)3240 3580 y Fo(with)f Fj(j)p Fk(j)6 b Fj(j)27 b(\024)h Fk(n)p Fo(.)0 3818 y Fa(Lemma)37 b(5.1)49 b Fe(In)34 b Fk(d)h Fe(dimensions,)244 4021 y Fo(\()p Fk(x)22 b Fj(\000)h Fk(a)p Fo(\))548 3979 y Fh(m)631 4021 y Fk(P)694 4036 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)912 4021 y Fo(=)44 b Fk(P)1095 4036 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)p Fi(+)p Fg(j)p Fh(m)p Fg(j)1441 4021 y Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\))1746 3979 y Fh(m)1829 4021 y Fk(P)1892 4036 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)2065 4021 y Fk(;)1607 b Fo(\(5.1\))0 4223 y Fe(and)244 4486 y Fj(k)17 b Fo(\()p Fk(x)22 b Fj(\000)g Fk(a)p Fo(\))614 4444 y Fh(m)698 4486 y Fk(P)761 4501 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)951 4486 y Fj(k)62 b(\024)1203 4389 y Fd(\020)1269 4396 y Fj(p)p 1352 4396 104 4 v 90 x Fo(2\026)-50 b Fk(h)35 b(d)17 b Fj(k)g Fk(A)g Fj(k)1783 4389 y Fd(\021)1830 4412 y Fg(j)p Fh(m)p Fg(j)1988 4339 y Fd( )2080 4418 y Fo(\()p Fk(n)22 b Fo(+)g Fj(j)p Fk(m)p Fj(j)p Fo(\)!)p 2080 4462 423 4 v 2248 4554 a Fk(n)p Fo(!)2529 4339 y Fd(!)2594 4362 y Fi(1)p Fh(=)p Fi(2)2721 4486 y Fk(:)951 b Fo(\(5.2\))0 4893 y Fa(Pro)s(of:)97 b Fo(F)-8 b(orm)m(ula)31 b(\(2.22\))h(of)g([10)o(])h(states)h(that)369 5194 y(\()p Fk(x)462 5209 y Fh(i)513 5194 y Fj(\000)22 b Fk(a)663 5209 y Fh(i)692 5194 y Fo(\))60 b(=)926 5030 y Fd(s)p 1009 5030 76 4 v 1019 5127 a Fo(\026)-50 b Fk(h)p 1019 5171 56 4 v 1022 5263 a Fo(2)1133 5048 y Fd( )1215 5111 y(X)1257 5286 y Fh(p)1368 5194 y Fk(A)1441 5209 y Fh(i)12 b(p)1534 5194 y Fj(A)1614 5209 y Fh(p)1653 5194 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))2215 5153 y Fg(\003)2292 5194 y Fo(+)2407 5111 y Fd(X)2449 5286 y Fh(p)p 2560 5116 149 4 v 2560 5194 a Fk(A)2633 5209 y Fh(i)12 b(p)2725 5194 y Fj(A)2805 5209 y Fh(p)2844 5194 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))3423 5048 y Fd(!)3504 5194 y Fk(:)1901 5506 y Fo(12)p eop %%Page: 13 13 13 12 bop 0 -43 a Fo(Note)39 b(that)g(the)g(righ)m(t)f(hand)h(side)g (con)m(tains)g(2)p Fk(d)f Fo(terms.)62 b(Supp)s(ose)40 b Fk(v)j Fo(is)38 b(an)m(y)h(v)m(ector)h(in)e(the)i(range)e(of)0 77 y Fk(P)63 93 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)236 77 y Fo(.)44 b(Then)33 b(using)g(form)m(ulas)e(\(2.8\))h(and)h(\(2.9\))f (of)g([10)o(],)h(w)m(e)h(easily)e(deduce)i(that)244 252 y Fj(k)17 b(A)391 267 y Fh(p)430 252 y Fo(\()p Fk(A;)g(B)5 b(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a;)g(\021)t Fo(\))991 211 y Fg(\003)1047 252 y Fk(v)t Fj(k)82 b(\024)1391 169 y(p)p 1474 169 228 4 v 83 x Fk(n)22 b Fo(+)g(1)32 b Fj(k)17 b Fk(v)j Fj(k)p Fk(;)283 439 y Fj(k)d(A)430 454 y Fh(p)469 439 y Fo(\()p Fk(A;)g(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a;)g(\021)t Fo(\))g Fk(v)t Fj(k)81 b(\024)1391 363 y(p)p 1474 363 59 4 v 76 x Fk(n)32 b Fj(k)17 b Fk(v)j Fj(k)p Fk(;)0 603 y Fo(and)33 b(that)f(b)s(oth)g Fj(A)711 618 y Fh(p)751 603 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a;)g(\021)t Fo(\))1312 567 y Fg(\003)1368 603 y Fk(v)36 b Fo(and)d Fj(A)1721 618 y Fh(p)1760 603 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a;)g(\021)t Fo(\))g Fk(v)36 b Fo(b)s(elong)31 b(to)h(the)h(range)g(of)f Fk(P)3459 619 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)p Fi(+1)3723 603 y Fo(.)146 723 y(It)h(follo)m(ws)e (immediately)f(that)1071 897 y Fj(k)17 b Fo(\()p Fk(x)1231 912 y Fh(i)1281 897 y Fj(\000)23 b Fk(a)1432 912 y Fh(i)1460 897 y Fo(\))17 b Fk(P)1578 913 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)1767 897 y Fj(k)60 b(\024)1983 808 y(p)p 2066 808 104 4 v 89 x Fo(2\026)-50 b Fk(h)32 b(d)17 b Fj(k)g Fk(A)g Fj(k)2494 814 y(p)p 2575 814 228 4 v 2575 897 a Fk(n)22 b Fo(+)g(1)p Fk(;)0 1071 y Fo(and)33 b(that)f(\()p Fk(x)494 1086 y Fh(i)545 1071 y Fj(\000)22 b Fk(a)695 1086 y Fh(i)724 1071 y Fo(\))17 b Fk(P)842 1087 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)1059 1071 y Fo(=)44 b Fk(P)1242 1087 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)p Fi(+1)1522 1071 y Fo(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\))17 b Fk(P)1906 1087 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)2079 1071 y Fo(.)146 1192 y(The)34 b(lemma)c(follo)m (ws)h(from)h(these)i(t)m(w)m(o)f(results)g(b)m(y)g(a)g(simple)e (induction.)p 3100 1192 67 84 v 146 1487 a(The)46 b(conclusion)f(to)g (the)g(next)h(lemma)d(con)m(tains)i(the)h(binomial)41 b(co)s(e\016cien)m(ts)3211 1341 y Fd( )3318 1426 y Fk(k)25 b Fj(\000)e Fo(1)3320 1546 y Fk(p)g Fj(\000)f Fo(1)3584 1341 y Fd(!)3650 1487 y Fo(.)81 b(F)-8 b(or)0 1664 y Fk(k)31 b Fo(=)c(1)33 b(and)f Fk(p)c Fo(=)f(1)33 b(w)m(e)g(de\014ne)h (this)e(to)h(b)s(e)f(1.)0 1903 y Fa(Lemma)37 b(5.2)49 b Fe(Supp)-5 b(ose)34 b Fk(V)57 b Fe(satis\014es)34 b(the)h(hyp)-5 b(otheses)34 b(of)g(The)-5 b(or)g(em)34 b(3.1.)146 2023 y(Fix)h Fk(T)48 b Fe(and)34 b(cho)-5 b(ose)34 b(a)h(classic)-5 b(al)34 b(orbit)g Fk(a)p Fo(\()p Fk(t)p Fo(\))i Fe(for)e Fo(0)28 b Fj(\024)g Fk(t)g Fj(\024)g Fk(T)14 b Fe(.)44 b(The)35 b(hyp)-5 b(otheses)34 b(guar)-5 b(ante)g(e)34 b(that)244 2249 y Fk(D)325 2264 y Fi(1)427 2249 y Fo(=)62 b(max)798 2103 y Fd(\()900 2249 y Fo(1)p Fk(;)202 b Fo(sup)1027 2332 y Fi(0)p Fg(\024j)p Fh(n)p Fg(j)p Fh(;)34 b Fi(0)p Fg(\024)p Fh(t)p Fg(\024)p Fh(T)1527 2249 y Fk(\016)1574 2208 y Fg(j)p Fh(n)p Fg(j)1705 2182 y Fj(j)16 b Fo(\()p Fk(D)1871 2146 y Fh(n)1918 2182 y Fk(V)21 b Fo(\)\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))c Fj(j)p 1705 2226 612 4 v 1968 2317 a Fk(n)p Fo(!)2362 2103 y Fd(\))3699 2249 y Fo(\(5.3\))0 2478 y Fe(and)244 2679 y Fk(D)325 2694 y Fi(2)427 2679 y Fo(=)62 b(max)798 2533 y Fd(\()900 2679 y Fo(1)p Fk(;)106 b Fo(sup)1062 2758 y Fi(0)p Fg(\024)p Fh(t)p Fg(\024)p Fh(T)1335 2592 y Fj(p)p 1418 2592 49 4 v 87 x Fo(2)35 b Fk(d)17 b(\016)1617 2638 y Fg(\000)p Fi(1)1745 2679 y Fj(k)g Fk(A)p Fo(\()p Fk(t)p Fo(\))g Fj(k)2097 2533 y Fd(\))3699 2679 y Fo(\(5.4\))0 2900 y Fe(ar)-5 b(e)35 b(\014nite.)146 3077 y(We)i(de\014ne)e Fk(D)691 3092 y Fi(3)797 3077 y Fo(=)940 2931 y Fd( )1048 3016 y Fk(d)22 b Fo(+)g(2)1047 3137 y Fk(d)g Fj(\000)h Fo(1)1310 2931 y Fd(!)1376 3077 y Fe(,)36 b(which)g(is)g(the)g(numb)-5 b(er)36 b(of)g(multi-indic)-5 b(es)35 b Fk(m)i Fe(with)f Fj(j)p Fk(m)p Fj(j)30 b Fo(=)g(3)36 b Fe(in)g Fk(d)0 3254 y Fe(dimensions.)146 3375 y(Supp)-5 b(ose)36 b Fk(c)561 3390 y Fi(0)600 3375 y Fo(\(0\))g Fe(is)h(a)f(normalize)-5 b(d)35 b(ve)-5 b(ctor)36 b(with)g Fk(c)1994 3390 y Fi(0)p Fh(;j)2086 3375 y Fo(\(0\))g Fe(non-zer)-5 b(o)35 b(only)h(for)g Fj(j)p Fk(j)6 b Fj(j)30 b(\024)h Fk(J)9 b Fe(,)37 b(and)f(supp)-5 b(ose)0 3495 y Fk(c)42 3510 y Fh(k)r(;j)137 3495 y Fo(\(0\))44 b(=)g(0)34 b Fe(for)g(al)5 b(l)33 b Fk(j)40 b Fe(when)34 b Fk(k)d Fj(\025)d Fo(1)p Fe(.)44 b(L)-5 b(et)34 b Fk(c)1650 3510 y Fh(k)r(;j)1745 3495 y Fo(\()p Fk(t)p Fo(\))g Fe(b)-5 b(e)34 b(the)g(solution)g(to)g(\(4.8\),)g(\(4.9\),)f(.)15 b(.)g(.)g(,)34 b(\(4.11\),)f(with)0 3615 y(these)i(initial)f(c)-5 b(onditions.)44 b(Then)34 b(for)g Fk(t)45 b Fj(2)g Fo([0)p Fk(;)33 b(T)14 b Fo(])p Fe(,)34 b(we)h(have)244 3779 y Fk(c)286 3794 y Fi(0)p Fh(;j)377 3779 y Fo(\()p Fk(t)p Fo(\))63 b(=)f(0)199 b Fe(whenever)f Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)n(;)1861 b Fo(\(5.5\))244 3955 y Fj(k)17 b Fk(c)353 3970 y Fi(0)392 3955 y Fo(\()p Fk(t)p Fo(\))g Fj(k)62 b(\024)h Fk(D)853 3970 y Fi(1)892 3955 y Fk(;)2780 b Fo(\(5.6\))0 4131 y Fe(and)34 b(for)h Fk(k)c Fj(\025)d Fo(1)p Fe(,)244 4364 y Fk(c)286 4379 y Fh(k)329 4364 y Fo(\()p Fk(t)p Fo(\))62 b(=)684 4256 y Fh(k)643 4281 y Fd(X)641 4463 y Fh(p)p Fi(=1)818 4364 y Fk(c)860 4313 y Fi([)p Fh(p)p Fi(])860 4390 y Fh(k)939 4364 y Fo(\()p Fk(t)p Fo(\))p Fk(;)2622 b Fo(\(5.7\))0 4605 y Fe(wher)-5 b(e)244 4769 y Fk(c)286 4718 y Fi([)p Fh(p)p Fi(])286 4794 y Fh(k)r(;j)380 4769 y Fo(\()p Fk(t)p Fo(\))63 b(=)g(0)199 b Fe(whenever)f Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)g(J)k Fo(+)22 b Fk(k)k Fo(+)c(2)p Fk(p;)1454 b Fo(\(5.8\))0 4933 y Fe(and)244 5158 y Fj(k)17 b Fk(c)353 5107 y Fi([)p Fh(p)p Fi(])353 5183 y Fh(k)431 5158 y Fo(\()p Fk(t)p Fo(\))g Fj(k)62 b(\024)811 5011 y Fd( )918 5097 y Fk(k)26 b Fj(\000)c Fo(1)921 5217 y Fk(p)g Fj(\000)h Fo(1)1185 5011 y Fd(!)1302 5158 y Fk(D)1386 5110 y Fh(p)1383 5179 y Fi(1)1442 5158 y Fk(D)1526 5110 y Fh(k)r Fi(+2)p Fh(p)1523 5179 y Fi(2)1710 5158 y Fk(D)1794 5116 y Fh(k)1791 5182 y Fi(3)1888 5011 y Fd( )1981 5090 y Fo(\()p Fk(J)31 b Fo(+)22 b Fk(k)j Fo(+)d(2)p Fk(p)p Fo(\)!)p 1981 5134 559 4 v 2215 5226 a Fk(J)9 b Fo(!)2565 5011 y Fd(!)2631 5034 y Fi(1)p Fh(=)p Fi(2)2803 5090 y Fk(t)2838 5054 y Fh(p)p 2803 5134 77 4 v 2803 5226 a Fk(p)p Fo(!)2889 5158 y Fk(:)783 b Fo(\(5.9\))1901 5506 y(13)p eop %%Page: 14 14 14 13 bop 0 -43 a Fa(Pro)s(of:)97 b Fo(The)34 b(\014niteness)g(of)e Fk(D)1217 -28 y Fi(1)1289 -43 y Fo(and)h Fk(D)1560 -28 y Fi(2)1631 -43 y Fo(is)g(standard.)146 77 y(The)h(conclusions)e (\(5.5\))g(and)h(\(5.6\))f(are)h(trivial.)146 198 y(W)-8 b(e)23 b(assume)f Fk(t)28 b Fj(2)g Fo([0)p Fk(;)33 b(T)14 b Fo(],)24 b(and)e(let)f Fk(X)8 b Fo(\()p Fk(t)p Fo(\))22 b(denote)g(the)g(formal)e(v)m(ector)j(whose)g(en)m(tries)f Fk(X)3300 213 y Fh(i)3328 198 y Fo(\()p Fk(t)p Fo(\))g(denote)h(the)0 318 y(in\014nite)31 b(matrix)f(that)h(represen)m(ts)j(\026)-50 b Fk(h)1369 275 y Fg(\000)p Fi(1)p Fh(=)p Fi(2)1551 318 y Fo(\()p Fk(x)1644 333 y Fh(i)1693 318 y Fj(\000)20 b Fk(a)1841 333 y Fh(i)1870 318 y Fo(\()p Fk(t)p Fo(\)\))31 b(in)g(the)h(basis)g Fj(f)17 b Fk(\036)2694 333 y Fh(j)2730 318 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g Fj(\001)p Fo(\))g Fj(g)p Fo(.)146 438 y(F)-8 b(rom)31 b(\(4.9\))h(w)m(e)i(ha)m(v)m(e)758 713 y Fk(i)29 b Fo(_)-39 b Fk(c)850 728 y Fi(1)889 713 y Fo(\()p Fk(t)p Fo(\))60 b(=)g Fk(K)1279 728 y Fi(3)1319 713 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)1489 728 y Fi(0)1528 713 y Fo(\()p Fk(t)p Fo(\))60 b(=)1871 630 y Fd(X)1835 818 y Fg(j)p Fh(m)p Fg(j)p Fi(=3)2070 646 y Fo(\()p Fk(D)2192 610 y Fh(m)2258 646 y Fk(V)22 b Fo(\)\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 2070 690 543 4 v 2285 782 a Fk(m)p Fo(!)2640 713 y Fk(X)8 b Fo(\()p Fk(t)p Fo(\))2840 672 y Fh(m)2923 713 y Fk(c)2965 728 y Fi(0)3004 713 y Fo(\()p Fk(t)p Fo(\))p Fk(:)0 1052 y Fo(W)-8 b(e)33 b(in)m(tegrate)f(to)g(obtain)g Fk(c)1042 1067 y Fi(1)1081 1052 y Fo(\()p Fk(t)p Fo(\))45 b(=)f Fk(c)1399 1001 y Fi([1])1399 1074 y(1)1477 1052 y Fo(\()p Fk(t)p Fo(\).)g(Lemma)31 b(5.1,)h(\(5.5\),)h(and)f(\(5.6\))g (imply)f(t)m(w)m(o)i(conclusions:)244 1269 y Fk(c)286 1218 y Fi([1])286 1292 y(1)p Fh(;j)377 1269 y Fo(\()p Fk(t)p Fo(\))61 b(=)f(0)194 b(whenev)m(er)k Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)j Fo(+)22 b(3)p Fk(;)1640 b Fo(\(5.10\))0 1472 y(and)244 1737 y Fj(k)17 b Fk(c)353 1686 y Fi([1])353 1758 y(1)431 1737 y Fo(\()p Fk(t)p Fo(\))g Fj(k)60 b(\024)g Fk(D)887 1752 y Fi(1)943 1737 y Fk(D)1027 1696 y Fi(3)1024 1761 y(2)1083 1737 y Fk(D)1164 1752 y Fi(3)1253 1591 y Fd( )1345 1669 y Fo(\()p Fk(J)31 b Fo(+)22 b(3\)!)p 1345 1713 336 4 v 1467 1805 a Fk(J)9 b Fo(!)1707 1591 y Fd(!)1772 1614 y Fi(1)p Fh(=)p Fi(2)1931 1737 y Fk(t;)1658 b Fo(\(5.11\))0 2077 y(where)34 b(the)g(factor)730 1931 y Fd( )838 2016 y Fk(d)22 b Fo(+)g(2)838 2136 y Fk(d)f Fj(\000)i Fo(1)1100 1931 y Fd(!)1199 2077 y Fo(is)33 b(the)g(n)m(um)m(b)s(er)h(of)e(m)m(ulti-indices)e Fk(m)k Fo(with)e Fj(j)p Fk(m)p Fj(j)d Fo(=)f(3)33 b(in)f Fk(d)h Fo(dimensions.)0 2260 y(This)g(pro)m(v)m(es)h(\(5.7\),)e(\(5.8\),)g (and)h(\(5.9\))f(for)g Fk(k)f Fo(=)c(1.)146 2380 y(F)-8 b(or)32 b Fk(k)f Fo(=)c(2,)33 b(w)m(e)g(ha)m(v)m(e)h(from)e(\(4.10\),) 244 2584 y Fk(i)e Fo(_)-40 b Fk(c)336 2599 y Fi(2)375 2584 y Fo(\()p Fk(t)p Fo(\))83 b(=)g Fk(K)811 2599 y Fi(4)851 2584 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)1021 2599 y Fi(0)1060 2584 y Fo(\()p Fk(t)p Fo(\))54 b(+)h Fk(K)1439 2599 y Fi(3)1478 2584 y Fo(\()p Fk(t)p Fo(\))17 b Fk(c)1648 2599 y Fi(1)1688 2584 y Fo(\()p Fk(t)p Fo(\))569 2817 y(=)764 2734 y Fd(X)728 2922 y Fg(j)p Fh(m)p Fg(j)p Fi(=4)963 2750 y Fo(\()p Fk(D)1085 2713 y Fh(m)1151 2750 y Fk(V)22 b Fo(\)\()p Fk(a)p Fo(\()p Fk(t)p 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y Fk(c)286 2549 y Fi([2])286 2623 y(3)p Fh(;j)377 2600 y Fo(\()p Fk(t)p Fo(\))61 b(=)f(0)194 b(whenev)m(er)k Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)j Fo(+)22 b(7)p Fk(;)1640 b Fo(\(5.18\))244 2847 y Fk(c)286 2796 y Fi([3])286 2870 y(3)p Fh(;j)377 2847 y Fo(\()p Fk(t)p Fo(\))61 b(=)f(0)194 b(whenev)m(er)k Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)j Fo(+)22 b(9)p Fk(;)1640 b Fo(\(5.19\))244 3167 y Fj(k)17 b Fk(c)353 3116 y Fi([1])353 3189 y(3)431 3167 y Fo(\()p Fk(t)p Fo(\))g Fj(k)60 b(\024)806 3021 y Fd( )914 3106 y Fk(d)22 b Fo(+)g(4)914 3226 y Fk(d)f Fj(\000)i Fo(1)1176 3021 y Fd(!)1259 3167 y Fk(D)1340 3182 y Fi(1)1396 3167 y Fk(D)1480 3126 y Fi(5)1477 3192 y(2)1552 3021 y Fd( )1644 3100 y Fo(\()p Fk(J)32 b Fo(+)22 b(5\)!)p 1644 3144 336 4 v 1767 3235 a Fk(J)9 b Fo(!)2006 3021 y Fd(!)2072 3044 y Fi(1)p Fh(=)p Fi(2)2231 3167 y Fk(t;)1358 b Fo(\(5.20\))244 3551 y Fj(k)17 b Fk(c)353 3500 y Fi([2])353 3572 y(3)431 3551 y Fo(\()p Fk(t)p Fo(\))g Fj(k)60 b(\024)806 3404 y Fd( )914 3490 y Fk(d)22 b 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a(3)2823 5140 y Fk(d)g Fo(+)g(1)p 2823 5184 V 2908 5276 a(2)3069 5061 y Fd(#)3117 5084 y Fi(2)3217 5207 y Fj(\025)61 b Fo(1)p Fk(:)1901 5506 y Fo(15)p eop %%Page: 16 16 16 15 bop 146 -43 a Fo(No)m(w)45 b(supp)s(ose)h(inductiv)m(ely)e(that)h (the)g(lemma)d(is)i(true)h(for)f(all)e Fk(k)51 b Fj(\024)e Fk(q)t Fo(,)e(for)d(some)g Fk(q)52 b Fj(\025)c Fo(2.)79 b(By)0 77 y(in)m(tegrating)31 b(\(4.11\))h(with)g Fk(n)c Fo(=)f Fk(q)f Fo(+)c(1,)33 b(w)m(e)g(can)g(decomp)s(ose)244 348 y Fk(c)286 363 y Fh(q)r Fi(+1)414 348 y Fo(\()p Fk(t)p Fo(\))60 b(=)g Fk(c)763 297 y Fi([1])763 369 y Fh(q)r Fi(+1)892 348 y Fo(\()p Fk(t)p Fo(\))38 b(+)1206 234 y Fh(q)1163 265 y Fd(X)1156 446 y Fh(n)p Fi(=1)1322 240 y Fh(n)p Fi(+1)1329 265 y Fd(X)1326 446 y Fh(p)p Fi(=2)1488 348 y Fk(d)p Fo([)p Fk(q)t(;)17 b(n;)g(p)p Fo(]\()p Fk(t)p Fo(\))p Fk(;)1678 b Fo(\(5.23\))0 623 y(where)244 841 y Fk(c)286 790 y Fi([1])286 863 y Fh(q)r Fi(+1)414 841 y Fo(\()p Fk(t)p Fo(\))60 b(=)g Fj(\000)17 b Fk(i)882 724 y Fd(Z)965 750 y Fh(t)928 913 y Fi(0)1028 841 y Fk(K)1111 856 y Fh(q)r Fi(+3)1239 841 y Fo(\()p Fk(s)p Fo(\))g Fk(c)1420 856 y Fi(0)1459 841 y Fo(\()p Fk(s)p Fo(\))g Fk(ds;)1929 b Fo(\(5.24\))244 1154 y Fk(d)p Fo([)p Fk(q)t(;)17 b(n;)g(p)p Fo(]\()p Fk(t)p Fo(\))59 b(=)i Fj(\000)17 b Fk(i)1058 1037 y Fd(Z)1141 1063 y Fh(t)1104 1226 y Fi(0)1204 1154 y Fk(K)1287 1169 y Fh(q)r Fi(+3)p Fg(\000)p Fh(n)1513 1154 y Fo(\()p Fk(s)p Fo(\))g Fk(c)1694 1113 y Fi([)p Fh(p)p Fg(\000)p Fi(1])1694 1179 y Fh(n)1862 1154 y Fo(\()p Fk(s)p Fo(\))g Fk(ds;)1526 b Fo(\(5.25\))0 1410 y(for)32 b(1)27 b Fj(\024)i Fk(n)f Fj(\024)g Fk(q)36 b Fo(and)d(2)27 b Fj(\024)h Fk(p)g Fj(\024)g Fk(n)23 b Fo(+)f(1.)43 b(W)-8 b(e)33 b(in)m(terc)m(hange)g(the)g(sums)g(in)f (\(5.23\))g(to)g(obtain)244 1686 y Fk(c)286 1701 y Fh(q)r Fi(+1)414 1686 y Fo(\()p Fk(t)p Fo(\))60 b(=)g Fk(c)763 1635 y Fi([1])763 1708 y Fh(q)r Fi(+1)892 1686 y Fo(\()p Fk(t)p Fo(\))38 b(+)1157 1573 y Fh(q)r Fi(+1)1159 1603 y Fd(X)1156 1784 y Fh(p)p Fi(=2)1315 1686 y Fk(c)1357 1635 y Fi([)p Fh(p)p Fi(])1357 1708 y Fh(q)r Fi(+1)1485 1686 y Fo(\()p Fk(t)p Fo(\))p Fk(;)2028 b Fo(\(5.26\))0 1961 y(where)244 2198 y Fk(c)286 2147 y Fi([)p Fh(p)p Fi(])286 2220 y Fh(q)r Fi(+1)414 2198 y Fo(\()p Fk(t)p Fo(\))60 b(=)816 2085 y Fh(q)773 2115 y Fd(X)721 2297 y Fh(n)p Fi(=)p Fh(p)p Fg(\000)p Fi(1)978 2198 y Fk(d)p Fo([)p Fk(q)t(;)17 b(n;)g(p)p Fo(]\()p Fk(t)p Fo(\))p Fk(;)2188 b Fo(\(5.27\))0 2479 y(for)32 b(2)27 b Fj(\024)i Fk(p)e Fj(\024)h Fk(q)e Fo(+)c(1.)44 b(This)32 b(establishes)h(\(5.7\)) f(for)g Fk(k)f Fo(=)d Fk(q)e Fo(+)c(1.)146 2599 y(The)40 b(induction)e(h)m(yp)s(otheses,)43 b(form)m(ulas)37 b(\(4.6\),)j (\(5.24\),)g(\(5.25\),)f(\(5.27\),)h(and)f(Lemma)e(5.1)i(imply)0 2720 y(\(5.8\))32 b(for)g Fk(k)f Fo(=)c Fk(q)f Fo(+)c(1,)33 b(as)g(w)m(ell)e(as)i(the)g(t)m(w)m(o)g(inequalities)244 3004 y Fj(k)17 b Fk(c)353 2953 y Fi([1])353 3026 y Fh(q)r Fi(+1)480 3004 y Fo(\()p Fk(t)p Fo(\))g Fj(k)60 b(\024)856 2858 y Fd( )963 2943 y Fk(d)22 b Fo(+)g Fk(q)k 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5251 y Fi(1)781 5236 y Fo(\()p Fj(\001)p Fk(;)g(t)p Fo(\))g Fk(\030)1024 5251 y Fh(l)1048 5236 y Fo(\()p Fj(\001)p Fk(;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))17 b Fj(k)g Fk(dt)59 b Fj(\024)h Fk(\013)1758 5251 y Fi(1)1815 5236 y Fk(\014)1876 5194 y Fh(l)1870 5260 y Fi(1)1926 5236 y Fk(l)1957 5194 y Fh(l)q(=)p Fi(2)2070 5236 y Fo(\026)-50 b Fk(h)2125 5193 y Fh(l)q(=)p Fi(2)2222 5236 y Fk(:)1450 b Fo(\(6.2\))1901 5506 y(18)p eop %%Page: 19 19 19 18 bop 0 -43 a Fo(If)38 b(this)g(can)h(b)s(e)g(established,)h(w)m(e) f(c)m(ho)s(ose)h Fk(G)1699 -28 y Fi(1)1776 -43 y Fo(=)d Fk(\014)1950 -84 y Fg(\000)p Fi(2)1944 -21 y(1)2044 -43 y Fo(.)61 b(Then)40 b(0)d Fk(<)h(g)i(<)e(G)2871 -28 y Fi(1)2949 -43 y Fo(and)g Fk(l)57 b Fo(=)d([)-18 b([)17 b Fk(g)t(=)p Fo(\026)-50 b Fk(h)16 b Fo(])-18 b(])39 b(imply)0 77 y Fk(\014)61 41 y Fi(2)55 102 y(1)117 77 y Fk(g)60 b Fo(=)e Fk(e)403 41 y Fg(\000)p Fh(!)508 77 y Fo(,)43 b(with)d Fk(!)k(>)d Fo(0.)66 b(Since)41 b Fk(\013)1497 92 y Fi(1)1553 77 y Fo(\()p Fk(\014)1652 41 y Fi(2)1646 102 y(1)1707 77 y Fk(l)19 b Fo(\026)-50 b Fk(h)p Fo(\))1848 41 y Fh(l)q(=)p Fi(2)2002 77 y Fo(=)58 b Fk(\013)2198 92 y Fi(1)2254 77 y Fk(e)2299 41 y Fg(\000)p Fh(!)r(g)r(=)p Fi(\(2\026)-35 b Fh(h)q Fi(\))2606 77 y Fo(,)43 b(this)d(implies)d(the) k(lemma)d(with)0 198 y Fk(C)70 213 y Fi(1)137 198 y Fo(=)27 b Fk(\013)302 213 y Fi(1)374 198 y Fo(and)33 b Fk(\015)615 213 y Fi(1)682 198 y Fo(=)27 b Fk(!)t(g)t(=)p Fo(2.)146 318 y(T)-8 b(o)33 b(pro)m(v)m(e)h(\(6.2\),)e(w)m(e)i(note)e(\014rst)h (that)g(our)f(h)m(yp)s(otheses)j(imply)c(the)i(\014niteness)h(of)244 576 y Fk(D)325 591 y Fi(4)424 576 y Fo(=)402 b(sup)561 659 y Fg(j)p Fh(n)p Fg(j\025)p Fi(0)p Fh(;)31 b Fi(0)p Fg(\024)p Fh(t)p Fg(\024)p Fh(T)6 b(;)33 b Fg(j)p Fh(x)p Fg(\000)p Fh(a)p Fi(\()p Fh(t)p Fi(\))p Fg(j\024)p Fh(b)1440 576 y Fk(\016)1487 535 y Fg(j)p Fh(n)p Fg(j)1616 509 y Fj(j)16 b Fo(\()p Fk(D)1782 473 y Fh(n)1829 509 y Fk(V)21 b Fo(\)\()p Fk(x)p Fo(\))c Fj(j)p 1616 553 505 4 v 1826 645 a Fk(n)p Fo(!)2131 576 y Fk(:)1541 b Fo(\(6.3\))0 850 y(W)-8 b(e)33 b(use)h(this,)e(\(4.13\),)g(and)g(\(4.14\))g(to)g (see)i(that)487 1054 y Fj(k)17 b Fk(\037)615 1069 y Fi(1)654 1054 y Fo(\()p Fk(x;)g(t)p Fo(\))g Fk(\030)924 1069 y Fh(l)950 1054 y Fo(\()p Fk(x;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))17 b Fj(k)327 1341 y(\024)487 1191 y Fd(\015)487 1241 y(\015)487 1291 y(\015)487 1341 y(\015)487 1390 y(\015)575 1233 y Fh(l)q Fg(\000)p Fi(1)571 1258 y Fd(X)567 1442 y Fh(k)r Fi(=0)866 1258 y Fd(X)745 1446 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1140 1341 y Fo(\026)-50 b Fk(h)1195 1298 y Fh(k)r(=)p Fi(2)1341 1341 y Fk(\037)1402 1356 y Fi(1)1442 1341 y Fo(\()p Fk(x;)17 b(t)p Fo(\))1679 1273 y(\()p Fk(D)1801 1237 y Fh(m)1867 1273 y Fk(V)k Fo(\)\()p Fk(\020)2064 1288 y Fh(m)2130 1273 y Fo(\()p Fk(x;)c(a)p Fo(\()p Fk(t)p Fo(\)\))p 1679 1317 790 4 v 2017 1409 a Fk(m)p Fo(!)2494 1341 y(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2909 1299 y Fh(m)1853 1653 y Fj(\002)2109 1570 y Fd(X)2045 1778 y Fg(j)p Fh(j)t Fg(j\024)2182 1768 y Fd(e)2173 1778 y Fh(J)t Fi(\()p Fh(l)q Fi(\))2326 1653 y Fk(c)2368 1668 y Fh(k)r(;j)2463 1653 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\036)2649 1668 y Fh(j)2685 1653 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3806 1504 y Fd(\015)3806 1554 y(\015)3806 1603 y(\015)3806 1653 y(\015)3806 1703 y(\015)327 2049 y Fj(\024)496 1941 y Fh(l)q Fg(\000)p Fi(1)492 1965 y Fd(X)487 2150 y Fh(k)r Fi(=0)649 2049 y Fo(\026)-50 b Fk(h)704 2006 y Fh(k)r(=)p Fi(2)834 2049 y Fk(D)915 2064 y Fi(4)971 2049 y Fk(\016)1018 2007 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2+)p Fh(k)878 2341 y Fj(\002)1190 2258 y Fd(X)1069 2446 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1464 2191 y Fd(\015)1464 2241 y(\015)1464 2291 y(\015)1464 2341 y(\015)1464 2391 y(\015)1527 2341 y Fo(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\()p Fk(t)p Fo(\)\))1942 2300 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Fk(l)e Fj(\025)e Fo(1.)1901 5506 y(19)p eop %%Page: 20 20 20 19 bop 146 -43 a Fo(F)-8 b(or)35 b Fk(k)h Fj(\025)d Fo(1,)j(w)m(e)g(write)f(the)h(in)m(tegral)e(on)h(the)h(righ)m(t)f(hand) h(side)f(of)g(\(6.4\))g(as)h(a)f(sum)g(of)g Fk(k)k Fo(terms)c(b)m(y)0 77 y(emplo)m(ying)c(\(5.7\).)43 b(By)33 b(\(5.8\),)f(\(5.9\),)g(and)h (Lemma)e(5.1,)i(the)g Fk(p)2350 41 y Fi(th)2453 77 y Fo(in)m(tegrand)f(satis\014es)487 280 y Fk(\016)534 239 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2+)p Fh(k)953 197 y Fd(X)832 385 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1227 180 y Fd(\015)1227 230 y(\015)1227 280 y(\015)17 b Fk(X)8 b Fo(\()p Fk(t)p Fo(\))1490 239 y Fh(m)1573 280 y Fk(c)1615 229 y Fi([)p Fh(p)p Fi(])1615 305 y Fh(k)1694 280 y Fo(\()p Fk(t)p Fo(\))1822 180 y Fd(\015)1822 230 y(\015)1822 280 y(\015)327 616 y Fj(\024)83 b Fk(D)571 575 y Fh(l)q Fi(+2)p Fg(\000)p Fh(k)568 641 y Fi(2)830 470 y Fd( )937 555 y Fk(d)22 b Fo(+)g Fk(l)i Fo(+)e(1)g Fj(\000)h Fk(k)1100 675 y(d)f Fj(\000)g Fo(1)1525 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b(=)g(4)p Fk(n)22 b Fo(+)g(2)p Fk(q)k Fo(+)c Fk(d)p Fe(.)0 4818 y Fa(Pro)s(of:)97 b Fo(W)-8 b(e)37 b(mimic)d(the)j(pro)s(of)e(of)h (Lemma)f(3.1)h(of)g([11].)55 b(The)37 b(\014rst)g(step)g(is)f(to)g(sho) m(w)i(that)e Fk(g)t Fo(\()p Fk(r)s Fo(\))69 b(=)0 4938 y Fk(r)47 4902 y Fh(\014)130 4938 y Fk(L)196 4902 y Fh(\014)196 4963 y(n)243 4938 y Fo(\()p Fk(r)328 4902 y Fi(2)367 4938 y Fo(\))j Fk(e)522 4902 y Fg(\000)p Fh(r)611 4878 y Fc(2)645 4902 y Fh(=)p Fi(2)756 4938 y Fo(cannot)36 b(v)-5 b(anish)36 b(in)f(the)h(classically)e(forbidden)i(region)f Fk(r)2912 4902 y Fi(2)3001 4938 y Fk(>)50 b Fo(4)p Fk(n)24 b Fo(+)g(2)p Fk(q)k Fo(+)d Fk(d)p Fo(.)53 b(This)0 5058 y(function)32 b(v)-5 b(anishes)33 b(at)f(in\014nit)m(y)g(and)h(is)f(a)h (non-trivial)c(solution)i(to)h(an)h(equation)f(of)g(the)h(form)1374 5257 y Fj(\000)17 b Fk(g)1519 5216 y Fg(00)1561 5257 y Fo(\()p Fk(r)s Fo(\))54 b(+)h Fk(w)s Fo(\()p Fk(r)s Fo(\))17 b Fk(g)t Fo(\()p Fk(r)s Fo(\))58 b(=)i(0)p Fk(;)1901 5506 y Fo(21)p eop %%Page: 22 22 22 21 bop 0 -43 a Fo(where)39 b Fk(w)s Fo(\()p Fk(r)s Fo(\))34 b Fk(>)i Fo(0)h(for)g Fk(r)916 -79 y Fi(2)1008 -43 y Fk(>)52 b Fo(4)p Fk(n)25 b Fo(+)h(2)p Fk(q)j Fo(+)c Fk(d)p Fo(.)58 b(F)-8 b(rom)36 b(this)h(di\013eren)m(tial)e(equation)i (w)m(e)i(conclude)f(that)f Fk(g)0 77 y Fo(and)29 b Fk(g)237 41 y Fg(00)308 77 y Fo(ha)m(v)m(e)i(the)f(same)f(sign)f(in)h(this)g (region.)41 b(By)30 b(standard)g(uniqueness)h(theorems,)f Fk(g)j Fo(and)c Fk(g)3561 41 y Fg(0)3613 77 y Fo(cannot)0 198 y(b)s(oth)g(v)-5 b(anish)28 b(at)h(the)g(same)g(p)s(oin)m(t.)42 b(T)-8 b(o)29 b(obtain)f(a)g(con)m(tradition,)h(supp)s(ose)h Fk(g)i Fo(has)d(a)g(zero)g(at)g(some)g(p)s(oin)m(t)0 318 y Fk(r)44 333 y Fi(1)118 318 y Fo(with)34 b Fk(r)389 282 y Fi(2)386 343 y(1)476 318 y Fk(>)47 b Fo(4)p Fk(n)24 b Fo(+)f(2)p Fk(q)k Fo(+)c Fk(d)p Fo(.)49 b(Since)35 b Fk(g)j Fo(v)-5 b(anishes)35 b(at)f(in\014nit)m(y)-8 b(,)35 b(the)g(mean)f(v)-5 b(alue)34 b(theorem)g(guaran)m(tees)0 438 y(that)d Fk(g)261 402 y Fg(0)283 438 y 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Fk(x)p Fo(\))47 b(=)f(1,)34 b(the)h(lemma)c(is)j(true)g(for)g Fk(n)c Fo(=)g(0.)48 b(W)-8 b(e)0 1040 y(no)m(w)33 b(assume)g Fk(n)45 b Fj(\025)g Fo(1)32 b(and)h(that)f(the)h(lemma)d(has)j(b)s(een) h(established)e(for)g Fk(L)2848 993 y Fh(\014)2848 1062 y(n)p Fg(\000)p Fi(1)2986 1040 y Fo(\()p Fk(x)p Fo(\).)146 1161 y(Our)h(non-v)-5 b(anishing)31 b(result)h(and)h(\(6.12\))f(imply) 1376 1365 y Fk(L)1442 1329 y Fh(\014)1442 1390 y(n)1489 1365 y Fo(\()p Fk(x)p Fo(\))h Fk(n)p Fo(!)p 1365 1409 385 4 v 1365 1501 a(\()p Fj(\000)p Fo(1\))1567 1472 y Fh(n)1647 1501 y Fk(x)1702 1472 y Fh(n)1819 1433 y Fo(=)60 b(1)55 b Fj(\000)g Fk(B)2265 1448 y Fh(\014)s(;)11 b(n)2386 1433 y Fo(\()p Fk(x)p Fo(\))p Fk(;)0 1698 y Fo(where)34 b Fk(B)356 1713 y Fh(\014)s(;)11 b(n)477 1698 y Fo(\()p Fk(x)p Fo(\))45 b(=)f Fk(O)s Fo(\(1)p Fk(=x)p Fo(\))32 b(for)g(large)f Fk(x)p Fo(,)i(and)g Fk(B)1878 1713 y Fh(\014)s(;)11 b(n)1999 1698 y Fo(\()p Fk(x)p Fo(\))45 b Fk(>)f Fj(\000)p Fo(1,)33 b(for)f Fk(x)45 b(>)f Fo(4)p Fk(n)22 b Fo(+)g(2)p Fk(\014)27 b Fo(+)22 b(2.)146 1818 y(Using)33 b(recurrence)h(relation)d(8.971.3)g(of)h([6],)h(w)m(e)h(ha)m (v)m(e)254 2019 y Fk(d)p 254 2063 107 4 v 254 2154 a(dx)387 2086 y(B)461 2101 y Fh(\014)s(;)11 b(n)582 2086 y Fo(\()p Fk(x)p Fo(\))83 b(=)965 2019 y Fk(x)17 b(L)1103 1982 y Fh(\014)1103 2043 y(n)1151 1963 y Fg(0)1174 2019 y Fo(\()p Fk(x)p Fo(\))39 b Fj(\000)g Fk(n)17 b(L)1601 1982 y Fh(\014)1601 2043 y(n)1649 2019 y Fo(\()p Fk(x)p Fo(\))p 965 2063 815 4 v 1276 2154 a Fk(x)1331 2126 y Fh(n)p Fi(+1)1914 2019 y Fk(n)p Fo(!)p 1832 2063 250 4 v 1832 2154 a(\()p Fj(\000)p Fo(1\))2034 2126 y Fh(n)796 2373 y Fo(=)83 b Fj(\000)1075 2306 y Fo(\()p Fk(n)22 b Fo(+)g Fk(\014)6 b Fo(\))17 b Fk(L)1473 2259 y Fh(\014)1473 2328 y(n)p Fg(\000)p Fi(1)1610 2306 y Fo(\()p Fk(x)p Fo(\))p 1075 2350 667 4 v 1312 2442 a Fk(x)1367 2413 y Fh(n)p Fi(+1)1876 2306 y Fk(n)p Fo(!)p 1794 2350 250 4 v 1794 2442 a(\()p Fj(\000)p Fo(1\))1996 2413 y Fh(n)2053 2373 y Fk(:)0 2642 y Fo(By)31 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Fo(\))45 b Fe(by)f(\(4.12\).)72 b(F)-7 b(or)43 b(\014xe)-5 b(d)43 b Fk(T)59 b(>)45 b Fo(0)f Fe(and)g Fk(b)h(>)g Fo(0)p Fe(,)h(ther)-5 b(e)44 b(exists)g Fk(G)3009 3377 y Fi(2)3094 3362 y Fk(>)g Fo(0)p Fe(,)j(such)d(that)g(for)0 3483 y(e)-5 b(ach)44 b Fk(g)49 b Fj(2)d Fo(\(0)p Fk(;)33 b(G)661 3498 y Fi(2)700 3483 y Fo(\))p Fe(,)47 b(ther)-5 b(e)44 b(exist)g Fk(C)1373 3498 y Fi(2)1457 3483 y Fe(and)g Fk(\015)1707 3498 y Fi(2)1792 3483 y Fk(>)h Fo(0)p Fe(,)i(such)d(that)h(if)f Fk(l)j Fe(is)d(chosen)f(to)i(dep)-5 b(end)44 b(on)g Fo(\026)-50 b Fk(h)44 b Fe(as)0 3603 y Fk(l)r Fo(\(\026)-50 b Fk(h)p Fo(\))45 b(=)f([)-18 b([)17 b Fk(g)t(=)p Fo(\026)-50 b Fk(h)15 b Fo(])-18 b(])p Fe(,)35 b(and)g Fo(\026)-50 b Fk(h)35 b Fe(is)f(su\016ciently)h(smal)5 b(l,)34 b(then)244 3846 y Fo(\026)-50 b Fk(h)299 3804 y Fg(\000)p Fi(1)445 3729 y Fd(Z)528 3756 y Fh(T)491 3918 y Fi(0)616 3846 y Fj(k)17 b Fk(\037)744 3861 y Fi(2)783 3846 y Fo(\()p Fj(\001)p Fk(;)g(t)p Fo(\))g Fk(\030)1026 3862 y Fh(l)q 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y Fo(If)33 b Fk(z)f Fj(2)c Fk(C)339 4644 y Fh(\016)377 4629 y Fo(\()p Fk(\020)8 b Fo(\()p Fk(x;)17 b(a)p Fo(\)\),)32 b(then,)h(for)f(all)e Fk(j)k Fo(=)28 b(1)p Fk(;)17 b Fo(2)p Fk(;)g Fj(\001)g(\001)g(\001)c Fk(;)k(d)p Fo(,)1044 4838 y Fj(j)p Fk(z)1117 4853 y Fh(j)1154 4838 y Fj(j)44 b(\024)h Fk(\016)26 b Fo(+)c Fj(j)p Fk(\020)1586 4853 y Fh(j)1621 4838 y Fo(\()p Fk(x;)17 b(a)p Fo(\))p Fj(j)45 b(\024)f Fk(\016)27 b Fo(+)22 b Fj(j)p Fk(a)2288 4853 y Fh(j)2324 4838 y Fj(j)g Fo(+)g Fj(j)p Fk(x)2555 4853 y Fh(j)2614 4838 y Fj(\000)g Fk(a)2764 4853 y Fh(j)2801 4838 y Fj(j)p Fk(:)0 5048 y Fo(Using)j(this)h(and)g(applying)f(\()p Fk(b)8 b Fo(+)g Fk(c)p Fo(\))1277 5011 y Fi(2)1345 5048 y Fj(\024)28 b Fo(2\()p Fk(b)1578 5011 y Fi(2)1626 5048 y Fo(+)8 b Fk(c)1752 5011 y Fi(2)1792 5048 y Fo(\))26 b(sev)m(eral)g(times,)h(w)m(e)g(see)g(that)e Fk(z)33 b Fj(2)28 b Fk(C)3186 5063 y Fh(\016)3223 5048 y Fo(\()p Fk(\020)8 b Fo(\()p Fk(x;)17 b(a)p Fo(\)\))25 b(implies)911 5257 y Fj(j)17 b Fk(V)j Fo(\()p Fk(z)t Fo(\))d Fj(j)45 b(\024)f Fk(M)g Fo(exp)q(\(2)p Fk(\034)11 b Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\))2101 5216 y Fi(2)2140 5257 y Fo(\))33 b(exp)q(\(4)p Fk(\034)11 b Fo(\()p Fk(\016)2585 5216 y Fi(2)2625 5257 y Fk(d)22 b Fo(+)g Fk(a)2847 5216 y Fi(2)2886 5257 y Fo(\)\))p Fk(:)1901 5506 y Fo(22)p eop %%Page: 23 23 23 22 bop 0 -11 a Fo(Hence,)34 b(writing)695 -79 y(1)p 663 -34 113 4 v 663 57 a Fk(m)p Fo(!)802 -11 y Fk(D)886 -52 y Fh(m)952 -11 y Fk(V)22 b Fo(\()p Fk(\020)8 b Fo(\()p Fk(x;)17 b(a)p Fo(\)\))32 b(as)g(a)h Fk(d)p Fo({dimensional)c(Cauc)m(h) m(y)35 b(in)m(tegral,)c(w)m(e)i(obtain)f(the)h(b)s(ound)678 223 y(1)p 664 267 77 4 v 664 359 a Fk(p)p Fo(!)783 291 y Fj(j)p Fk(D)895 250 y Fh(p)934 291 y Fk(V)22 b Fo(\()p Fk(\020)8 b Fo(\()p Fk(x;)17 b(a)p Fo(\)\))p Fj(j)59 b(\024)h Fk(M)1738 223 y Fo(exp)q(\(4)p Fk(\034)11 b Fo(\()p Fk(\016)2112 187 y Fi(2)2151 223 y Fk(d)22 b Fo(+)g Fk(a)2373 187 y Fi(2)2413 223 y Fo(\)\))p 1738 267 751 4 v 2050 359 a Fk(\016)2097 330 y Fg(j)p Fh(p)p Fg(j)2548 291 y Fo(exp)q(\(2)p Fk(\034)11 b Fo(\()p Fk(x)22 b Fj(\000)h Fk(a)p Fo(\))3141 250 y Fi(2)3181 291 y Fo(\))p Fk(;)0 540 y Fo(where)39 b Fk(\020)8 b Fo(\()p Fk(x;)17 b(a)p Fo(\))38 b(is)f(an)m(y)i(v)-5 b(alue)38 b(b)s(et)m(w)m(een)i Fk(x)e Fo(and)h Fk(a)p Fo(.)60 b(Th)m(us,)41 b(for)d(0)f Fj(\024)g Fk(t)h Fj(\024)f Fk(T)14 b Fo(,)39 b(there)g(exists)g(a)f (constan)m(t)0 661 y Fk(M)94 676 y Fi(1)134 661 y Fo(,)32 b(suc)m(h)i(that)267 822 y(1)p 254 866 77 4 v 254 957 a Fk(p)p Fo(!)372 889 y Fj(j)p Fo(\()p Fk(D)522 848 y Fh(p)561 889 y Fk(V)22 b Fo(\)\()p Fk(\020)759 904 y Fh(p)798 889 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\)\))p Fj(j)60 b(\024)1409 822 y Fk(M)1503 837 y Fi(1)p 1409 866 134 4 v 1413 957 a Fk(\016)1460 929 y Fg(j)p Fh(p)p Fg(j)1602 889 y Fo(exp)q(\(2)p Fk(\034)11 b Fo(\()p Fk(x)22 b Fj(\000)h Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2306 848 y Fi(2)2346 889 y Fo(\))p Fk(:)1240 b Fo(\(6.16\))146 1125 y(W)-8 b(e)33 b(use)h(this,)e(\(4.13\),)g(\(4.14\),)g(and)h (\(5.33\))e(to)i(see)g(that)487 1311 y Fj(k)17 b Fk(\037)615 1326 y Fi(2)654 1311 y Fo(\()p Fk(x;)g(t)p Fo(\))g Fk(\030)924 1326 y Fh(l)950 1311 y Fo(\()p Fk(x;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))17 b Fj(k)327 1581 y(\024)487 1432 y Fd(\015)487 1481 y(\015)487 1531 y(\015)487 1581 y(\015)487 1631 y(\015)575 1473 y Fh(l)q Fg(\000)p Fi(1)571 1498 y Fd(X)567 1682 y Fh(k)r Fi(=0)729 1581 y Fo(\026)-50 b Fk(h)784 1538 y Fh(k)r(=)p Fi(2)930 1581 y Fk(\037)991 1596 y Fi(2)1031 1581 y Fo(\()p Fk(x;)17 b(t)p Fo(\))1382 1498 y Fd(X)1274 1686 y Fg(j)p Fh(p)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1652 1514 y Fo(\()p Fk(D)1774 1477 y Fh(p)1814 1514 y Fk(V)k Fo(\)\()p Fk(\020)2011 1529 y Fh(p)2050 1514 y Fo(\()p Fk(x;)c(a)p Fo(\()p Fk(t)p Fo(\)\))p 1652 1558 736 4 v 1982 1649 a Fk(p)p Fo(!)2414 1581 y(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2829 1540 y Fh(p)1788 1894 y Fj(\002)2070 1811 y Fd(X)1980 1999 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2313 1894 y Fk(c)2355 1909 y Fh(k)r(;j)2449 1894 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\036)2635 1909 y Fh(j)2671 1894 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3793 1744 y Fd(\015)3793 1794 y(\015)3793 1844 y(\015)3793 1894 y(\015)3793 1944 y(\015)327 2269 y Fj(\024)496 2161 y Fh(l)q Fg(\000)p Fi(1)492 2186 y Fd(X)487 2371 y Fh(k)r Fi(=0)649 2269 y Fo(\026)-50 b Fk(h)704 2226 y Fh(k)r(=)p Fi(2)834 2269 y Fk(M)928 2284 y Fi(1)985 2269 y Fk(\016)1032 2228 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2+)p Fh(k)1437 2186 y Fd(X)1329 2374 y Fg(j)p Fh(p)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1698 2120 y Fd(\015)1698 2170 y(\015)1698 2219 y(\015)1698 2269 y(\015)1698 2319 y(\015)1760 2269 y Fk(\037)1821 2284 y Fi(2)1861 2269 y Fo(\()p Fk(x;)17 b(t)p Fo(\))33 b(exp)q(\(2)p Fk(\034)11 b Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2809 2228 y Fi(2)2848 2269 y Fo(\))17 b(\()p Fk(x)22 b Fj(\000)h Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3318 2228 y Fh(p)1788 2582 y Fj(\002)2070 2499 y Fd(X)1980 2687 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2313 2582 y Fk(c)2355 2597 y Fh(k)r(;j)2449 2582 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\036)2635 2597 y Fh(j)2671 2582 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3793 2432 y Fd(\015)3793 2482 y(\015)3793 2532 y(\015)3793 2582 y(\015)3793 2632 y(\015)327 2957 y Fj(\024)83 b Fk(M)581 2972 y Fi(1)638 2957 y Fk(\016)685 2916 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)881 2849 y Fh(l)q Fg(\000)p Fi(1)877 2874 y Fd(X)872 3059 y Fh(k)r Fi(=0)1034 2957 y Fo(\026)-50 b Fk(h)1089 2914 y Fh(k)r(=)p Fi(2)1219 2957 y Fk(\016)1266 2916 y Fh(k)1415 2874 y Fd(X)1325 3062 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1659 2957 y Fj(j)p Fk(c)1729 2972 y Fh(k)r(;j)1823 2957 y Fo(\()p Fk(t)p Fo(\))p Fj(j)2102 2874 y Fd(X)1995 3062 y Fg(j)p Fh(p)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)2363 2808 y Fd(\015)2363 2858 y(\015)2363 2907 y(\015)2363 2957 y(\015)2363 3007 y(\015)2426 2957 y Fk(\037)2487 2972 y Fi(2)2526 2957 y Fo(\()p Fk(x;)17 b(t)p Fo(\))g Fk(e)2798 2916 y Fi(2)p Fh(\034)8 b Fi(\()p Fh(x)p Fg(\000)p Fh(a)p Fi(\()p Fh(t)p Fi(\)\))3137 2893 y Fc(2)3195 2957 y Fo(\()p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3611 2916 y Fh(p)2341 3270 y Fj(\002)130 b Fk(\036)2606 3285 y Fh(j)2643 3270 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3765 3121 y Fd(\015)3764 3170 y(\015)3764 3220 y(\015)3764 3270 y(\015)3764 3320 y(\015)3810 3270 y Fk(:)3651 3519 y Fo(\(6.17\))0 3705 y(Note)33 b(that)f(\(5.33\))g(has)h (b)s(een)g(used)h(to)e(replace)h Fj(j)p Fk(j)6 b Fj(j)27 b(\024)2065 3672 y Fd(e)2040 3705 y Fk(J)9 b Fo(\()p Fk(l)r Fo(\))33 b(with)f Fj(j)p Fk(j)6 b Fj(j)27 b(\024)h Fk(J)j Fo(+)22 b(3)p Fk(k)s Fo(.)146 3825 y(The)34 b(norm)e(in)f(the)i (\014nal)f(expression)i(of)e(\(6.17\))g(equals)244 3924 y Fd(\015)244 3973 y(\015)244 4023 y(\015)17 b Fk(\037)368 4039 y Fg(f)12 b Fh(z)s Fi(:)f Fg(j)p Fh(z)s Fg(j)p Fh(>b)f Fg(g)693 4023 y Fo(\()p Fk(x)p Fo(\))33 b(exp)q(\(2)p Fk(\034)11 b(x)1201 3982 y Fi(2)1241 4023 y Fo(\))17 b Fk(x)1351 3982 y Fh(p)1408 4023 y Fk(\036)1466 4038 y Fh(j)1502 4023 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b Fo(0)p Fk(;)g Fo(0)p Fk(;)g(x)p Fo(\))2397 3924 y Fd(\015)2396 3973 y(\015)2396 4023 y(\015)f Fk(;)1166 b Fo(\(6.18\))0 4221 y(where)34 b Fj(j)p Fk(p)p Fj(j)27 b Fo(=)g Fk(l)e Fo(+)d(2)g Fj(\000)g Fk(k)s Fo(.)146 4342 y(W)-8 b(e)39 b(assume)g(that)f(\026)-50 b Fk(h)38 b Fo(is)f(su\016cien)m(tly)i(small)d(that)i(4)p Fk(\034)11 b Fo(\026)-50 b Fk(h)q Fj(j)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)2446 4306 y Fi(2)2538 4342 y Fk(<)54 b Fo(2)p Fk(=)p Fo(3.)60 b(Then)39 b(the)f(square)i(of)e(the)0 4462 y(quan)m(tit)m(y)33 b(\(6.18\))f(equals)487 4648 y(2)536 4607 y Fg(\000j)p Fh(j)t Fg(j)683 4648 y Fo(\()p Fk(j)6 b Fo(!\))832 4607 y Fg(\000)p Fi(1)943 4648 y Fk(\031)1002 4607 y Fg(\000)p Fh(d=)p Fi(2)1185 4648 y Fj(j)17 b Fo(det)33 b Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)1610 4607 y Fg(\000)p Fi(1)1720 4648 y Fo(\026)-50 b Fk(h)1776 4605 y Fg(\000)p Fh(d=)p Fi(2)1268 4864 y Fj(\002)1459 4747 y Fd(Z)1506 4936 y Fg(j)p Fh(x)p Fg(j)p Fh(>b)1690 4864 y Fk(x)1745 4823 y Fi(2)p Fh(p)1821 4864 y Fk(e)1866 4823 y Fi(4)p Fh(\034)8 b(x)1980 4800 y Fc(2)2035 4864 y Fj(jH)2147 4879 y Fh(j)2184 4864 y Fo(\()p Fk(A)p Fo(;)33 b Fj(j)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)2595 4823 y Fg(\000)p Fi(1)2705 4864 y Fo(\026)-50 b Fk(h)2761 4822 y Fg(\000)p Fi(1)p Fh(=)p Fi(2)2925 4864 y Fk(x)p Fo(\))p Fj(j)3046 4823 y Fi(2)3103 4864 y Fk(e)3148 4823 y Fg(\000j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)3375 4800 y Fb(\000)p Fc(2)3458 4823 y Fh(x)3498 4800 y Fc(2)3532 4823 y Fh(=)p Fi(\026)-35 b Fh(h)3629 4864 y Fk(dx)327 5214 y Fj(\024)497 5146 y Fo(\(\026)-50 b Fk(h)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)891 5110 y Fi(2)930 5146 y Fo(\))968 5094 y Fg(j)p Fh(p)p Fg(j)p 497 5191 550 4 v 534 5282 a Fo(2)583 5253 y Fg(j)p Fh(j)t Fg(j)675 5282 y Fo(\()p Fk(j)6 b Fo(!\))17 b Fk(\031)900 5253 y Fh(d=)p Fi(2)1106 5097 y Fd(Z)1152 5285 y Fi(\026)-35 b Fh(h)1193 5260 y Fc(1)p Ff(=)p Fc(2)1290 5285 y Fg(j)11 b(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)h Fh(y)i Fg(j)p Fh(>b)1696 5214 y Fj(j)p Fk(y)t Fj(j)1804 5173 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1934 5214 y Fj(jH)2046 5229 y Fh(j)2082 5214 y Fo(\()p Fk(A)p Fo(;)33 b Fk(y)t Fo(\))p Fj(j)2371 5173 y Fi(2)2426 5214 y Fk(e)2471 5173 y Fi(\(4)p Fh(\034)8 b Fi(\026)-35 b Fh(h)q Fg(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)2786 5149 y Fc(2)2821 5173 y Fg(\000)p Fi(1\))p Fh(y)2975 5149 y Fc(2)3031 5214 y Fk(dy)t(:)1901 5506 y Fo(23)p eop %%Page: 24 24 24 23 bop 327 27 a Fj(\024)497 -40 y Fo(\(\026)-50 b Fk(h)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)891 -76 y Fi(2)930 -40 y Fo(\))968 -93 y Fg(j)p Fh(p)p Fg(j)p 497 4 550 4 v 534 95 a Fo(2)583 67 y Fg(j)p Fh(j)t Fg(j)675 95 y Fo(\()p Fk(j)6 b Fo(!\))17 b Fk(\031)900 67 y Fh(d=)p Fi(2)1106 -90 y Fd(Z)1152 99 y Fi(\026)-35 b Fh(h)1193 73 y Fc(1)p Ff(=)p Fc(2)1290 99 y Fg(j)11 b(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)h Fh(y)i Fg(j)p Fh(>b)1696 27 y Fj(j)p Fk(y)t Fj(j)1804 -14 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1934 27 y Fj(jH)2046 42 y Fh(j)2082 27 y Fo(\()p Fk(A)p Fo(;)33 b Fk(y)t Fo(\))p Fj(j)2371 -14 y Fi(2)2426 27 y Fk(e)2471 -14 y Fg(\000)p Fh(y)2563 -38 y Fc(2)2598 -14 y Fi(2)p Fh(=)p Fi(3)2725 27 y Fk(dy)327 382 y Fj(\024)83 b Fo(e)531 340 y Fg(\000)p Fh(b)616 317 y Fc(2)650 340 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)949 317 y Fc(2)987 340 y Fi(\026)-35 b Fh(h)o Fi(\))1069 314 y Fo(\(\026)-50 b Fk(h)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1463 278 y Fi(2)1502 314 y Fo(\))1540 261 y Fg(j)p Fh(p)p Fg(j)p 1069 358 V 1106 450 a Fo(2)1155 421 y Fg(j)p Fh(j)t Fg(j)1247 450 y Fo(\()p Fk(j)6 b Fo(!\))17 b Fk(\031)1472 421 y Fh(d=)p Fi(2)1678 264 y Fd(Z)1724 453 y Fi(\026)-35 b Fh(h)1765 427 y Fc(1)p Ff(=)p Fc(2)1862 453 y Fg(j)11 b(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)h Fh(y)i Fg(j)p Fh(>b)2268 382 y Fj(j)p Fk(y)t Fj(j)2376 340 y Fi(2)p Fg(j)p Fh(p)p Fg(j)2505 382 y Fj(jH)2617 397 y Fh(j)2654 382 y Fo(\()p Fk(A)p Fo(;)33 b Fk(y)t Fo(\))p Fj(j)2943 340 y Fi(2)2998 382 y Fk(e)3043 340 y Fg(\000)p Fh(y)3135 317 y Fc(2)3170 340 y Fi(1)p Fh(=)p Fi(2)3296 382 y Fk(dy)t(:)225 b Fo(\(6.19\))146 725 y(By)39 b(form)m(ula)e(\(3.7\))h(of)g([11],)i(\012)1314 740 y Fh(j)1351 725 y Fo(\()p Fk(y)t Fo(\))76 b(=)1707 573 y Fd(s)p 1790 573 421 4 v 1976 657 a Fo(1)p 1800 701 401 4 v 1800 793 a(2)1849 764 y Fg(j)p Fh(j)t Fg(j)1941 793 y Fk(j)6 b Fo(!)17 b Fk(\031)2090 764 y Fh(d=)p Fi(2)2250 725 y Fj(H)2334 740 y Fh(j)2370 725 y Fo(\()p Fk(A)p Fo(;)34 b Fk(y)t Fo(\))j Fk(e)2714 683 y Fg(\000)p Fh(y)2806 660 y Fc(2)2841 683 y Fh(=)p Fi(2)2955 725 y Fo(is)h(a)g(normalized)f (eigen-)0 906 y(function)j(of)g Fj(\000)p Fo(\001)667 921 y Fh(y)753 906 y Fo(+)k Fk(y)925 870 y Fi(2)1004 906 y Fo(with)c(eigen)m(v)-5 b(alue)39 b(2)p Fj(j)p Fk(j)6 b Fj(j)27 b Fo(+)g Fk(d)p Fo(.)67 b(Th)m(us,)44 b(in)39 b(spherical)h(co)s(ordinates,)i(it)d(can)i(b)s(e)0 1026 y(written)32 b(as)244 1230 y(\012)314 1245 y Fh(j)351 1230 y Fo(\()p Fk(y)t Fo(\))59 b(=)901 1146 y Fd(X)674 1335 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)1264 1230 y Fk(d)1315 1245 y Fh(j;q)r(;n;m)1562 1230 y Fk( )1625 1245 y Fh(q)r(;n;m)1808 1230 y Fo(\()p Fk(r)m(;)17 b(!)t Fo(\))p Fk(;)1590 b Fo(\(6.20\))0 1542 y(where)509 1459 y Fd(X)282 1647 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)872 1542 y Fj(j)p Fk(d)951 1557 y Fh(j;q)r(;n;m)1181 1542 y Fj(j)1209 1501 y Fi(2)1308 1542 y Fo(=)60 b(1.)146 1773 y(W)-8 b(e)30 b(ultimately)d(c)m(ho)s(ose) k Fk(l)46 b Fo(=)e([)-18 b([)17 b Fk(g)t(=)p Fo(\026)-50 b Fk(h)16 b Fo(])-18 b(],)31 b(with)e(0)f Fk(<)f(g)k(<)d(G)2255 1788 y Fi(2)2294 1773 y Fo(.)43 b(Since)2640 1741 y Fd(e)2615 1773 y Fk(J)10 b Fo(\()p Fk(l)r Fo(\))44 b(=)g Fk(J)25 b Fo(+)16 b(3)p Fk(l)i Fj(\000)e Fo(3,)31 b(there)f(exists)0 1893 y Fk(C)70 1908 y Fi(3)109 1893 y Fo(,)39 b(suc)m(h)g(that)f(\026) -50 b Fk(h)36 b(<)h Fo(1)g(implies)1268 1861 y Fd(e)1243 1893 y Fk(J)9 b Fo(\()p Fk(l)r Fo(\))37 b Fj(\024)g Fk(C)1634 1908 y Fi(3)1673 1893 y Fk(=)p Fo(\026)-50 b Fk(h)p Fo(.)59 b(By)38 b(c)m(ho)s(osing)f Fk(G)2499 1908 y Fi(2)2576 1893 y Fo(su\016cien)m(tly)h(small,)f(w)m(e)i(also)e(ha)m(v)m(e)25 1981 y Fd(e)0 2014 y Fk(J)9 b Fo(\()p Fk(l)r Fo(\))48 b Fk(<)g Fo(\()p Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)664 1977 y Fg(\000)p Fi(2)758 2014 y Fk(b)799 1977 y Fi(2)862 2014 y Fj(\000)24 b Fo(1\))p Fk(=)p Fo(\(2\026)-50 b Fk(h)p Fo(\))34 b(for)g(0)e Fj(\024)f Fk(t)h Fj(\024)f Fk(T)49 b Fo(and)35 b(small)d(\026)-50 b Fk(h)p Fo(.)50 b(Th)m(us,)37 b(the)e(relev)-5 b(an)m(t)34 b(v)-5 b(alues)35 b(of)f Fk(j)41 b Fo(in)0 2154 y(\(6.19\))32 b(satisfy)586 2054 y Fd(q)p 669 2054 321 4 v 100 x Fo(2)p Fj(j)p Fk(j)6 b Fj(j)22 b Fo(+)g Fk(d)44 b(<)g Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1439 2117 y Fg(\000)p Fi(1)1533 2154 y Fk(b)p Fo(\026)-50 b Fk(h)1630 2111 y Fg(\000)p Fi(1)p Fh(=)p Fi(2)1794 2154 y Fo(.)146 2354 y(Lemma)29 b(6.2)h(sho)m(ws)i (that)1135 2230 y Fd(\014)1135 2280 y(\014)1135 2329 y(\014)1135 2379 y(\014)1179 2354 y Fk(r)1226 2313 y Fh(q)1280 2354 y Fk(L)1346 2294 y Fh(q)r Fi(+)1445 2267 y Ff(d)p 1445 2279 33 4 v 1446 2320 a Fc(2)1487 2294 y Fg(\000)p Fi(1)1346 2362 y Fh(n)1582 2354 y Fo(\()p Fk(r)1667 2313 y Fi(2)1706 2354 y Fo(\))1761 2230 y Fd(\014)1760 2280 y(\014)1760 2329 y(\014)1760 2379 y(\014)1846 2354 y Fj(\024)1991 2287 y Fk(r)2038 2251 y Fi(2)p Fh(n)p Fi(+)p Fh(q)p 1991 2331 218 4 v 2058 2423 a Fk(n)p Fo(!)2249 2354 y(whenev)m(er)h Fk(r)2722 2318 y Fi(2)2820 2354 y Fk(>)57 b Fo(4)p Fk(n)18 b Fo(+)g(2)p Fk(q)j Fo(+)d Fk(d)27 b Fo(=)h(2)p Fj(j)p Fk(j)6 b Fj(j)17 b Fo(+)g Fk(d)p Fo(.)0 2512 y(So,)33 b(w)m(e)g(see)h(that)e(\(6.19\))g(is)g(b)s (ounded)h(b)m(y)169 2836 y(\026)-50 b Fk(h)224 2793 y Fg(j)p Fh(p)p Fg(j)320 2836 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)604 2795 y Fi(2)p Fg(j)p Fh(p)p Fg(j)751 2719 y Fd(Z)797 2907 y Fh(S)844 2888 y Ff(d)p Fb(\000)p Fc(1)979 2719 y Fd(Z)1025 2907 y Fh(r)r(>)1231 2880 y Ff(b)p 1124 2892 242 4 v 1124 2951 a Fb(k)p Ff(A)p Fb(k)p Fc(\026)-32 b Ff(h)1268 2933 y Fc(1)p Ff(=)p Fc(2)1428 2836 y Fk(r)1475 2795 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1622 2661 y Fd(\014)1622 2711 y(\014)1622 2761 y(\014)1622 2811 y(\014)1622 2861 y(\014)1622 2910 y(\014)1893 2753 y(X)1667 2941 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2224 2836 y Fk(d)2275 2851 y Fh(j;q)r(;n;m)2505 2836 y Fk( )2568 2851 y Fh(q)r(;n;m)2751 2836 y Fo(\()p Fk(r)m(;)17 b(!)t Fo(\))2994 2661 y Fd(\014)2992 2711 y(\014)2992 2761 y(\014)2992 2811 y(\014)2992 2861 y(\014)2992 2910 y(\014)3020 2688 y Fi(2)3093 2836 y Fk(e)3138 2795 y Fh(r)3172 2771 y Fc(2)3206 2795 y Fh(=)p Fi(2)3298 2836 y Fk(r)3345 2795 y Fh(d)p Fg(\000)p Fi(1)3475 2836 y Fk(dr)i(d!)t(:)0 3143 y Fo(W)-8 b(e)35 b(in)m(terc)m(hange)h(the)g(sum)f(and)g(in)m (tegrals)f(and)h(apply)f(the)i(Sc)m(h)m(w)m(artz)h(inequalit)m(y)d(to)h (the)g(sum.)51 b(This)0 3263 y(sho)m(ws)34 b(that)f(\(6.19\))e(is)h(b)s (ounded)i(b)m(y)486 3516 y(e)529 3475 y Fg(\000)p Fh(b)614 3452 y Fc(2)649 3475 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)948 3452 y Fc(2)985 3475 y Fi(\026)-35 b Fh(h)p Fi(\))1074 3516 y Fo(\026)-50 b Fk(h)1129 3473 y Fg(j)p Fh(p)p Fg(j)1225 3516 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1509 3475 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1883 3433 y Fd(X)1656 3621 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2475 3449 y Fo(2)p Fk(n)p Fo(!)p 2273 3493 539 4 v 2273 3588 a(\000\()p Fk(q)26 b Fo(+)2549 3549 y Fh(d)p 2549 3565 37 4 v 2550 3623 a Fi(2)2617 3588 y Fo(+)c Fk(n)p Fo(\))974 3854 y Fj(\002)1165 3736 y Fd(Z)1248 3763 y Fg(1)1328 3898 y Ff(b)p 1221 3910 242 4 v 1221 3969 a Fb(k)p Ff(A)p Fb(k)p Fc(\026)-32 b Ff(h)1366 3951 y Fc(1)p Ff(=)p Fc(2)1493 3854 y Fk(r)1540 3812 y Fh(d)p Fg(\000)p Fi(1+2)p Fg(j)p Fh(p)p Fg(j)p Fi(+2)p Fh(q)1993 3729 y Fd(\014)1993 3779 y(\014)1993 3829 y(\014)1993 3879 y(\014)2037 3854 y Fk(L)2103 3793 y Fh(q)r Fi(+)2202 3766 y Ff(d)p 2202 3778 33 4 v 2203 3819 a Fc(2)2244 3793 y Fg(\000)p Fi(1)2103 3861 y Fh(n)2338 3854 y Fo(\()p Fk(r)2423 3812 y Fi(2)2462 3854 y Fo(\))2517 3729 y Fd(\014)2517 3779 y(\014)2517 3829 y(\014)2517 3879 y(\014)2545 3755 y Fi(2)2617 3854 y Fk(e)2662 3812 y Fg(\000)p Fh(r)2751 3789 y Fc(2)2786 3812 y Fh(=)p Fi(2)2877 3854 y Fk(dr)3008 3736 y Fd(Z)3054 3925 y Fh(S)3101 3906 y Ff(d)p Fb(\000)p Fc(1)3252 3854 y Fj(j)17 b Fk(Y)3354 3869 y Fh(q)r(;m)3473 3854 y Fo(\()p Fk(!)t Fo(\))g Fj(j)3658 3805 y Fi(2)3730 3854 y Fk(d!)327 4209 y Fo(=)83 b(e)529 4168 y Fg(\000)p Fh(b)614 4144 y Fc(2)649 4168 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)948 4144 y Fc(2)985 4168 y Fi(\026)-35 b Fh(h)p Fi(\))1074 4209 y Fo(\026)-50 b Fk(h)1129 4166 y Fg(j)p Fh(p)p Fg(j)1225 4209 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1509 4168 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1883 4126 y Fd(X)1656 4314 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2475 4142 y Fo(2)p Fk(n)p Fo(!)p 2273 4186 539 4 v 2273 4281 a(\000\()p Fk(q)26 b Fo(+)2549 4242 y Fh(d)p 2549 4258 37 4 v 2550 4315 a Fi(2)2617 4281 y Fo(+)c Fk(n)p Fo(\))1364 4571 y Fj(\002)1555 4454 y Fd(Z)1638 4481 y Fg(1)1718 4616 y Ff(b)p 1611 4628 242 4 v 1611 4687 a Fb(k)p Ff(A)p Fb(k)p Fc(\026)-32 b Ff(h)1756 4669 y Fc(1)p Ff(=)p Fc(2)1883 4571 y Fk(r)1930 4530 y Fh(d)p Fg(\000)p Fi(1+2)p Fg(j)p Fh(p)p Fg(j)p Fi(+2)p Fh(q)2366 4447 y Fd(\014)2366 4497 y(\014)2366 4547 y(\014)2366 4596 y(\014)2394 4571 y Fk(L)2460 4511 y Fh(q)r Fi(+)2559 4484 y Ff(d)p 2559 4496 33 4 v 2560 4537 a Fc(2)2601 4511 y Fg(\000)p Fi(1)2460 4579 y Fh(n)2695 4571 y Fo(\()p Fk(r)2780 4530 y Fi(2)2819 4571 y Fo(\))2857 4447 y Fd(\014)2857 4497 y(\014)2857 4547 y(\014)2857 4596 y(\014)2885 4473 y Fi(2)2941 4571 y Fk(e)2986 4530 y Fg(\000)p Fh(r)3075 4507 y Fc(2)3110 4530 y Fh(=)p Fi(2)3201 4571 y Fk(dr)m(:)3651 4848 y Fo(\(6.21\))0 5052 y(By)36 b(reducing)f(the)h(v)-5 b(alue)35 b(of)f Fk(G)1170 5067 y Fi(2)1245 5052 y Fo(if)g(necessary)-8 b(,)38 b(w)m(e)f(can)e(ensure)i(that)e(the)h(h)m(yp)s(otheses)i(of)c (Lemma)g(6.2)0 5172 y(are)e(satis\014ed)g(in)f(the)h(in)m(tegration)f (region)f(in)h(the)i(righ)m(t)e(hand)h(side)g(of)f(\(6.21\).)43 b(So,)32 b(Lemma)e(6.2)h(sho)m(ws)1901 5506 y(24)p eop %%Page: 25 25 25 24 bop 0 -43 a Fo(that)32 b(the)h(in)m(tegral)e(satis\014es)487 91 y Fd(Z)570 118 y Fg(1)651 253 y Ff(b)p 543 265 242 4 v 543 324 a Fb(k)p Ff(A)p Fb(k)p Fc(\026)-32 b Ff(h)688 306 y Fc(1)p Ff(=)p Fc(2)832 209 y Fk(r)879 167 y Fh(d)p Fg(\000)p Fi(1+2)p Fg(j)p Fh(p)p Fg(j)p Fi(+2)p Fh(q)1332 84 y Fd(\014)1332 134 y(\014)1332 184 y(\014)1332 233 y(\014)1376 209 y Fk(L)1442 148 y Fh(q)r Fi(+)1541 121 y Ff(d)p 1541 133 33 4 v 1542 174 a Fc(2)1583 148 y Fg(\000)p Fi(1)1442 216 y Fh(n)1677 209 y Fo(\()p Fk(r)1762 167 y Fi(2)1801 209 y Fo(\))1856 84 y Fd(\014)1856 134 y(\014)1856 184 y(\014)1856 233 y(\014)1884 110 y Fi(2)1956 209 y Fk(e)2001 167 y Fg(\000)p Fh(r)2090 144 y Fc(2)2125 167 y Fh(=)p Fi(2)2216 209 y Fk(dr)327 542 y Fj(\024)573 475 y Fo(1)p 497 519 201 4 v 497 611 a(\()p Fk(n)p Fo(!\))658 582 y Fi(2)741 425 y Fd(Z)824 452 y Fg(1)904 587 y Ff(b)p 797 599 242 4 v 797 657 a Fb(k)p Ff(A)p Fb(k)p Fc(\026)g Ff(h)942 640 y Fc(1)p Ff(=)p Fc(2)1086 542 y Fk(r)1133 501 y Fi(4)p Fh(n)p Fi(+)p Fh(d)p Fg(\000)p Fi(1+2)p Fg(j)p Fh(p)p Fg(j)p Fi(+2)p Fh(q)1702 542 y Fk(e)1747 501 y Fg(\000)p Fh(r)1836 478 y Fc(2)1870 501 y Fh(=)p Fi(2)1962 542 y Fk(dr)327 910 y Fj(\024)497 843 y Fo(2)546 806 y Fi(2)p Fh(n)p Fi(+)689 779 y Ff(d)p 689 791 33 4 v 690 832 a Fc(2)731 806 y Fi(+)p Fh(q)r Fi(+)p Fg(j)p Fh(p)p Fg(j)p 497 887 457 4 v 625 979 a Fo(\()p Fk(n)p Fo(!\))786 950 y Fi(2)997 793 y Fd(Z)1080 820 y Fg(1)1043 982 y Fi(0)1188 910 y Fk(z)1237 869 y Fi(4)p Fh(n)p Fi(+)p Fh(d)p Fg(\000)p Fi(1+2)p Fh(q)r Fi(+2)p Fg(j)p Fh(p)p Fg(j)1807 910 y Fk(e)1852 869 y Fg(\000)p Fh(z)1943 846 y Fc(2)1998 910 y Fk(dz)328 1244 y Fo(=)497 1177 y(2)546 1140 y Fi(2)p Fh(n)p Fi(+)689 1113 y Ff(d)p 689 1125 33 4 v 690 1166 a Fc(2)731 1140 y Fi(+)p Fh(q)r Fi(+)p Fg(j)p Fh(p)p Fg(j\000)p Fi(1)p 497 1221 547 4 v 670 1313 a Fo(\()p Fk(n)p Fo(!\))831 1284 y Fi(2)1070 1244 y Fo(\000\(2)p Fk(n)22 b Fo(+)1406 1177 y Fk(d)p 1406 1221 51 4 v 1407 1313 a Fo(2)1489 1244 y(+)g Fk(q)k Fo(+)c Fj(j)p Fk(p)p Fj(j)p Fo(\))p Fk(:)0 1504 y Fo(So,)33 b(\(6.21\))e(is)h(b)s(ounded)i(b)m(y)378 1802 y(e)422 1761 y Fg(\000)p Fh(b)507 1738 y Fc(2)541 1761 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)840 1738 y Fc(2)878 1761 y Fi(\026)-35 b Fh(h)o Fi(\))950 1802 y Fo(\026)-50 b Fk(h)1005 1760 y Fg(j)p Fh(p)p Fg(j)1100 1802 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1384 1761 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1759 1719 y Fd(X)1532 1908 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2164 1735 y Fo(2)2213 1698 y Fi(2)p Fh(n)p Fi(+)2356 1671 y Ff(d)p 2356 1683 33 4 v 2357 1724 a Fc(2)2398 1698 y Fi(+)p Fh(q)r Fi(+)p Fg(j)p Fh(p)p Fg(j)p 2164 1779 457 4 v 2350 1871 a Fk(n)p Fo(!)2673 1732 y(\000\(2)p Fk(n)22 b Fo(+)3009 1692 y Fh(d)p 3009 1708 37 4 v 3010 1766 a Fi(2)3078 1732 y Fo(+)g Fk(q)j Fo(+)d Fj(j)p Fk(p)p Fj(j)p Fo(\))p 2673 1779 812 4 v 2810 1874 a(\000\()p Fk(q)k Fo(+)3086 1835 y Fh(d)p 3086 1851 37 4 v 3087 1909 a Fi(2)3154 1874 y Fo(+)c Fk(n)p Fo(\))3495 1802 y Fk(:)0 2115 y Fo(W)-8 b(e)33 b(use)h(\(6.10\))d(to)i(estimate)e(the)i(sum)g(o)m(v)m(er)g Fk(m)28 b Fj(\024)h Fk(m)2015 2130 y Fh(q)2085 2115 y Fo(and)k(b)s(ound)g(this)f(b)m(y)486 2400 y Fk(C)556 2415 y Fh(d)613 2400 y Fo(e)656 2359 y Fg(\000)p Fh(b)741 2336 y Fc(2)776 2359 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1075 2336 y Fc(2)1112 2359 y Fi(\026)-35 b Fh(h)p Fi(\))1201 2400 y Fo(\026)-50 b Fk(h)1256 2357 y Fg(j)p Fh(p)p Fg(j)1352 2400 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1636 2359 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1969 2317 y Fd(X)1783 2505 y Fg(f)p Fh(n;q)r Fi(:)11 b Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(j\000)p Fi(2)p Fh(n)p Fg(g)2308 2400 y Fk(e)2353 2359 y Fh(\013)2398 2371 y Ff(d)2434 2359 y Fh(q)2515 2333 y Fo(2)2564 2296 y Fg(j)p Fh(j)t Fg(j)p Fi(+)2702 2269 y Ff(d)p 2700 2281 33 4 v 2701 2322 a Fc(2)2742 2296 y Fi(+)p Fg(j)p Fh(p)p Fg(j)p 2515 2377 361 4 v 2653 2469 a Fk(n)p Fo(!)2928 2329 y(\000\()p Fj(j)p Fk(j)6 b Fj(j)21 b Fo(+)3258 2290 y Fh(d)p 3258 2306 37 4 v 3259 2364 a Fi(2)3327 2329 y Fo(+)h Fj(j)p Fk(p)p Fj(j)p Fo(\))p 2928 2377 639 4 v 2978 2472 a(\000\()p Fk(q)k Fo(+)3254 2433 y Fh(d)p 3254 2449 37 4 v 3255 2507 a Fi(2)3323 2472 y Fo(+)c Fk(n)p Fo(\))327 2733 y(=)83 b Fk(C)556 2748 y Fh(d)596 2733 y Fo(e)639 2692 y Fg(\000)p Fh(b)724 2669 y Fc(2)759 2692 y Fh(=)p Fi(\(6)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1058 2669 y Fc(2)1095 2692 y Fi(\026)-35 b Fh(h)p Fi(\))1184 2733 y Fo(\026)-50 b Fk(h)1239 2690 y Fg(j)p Fh(p)p Fg(j)1335 2733 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1619 2692 y Fi(2)p Fg(j)p Fh(p)p Fg(j)1750 2733 y Fo(2)1799 2692 y Fg(j)p Fh(j)t Fg(j)p Fi(+)1937 2665 y Ff(d)p 1935 2677 33 4 v 1936 2718 a Fc(2)1977 2692 y Fi(+)p Fg(j)p Fh(p)p Fg(j)2127 2733 y Fk(e)2172 2692 y Fh(\013)2217 2704 y Ff(d)2254 2692 y Fg(j)p Fh(j)t Fg(j)2346 2733 y Fo(\000\()p Fj(j)p Fk(j)6 b Fj(j)21 b Fo(+)2676 2666 y Fk(d)p 2676 2710 51 4 v 2677 2802 a Fo(2)2759 2733 y(+)h Fj(j)p Fk(p)p Fj(j)p Fo(\))2144 3014 y Fj(\002)2396 2931 y Fd(X)2336 3120 y Fh(n)p Fg(\024j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)2831 2947 y Fk(e)2876 2911 y Fg(\000)p Fi(2)p Fh(\013)3011 2923 y Ff(d)3048 2911 y Fh(n)p 2635 2991 657 4 v 2635 3086 a Fk(n)p Fo(!)33 b(\000\()p Fk(q)25 b Fo(+)3028 3047 y Fh(d)p 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Ff(d)1901 5068 y Fg(j)p Fh(j)t Fg(j)2035 5038 y Fo(\000\()p Fj(j)p Fk(j)6 b Fj(j)22 b Fo(+)2366 4999 y Fh(d)p 2366 5015 37 4 v 2367 5072 a Fi(2)2434 5038 y Fo(+)g Fj(j)p Fk(p)p Fj(j)p Fo(\))p 2035 5086 639 4 v 2291 5177 a Fj(j)p Fk(j)6 b Fj(j)p Fo(!)2684 5109 y Fk(:)940 b Fo(\(6.23\))1901 5506 y(25)p eop %%Page: 26 26 26 25 bop 0 -43 a Fo(This)30 b(quan)m(tit)m(y)g(b)s(ounds)g(\(6.19\),)f (whic)m(h,)i(in)e(turn,)h(b)s(ounds)g(the)g(square)h(of)e(\(6.18\).)42 b(T)-8 b(erms)29 b(of)g(the)h(form)0 77 y(\(6.18\))i(o)s(ccur)h(in)e (\(6.17\).)43 b(Putting)32 b(this)g(all)f(together,)i(w)m(e)g(see)h (that)e(\(6.17\))g(is)g(b)s(ounded)i(b)m(y)410 356 y Fk(M)504 371 y Fi(1)560 356 y Fk(C)637 315 y Fg(0000)718 304 y Fi(1)p Fh(=)p Fi(2)845 356 y Fo(e)888 315 y Fg(\000)p Fh(b)973 291 y Fc(2)1008 315 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1342 291 y Fc(2)1379 315 y Fi(\026)-35 b Fh(h)p Fi(\))1468 356 y Fk(\016)1515 315 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1728 248 y Fh(l)q Fg(\000)p Fi(1)1724 273 y Fd(X)1719 457 y Fh(k)r Fi(=0)1881 356 y Fo(\026)-50 b Fk(h)1936 313 y Fh(k)r(=)p Fi(2)2066 356 y Fk(\016)2113 315 y Fh(k)2279 273 y Fd(X)2189 461 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2532 288 y Fj(j)p Fk(c)2602 303 y Fh(k)r(;j)2696 288 y Fo(\()p Fk(t)p Fo(\))p Fj(j)p 2532 333 304 4 v 2578 349 a Fd(q)p 2661 349 129 4 v 100 x Fj(j)p Fk(j)g Fj(j)p Fo(!)2862 356 y Fk(e)2907 315 y Fh(\014)2947 327 y Ff(d)2983 315 y Fg(j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)1093 732 y Fj(\002)1490 649 y Fd(X)1382 837 y Fg(j)p Fh(p)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1750 732 y Fo(\026)-50 b Fk(h)1805 689 y Fg(j)p Fh(p)p Fg(j)p Fh(=)p Fi(2)1987 732 y Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2271 691 y Fg(j)p Fh(p)p Fg(j)2383 732 y Fo(2)2432 691 y Fg(j)p Fh(p)p Fg(j)p Fh(=)p Fi(2)2613 568 y Fd(s)p 2696 568 654 4 v 164 x Fo(\000\()p Fj(j)p Fk(j)6 b Fj(j)22 b Fo(+)3027 665 y Fk(d)p 3027 709 51 4 v 3028 801 a Fo(2)3109 732 y(+)g Fj(j)p Fk(p)p Fj(j)p Fo(\))16 b Fk(:)258 b Fo(\(6.24\))0 1103 y(The)29 b(n)m(um)m(b)s(er)g(of)e(terms)i(that)f(o)s(ccur)g(in)f (the)i(\014nal)f(sum)g(of)f(this)h(expression)i(is)2925 957 y Fd( )3033 1042 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)3195 1163 y Fk(d)g Fj(\000)h Fo(1)3621 957 y Fd(!)3687 1103 y Fo(,)29 b(and)0 1286 y(the)k(terms)g(in)e(that)i (sum)f(are)h(increasing.)42 b(Th)m(us,)35 b(\(6.24\))c(is)i(b)s(ounded) g(b)m(y)486 1564 y Fk(M)580 1579 y Fi(1)636 1564 y Fk(C)713 1523 y Fg(0000)794 1513 y Fi(1)p Fh(=)p Fi(2)920 1564 y Fo(e)964 1523 y Fg(\000)p Fh(b)1049 1500 y Fc(2)1083 1523 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1417 1500 y Fc(2)1455 1523 y Fi(\026)-35 b Fh(h)p Fi(\))1544 1564 y Fk(\016)1591 1523 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1804 1456 y Fh(l)q Fg(\000)p Fi(1)1800 1481 y Fd(X)1795 1666 y Fh(k)r Fi(=0)1957 1564 y Fo(\026)-50 b Fk(h)2012 1522 y Fh(k)r(=)p Fi(2)2142 1564 y Fk(\016)2189 1523 y Fh(k)2355 1481 y Fd(X)2265 1670 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2608 1497 y Fj(j)p Fk(c)2678 1512 y Fh(k)r(;j)2772 1497 y Fo(\()p Fk(t)p Fo(\))p Fj(j)p 2608 1541 304 4 v 2654 1558 a Fd(q)p 2737 1558 129 4 v 100 x Fj(j)p Fk(j)g Fj(j)p Fo(!)2954 1564 y Fk(e)2999 1523 y Fh(\014)3039 1535 y Ff(d)3075 1523 y Fg(j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)583 1908 y Fj(\002)873 1762 y Fd( )980 1847 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)1143 1967 y Fk(d)f Fj(\000)i Fo(1)1568 1762 y Fd(!)1650 1811 y(\020)1700 1908 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2121 1867 y Fi(2)2160 1811 y Fd(\021)2210 1834 y Fi(\()p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)r Fi(\))p Fh(=)p Fi(2)2578 1744 y Fd(s)p 2661 1744 942 4 v 2678 1908 a Fo(\000\()p Fj(j)p Fk(j)6 b Fj(j)21 b Fo(+)3008 1840 y Fk(d)p 3008 1884 51 4 v 3009 1976 a Fo(2)3091 1908 y(+)h Fk(l)i Fo(+)e(2)g Fj(\000)h Fk(k)s Fo(\))327 2203 y(=)83 b Fk(M)580 2218 y Fi(1)636 2203 y Fk(C)713 2162 y Fg(0000)794 2151 y Fi(1)p Fh(=)p Fi(2)920 2203 y Fo(e)964 2162 y Fg(\000)p Fh(b)1049 2138 y Fc(2)1083 2162 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1417 2138 y Fc(2)1455 2162 y Fi(\026)-35 b Fh(h)p Fi(\))1544 2203 y Fk(\016)1591 2162 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1795 2107 y Fd(\020)1845 2203 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2266 2162 y Fi(2)2305 2107 y Fd(\021)2370 2102 y Ff(l)p 2365 2114 31 4 v 2365 2156 a Fc(2)2405 2130 y Fi(+1)2557 2095 y Fh(l)q Fg(\000)p Fi(1)2553 2120 y Fd(X)2549 2304 y Fh(k)r Fi(=0)2711 2203 y Fk(\016)2758 2162 y Fh(k)2833 2107 y Fd(\020)2883 2203 y Fo(2)h Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)3233 2162 y Fi(2)3272 2107 y Fd(\021)3321 2130 y Fg(\000)p Fh(k)r(=)p Fi(2)583 2517 y Fj(\002)873 2370 y Fd( )980 2456 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)1143 2576 y Fk(d)f Fj(\000)i Fo(1)1568 2370 y Fd(!)1740 2434 y(X)1650 2622 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1994 2449 y Fj(j)p Fk(c)2064 2464 y Fh(k)r(;j)2158 2449 y Fo(\()p Fk(t)p Fo(\))p Fj(j)p 1994 2493 304 4 v 2039 2510 a Fd(q)p 2122 2510 129 4 v 100 x Fj(j)p Fk(j)g Fj(j)p Fo(!)2339 2517 y Fk(e)2384 2475 y Fh(\014)2424 2487 y Ff(d)2460 2475 y Fg(j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)2639 2353 y Fd(s)p 2722 2353 942 4 v 2739 2517 a Fo(\000\()p Fj(j)p Fk(j)g Fj(j)21 b Fo(+)3069 2449 y Fk(d)p 3069 2493 51 4 v 3070 2585 a Fo(2)3152 2517 y(+)h Fk(l)i Fo(+)e(2)g Fj(\000)h Fk(k)s Fo(\))16 b Fk(:)0 2817 y Fo(Applying)32 b(the)g(Sc)m(h)m(w)m (artz)j(inequalit)m(y)c(to)h(the)h(sum)f(o)m(v)m(er)i Fk(j)6 b Fo(,)32 b(w)m(e)i(see)f(that)f(this)h(expression)g(is)f(b)s (ounded)0 2937 y(b)m(y)410 3210 y Fk(M)504 3225 y Fi(1)560 3210 y Fk(C)637 3169 y Fg(0000)718 3158 y Fi(1)p Fh(=)p Fi(2)845 3210 y Fo(e)888 3169 y Fg(\000)p Fh(b)973 3145 y Fc(2)1008 3169 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1342 3145 y Fc(2)1379 3169 y Fi(\026)-35 b Fh(h)p Fi(\))1468 3210 y Fk(\016)1515 3169 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1719 3114 y Fd(\020)1769 3210 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2190 3169 y Fi(2)2229 3114 y Fd(\021)2294 3110 y Ff(l)p 2289 3122 31 4 v 2289 3163 a Fc(2)2329 3137 y Fi(+1)2481 3102 y Fh(l)q Fg(\000)p Fi(1)2477 3127 y Fd(X)2473 3311 y Fh(k)r Fi(=0)2635 3210 y Fk(\016)2682 3169 y Fh(k)2757 3114 y Fd(\020)2807 3210 y Fo(2)h Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)3157 3169 y Fi(2)3196 3114 y Fd(\021)3245 3137 y Fg(\000)p Fh(k)r(=)p Fi(2)703 3551 y Fj(\002)992 3405 y Fd( )1099 3490 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)1262 3610 y Fk(d)g Fj(\000)g Fo(1)1687 3405 y Fd(!)1786 3551 y Fj(k)p Fk(c)1878 3566 y Fh(k)1921 3551 y Fo(\()p Fk(t)p Fo(\))p Fj(k)2099 3380 y Fd(0)2098 3529 y(@)2261 3468 y(X)2171 3656 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2514 3483 y Fk(e)2559 3447 y Fh(\014)2599 3459 y Ff(d)2635 3447 y Fg(j)p Fh(j)t Fg(j)p 2514 3528 197 4 v 2548 3619 a Fj(j)p Fk(j)g Fj(j)p Fo(!)2737 3551 y(\000\()p Fj(j)p Fk(j)g Fj(j)22 b Fo(+)3068 3483 y Fk(d)p 3068 3528 51 4 v 3069 3619 a Fo(2)3150 3551 y(+)g Fk(l)j Fo(+)d(2)g Fj(\000)g Fk(k)s Fo(\))3679 3380 y Fd(1)3679 3529 y(A)3752 3403 y Fi(1)p Fh(=)p Fi(2)3878 3551 y Fk(:)0 3922 y Fo(The)35 b(n)m(um)m(b)s(er)g(of)f(terms)g(that)g (o)s(ccur)g(in)g(the)g(\014nal)g(sum)g(of)g(this)f(expression)j(is)3010 3776 y Fd( )3117 3861 y Fk(J)31 b Fo(+)22 b(3)p Fk(k)j Fo(+)d Fk(d)3320 3981 y(d)3615 3776 y Fd(!)3681 3922 y Fo(,)35 b(and)0 4099 y(the)e(terms)g(in)e(that)i(sum)f(are)h (increasing.)42 b(Th)m(us,)35 b(the)e(expression)h(is)e(b)s(ounded)h(b) m(y)410 4372 y Fk(M)504 4387 y Fi(1)560 4372 y Fk(C)637 4331 y Fg(0000)718 4320 y Fi(1)p Fh(=)p Fi(2)845 4372 y Fo(e)888 4331 y Fg(\000)p Fh(b)973 4307 y Fc(2)1008 4331 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1342 4307 y Fc(2)1379 4331 y Fi(\026)-35 b Fh(h)p Fi(\))1468 4372 y Fk(\016)1515 4331 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1719 4276 y Fd(\020)1769 4372 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2190 4331 y Fi(2)2229 4276 y Fd(\021)2294 4271 y Ff(l)p 2289 4283 31 4 v 2289 4325 a Fc(2)2329 4299 y Fi(+1)2481 4264 y Fh(l)q Fg(\000)p Fi(1)2477 4289 y Fd(X)2473 4473 y Fh(k)r Fi(=0)2635 4372 y Fk(\016)2682 4331 y Fh(k)2757 4276 y Fd(\020)2807 4372 y Fo(2)h Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)3157 4331 y Fi(2)3196 4276 y Fd(\021)3245 4299 y Fg(\000)p Fh(k)r(=)p Fi(2)442 4692 y Fj(\002)536 4546 y Fd( )644 4631 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)807 4751 y Fk(d)f Fj(\000)i Fo(1)1232 4546 y Fd(!)1314 4692 y Fj(k)p Fk(c)1406 4707 y Fh(k)1449 4692 y Fo(\()p Fk(t)p Fo(\))p Fj(k)1627 4546 y Fd(" )1782 4631 y Fk(J)31 b Fo(+)22 b(3)p Fk(k)j Fo(+)d Fk(d)1985 4751 y(d)2281 4546 y Fd(!)2391 4624 y Fk(e)2436 4588 y Fh(\014)2476 4600 y Ff(d)2512 4588 y Fi(\()p Fh(J)6 b Fi(+3)p Fh(k)r Fi(\))p 2373 4668 390 4 v 2373 4760 a Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(k)s Fo(\)!)2772 4692 y(\000\()p Fk(J)31 b Fo(+)22 b(2)p Fk(k)j Fo(+)3287 4624 y Fk(d)p 3287 4668 51 4 v 3288 4760 a Fo(2)3370 4692 y(+)d Fk(l)i Fo(+)e(2\))3706 4546 y Fd(#)3754 4565 y Fi(1)p Fh(=)p Fi(2)3881 4692 y Fk(:)0 4957 y Fo(W)-8 b(e)33 b(no)m(w)g(apply)f(the)h(estimate)f(of)g Fj(k)p Fk(c)1401 4972 y Fh(k)1444 4957 y Fo(\()p Fk(t)p Fo(\))p Fj(k)g Fo(from)g(Corollary)e(5.3)j(to)f(b)s(ound)h(this)f(b)m(y)487 5209 y Fk(M)581 5224 y Fi(1)638 5209 y Fk(C)715 5168 y Fg(0000)795 5157 y Fi(1)p Fh(=)p Fi(2)922 5209 y Fo(e)965 5168 y Fg(\000)p Fh(b)1050 5144 y Fc(2)1085 5168 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1419 5144 y Fc(2)1457 5168 y Fi(\026)-35 b Fh(h)o Fi(\))1529 5209 y Fk(\016)1576 5168 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1780 5112 y Fd(\020)1830 5209 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2251 5168 y Fi(2)2290 5112 y Fd(\021)2355 5108 y Ff(l)p 2350 5120 31 4 v 2350 5161 a Fc(2)2390 5135 y Fi(+1)1901 5506 y Fo(26)p eop %%Page: 27 27 27 26 bop 683 30 a Fj(\002)882 -78 y Fh(l)q Fg(\000)p Fi(1)879 -53 y Fd(X)874 132 y Fh(k)r Fi(=0)1036 30 y Fk(\016)1083 -11 y Fh(k)1159 -66 y Fd(\020)1208 30 y Fo(2)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1558 -11 y Fi(2)1597 -66 y Fd(\021)1647 -43 y Fg(\000)p Fh(k)r(=)p Fi(2)1848 -116 y Fd( )1955 -31 y Fk(l)25 b Fj(\000)d Fk(k)j Fo(+)d Fk(d)g Fo(+)g(1)2118 90 y Fk(d)g Fj(\000)h Fo(1)2544 -116 y Fd(!)33 b( )2750 -31 y Fk(J)e Fo(+)22 b(3)p Fk(k)j Fo(+)d Fk(d)2953 90 y(d)3248 -116 y Fd(!)3314 -93 y Fi(1)p Fh(=)p Fi(2)878 386 y Fj(\002)98 b Fk(e)1098 345 y Fh(\014)1138 357 y Ff(d)1174 345 y Fi(\()p Fh(J)6 b Fi(+3)p Fh(k)r Fi(\))p Fh(=)p Fi(2)1503 319 y Fk(D)1587 283 y Fh(k)1584 343 y Fi(2)1630 319 y Fk(D)1714 283 y Fh(k)1711 343 y Fi(3)1756 319 y Fo(\(1)22 b(+)g Fk(D)2044 334 y Fi(1)2083 319 y Fk(D)2167 283 y Fi(2)2164 343 y(2)2206 319 y Fj(j)p Fk(t)p Fj(j)p Fo(\))2335 283 y Fh(k)p 1503 363 875 4 v 1900 454 a Fk(k)s Fo(!)2404 240 y Fd( )2480 315 y Fo(\000\()p Fk(J)31 b Fo(+)22 b(2)p Fk(k)j Fo(+)2995 276 y Fh(d)p 2995 292 37 4 v 2996 350 a Fi(2)3063 315 y Fo(+)d Fk(l)j Fo(+)d(2\))p 2480 363 920 4 v 2895 454 a Fk(J)9 b Fo(!)3409 240 y Fd(!)3475 256 y Fi(1)p Fh(=)p Fi(2)327 730 y Fj(\024)83 b Fk(M)581 745 y Fi(1)621 730 y Fk(C)698 689 y Fg(0000)779 679 y Fi(1)p Fh(=)p Fi(2)889 730 y Fo(e)932 689 y Fg(\000)p Fh(b)1017 666 y Fc(2)1052 689 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1386 666 y Fc(2)1424 689 y Fi(\026)-35 b Fh(h)o Fi(\))1496 730 y Fk(\016)1543 689 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1730 634 y Fd(\020)1780 730 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2201 689 y Fi(2)2240 634 y Fd(\021)2305 630 y Ff(l)p 2300 642 31 4 v 2300 683 a Fc(2)2340 657 y Fi(+1)2461 663 y Fk(e)2506 627 y Fh(\014)2546 639 y Ff(d)2582 627 y Fh(J)r(=)p Fi(2)p 2461 707 237 4 v 2493 725 a Fj(p)p 2576 725 91 4 v 85 x Fk(J)9 b Fo(!)683 1060 y Fj(\002)882 952 y Fh(l)q Fg(\000)p Fi(1)879 977 y Fd(X)874 1162 y Fh(k)r Fi(=0)1036 1060 y Fk(\016)1083 1019 y Fh(k)1142 964 y Fd(\020)1192 1060 y Fo(2)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1542 1019 y Fi(2)1581 964 y Fd(\021)1630 987 y Fg(\000)p Fh(k)r(=)p Fi(2)1815 914 y Fd( )1922 999 y Fk(l)24 b Fj(\000)f Fk(k)i Fo(+)d Fk(d)g Fo(+)g(1)2085 1119 y Fk(d)g Fj(\000)g Fo(1)2510 914 y Fd(!)33 b( )2717 999 y Fk(J)e Fo(+)22 b(3)p Fk(k)j Fo(+)d Fk(d)2920 1119 y(d)3215 914 y Fd(!)3281 937 y Fi(1)p Fh(=)p Fi(2)1073 1407 y Fj(\002)195 b Fk(e)1390 1366 y Fi(3)p Fh(\014)1465 1378 y Ff(d)1502 1366 y Fh(k)r(=)p Fi(2)1657 1340 y Fk(D)1741 1303 y Fh(k)1738 1364 y Fi(2)1784 1340 y Fk(D)1868 1303 y Fh(k)1865 1364 y Fi(3)1910 1340 y Fo(\(1)22 b(+)g Fk(D)2198 1355 y Fi(1)2237 1340 y Fk(D)2321 1303 y Fi(2)2318 1364 y(2)2360 1340 y Fj(j)p Fk(t)p Fj(j)p Fo(\))2489 1303 y Fh(k)p 1657 1384 875 4 v 2054 1475 a Fk(k)s Fo(!)2574 1243 y Fd(s)p 2657 1243 935 4 v 164 x Fo(\000\()p Fk(J)31 b Fo(+)22 b(2)p Fk(k)j Fo(+)3172 1340 y Fk(d)p 3172 1384 51 4 v 3173 1475 a Fo(2)3255 1407 y(+)d Fk(l)i Fo(+)e(2\))p Fk(:)3651 1597 y Fo(\(6.25\))0 1785 y(W)-8 b(e)33 b(no)m(w)g(emplo)m(y) f(the)h(follo)m(wing)d(inequalities)h(that)h(hold)g(for)g(some)g(n)m (um)m(b)s(ers)i Fk(D)3096 1800 y Fi(5)3168 1785 y Fo(and)e Fk(D)3438 1800 y Fi(6)3478 1785 y Fo(:)633 1883 y Fd( )740 1968 y Fk(l)25 b Fj(\000)d Fk(k)k Fo(+)c Fk(d)g Fo(+)g(1)903 2089 y Fk(d)g Fj(\000)h Fo(1)1329 1883 y Fd(!)1478 2030 y Fj(\024)1638 1883 y Fd( )1745 1968 y Fk(l)i Fo(+)d Fk(d)g Fo(+)g(1)1820 2089 y Fk(d)g Fj(\000)h Fo(1)2158 1883 y Fd(!)2240 2030 y Fk(;)723 2199 y Fd( )830 2284 y Fk(J)32 b Fo(+)22 b(3)p Fk(k)j Fo(+)d Fk(d)1033 2404 y(d)1329 2199 y Fd(!)1478 2345 y Fj(\024)1638 2199 y Fd( )1745 2284 y Fk(J)32 b Fo(+)22 b(3)p Fk(l)i Fj(\000)e Fo(3)g(+)g Fk(d)2022 2404 y(d)2392 2199 y Fd(!)2474 2345 y Fk(;)1125 2602 y(e)1170 2561 y Fi(3)p Fh(\014)1245 2573 y Ff(d)1281 2561 y Fh(k)r(=)p Fi(4)1478 2602 y Fj(\024)83 b Fk(e)1683 2561 y Fi(3)p Fh(\014)1758 2573 y Ff(d)1794 2561 y Fi(\()p Fh(l)q Fg(\000)p Fi(1\))p Fh(=)p Fi(4)2036 2602 y Fk(;)487 2780 y(D)571 2739 y Fh(k)568 2805 y Fi(2)630 2780 y Fk(D)714 2739 y Fh(k)711 2805 y Fi(3)773 2780 y Fo(\(1)22 b(+)g Fk(D)1061 2795 y Fi(1)1100 2780 y Fk(D)1184 2739 y Fi(2)1181 2805 y(2)1223 2780 y Fj(j)p Fk(t)p Fj(j)p Fo(\))1352 2739 y Fh(k)1478 2780 y Fj(\024)83 b Fk(D)1722 2739 y Fh(l)q Fg(\000)p Fi(1)1719 2805 y(5)1838 2780 y Fk(;)270 3000 y Fo(1)p 254 3044 82 4 v 254 3136 a Fk(k)s Fo(!)377 2903 y Fd(s)p 461 2903 935 4 v 461 3067 a Fo(\000\()p Fk(J)31 b Fo(+)22 b(2)p Fk(k)j Fo(+)976 3000 y Fk(d)p 976 3044 51 4 v 977 3136 a Fo(2)1058 3067 y(+)d Fk(l)j Fo(+)d(2\))83 b Fj(\024)1776 3000 y Fo(1)p 1648 3044 305 4 v 1648 3136 a(\()p Fk(l)24 b Fj(\000)f Fo(1\)!)1995 2903 y Fd(s)p 2078 2903 591 4 v 164 x Fo(\000\()p Fk(J)31 b Fo(+)23 b(3)p Fk(l)h Fo(+)2571 3000 y Fk(d)p 2571 3044 51 4 v 2572 3136 a Fo(2)2631 3067 y(\))p Fk(;)407 b Fo(and)462 3281 y Fh(l)q Fg(\000)p Fi(1)458 3306 y Fd(X)454 3491 y Fh(k)r Fi(=0)616 3389 y Fk(\016)663 3348 y Fh(k)739 3293 y Fd(\020)788 3389 y Fo(2)17 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)1138 3348 y Fi(2)1177 3293 y Fd(\021)1226 3316 y Fg(\000)p Fh(k)r(=)p Fi(2)1478 3389 y Fj(\024)83 b Fk(D)1722 3348 y Fh(l)1719 3414 y Fi(6)1759 3389 y Fk(:)0 3650 y Fo(W)-8 b(e)33 b(then)g(see)h(that)e(\(6.25\))g(is)g(b)s (ounded)h(b)m(y)410 3904 y Fk(M)504 3919 y Fi(1)544 3904 y Fk(C)621 3863 y Fg(0000)701 3852 y Fi(1)p Fh(=)p Fi(2)811 3904 y Fo(e)855 3863 y Fg(\000)p Fh(b)940 3840 y Fc(2)974 3863 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1308 3840 y Fc(2)1346 3863 y Fi(\026)-35 b Fh(h)p Fi(\))1435 3904 y Fk(\016)1482 3863 y Fg(\000)p Fh(l)q Fg(\000)p Fi(2)1686 3808 y Fd(\020)1736 3904 y Fo(2)17 b(\026)-50 b Fk(h)16 b Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)2157 3863 y Fi(2)2196 3808 y Fd(\021)2261 3804 y Ff(l)p 2256 3816 31 4 v 2256 3857 a Fc(2)2296 3831 y Fi(+1)2434 3837 y Fk(e)2479 3801 y Fh(\014)2519 3813 y Ff(d)2555 3801 y Fh(J)r(=)p Fi(2)p 2434 3881 237 4 v 2465 3899 a Fj(p)p 2548 3899 91 4 v 84 x Fk(J)9 b Fo(!)2713 3758 y Fd( )2821 3843 y Fk(l)24 b Fo(+)e Fk(d)g Fo(+)g(1)2895 3963 y Fk(d)g Fj(\000)h Fo(1)3233 3758 y Fd(!)605 4251 y Fj(\002)894 4105 y Fd( )1001 4190 y Fk(J)32 b Fo(+)22 b(3)p Fk(l)i Fj(\000)f Fo(3)e(+)h Fk(d)1278 4311 y(d)1648 4105 y Fd(!)1713 4128 y Fi(1)p Fh(=)p Fi(2)1840 4251 y Fo(e)1883 4210 y Fi(3)p Fh(\014)1958 4222 y Ff(d)1994 4210 y Fi(\()p Fh(l)q Fg(\000)p Fi(1\))p Fh(=)p Fi(4)2253 4251 y Fk(D)2337 4210 y Fh(l)q Fg(\000)p Fi(1)2334 4276 y(5)2469 4251 y Fk(D)2553 4210 y Fh(l)2550 4276 y Fi(6)2744 4184 y Fo(1)p 2616 4228 305 4 v 2616 4320 a(\()p Fk(l)i Fj(\000)f Fo(1\)!)2947 4087 y Fd(s)p 3030 4087 591 4 v 164 x Fo(\000\()p Fk(J)32 b Fo(+)22 b(3)p Fk(l)i Fo(+)3523 4184 y Fk(d)p 3523 4228 51 4 v 3524 4320 a Fo(2)3583 4251 y(\))p Fk(:)3651 4459 y Fo(\(6.26\))0 4647 y(W)-8 b(e)33 b(b)s(ound)g(this)f(expression)i(b)m(y)f(using)g(the)g(t)m(w)m (o)g(inequalities)478 4746 y Fd( )585 4831 y Fk(l)24 b Fo(+)e Fk(d)g Fo(+)g(1)660 4951 y Fk(d)g Fj(\000)g Fo(1)997 4746 y Fd(!)1146 4892 y Fj(\024)84 b Fo(\()p Fk(l)24 b Fo(+)e Fk(d)g Fo(+)g(1\))1754 4851 y Fh(d)p Fg(\000)p Fi(1)2274 4892 y Fo(and)244 5061 y Fd( )351 5146 y Fk(J)31 b Fo(+)22 b(3)p Fk(l)j Fj(\000)d Fo(3)g(+)g Fk(d)628 5267 y(d)997 5061 y Fd(!)1146 5207 y Fj(\024)84 b Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(l)i Fj(\000)f Fo(3)f(+)g Fk(d)p Fo(\))1988 5166 y Fh(d)2028 5207 y Fk(:)1901 5506 y Fo(27)p eop %%Page: 28 28 28 27 bop 0 -43 a Fo(Note)33 b(that)f(the)h(righ)m(t)f(hand)h(sides)g (of)f(these)i(inequalities)c(gro)m(w)j(p)s(olynomially)c(with)j Fk(l)r Fo(.)146 77 y(Since)38 b Fk(d)g Fo(is)f(\014xed,)j(w)m(e)f (conclude)f(that)g(\(6.26\),)g(and)g(hence,)j(\(6.17\))c(are)g(b)s (ounded)i(b)m(y)g(a)e(constan)m(t)0 198 y(times)244 460 y(e)287 418 y Fg(\000)p Fh(b)372 395 y Fc(2)407 418 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)741 395 y Fc(2)779 418 y Fi(\026)-35 b Fh(h)o Fi(\))902 392 y Fo(e)946 356 y Fh(\015)986 333 y Fb(0)1008 356 y Fh(J)p 893 436 174 4 v 893 454 a Fj(p)p 976 454 91 4 v 85 x Fk(J)9 b Fo(!)1093 460 y(\026)-50 b Fk(h)1148 417 y Fh(l)q(=)p Fi(2)1385 392 y Fo(e)1428 356 y Fh(\015)t(l)p 1287 436 305 4 v 1287 528 a Fo(\()p Fk(l)25 b Fj(\000)d Fo(1\)!)1635 296 y Fd(s)p 1718 296 591 4 v 164 x Fo(\000\()p Fk(J)31 b Fo(+)22 b(3)p Fk(l)i Fo(+)2210 392 y Fk(d)p 2210 436 51 4 v 2211 528 a Fo(2)2271 460 y(\))o Fk(;)1316 b Fo(\(6.27\))0 717 y(for)31 b(some)g(p)s(ositiv)m(e)f Fk(\015)5 b Fo(,)32 b(and)f Fk(\015)1109 681 y Fg(0)1132 717 y Fo(.)43 b(With)31 b Fk(J)40 b Fo(\014xed,)33 b(w)m(e)f(apply)f (Stirling's)e(form)m(ula)h(to)g(the)i(factorial)d(and)i(\000)0 838 y(function)h(to)g(b)s(ound)h(this)f(b)m(y)i(another)e(constan)m(t)i (times)806 1118 y(e)850 1077 y Fg(\000)p Fh(b)935 1054 y Fc(2)969 1077 y Fh(=)p Fi(\(12)p Fg(k)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(k)1303 1054 y Fc(2)1341 1077 y Fi(\026)-35 b Fh(h)p Fi(\))1446 1118 y Fo(\026)-50 b Fk(h)1501 1075 y Fh(l)q(=)p Fi(2)1630 1118 y Fk(e)1675 1077 y Fh(\015)t(l)1784 1051 y Fo(\(3)p Fk(l)r Fo(\))1940 1015 y Fi(3)p Fh(l)q(=)p Fi(2)p 1784 1095 288 4 v 1899 1187 a Fk(l)1930 1158 y Fh(l)2142 1118 y Fo(=)2278 1022 y Fd(\020)2327 1118 y Fk(e)2372 1077 y Fh(\015)2434 1118 y Fo(3)2483 1077 y Fi(3)p Fh(=)p Fi(2)2592 1022 y Fd(\021)2642 1045 y Fh(l)2701 1118 y Fo(\(\026)g Fk(h)17 b(l)r Fo(\))2880 1077 y Fh(l)q(=)p Fi(2+1)3067 1118 y Fk(:)0 1369 y Fo(By)39 b(c)m(ho)s(osing)g Fk(l)79 b Fo(=)e([)-18 b([)17 b Fk(g)t(=)p Fo(\026)-50 b Fk(h)16 b Fo(])-18 b(])40 b(for)e(some)h(su\016cien)m(tly)g(small)e Fk(g)42 b(>)c Fo(0,)i(this)f(is)f(b)s(ounded)i(b)m(y)f(a)g(constan)m(t) 0 1490 y(times)48 b Fk(e)322 1448 y Fg(\000)p Fh(\015)413 1457 y Fc(2)448 1448 y Fh(=)p Fi(\026)-35 b Fh(h)528 1490 y Fo(.)146 1610 y(This)33 b(implies)d(the)j(lemma.)p 1416 1610 67 84 v 146 1848 a(Theorem)27 b(3.1)f(follo)m(ws)g (immediately)d(from)j(Lemmas)f(\(6.1\))h(and)h(\(6.3\))f(with)g Fk(G)45 b Fo(=)f(min)15 b Fj(f)i Fk(G)3591 1863 y Fi(1)3630 1848 y Fk(;)33 b(G)3767 1863 y Fi(2)3823 1848 y Fj(g)p Fo(.)0 2181 y Fl(7)161 b(Lo)t(calization)56 b(Estimates)d(for)h(the)f (W)-13 b(a)l(v)l(e)52 b(P)l(ac)l(k)l(ets)0 2459 y Fo(In)26 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5131 y Fk(C)23 b(l)c Fo(\026)-50 b Fk(h)17 b Fo(e)2632 5090 y Fi(\(6)p Fh(\025)p Fi(+2)p Fh(v)r(\034)8 b Fi(\))p Fh(T)3001 5035 y Fd(\021)3050 5058 y Fh(l)q(=)p Fi(2)3164 5131 y Fk(:)508 b Fo(\(8.1\))1901 5506 y(29)p eop %%Page: 30 30 30 29 bop 0 -43 a Fo(Th)m(us,)34 b(if)e(w)m(e)h(c)m(ho)s(ose)h Fk(l)c Fo(=)d Fk(g)t Fo(\()p Fk(T)14 b Fo(\))p Fk(=)p Fo(\026)-50 b Fk(h)o Fo(,)33 b(then)g(\(8.1\))f(is)g(b)s(ounded)h(b)m (y)1007 213 y(\026)-50 b Fk(h)1062 170 y Fg(\000)p Fi(1)1173 213 y Fk(D)19 b Fo(e)1317 172 y Fi(\(2)p Fh(v)r(\034)8 b Fi(+3)p Fh(\025)p Fi(\))p Fh(T)1702 116 y Fd(\020)1768 213 y Fk(C)24 b(g)t Fo(\()p Fk(T)14 b Fo(\))j(e)2119 172 y Fi(\(6)p Fh(\025)p Fi(+2)p Fh(v)r(\034)8 b Fi(\))p Fh(T)2488 116 y Fd(\021)2537 139 y Fh(g)r Fi(\()p Fh(T)i Fi(\))p Fh(=)p Fi(\(2\026)-35 b Fh(h)r Fi(\))2866 213 y Fk(;)0 433 y Fo(so)33 b(that)f(w)m(e)i(need)1141 553 y Fk(g)t Fo(\()p Fk(T)14 b Fo(\))j(e)1398 512 y Fi(\(6)p Fh(\025)p Fi(+2)p Fh(v)r(\034)8 b Fi(\))p Fh(T)1778 553 y Fj(!)27 b Fo(0)33 b(and)f Fk(g)t Fo(\()p Fk(T)14 b Fo(\))p Fk(=)p Fo(\026)-50 b Fk(h)27 b Fj(!)g(1)p Fk(:)0 728 y Fo(These)34 b(demands)f(are)g(satis\014ed)g(b)m(y)g(the)g(c)m (hoices)244 931 y Fk(g)t Fo(\()p Fk(T)14 b Fo(\))43 b(=)h(e)649 890 y Fg(\000)p Fh(\024T)994 931 y Fo(and)195 b Fk(T)58 b Fo(=)45 b Fk(T)1653 890 y Fg(0)1692 931 y Fo(ln\(1)p Fk(=)p Fo(\026)-50 b Fk(h)o Fo(\))p Fk(;)1670 b Fo(\(8.2\))0 1134 y(pro)m(vided)244 1338 y(6)p Fk(\025)22 b Fo(+)g(2)p Fk(v)t(\034)55 b(<)44 b(\024)h(<)f Fo(1)p Fk(=T)1177 1297 y Fg(0)1199 1338 y Fk(:)2473 b Fo(\(8.3\))0 1541 y(Note)31 b(that)h(the)f(prefactor)h(in)e(\(8.1\))h(will)e(b)s(e)j(of)f (order)g(\026)-50 b Fk(h)2102 1498 y Fg(\000)p Fh(\027)2192 1507 y Fc(1)2231 1541 y Fo(,)31 b(for)g(some)g(\014nite)g Fk(\027)2976 1556 y Fi(1)3016 1541 y Fo(.)43 b(It)31 b(will)e(th)m(us)k(pla)m(y)e(no)0 1662 y(role)h(since)h(it)e(follo)m (ws)h(from)f(these)j(considerations)e(that)g(there)h(exists)h Fk(\015)2743 1677 y Fi(1)2810 1662 y Fk(>)27 b Fo(0,)32 b(suc)m(h)j(that)858 1925 y(\026)-50 b Fk(h)913 1882 y Fg(\000)p Fi(1)1024 1808 y Fd(Z)1107 1834 y Fh(T)1070 1997 y Fi(0)1195 1925 y Fj(k)p Fk(\037)1306 1940 y Fi(1)1346 1925 y Fo(\()p Fj(\001)p Fk(;)17 b(t)p Fo(\))p Fk(\030)1572 1940 y Fh(l)1597 1925 y Fo(\()p Fj(\001)p Fk(;)g(t;)g Fo(\026)-50 b Fk(h)o Fo(\))p Fj(k)17 b Fk(dt)60 b Fo(=)g Fk(O)s Fo(\(\026)-50 b Fk(h)2397 1882 y Fg(\000)p Fh(\027)2487 1891 y Fc(1)2542 1925 y Fo(e)2585 1884 y Fg(\000)p Fh(\015)2676 1893 y Fc(1)2711 1884 y Fh(=)p Fi(\026)-35 b Fh(h)2787 1858 y Fc(1)p Fb(\000)p Ff(\024T)2945 1837 y Fb(0)2977 1925 y Fo(\))p Fk(:)0 2179 y Fo(By)43 b(a)f(similar)e(argumen)m(t,)45 b(w)m(e)e(obtain)f(the)h(estimate)f(corresp)s(onding)g(to)h(\(6.15\))f (with)g(other)h(time{)0 2300 y(indep)s(enden)m(t)34 b(constan)m(ts)g Fk(D)1069 2263 y Fg(0)1124 2300 y Fo(and)f Fk(C)1391 2263 y Fg(0)1414 2300 y Fo(:)244 2547 y(\026)-50 b Fk(h)299 2504 y Fg(\000)p Fi(1)410 2429 y Fd(Z)493 2456 y Fh(T)456 2618 y Fi(0)581 2547 y Fj(k)p Fk(\037)692 2562 y Fi(2)732 2547 y Fo(\()p Fj(\001)p Fk(;)17 b(t)p Fo(\))p Fk(\030)958 2562 y Fh(l)983 2547 y Fo(\()p Fj(\001)p Fk(;)g(t;)g Fo(\026)-50 b Fk(h)o Fo(\))p Fj(k)17 b Fk(dt)59 b Fj(\024)i Fo(\026)-50 b Fk(h)1669 2504 y Fg(\000)p Fi(1)1780 2547 y Fo(e)1824 2505 y Fg(\000)p Fh(b)1909 2482 y Fc(2)1943 2505 y Fh(=)p Fi(\(12\026)-35 b Fh(h)q Fi(e)2149 2482 y Fc(2)p Ff(\025T)2264 2505 y Fi(\))2312 2547 y Fk(D)2396 2505 y Fg(0)2453 2450 y Fd(\020)2502 2547 y Fk(C)2579 2505 y Fg(0)2619 2547 y Fk(l)19 b Fo(\026)-50 b Fk(h)17 b Fo(e)2782 2505 y Fi(\(8)p Fh(\025)p Fi(+2)p Fh(v)r(\034)8 b Fi(\))p Fh(T)3134 2450 y Fd(\021)3184 2473 y Fh(l)q(=)p Fi(2)3297 2547 y Fk(:)375 b Fo(\(8.4\))0 2784 y(Inserting)37 b(our)f(c)m(hoices)i(\(8.2\))e(and)h(constrain)m(ts)g(\(8.3\))f(in)g (\(8.4\),)h(it)f(is)g(elemen)m(tary)h(to)f(see)i(that)e(there)0 2904 y(exists)d(p)s(ositiv)m(e)f Fk(\027)678 2919 y Fi(2)750 2904 y Fo(and)h Fk(\015)991 2919 y Fi(2)1030 2904 y Fo(,)g(suc)m(h)h (that)843 3168 y(\026)-50 b Fk(h)898 3125 y Fg(\000)p Fi(1)1009 3051 y Fd(Z)1092 3077 y Fh(T)1055 3240 y Fi(0)1180 3168 y Fj(k)p Fk(\037)1291 3183 y Fi(2)1330 3168 y Fo(\()p Fj(\001)p Fk(;)17 b(t)p Fo(\))p Fk(\030)1556 3183 y Fh(l)1581 3168 y Fo(\()p Fj(\001)p Fk(;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))p Fj(k)17 b Fk(dt)59 b Fo(=)h Fk(O)s Fo(\(\026)-50 b Fk(h)2382 3125 y Fg(\000)p Fh(\027)2472 3134 y Fc(2)2527 3168 y Fo(e)2570 3127 y Fg(\000)p Fh(\015)2661 3136 y Fc(2)2696 3127 y Fh(=)p Fi(\026)-35 b Fh(h)2772 3101 y Fc(1)p Fb(\000)p Fc(2)p Ff(\025T)2961 3080 y Fb(0)2992 3168 y Fo(\))p Fk(;)0 3430 y Fo(whic)m(h)33 b(pro)m(v)m(es)h(the)f (Theorem.)p 1393 3430 67 84 v 0 3669 a Fa(Pro)s(of)42 b(of)g(Theorem)g(3.4)p Fo(:)98 b(Considerations)36 b(similar)d(to)j (those)i(in)d(the)i(second)h(part)f(of)f(the)h(pro)s(of)0 3789 y(of)32 b(Theorem)h(3.3)f(sho)m(w)i(that)e(there)h(exists)h (constan)m(ts)g Fk(C)2154 3804 y Fi(0)2193 3789 y Fo(,)f Fk(C)2323 3804 y Fi(1)2394 3789 y Fo(indep)s(enden)m(t)h(of)e Fk(T)46 b Fo(suc)m(h)35 b(that)410 3993 y Fj(k)p Fk(\037)521 4008 y Fi(2)560 3993 y Fo(\()p Fj(\001)p Fk(;)17 b(t)p Fo(\))p Fk( )t Fo(\()p Fj(\001)p Fk(;)g(t;)g Fo(\026)-50 b Fk(h)o Fo(\))p Fj(k)410 4249 y(\024)556 4141 y Fh(l)q Fg(\000)p Fi(1)552 4166 y Fd(X)548 4350 y Fh(k)r Fi(=0)709 4249 y Fo(\026)g Fk(h)765 4206 y Fh(k)r(=)p Fi(2)894 4249 y Fj(k)p Fk(c)986 4264 y Fh(k)1029 4249 y Fo(\()p Fk(t)p Fo(\))p Fj(k)1223 4078 y Fd(0)1223 4227 y(@)1402 4166 y(X)1312 4354 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1645 4249 y Fj(k)p Fk(\037)1756 4264 y Fi(2)1796 4249 y Fo(\()p Fj(\001)p Fk(;)17 b(t)p Fo(\))g Fk(\036)2054 4264 y Fh(j)2089 4249 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g Fj(\001)p Fo(\))p Fj(k)3217 4208 y Fi(2)3272 4078 y Fd(1)3272 4227 y(A)3344 4101 y Fi(1)p Fh(=)p Fi(2)410 4587 y Fj(\024)61 b Fo(e)591 4546 y Fg(\000)p Fh(b)676 4523 y Fc(2)711 4546 y Fh(=)p Fi(\(12\026)-35 b Fh(h)q Fi(e)916 4523 y Fc(2)p Ff(\025T)1032 4546 y Fi(\))1097 4470 y Fh(g)r Fi(\()p Fh(T)10 b Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1145 4504 y Fd(X)1141 4689 y Fh(k)r Fi(=0)1347 4491 y Fd(\020)1397 4587 y Fo(\026)-50 b Fk(hk)s Fo(e)1550 4546 y Fi(\(2)p Fh(v)r(\034)8 b Fi(+6)p Fh(\025)p Fi(\))p Fh(T)1902 4587 y Fk(C)1972 4602 y Fi(0)2011 4491 y Fd(\021)2061 4514 y Fh(k)r(=)p Fi(2)2234 4587 y Fj(\024)61 b Fo(e)2415 4546 y Fg(\000)p Fh(b)2500 4523 y Fc(2)2535 4546 y Fh(=)p Fi(\(12\026)-35 b Fh(h)q Fi(e)2740 4523 y Fc(2)p Ff(\025T)2856 4546 y Fi(\))2950 4479 y Fg(1)2925 4504 y Fd(X)2921 4689 y Fh(k)r Fi(=0)3083 4587 y Fk(C)3160 4546 y Fh(k)3153 4612 y Fi(1)3202 4587 y Fk(;)0 4863 y Fo(where,)51 b(b)m(y)d(virtue)e (of)h(\(8.2\))f(and)g(\(8.3\),)k(w)m(e)e(can)e(tak)m(e)i Fk(C)2274 4878 y Fi(1)2364 4863 y Fk(<)k Fo(1.)85 b(So,)50 b(the)d(Theorem)g(holds)f(with)0 5007 y(exp)s(onen)m(tial)32 b(deca)m(y)i(of)e(order)h(e)1203 4970 y Fg(\000)p Fh(b)1288 4947 y Fc(2)1323 4970 y Fh(=)p Fi(\(12\026)-35 b Fh(h)1497 4945 y Fc(\(1)p Fb(\000)p Fc(2)p Ff(\025\024T)1746 4924 y Fb(0)1769 4945 y Fc(\))1797 4970 y Fi(\))1829 5007 y Fo(.)p 2095 5007 V 1901 5506 a(30)p eop %%Page: 31 31 31 30 bop 0 -43 a Fl(9)161 b(Scattering)53 b(Theory)0 235 y Fo(In)36 b(this)f(section)h(w)m(e)g(sho)m(w)h(our)e(appro)m (ximations)f(are)h(v)-5 b(alid)34 b(up)i(to)f(exp)s(onen)m(tially)g (small)e(corrections)0 355 y(in)f(a)g(scattering)g(framew)m(ork,)h(pro) m(vided)g(the)g(p)s(oten)m(tial)e(satis\014es)i(h)m(yp)s(othesis)h(D.)0 594 y Fa(Pro)s(of)j(of)h(Theorem)f(3.5)p Fo(:)98 b(First)31 b(note)i(that)f(equations)h(\(3.2\))f(together)h(with)468 807 y(e)511 766 y Fg(\000)p Fh(itH)673 775 y Fc(0)708 766 y Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)900 807 y Fk(\036)958 822 y Fh(j)995 807 y Fo(\()p Fk(A;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a;)g(\021)t(;)g(x)p Fo(\))60 b(=)g(e)1894 766 y Fh(it\021)1980 742 y Fc(2)2016 766 y Fh(=)p Fi(\(2\026)-35 b Fh(h)q Fi(\))2203 807 y Fk(\036)2261 822 y Fh(j)2297 807 y Fo(\()p Fk(A)22 b Fo(+)h Fk(tiB)5 b(;)17 b(B)5 b(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)22 b Fo(+)g Fk(t\021)t(;)17 b(\021)t(;)g(x)p Fo(\))0 1010 y(for)32 b(an)m(y)h Fk(j)h Fj(2)28 b Fa(N)589 968 y Fh(d)662 1010 y Fo(imply)i(that)j(as)f Fk(t)c Fj(!)g(\0061)p Fo(,)294 1214 y(e)337 1173 y Fh(itH)444 1182 y Fc(0)479 1173 y Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)687 1214 y Fo(e)731 1173 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)978 1214 y Fk(\036)1036 1229 y Fh(j)1073 1214 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))43 b Fj(!)h Fo(e)2409 1173 y Fh(iS)2476 1182 y Fb(\006)2528 1173 y Fh(=)p Fi(\026)-35 b Fh(h)2624 1214 y Fk(\036)2682 1229 y Fh(j)2719 1214 y Fo(\()p Fk(A)2830 1229 y Fg(\006)2889 1214 y Fk(;)17 b(B)3007 1229 y Fg(\006)3066 1214 y Fk(;)g Fo(\026)-50 b Fk(h)o(;)17 b(a)3259 1229 y Fg(\006)3318 1214 y Fk(;)g(\021)3410 1229 y Fg(\006)3469 1214 y Fk(;)g(x)p Fo(\))0 1418 y(with)32 b Fk(S)282 1433 y Fg(\000)369 1418 y Fo(=)27 b(0,)33 b(for)f(an)m(y)h Fk(j)h Fj(2)28 b Fa(N)1169 1376 y Fh(d)1210 1418 y Fo(.)43 b(Moreo)m(v)m(er,)35 b(using)d(\(3.2\))g(and)h(the)g(prop)s(ert)m(y)1081 1621 y(min)o(\()p Fj(j)p Fk(v)t Fj(j)p Fk(;)17 b Fo(1\))g Fj(h)p Fk(t)p Fj(i)58 b(\024)i(h)p Fk(tv)t Fj(i)g(\024)h Fo(max)o(\()p Fj(j)p Fk(v)t Fj(j)p Fk(;)17 b Fo(1\))g Fj(h)p Fk(t)p Fj(i)p Fk(;)0 1825 y Fo(for)39 b(an)m(y)h Fk(v)j Fj(2)d Fk(I)-19 b(R)642 1783 y Fh(d)722 1825 y Fo(and)40 b(an)m(y)g Fk(t)g Fj(2)g Fk(I)-19 b(R)q Fo(,)41 b(with)e Fj(h)p Fk(t)p Fj(i)56 b Fo(=)1988 1743 y Fj(p)p 2071 1743 244 4 v 82 x Fo(1)22 b(+)g Fk(t)2275 1796 y Fi(2)2315 1825 y Fo(,)41 b(w)m(e)g(get)e(the)h(existence)h(of)g(~)-51 b Fk(c)3460 1840 y Fi(0)3539 1825 y Fk(>)40 b Fo(0)f(and)2 1945 y(~)-51 b Fk(c)42 1960 y Fi(1)109 1945 y Fk(>)28 b Fo(0)k(dep)s(ending)h(on)f(the)h(asymptotic)f(data)g(\()p Fk(a)1879 1960 y Fg(\006)1938 1945 y Fk(;)17 b(\021)2030 1960 y Fg(\006)2089 1945 y Fo(\),)33 b(suc)m(h)h(that)244 2066 y Fd(\014)244 2116 y(\014)244 2166 y(\014)244 2216 y(\014)244 2265 y(\014)298 2148 y Fk(D)382 2112 y Fh(m)448 2148 y Fk(V)22 b Fo(\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 298 2192 468 4 v 475 2284 a Fk(m)p Fo(!)792 2066 y Fd(\014)792 2116 y(\014)792 2166 y(\014)792 2216 y(\014)792 2265 y(\014)880 2216 y Fj(\024)1065 2148 y Fo(~)-52 b Fk(c)1104 2163 y Fi(0)1163 2148 y Fo(~)h Fk(c)1203 2097 y Fg(j)p Fh(m)p Fg(j)1203 2170 y Fi(1)p 1027 2192 317 4 v 1027 2284 a Fj(h)p Fk(t)p Fj(i)1140 2255 y Fh(\014)s Fi(+)p Fg(j)p Fh(m)p Fg(j)3699 2216 y Fo(\(9.1\))0 2467 y(for)48 b(large)g(times.)91 b(This)49 b(estimate)f(together)h(with)g(\(3.3\))f (and)h(Lemma)e(5.1)i(yields)f(the)i(follo)m(wing)0 2587 y(estimate)32 b(on)g(the)h(op)s(erator)f Fk(K)1170 2602 y Fh(k)1213 2587 y Fo(\()p Fk(t)p Fo(\))17 b Fk(P)1404 2603 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)1609 2587 y Fo(de\014ned)34 b(in)e(\(4.6\):)244 2871 y Fj(k)p Fk(K)377 2886 y Fh(k)419 2871 y Fo(\()p Fk(t)p Fo(\))17 b Fk(P)610 2887 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)783 2871 y Fj(k)84 b Fo(=)1077 2697 y Fd(\015)1077 2747 y(\015)1077 2796 y(\015)1077 2846 y(\015)1077 2896 y(\015)1077 2946 y(\015)1177 2788 y(X)1139 2976 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(k)1378 2804 y Fk(D)1462 2768 y Fh(m)1528 2804 y Fk(V)21 b Fo(\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 1378 2848 468 4 v 1555 2940 a Fk(m)p Fo(!)1871 2871 y Fk(X)8 b Fo(\()p Fk(t)p Fo(\))2071 2830 y Fh(m)2154 2871 y Fk(P)2217 2887 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)2407 2697 y Fd(\015)2407 2747 y(\015)2407 2796 y(\015)2407 2846 y(\015)2407 2896 y(\015)2407 2946 y(\015)916 3241 y Fj(\024)1077 3120 y Fd(\022)1154 3181 y Fk(d)22 b Fj(\000)h Fo(1)f(+)g Fk(k)1242 3301 y(d)f Fj(\000)i Fo(1)1566 3120 y Fd(\023)1677 3074 y(s)p 1760 3074 356 4 v 1770 3174 a Fo(\()p Fk(n)f Fo(+)g Fk(k)s Fo(\)!)p 1770 3218 336 4 v 1895 3309 a Fk(n)p Fo(!)2160 3174 y(~)-51 b Fk(c)2200 3189 y Fi(0)2258 3174 y Fo(~)g Fk(c)2298 3138 y Fh(k)2298 3198 y Fi(2)p 2158 3218 183 4 v 2169 3309 a Fj(h)p Fk(t)p Fj(i)2282 3281 y Fh(\014)2367 3241 y Fk(;)1305 b Fo(\(9.2\))0 3486 y(where)32 b(~)-51 b Fk(c)320 3501 y Fi(2)389 3486 y Fo(dep)s(ends)30 b(on)f(the)g (asymptotic)g(data)f(\()p Fk(a)1864 3501 y Fg(\006)1923 3486 y Fk(;)17 b(\021)2015 3501 y Fg(\006)2074 3486 y Fk(;)g(A)2191 3501 y Fg(\006)2250 3486 y Fk(;)g(B)2368 3501 y Fg(\006)2427 3486 y Fo(\))29 b(and)g(the)g(binomial)d(co)s (e\016cien)m(t)k(giv)m(es)0 3607 y(the)j(n)m(um)m(b)s(er)g(of)f(m)m (ulti-indices)d(of)j(order)g Fk(k)s Fo(.)44 b(A)m(t)33 b(the)f(p)s(ossible)g(cost)h(of)f(an)g(increase)h(in)f(the)g(constan)m (ts,)0 3727 y(w)m(e)i(ma)m(y)e(assume)h(this)f(estimate)g(is)g(v)-5 b(alid)31 b(for)h(all)e Fk(t)e Fj(2)g Fk(I)-19 b(R)q Fo(.)146 3847 y(This)40 b(estimate)e(sho)m(ws)j(in)d(particular)f(that) i Fk(K)1937 3862 y Fh(k)1980 3847 y Fo(\()p Fk(t)p Fo(\))17 b Fk(P)2171 3863 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)2383 3847 y Fo(is)39 b(in)m(tegrable)f(in)g(time.)62 b(F)-8 b(rom)37 b(this,)k(it)0 3968 y(is)d(easy)i(to)f(c)m(hec)m(k)i(inductiv) m(ely)e(that)f(the)i(solutions)e Fk(c)2076 3983 y Fh(n)2123 3968 y Fo(\()p Fk(t)p Fo(\))h(to)f(the)h(equations)h(\(4.11\))e(ha)m(v) m(e)i(limits)c(as)0 4088 y Fj(j)p Fk(t)p Fj(j)27 b(!)h(1)p Fo(.)146 4209 y(The)37 b(asymptotic)e(v)-5 b(alues)36 b(of)g(the)g(co)s(e\016cien)m(ts)h Fk(c)1974 4224 y Fh(n)2021 4209 y Fo(\()p Fk(t)p Fo(\))f(at)g(in\014nit)m(y)f(allo)m(w)g(us)h(to)g (de\014ne)h(the)f(asymp-)0 4329 y(totic)30 b(states)i(\010)572 4344 y Fg(\006)631 4329 y Fo(\()p Fk(A)742 4344 y Fg(\006)801 4329 y Fk(;)17 b(B)919 4344 y Fg(\006)978 4329 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)1172 4344 y Fg(\006)1231 4329 y Fk(;)g(\021)1323 4344 y Fg(\006)1382 4329 y Fk(;)g(x)p Fo(\))31 b(b)m(y)g(\(3.6\))g(with)f(initial)d(conditions)j(at)g Fj(\0001)h Fo(c)m(haracterized)h(b)m(y)0 4449 y(arbitrary)g(normalized) e(co)s(e\016cien)m(ts)k(that)e(satisfy)244 4639 y Fk(c)286 4654 y Fi(0)p Fh(;j)377 4639 y Fo(\()p Fj(\0001)p Fo(\))83 b(=)g(0)p Fk(;)114 b Fo(for)32 b Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)n(;)114 b Fo(and)489 4784 y Fk(c)531 4799 y Fh(n;j)713 4784 y Fo(=)83 b(0)p Fk(;)114 b Fo(for)32 b Fk(n)c Fo(=)g(1)p Fk(;)17 b Fo(2)p Fk(;)g Fj(\001)g(\001)g(\001)c Fk(;)50 b Fo(and)32 b(all)f Fk(j)i Fj(2)c Fa(N)2375 4742 y Fh(d)2416 4784 y Fk(:)1256 b Fo(\(9.3\))0 4973 y(Th)m(us,)34 b(our)f(appro)m(ximate)e(solution)517 5177 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))60 b(=)g Fk(e)1134 5136 y Fh(iS)t Fi(\()p Fh(t)p Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)1554 5094 y Fd(X)1382 5282 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(g)r(=)p Fi(\026)-35 b Fh(h)o Fg(\000)p Fi(3)1879 5177 y Fk(c)1921 5192 y Fh(j)1958 5177 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(\036)2243 5192 y Fh(j)2278 5177 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))1901 5506 y(31)p eop %%Page: 32 32 32 31 bop 0 -43 a Fo(has)33 b(the)g(asymptotic)f(prop)s(ert)m(y)h(as)g Fk(t)28 b Fj(!)f(\0061)p Fo(,)244 160 y(e)287 119 y Fh(itH)394 128 y Fc(0)429 119 y Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)622 160 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))44 b Fj(!)g Fo(\010)1256 175 y Fg(\006)1315 160 y Fo(\()p Fk(A)1426 175 y Fg(\006)1485 160 y Fk(;)17 b(B)1603 175 y Fg(\006)1662 160 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)1856 175 y Fg(\006)1915 160 y Fk(;)g(\021)2007 175 y Fg(\006)2066 160 y Fk(;)g(x)p Fo(\))p Fk(:)1469 b Fo(\(9.4\))0 364 y(W)-8 b(e)33 b(pro)m(v)m(e)h(b)s (elo)m(w)e(that)486 567 y Fj(k)p Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))38 b Fj(\000)59 b Fo(lim)1067 617 y Fh(s)p Fg(!1)1274 567 y Fo(e)1317 526 y Fh(i)p Fi(\()p Fh(t)p Fg(\000)p Fh(s)p Fi(\))p Fh(H)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)1764 567 y Fo(e)1808 526 y Fh(isH)1923 535 y Fc(0)1957 526 y Fi(\(\026)g Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)2149 567 y Fo(\010)2219 582 y Fg(\000)2279 567 y Fo(\()p Fk(A)2390 582 y Fg(\000)2449 567 y Fk(;)17 b(B)2567 582 y Fg(\000)2626 567 y Fk(;)g Fo(\026)-50 b Fk(h)o(;)17 b(a)2819 582 y Fg(\000)2878 567 y Fk(;)g(\021)2970 582 y Fg(\000)3029 567 y Fk(;)g(x)p Fo(\))p Fj(k)327 774 y Fo(=)83 b Fj(k)p Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))38 b Fj(\000)i Fo(e)1110 733 y Fh(itH)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)1415 774 y Fo(\012)1485 733 y Fi(+)1544 774 y Fo(\(\026)-50 b Fk(h)p Fo(\))17 b(\010)1762 789 y Fg(\000)1821 774 y Fo(\()p Fk(A)1932 789 y Fg(\000)1992 774 y Fk(;)g(B)2110 789 y Fg(\000)2168 774 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)2362 789 y Fg(\000)2421 774 y Fk(;)g(\021)2513 789 y Fg(\000)2572 774 y Fk(;)g(x)p Fo(\))p Fj(k)327 961 y Fo(=)83 b Fk(O)s Fo(\(e)644 920 y Fg(\000)p Fh(\015)t(=)p Fi(\026)-35 b Fh(h)820 961 y Fo(\))p Fk(;)0 1164 y Fo(uniformly)30 b(for)i Fk(t)c Fj(2)g Fk(I)-19 b(R)q Fo(.)44 b(Th)m(us,)34 b(making)d(use)j(of)e(\(9.4\),)g(w)m(e)i(ha)m(v)m(e)529 1368 y(lim)486 1422 y Fh(t)p Fg(!)p Fi(+)p Fg(1)741 1368 y Fj(k)p Fo(e)834 1327 y Fh(itH)941 1336 y Fc(0)976 1327 y Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)1168 1368 y Fk( )t Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))39 b Fj(\000)g Fo(e)1742 1327 y Fh(itH)1849 1336 y Fc(0)1884 1327 y Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)2077 1368 y Fo(e)2120 1327 y Fh(itH)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)2425 1368 y Fo(\012)2495 1327 y Fi(+)2554 1368 y Fo(\(\026)-50 b Fk(h)p Fo(\))17 b(\010)2772 1383 y Fg(\000)2832 1368 y Fo(\()p Fk(A)2943 1383 y Fg(\000)3002 1368 y Fk(;)g(B)3120 1383 y Fg(\000)3178 1368 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)3372 1383 y Fg(\000)3431 1368 y Fk(;)g(\021)3523 1383 y Fg(\000)3582 1368 y Fk(;)g(x)p Fo(\))p Fj(k)327 1577 y Fo(=)83 b Fj(k)p Fo(\010)606 1592 y Fi(+)665 1577 y Fo(\()p Fk(A)776 1592 y Fi(+)835 1577 y Fk(;)17 b(B)953 1592 y Fi(+)1012 1577 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)1206 1592 y Fi(+)1265 1577 y Fk(;)g(\021)1357 1592 y Fi(+)1416 1577 y Fk(;)g(x)p Fo(\))38 b Fj(\000)i Fo(\012)1778 1536 y Fg(\000)1837 1577 y Fo(\(\026)-50 b Fk(h)p Fo(\))1968 1536 y Fg(\003)2024 1577 y Fo(\012)2094 1536 y Fi(+)2154 1577 y Fo(\(\026)g Fk(h)p Fo(\))17 b(\010)2372 1592 y Fg(\000)2431 1577 y Fo(\()p Fk(A)2542 1592 y Fg(\000)2601 1577 y Fk(;)g(B)2719 1592 y Fg(\000)2778 1577 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)2972 1592 y Fg(\000)3031 1577 y Fk(;)g(\021)3123 1592 y Fg(\000)3181 1577 y Fk(;)g(x)p Fo(\))p Fj(k)327 1747 y Fo(=)83 b Fj(k)p Fo(\010)606 1762 y Fi(+)665 1747 y Fo(\()p Fk(A)776 1762 y Fi(+)835 1747 y Fk(;)17 b(B)953 1762 y Fi(+)1012 1747 y Fk(;)g Fo(\026)-50 b Fk(h;)17 b(a)1206 1762 y Fi(+)1265 1747 y Fk(;)g(\021)1357 1762 y Fi(+)1416 1747 y Fk(;)g(x)p Fo(\))38 b Fj(\000)i Fk(S)6 b Fo(\(\026)-50 b Fk(h)p Fo(\))17 b(\010)1992 1762 y Fg(\000)2051 1747 y Fo(\()p Fk(A)2162 1762 y Fg(\000)2221 1747 y Fk(;)g(B)2339 1762 y Fg(\000)2398 1747 y Fk(;)g Fo(\026)-50 b Fk(h)o(;)17 b(a)2591 1762 y Fg(\000)2650 1747 y Fk(;)g(\021)2742 1762 y Fg(\000)2801 1747 y Fk(;)g(x)p Fo(\))p Fj(k)327 1934 y Fo(=)83 b Fk(O)s Fo(\(e)644 1893 y Fg(\000)p Fh(\015)t(=)p Fi(\026)-35 b Fh(h)820 1934 y Fo(\))p Fk(:)146 2137 y Fo(Hence,)36 b(w)m(e)g(need)f(only)f(sho)m(w)h(that)f(the)h(estimate)e (on)i Fk(\030)2255 2152 y Fh(l)2280 2137 y Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))35 b(corresp)s(onding)f(to)g(our)g (appro)m(xi-)0 2258 y(mation)f(yields)i(an)f(exp)s(onen)m(tially)g (small)f(correction)i(term)f(after)h(c)m(ho)s(osing)f Fk(l)g Fo(=)e Fk(g)t(=)p Fo(\026)-50 b Fk(h)34 b Fo(for)g(su\016cien)m (tly)0 2378 y(small)c Fk(g)t Fo(,)i(uniformly)f(for)h Fk(t)27 b Fj(2)i Fk(I)-19 b(R)p Fo(.)146 2499 y(W)-8 b(e)33 b(mimic)d(Section)i(5)h(to)f(get)g(estimates)h(on)f(the)h(co)s (e\016cien)m(ts)244 2772 y Fk(c)286 2787 y Fh(k)329 2772 y Fo(\()p Fk(t)p Fo(\))60 b(=)679 2664 y Fh(k)639 2689 y Fd(X)636 2870 y Fh(p)p Fi(=1)811 2772 y Fk(c)853 2721 y Fi([)p Fh(p)p Fi(])853 2797 y Fh(k)932 2772 y Fo(\()p Fk(t)p Fo(\))2656 b(\(9.5\))0 3052 y(starting)32 b(with)298 3272 y Fk(c)340 3287 y Fi(0)p Fh(;j)431 3272 y Fo(\()p Fk(t)p Fo(\))c(=)f Fk(c)715 3287 y Fi(0)p Fh(;j)807 3272 y Fo(\()p Fj(\0001)p Fo(\))p Fk(;)212 b(c)1341 3287 y Fi(0)p Fh(;j)1432 3272 y Fo(\()p Fj(\0001)p Fo(\))28 b(=)f(0)98 b(if)31 b Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)n(;)212 b Fo(and)195 b Fj(k)p Fk(c)3020 3287 y Fi(0)3059 3272 y Fo(\()p Fj(\0001)p Fo(\))p Fj(k)44 b Fo(=)g(1)p Fk(:)0 3508 y Fo(W)-8 b(e)40 b(note)g(that)g(the)g(n)m(um)m(b)s(er)g(of)g (comp)s(onen)m(ts)g(of)g(the)g(v)m(ectors)h Fk(c)2492 3457 y Fi([)p Fh(p)p Fi(])2492 3533 y Fh(k)2571 3508 y Fo(\()p Fk(t)p Fo(\))f(is)f(the)i(same)e(as)h(in)g(\(5.8\))f(and)0 3628 y(that)32 b(the)g(com)m(binatorics)f(asso)s(ciated)h(with)g(the)g Fk(n)g Fo(and)h Fk(p)f Fo(dep)s(endence)i(of)e(the)g(estimates)g(is)g (iden)m(tical)0 3749 y(to)g(that)h(p)s(erformed)f(in)f(Section)i(5.)146 3911 y(Hence,)h(with)f Fk(D)767 3926 y Fi(3)834 3911 y Fo(=)937 3790 y Fd(\022)1016 3851 y Fk(d)22 b Fo(+)g(2)1015 3972 y Fk(d)g Fj(\000)g Fo(1)1253 3790 y Fd(\023)1314 3911 y Fo(,)32 b(at)h(\014rst)g(order)f(w)m(e)i(ha)m(v)m(e)736 4278 y Fj(k)p Fk(c)828 4293 y Fi(1)867 4278 y Fo(\()p Fk(t)p Fo(\))p Fj(k)60 b Fo(=)h Fj(k)p Fk(c)1317 4227 y Fi([1])1317 4300 y(1)1395 4278 y Fo(\()p Fk(t)p Fo(\))p Fj(k)f(\024)h Fk(D)1835 4293 y Fi(3)1907 4111 y Fd(s)p 1990 4111 355 4 v 2000 4210 a Fo(\()p Fk(J)31 b Fo(+)22 b(3\)!)p 2000 4255 336 4 v 2122 4346 a Fk(J)9 b Fo(!)2379 4278 y(~)-51 b Fk(c)2419 4293 y Fi(0)2477 4278 y Fo(~)g Fk(c)2517 4237 y Fi(3)2517 4303 y(2)2606 4161 y Fd(Z)2689 4187 y Fh(t)2652 4349 y Fg(\0001)2798 4278 y Fj(h)p Fk(s)p Fj(i)2922 4237 y Fg(\000)p Fh(\014)3040 4278 y Fk(ds:)0 4566 y Fo(A)m(t)33 b(second)h(order,)f(w)m(e)g(obtain)f Fk(c)1227 4581 y Fi(2)1266 4566 y Fo(\()p Fk(t)p Fo(\))c(=)f Fk(c)1550 4515 y Fi([1])1550 4587 y(2)1629 4566 y Fo(\()p Fk(t)p Fo(\))22 b(+)g Fk(c)1902 4515 y Fi([2])1902 4587 y(2)1981 4566 y Fo(\()p Fk(t)p Fo(\),)33 b(where)244 4855 y Fj(k)p Fk(c)336 4804 y Fi([1])336 4877 y(2)414 4855 y Fo(\()p Fk(t)p Fo(\))p Fj(k)83 b(\024)h Fk(D)903 4814 y Fi(2)900 4880 y(3)974 4688 y Fd(s)p 1057 4688 355 4 v 1067 4788 a Fo(\()p Fk(J)32 b Fo(+)22 b(4\)!)p 1067 4832 336 4 v 1190 4924 a Fk(J)9 b Fo(!)1447 4855 y(~)-51 b Fk(c)1487 4870 y Fi(0)1545 4855 y Fo(~)g Fk(c)1585 4814 y Fi(4)1585 4880 y(2)1658 4738 y Fd(Z)1741 4764 y Fh(t)1704 4927 y Fg(\0001)1850 4855 y Fj(h)p Fk(s)p Fj(i)1974 4814 y Fg(\000)p Fh(\014)2092 4855 y Fk(ds)195 b Fo(and)244 5182 y Fj(k)p Fk(c)336 5131 y Fi([2])336 5204 y(2)414 5182 y Fo(\()p Fk(t)p Fo(\))p Fj(k)83 b(\024)h Fk(D)903 5141 y Fi(2)900 5207 y(3)974 5016 y Fd(s)p 1057 5016 355 4 v 1067 5115 a Fo(\()p Fk(J)32 b Fo(+)22 b(6\)!)p 1067 5159 336 4 v 1190 5251 a Fk(J)9 b Fo(!)1447 5182 y(~)-51 b Fk(c)1487 5141 y Fi(2)1487 5207 y(0)1545 5182 y Fo(~)g Fk(c)1585 5141 y Fi(6)1585 5207 y(2)1658 5065 y Fd(Z)1741 5091 y Fh(t)1704 5254 y Fg(\0001)1867 5182 y Fk(ds)1964 5197 y Fi(1)2019 5182 y Fj(h)p Fk(s)2104 5197 y Fi(1)2143 5182 y Fj(i)2182 5141 y Fg(\000)p Fh(\014)2317 5065 y Fd(Z)2400 5091 y Fh(s)2433 5100 y Fc(1)2363 5254 y Fg(\0001)2526 5182 y Fk(ds)2623 5197 y Fi(2)2679 5182 y Fj(h)p Fk(s)2764 5197 y Fi(2)2803 5182 y Fj(i)2842 5141 y Fg(\000)p Fh(\014)2944 5182 y Fk(:)1901 5506 y Fo(32)p eop %%Page: 33 33 33 32 bop 0 -42 a Fo(A)m(t)33 b(third)f(order,)h(w)m(e)g(obtain)f Fk(c)1156 -27 y Fi(3)1195 -42 y Fo(\()p Fk(t)p Fo(\))c(=)f Fk(c)1479 -93 y Fi([1])1479 -20 y(3)1558 -42 y Fo(\()p Fk(t)p Fo(\))22 b(+)g Fk(c)1831 -93 y Fi([2])1831 -20 y(3)1910 -42 y Fo(\()p Fk(t)p Fo(\))g(+)g Fk(c)2183 -93 y Fi([3])2183 -20 y(3)2262 -42 y Fo(\()p Fk(t)p Fo(\),)32 b(where)244 231 y Fj(k)p Fk(c)336 180 y Fi([1])336 253 y(3)414 231 y Fo(\()p Fk(t)p Fo(\))p Fj(k)83 b(\024)h Fk(D)903 190 y Fi(3)900 256 y(3)974 65 y Fd(s)p 1057 65 355 4 v 1067 164 a Fo(\()p Fk(J)32 b Fo(+)22 b(5\)!)p 1067 208 336 4 v 1190 300 a Fk(J)9 b Fo(!)1447 231 y(~)-51 b Fk(c)1487 246 y Fi(0)1545 231 y Fo(~)g Fk(c)1585 190 y Fi(5)1585 256 y(2)1658 114 y Fd(Z)1741 140 y Fh(t)1704 303 y Fg(\0001)1850 231 y Fj(h)p Fk(s)p Fj(i)1974 190 y Fg(\000)p Fh(\014)2092 231 y Fk(ds;)244 558 y Fj(k)p Fk(c)336 508 y Fi([2])336 580 y(3)414 558 y Fo(\()p Fk(t)p Fo(\))p Fj(k)83 b(\024)h Fo(2)17 b Fk(D)969 517 y Fi(3)966 583 y(3)1040 392 y Fd(s)p 1123 392 355 4 v 1133 491 a Fo(\()p Fk(J)31 b Fo(+)22 b(7\)!)p 1133 535 336 4 v 1255 627 a Fk(J)9 b Fo(!)1512 558 y(~)-51 b Fk(c)1552 517 y Fi(2)1552 583 y(0)1610 558 y Fo(~)g Fk(c)1650 517 y Fi(7)1650 583 y(2)1723 441 y Fd(Z)1806 468 y Fh(t)1769 630 y Fg(\0001)1932 558 y Fk(ds)2029 573 y Fi(1)2085 558 y Fj(h)p Fk(s)2170 573 y Fi(1)2209 558 y Fj(i)2248 517 y Fg(\000)p Fh(\014)2383 441 y Fd(Z)2466 468 y Fh(s)2499 477 y Fc(1)2429 630 y Fg(\0001)2592 558 y Fk(ds)2689 573 y Fi(2)2744 558 y Fj(h)p Fk(s)2829 573 y Fi(2)2868 558 y Fj(i)2907 517 y Fg(\000)p Fh(\014)3009 558 y Fk(;)212 b Fo(and)244 844 y Fj(k)p Fk(c)336 793 y Fi([3])336 866 y(3)414 844 y Fo(\()p Fk(t)p Fo(\))p Fj(k)83 b(\024)h Fk(D)903 803 y Fi(3)900 869 y(3)974 677 y Fd(s)p 1057 677 355 4 v 1067 777 a Fo(\()p Fk(J)32 b Fo(+)22 b(9\)!)p 1067 821 336 4 v 1190 912 a Fk(J)9 b Fo(!)1447 844 y(~)-51 b Fk(c)1487 803 y Fi(3)1487 869 y(0)1545 844 y Fo(~)g Fk(c)1585 803 y Fi(9)1585 869 y(2)1641 727 y Fd(Z)1724 753 y Fh(t)1687 916 y Fg(\0001)1833 844 y Fk(ds)1930 859 y Fi(1)1986 844 y Fj(h)p Fk(s)2071 859 y Fi(1)2110 844 y Fj(i)2149 803 y Fg(\000)p Fh(\014)2268 727 y Fd(Z)2351 753 y Fh(s)2384 762 y Fc(1)2314 916 y Fg(\0001)2460 844 y Fk(ds)2557 859 y Fi(2)2613 844 y Fj(h)p Fk(s)2698 859 y Fi(2)2737 844 y Fj(i)2776 803 y Fg(\000)p Fh(\014)2894 727 y Fd(Z)2977 753 y Fh(s)3010 762 y Fc(2)2940 916 y Fg(\0001)3086 844 y Fk(ds)3183 859 y Fi(3)3239 844 y Fj(h)p Fk(s)3324 859 y Fi(3)3363 844 y Fj(i)3402 803 y Fg(\000)p Fh(\014)3504 844 y Fk(:)0 1067 y Fo(Using)32 b(the)h(iden)m(tit)m(y)307 1193 y Fd(Z)390 1219 y Fh(t)353 1381 y Fg(\0001)516 1310 y Fk(ds)613 1325 y Fi(1)668 1310 y Fj(h)p Fk(s)753 1325 y Fi(1)792 1310 y Fj(i)831 1269 y Fg(\000)p Fh(\014)950 1193 y Fd(Z)1033 1219 y Fh(s)1066 1228 y Fc(1)996 1381 y Fg(\0001)1159 1310 y Fk(ds)1256 1325 y Fi(2)1311 1310 y Fj(h)p Fk(s)1396 1325 y Fi(2)1436 1310 y Fj(i)1475 1269 y Fg(\000)p Fh(\014)1610 1310 y Fj(\001)17 b(\001)g(\001)1759 1193 y Fd(Z)1842 1219 y Fh(s)1875 1228 y Ff(n)p Fb(\000)p Fc(1)1805 1381 y Fg(\0001)2033 1310 y Fk(ds)2130 1325 y Fh(n)2193 1310 y Fj(h)p Fk(s)2278 1325 y Fh(n)2325 1310 y Fj(i)2364 1269 y Fg(\000)p Fh(\014)2526 1310 y Fo(=)2690 1242 y(1)p 2672 1286 86 4 v 2672 1378 a Fk(n)p Fo(!)2816 1188 y Fd(\022)2894 1193 y(Z)2977 1219 y Fh(t)2940 1381 y Fg(\0001)3086 1310 y Fj(h)p Fk(s)p Fj(i)3210 1269 y Fg(\000)p Fh(\014)3328 1310 y Fk(ds)3442 1188 y Fd(\023)3503 1204 y Fh(n)3566 1310 y Fk(;)0 1552 y Fo(w)m(e)34 b(get)e(estimates)g(iden)m(tical)f(to) i(\(5.11\),)e(\(5.14\),)h(\(5.15\),)g(\(5.20\),)g(\(5.21\),)g (\(5.22\).)146 1672 y(It)k(is)f(easy)h(to)f(c)m(hec)m(k)j(that)d(the)h (induction)e(can)i(b)s(e)f(carried)g(out)h(exactly)f(as)h(in)f(Section) g(5)g(to)g(giv)m(e)0 1793 y(an)d(analog)f(of)h(Corollary)f(5.3)i(that)f (states)0 1999 y Fa(Lemma)37 b(9.1)49 b Fe(Assume)h(the)g(de)-5 b(c)g(ay)49 b(hyp)-5 b(othesis)50 b(D.)f(The)g(exp)-5 b(ansion)49 b(c)-5 b(o)g(e\016cients)48 b(\(9.5\))h(satisfying)0 2120 y(\(9.3\))34 b(ob)-5 b(ey)35 b(the)g(fol)5 b(lowing)33 b(estimates:)328 2307 y Fk(c)370 2322 y Fh(k)r(;j)464 2307 y Fo(\()p Fk(t)p Fo(\))84 b(=)g(0)p Fk(;)151 b Fe(whenever)133 b Fj(j)p Fk(j)6 b Fj(j)27 b Fk(>)h(J)j Fo(+)22 b(3)p Fk(k)s(;)244 2589 y Fj(k)p Fk(c)336 2538 y Fi([)p Fh(p)p Fi(])336 2614 y Fh(k)415 2589 y Fo(\()p Fk(t)p Fo(\))p Fj(k)82 b(\024)819 2468 y Fd(\022)897 2529 y Fk(k)25 b Fj(\000)e Fo(1)899 2650 y Fk(p)f Fj(\000)h Fo(1)1138 2468 y Fd(\023)1251 2589 y Fk(D)1335 2548 y Fh(k)1332 2614 y Fi(3)1412 2423 y Fd(s)p 1495 2423 410 4 v 1505 2522 a Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(k)s Fo(\)!)p 1505 2566 390 4 v 1654 2658 a Fk(J)9 b Fo(!)1941 2589 y(~)-51 b Fk(c)1981 2542 y Fh(p)1981 2611 y Fi(0)2057 2589 y Fo(~)g Fk(c)2097 2542 y Fh(k)r Fi(+2)p Fh(p)2097 2611 y Fi(2)2310 2407 y Fd(\020)2360 2432 y(R)2415 2458 y Fh(t)2399 2528 y Fg(\0001)2545 2503 y Fk(ds)17 b Fj(h)p Fk(s)p Fj(i)2783 2467 y Fg(\000)p Fh(\014)2884 2407 y Fd(\021)2934 2430 y Fh(p)p 2310 2566 663 4 v 2601 2658 a Fk(k)s Fo(!)0 2819 y Fe(for)35 b Fk(p)27 b Fj(\024)h Fk(k)s Fe(,)35 b(and)319 3101 y Fj(k)p Fk(c)411 3116 y Fh(k)453 3101 y Fo(\()p Fk(t)p Fo(\))p Fj(k)44 b(\024)780 2934 y Fd(s)p 863 2934 410 4 v 873 3033 a Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(k)s Fo(\)!)p 873 3077 390 4 v 1023 3169 a Fk(J)9 b Fo(!)1319 3033 y(~)-51 b Fk(c)1359 2997 y Fh(k)1359 3058 y Fi(2)1418 3033 y Fk(D)1502 2997 y Fh(k)1499 3058 y Fi(3)p 1317 3077 228 4 v 1390 3169 a Fk(k)s Fo(!)1588 2979 y Fd(\022)1649 3101 y Fo(1)22 b(+)i(~)-51 b Fk(c)1860 3116 y Fi(0)1901 3101 y Fo(~)g Fk(c)1941 3059 y Fi(2)1941 3125 y(2)1997 2983 y Fd(Z)2080 3010 y Fh(t)2043 3172 y Fg(\0001)2190 3101 y Fk(ds)17 b Fj(h)p Fk(s)p Fj(i)2428 3059 y Fg(\000)p Fh(\014)2528 2979 y Fd(\023)2590 2995 y Fh(k)r Fg(\000)p Fi(1)2776 3101 y Fo(~)-51 b Fk(c)2816 3116 y Fi(0)2892 3101 y Fo(~)g Fk(c)2932 3059 y Fi(2)2932 3125 y(2)3023 2983 y Fd(Z)3106 3010 y Fh(t)3069 3172 y Fg(\0001)3215 3101 y Fk(ds)17 b Fj(h)p Fk(s)p Fj(i)3453 3059 y Fg(\000)p Fh(\014)3554 3101 y Fk(:)146 3420 y Fo(Our)29 b(next)h(task)g(is)e(to)h(estimate)f(the)i (norm)e(of)g Fk(\030)1928 3435 y Fh(l)1954 3420 y Fo(\()p Fk(x;)17 b(t)p Fo(\).)42 b(W)-8 b(e)30 b(again)d(consider)j(separately) f(the)h(errors)0 3540 y(near)j(the)g(classical)e(orbit)h(and)h(those)g (far)f(from)g(the)h(orbit.)42 b(Let)33 b Fk(b)p Fo(\()p Fk(t)p Fo(\))h(b)s(e)e(a)h(real)f(v)-5 b(alued)32 b(function)g(that)0 3660 y(satis\014es)254 3810 y Fj(h)p Fk(a)p Fo(\()p Fk(t)p Fo(\))p Fj(i)p 254 3854 240 4 v 349 3946 a Fo(4)564 3878 y Fj(\024)60 b Fk(b)p Fo(\()p Fk(t)p Fo(\))h Fj(\024)1062 3810 y(h)p Fk(a)p Fo(\()p Fk(t)p Fo(\))p Fj(i)p 1062 3854 V 1157 3946 a Fo(2)1312 3878 y Fk(;)2360 b Fo(\(9.6\))0 4096 y(for)33 b(all)f Fk(t)e Fj(2)g Fk(I)-19 b(R)q Fo(.)48 b(W)-8 b(e)34 b(de\014ne)h Fk(\037)1135 4111 y Fi(1)1174 4096 y Fo(\()p Fk(x;)17 b(t)p Fo(\))34 b(to)g(b)s(e)g(the)g(c)m (haracteristic)g(function)f(of)h Fj(f)p Fk(x)46 b Fo(:)h Fj(j)p Fk(x)23 b Fj(\000)g Fk(a)p Fo(\()p Fk(t)p Fo(\))p Fj(j)30 b(\024)h Fk(b)p Fo(\()p Fk(t)p Fo(\))p Fj(g)0 4216 y Fo(and)i Fk(\037)251 4231 y Fi(2)290 4216 y Fo(\()p Fk(x;)17 b(t)p Fo(\))28 b(=)g(1)22 b Fj(\000)g Fk(\037)863 4231 y Fi(1)903 4216 y Fo(\()p Fk(x;)17 b(t)p Fo(\).)43 b(Then,)34 b(for)e(some)h(constan)m(ts)j(~)-51 b Fk(c)2333 4231 y Fi(3)2405 4216 y Fo(and)34 b(~)-51 b Fk(c)2636 4231 y Fi(4)2708 4216 y Fo(and)33 b(an)m(y)g Fk(t)28 b Fj(2)g Fk(I)-19 b(R)q Fo(,)33 b(w)m(e)g(ha)m(v)m(e)244 4335 y Fd(\014)244 4385 y(\014)244 4435 y(\014)244 4484 y(\014)244 4534 y(\014)298 4417 y Fk(D)382 4381 y Fh(m)448 4417 y Fk(V)22 b Fo(\()p Fk(\020)608 4432 y Fh(m)674 4417 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\)\))p 298 4461 752 4 v 617 4553 a Fk(m)p Fo(!)1076 4484 y Fk(\037)1137 4499 y Fi(1)1176 4484 y Fo(\()p Fk(x;)g(t)p Fo(\)\))1441 4335 y Fd(\014)1441 4385 y(\014)1441 4435 y(\014)1441 4484 y(\014)1441 4534 y(\014)1552 4484 y Fj(\024)1967 4417 y Fk(v)2014 4432 y Fi(0)2086 4417 y Fk(v)2137 4366 y Fg(j)p Fh(m)p Fg(j)2133 4439 y Fi(1)p 1722 4461 766 4 v 1722 4553 a Fj(h)p Fk(\020)1804 4568 y Fh(m)1870 4553 y Fo(\()p Fk(x;)g(a)p Fo(\()p Fk(t)p Fo(\)\)\))p Fj(i)2284 4524 y Fh(\014)s Fi(+)p Fg(j)p Fh(m)p Fg(j)2530 4484 y Fk(\037)2591 4499 y Fi(1)2631 4484 y Fo(\()p Fk(x;)g(t)p Fo(\))1552 4819 y Fj(\024)1752 4751 y Fo(~)-51 b Fk(c)1792 4766 y Fi(3)1866 4751 y Fo(~)g Fk(c)1906 4700 y Fg(j)p Fh(m)p Fg(j)1906 4773 y Fi(4)p 1722 4795 317 4 v 1722 4887 a Fj(h)p Fk(t)p Fj(i)1835 4858 y Fh(\014)s Fi(+)p Fg(j)p Fh(m)p Fg(j)2049 4819 y Fk(;)0 5056 y Fo(since)33 b(for)f(large)f(times)h(on)h(the)g(supp)s(ort)g(of)f Fk(\037)1724 5071 y Fi(1)1763 5056 y Fo(,)1410 5257 y Fj(j)p Fk(\020)1481 5272 y Fh(m)1547 5257 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\)\))p Fj(j)60 b(\025)h(j)p Fk(a)p Fo(\()p Fk(t)p Fo(\))p Fj(j)p Fk(=)p Fo(4)p Fk(:)1901 5506 y Fo(33)p eop %%Page: 34 34 34 33 bop 0 -43 a Fo(Therefore,)34 b(for)e(some)g(constan)m(ts)k(~)-51 b Fk(c)1336 -28 y Fi(5)1408 -43 y Fo(and)35 b(~)-51 b Fk(c)1640 -28 y Fi(6)1679 -43 y Fo(,)546 98 y Fd(\015)546 148 y(\015)546 198 y(\015)546 248 y(\015)546 298 y(\015)609 248 y Fk(\037)670 263 y Fi(1)709 248 y Fo(\()p Fk(x;)17 b(t)p Fo(\))946 181 y Fk(D)1030 144 y Fh(m)1096 181 y Fk(V)22 b Fo(\()p Fk(\020)1256 196 y Fh(m)1322 181 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\)\))p 946 225 752 4 v 1265 316 a Fk(m)p Fo(!)1724 248 y Fk(X)8 b Fo(\()p Fk(t)p Fo(\))1924 207 y Fh(m)2007 248 y Fk(P)2070 263 y Fg(j)p Fh(j)t Fg(j\024)p Fh(n)2260 98 y Fd(\015)2260 148 y(\015)2260 198 y(\015)2260 248 y(\015)2260 298 y(\015)2366 248 y Fj(\024)2504 81 y Fd(s)p 2587 81 442 4 v 2597 181 a Fo(\()p Fk(n)22 b Fo(+)g Fj(j)p Fk(m)p Fj(j)p Fo(\)!)p 2597 225 423 4 v 2765 316 a Fk(n)p Fo(!)3073 181 y(~)-51 b Fk(c)3113 196 y Fi(5)3171 181 y Fo(~)g Fk(c)3211 130 y Fg(j)p Fh(m)p Fg(j)3211 202 y Fi(6)p 3071 225 246 4 v 3114 316 a Fj(h)p Fk(t)p Fj(i)3227 287 y Fh(\014)3327 248 y Fk(;)0 515 y Fo(for)32 b(an)m(y)h Fk(t)28 b Fj(2)g Fk(I)-19 b(R)q Fo(.)44 b(W)-8 b(e)33 b(no)m(w)g(mimic)c(the)k (manipulations)d(p)s(erformed)i(in)g(Section)g(6)h(to)f(get)487 710 y Fj(k)17 b Fk(\037)615 725 y Fi(1)654 710 y Fo(\()p Fk(x;)g(t)p Fo(\))g Fk(\030)924 725 y Fh(l)950 710 y Fo(\()p Fk(x;)g(t;)g Fo(\026)-50 b Fk(h)p Fo(\))17 b Fj(k)2373 b Fo(\(9.7\))327 980 y Fj(\024)487 831 y Fd(\015)487 881 y(\015)487 931 y(\015)487 980 y(\015)487 1030 y(\015)575 873 y Fh(l)q Fg(\000)p Fi(1)571 897 y Fd(X)567 1082 y Fh(k)r Fi(=0)850 897 y Fd(X)729 1086 y Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)1124 980 y Fo(\026)-50 b Fk(h)1179 938 y Fh(k)r(=)p Fi(2)1325 980 y Fk(\037)1386 995 y Fi(1)1426 980 y Fo(\()p Fk(x;)17 b(t)p Fo(\))1679 913 y(\()p Fk(D)1801 877 y Fh(m)1867 913 y Fk(V)k Fo(\)\()p Fk(\020)2064 928 y Fh(m)2130 913 y Fo(\()p Fk(x;)c(a)p Fo(\()p Fk(t)p Fo(\)\))p 1679 957 790 4 v 2017 1049 a Fk(m)p Fo(!)2510 980 y(\026)-50 b Fk(h)2565 938 y Fg(j)p Fh(m)p Fg(j)p Fh(=)p Fi(2)2758 980 y Fo(\()p Fk(x)23 b Fj(\000)f Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3173 939 y Fh(m)3257 980 y Fo(\026)-50 b Fk(h)3312 938 y Fg(\000j)p Fh(m)p Fg(j)p Fh(=)p Fi(2)1268 1293 y Fj(\002)98 b Fk(P)1506 1309 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1949 1210 y Fd(X)1859 1398 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)g Fi(+3)p Fh(k)2192 1293 y Fk(c)2234 1308 y Fh(k)r(;j)2329 1293 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\036)2515 1308 y Fh(j)2551 1293 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3673 1144 y Fd(\015)3672 1194 y(\015)3672 1243 y(\015)3672 1293 y(\015)3672 1343 y(\015)327 1669 y Fj(\024)496 1561 y Fh(l)q Fg(\000)p Fi(1)492 1586 y Fd(X)487 1770 y Fh(k)r Fi(=0)665 1669 y Fo(\026)-50 b Fk(h)720 1626 y Fi(\()p Fh(l)q Fi(+2\))p Fh(=)p Fi(2)995 1547 y Fd(\022)1073 1608 y Fk(d)22 b Fj(\000)g Fo(1)g(+)g Fk(l)j Fo(+)d(2)g Fj(\000)g Fk(k)1321 1729 y(d)g Fj(\000)g Fo(1)1807 1547 y Fd(\023)1917 1669 y Fj(k)p Fk(c)2009 1684 y Fh(k)2052 1669 y Fo(\()p Fk(t)p Fo(\))p Fj(k)878 1961 y(\002)400 b Fo(max)1167 2025 y Fg(f)p Fh(m)12 b Fi(:)32 b Fg(j)p Fh(m)p Fg(j)p Fi(=)p Fh(l)q Fi(+2)p Fg(\000)p Fh(k)r Fg(g)1775 1812 y Fd(\015)1775 1861 y(\015)1775 1911 y(\015)1775 1961 y(\015)1775 2011 y(\015)1837 1961 y Fk(\037)1898 1976 y Fi(1)1938 1961 y Fo(\()p Fk(x;)17 b(t)p Fo(\))2175 1894 y Fk(D)2259 1858 y Fh(m)2325 1894 y Fk(V)22 b Fo(\()p Fk(\020)2485 1909 y Fh(m)2551 1894 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\)\))p 2175 1938 752 4 v 2494 2029 a Fk(m)p Fo(!)2953 1961 y Fk(X)8 b Fo(\()p Fk(t)p Fo(\))3153 1920 y Fh(m)3236 1961 y Fk(P)3299 1977 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)e Fi(+3)p Fh(k)3619 1812 y Fd(\015)3619 1861 y(\015)3619 1911 y(\015)3619 1961 y(\015)3619 2011 y(\015)327 2267 y Fj(\024)496 2159 y Fh(l)q Fg(\000)p Fi(1)492 2184 y Fd(X)487 2369 y Fh(k)r Fi(=0)649 2267 y Fo(\026)-50 b Fk(h)704 2224 y Fi(\()p Fh(l)q Fi(+2\))p Fh(=)p Fi(2)979 2146 y Fd(\022)1057 2207 y Fk(d)22 b Fj(\000)g Fo(1)g(+)g Fk(l)j Fo(+)d(2)g Fj(\000)g Fk(k)1305 2328 y(d)g Fj(\000)h Fo(1)1791 2146 y Fd(\023)1869 2101 y(s)p 1952 2101 730 4 v 1962 2200 a Fo(\()p Fk(J)31 b Fo(+)22 b(2)p Fk(k)j Fo(+)d Fk(l)i Fo(+)e(2\)!)p 1962 2244 710 4 v 2271 2336 a Fk(J)9 b Fo(!)2726 2200 y(~)-51 b Fk(c)2766 2215 y Fi(5)2824 2200 y Fo(~)g Fk(c)2864 2159 y Fh(l)q Fi(+2)p Fg(\000)p Fh(k)2864 2222 y Fi(6)p 2724 2244 350 4 v 2818 2336 a Fj(h)p Fk(t)p Fj(i)2931 2307 y Fh(\014)1073 2562 y Fj(\002)1260 2494 y Fo(~)g Fk(c)1300 2458 y Fh(k)1300 2519 y Fi(2)1359 2494 y Fk(D)1443 2458 y Fh(k)1440 2519 y Fi(3)p 1258 2539 228 4 v 1331 2630 a Fk(k)s Fo(!)1512 2441 y Fd(\022)1573 2562 y Fo(1)22 b(+)i(~)-51 b Fk(c)1784 2577 y Fi(0)1825 2562 y Fo(~)g Fk(c)1865 2521 y Fi(2)1865 2587 y(2)1921 2445 y Fd(Z)2004 2471 y Fh(t)1967 2633 y Fg(\0001)2114 2562 y Fk(ds)17 b Fj(h)p Fk(s)p Fj(i)2352 2521 y Fg(\000)p Fh(\014)2452 2441 y Fd(\023)2514 2456 y Fh(k)r Fg(\000)p Fi(1)2682 2562 y Fo(~)-51 b Fk(c)2722 2577 y Fi(0)2780 2562 y Fo(~)g Fk(c)2820 2521 y Fi(2)2820 2587 y(2)2892 2445 y Fd(Z)2975 2471 y Fh(t)2938 2633 y Fg(\0001)3085 2562 y Fk(ds)17 b Fj(h)p Fk(s)p Fj(i)3323 2521 y Fg(\000)p Fh(\014)3423 2562 y Fk(:)0 2798 y Fo(Making)32 b(use)i(of)e(\(5.30\),)g(the)h (de\014nition)e(of)h Fk(D)1734 2813 y Fi(3)1774 2798 y Fo(,)g(and)h(in)m(tro)s(ducing)e(another)i(constan)m(t)632 3041 y(~)-51 b Fk(c)672 3056 y Fi(7)771 3041 y Fo(=)907 2944 y Fd(\020)973 3041 y Fo(1)39 b(+)i(~)-51 b Fk(c)1218 3056 y Fi(0)1276 3041 y Fo(~)g Fk(c)1316 3000 y Fi(2)1316 3065 y(2)1372 3041 y Fk(I)1439 2944 y Fd(\021)1507 3041 y Fo(~)g Fk(c)1547 3056 y Fi(2)1587 3041 y Fk(=)r Fo(~)g Fk(c)1678 3056 y Fi(6)1717 3041 y Fk(;)244 b 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Fi(2)1750 5137 y Fo(\()p Fk(x;)17 b(t)p Fo(\))p Fk(\030)2003 5152 y Fh(l)2028 5137 y Fo(\()p Fk(x;)g(t)p Fo(\))p Fj(k)p Fo(,)41 b(whic)m(h)e(is)f(a)h(little)d(bit)i(more)g(elab)s(orate.)0 5257 y(The)d(di\016cult)m(y)f(stems)h(from)e(a)h(lac)m(k)g(of)g (su\016cien)m(t)h(information)c(on)j(the)h(p)s(osition)d(of)i Fk(\020)3291 5272 y Fh(m)3357 5257 y Fo(\()p Fk(x;)17 b(a)p Fo(\()p Fk(t)p Fo(\)\).)49 b(So,)1901 5506 y(34)p eop %%Page: 35 35 35 34 bop 0 -43 a Fo(instead)33 b(of)f(the)h(usual)f(T)-8 b(a)m(ylor)32 b(series)h(error)g(form)m(ula,)e(w)m(e)i(use)h(the)f (de\014nition)911 210 y Fk(W)1017 159 y Fi(\()p Fh(q)r Fi(\))1003 239 y Fh(a)p Fi(\()p Fh(t)p Fi(\))1125 210 y Fo(\()p Fk(x)p Fo(\))28 b(=)g Fk(V)21 b Fo(\()p Fk(x)p Fo(\))39 b Fj(\000)1788 127 y Fd(X)1753 315 y Fg(j)p Fh(m)p Fg(j\024)p Fh(q)1986 143 y Fk(D)2070 106 y Fh(m)2136 143 y Fk(V)22 b Fo(\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))p 1986 187 468 4 v 2163 278 a Fk(m)p Fo(!)2480 210 y(\()p Fk(x)g Fj(\000)h Fk(a)p Fo(\()p Fk(t)p 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Fi(2)2746 955 y Fk(dx)455 1210 y Fo(=)614 1093 y Fd(Z)660 1282 y Fg(j)p Fh(z)s Fg(j\025)p Fh(b)p Fi(\()p Fh(t)p Fi(\))938 1210 y Fj(j)p Fk(\036)1024 1225 y Fh(j)1059 1210 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b Fo(0)p Fk(;)g Fo(0)p Fk(;)g(z)t Fo(\))p Fj(j)1959 1169 y Fi(2)2014 1210 y Fk(V)22 b Fo(\()p Fk(z)27 b Fo(+)22 b Fk(a)p Fo(\()p Fk(t)p Fo(\)\))2501 1169 y Fi(2)2557 1210 y Fk(dz)t(:)0 1445 y Fo(W)-8 b(e)33 b(use)h(the)f(form)m(ula)554 1697 y Fj(j)p Fk(\036)640 1712 y Fh(j)676 1697 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b Fo(0)p Fk(;)g Fo(0)p Fk(;)g(z)t Fo(\))p Fj(j)59 b Fo(=)1781 1629 y(e)1824 1593 y Fg(\000)p Fi(\()p Fg(j)p Fh(A)p Fg(j)1999 1570 y Fb(\000)p Fc(1)2081 1593 y Fi(\()p Fh(t)p Fi(\))p Fh(z)s Fi(\))2223 1570 y Fc(2)2259 1593 y Fh(=)p Fi(2\026)-35 b Fh(h)2391 1629 y Fj(jH)2503 1644 y Fh(j)2540 1629 y Fo(\()p Fk(A)p Fo(;)17 b 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1252 2743 1568 4 v 1532 2846 a Fk(j)6 b Fo(!)17 b(2)1671 2817 y Fg(j)p Fh(j)t Fg(j)1763 2846 y Fk(\031)1822 2817 y Fh(d=)p Fi(2)1949 2846 y Fj(j)g Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)g Fo(\026)-50 b Fk(h)2429 2803 y Fh(d=)p Fi(2)2862 2767 y Fk(V)22 b Fo(\()p Fk(z)k Fo(+)c Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3348 2725 y Fi(2)3405 2767 y Fk(dz)t(;)0 3055 y Fo(for)36 b(some)h(\014nite,)h(p)s(ositiv)m(e)1046 3029 y(~)1049 3055 y Fk(b)q Fo(.)57 b(Note)37 b(that)f(this)h(estimate)f(has)i(a)e (uniform)g(exp)s(onen)m(tially)f(decreasing)0 3175 y(prefactor.)146 3296 y(As)e(in)f(Section)h(6,)f(w)m(e)i(use)f(spherical)f(co)s (ordinates)g(and)h(the)g(decomp)s(osition)e(\(6.20\))1042 3499 y(\012)1112 3514 y Fh(j)1149 3499 y Fo(\()p Fk(y)t Fo(\))59 b(=)1699 3416 y Fd(X)1472 3605 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2062 3499 y Fk(d)2113 3514 y Fh(j;q)r(;n;m)2360 3499 y Fk( )2423 3514 y Fh(q)r(;n;m)2606 3499 y Fo(\()p Fk(r)m(;)17 b(!)t Fo(\))p Fk(;)0 3880 y Fo(where)512 3797 y Fd(X)286 3985 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)876 3880 y Fj(j)p Fk(d)955 3895 y Fh(j;q)r(;n;m)1185 3880 y Fj(j)1213 3838 y Fi(2)1322 3880 y Fo(=)70 b(1)36 b(and)h(\012)1817 3895 y Fh(j)1854 3880 y Fo(\()p Fk(y)t Fo(\))69 b(=)2197 3728 y Fd(s)p 2280 3728 421 4 v 2466 3812 a Fo(1)p 2290 3856 401 4 v 2290 3948 a(2)2339 3919 y Fg(j)p Fh(j)t Fg(j)2431 3948 y Fk(j)6 b Fo(!)17 b Fk(\031)2580 3919 y Fh(d=)p Fi(2)2737 3880 y Fj(H)2821 3895 y Fh(j)2858 3880 y Fo(\()p Fk(A)p Fo(;)33 b Fk(y)t Fo(\))17 b(e)3178 3838 y Fg(\000)p Fh(y)3270 3815 y Fc(2)3305 3838 y Fh(=)p Fi(2)3380 3880 y Fo(.)55 b(This)36 b(leads)0 4087 y(to)c(the)h (estimate)212 4338 y Fk(N)300 4297 y Fi(2)400 4338 y Fj(\024)61 b Fo(e)581 4297 y Fg(\000)633 4279 y Fi(~)636 4297 y Fh(b=)p Fi(\026)-35 b Fh(h)990 4255 y Fd(X)763 4443 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)1370 4221 y Fd(Z)1416 4409 y Fg(j)p Fh(z)s Fg(j\025)p Fh(b)p Fi(\()p Fh(t)p Fi(\))1704 4270 y Fo(e)1747 4234 y Fi(+\()p Fg(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)2001 4211 y Fb(\000)p Fc(1)2084 4234 y Fh(z)s Fi(\))2147 4211 y Fc(2)2182 4234 y Fh(=)p Fi(2\026)-35 b Fh(h)2314 4270 y Fj(j)17 b Fk( )2422 4285 y Fh(q)r(;n;m)2604 4270 y Fo(\()p Fk(r)2686 4285 y Fh(z)2726 4270 y Fk(;)g(!)2831 4285 y Fh(z)2870 4270 y Fo(\))p Fj(j)2936 4234 y Fi(2)p 1704 4315 1272 4 v 2044 4417 a Fj(j)g Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)h Fo(\026)-50 b Fk(h)2524 4374 y Fh(d=)p Fi(2)3018 4338 y Fk(V)21 b Fo(\()p Fk(z)27 b Fo(+)22 b Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3504 4297 y Fi(2)3560 4338 y Fk(dz)t(;)0 4652 y Fo(where)34 b(the)f(spherical)f(co)s(ordinates)g(\()p Fk(r)1457 4667 y Fh(z)1497 4652 y Fk(;)17 b(!)1602 4667 y Fh(z)1641 4652 y Fo(\))32 b(describ)s(e)h(the)g(v)m(ector)h(\026)-50 b Fk(h)2604 4609 y Fi(1)p Fh(=)p Fi(2)2731 4652 y Fj(j)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)2971 4616 y Fg(\000)p Fi(1)3081 4652 y Fk(z)t Fo(.)146 4773 y(W)-8 b(e)42 b(c)m(ho)s(ose)h Fk(p)g(>)g Fo(2,)h(suc)m(h)f(that)e Fk(d=\014)65 b(<)59 b(p)h(<)g(d)p Fo(,)43 b(and)f(de\014ne)h Fk(s)g(>)g Fo(2)e(b)m(y)i(1)p Fk(=s)h Fo(+)h(1)p Fk(=p)59 b Fo(=)h(1)p Fk(=)p Fo(2.)0 4893 y(Applying)32 b(H\177)-49 b(older's)32 b(inequalit)m(y)-8 b(,)32 b(w)m(e)h(get)g(the)g(b)s(ound)73 5177 y Fk(N)161 5136 y Fi(2)262 5177 y Fj(\024)601 5110 y Fo(e)644 5073 y Fg(\000)696 5056 y Fi(~)699 5073 y Fh(b=)p Fi(\026)-35 b Fh(h)p 409 5154 592 4 v 409 5256 a Fj(j)17 b Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)g Fo(\026)-50 b Fk(h)889 5213 y Fh(d=)p Fi(2)1254 5094 y Fd(X)1027 5282 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)1617 5177 y Fj(k)p Fk(V)21 b Fj(k)1795 5136 y Fi(2)1795 5202 y Fh(p)1852 5031 y Fd( )1917 5060 y(Z)1963 5248 y Fg(j)p Fh(z)s Fg(j\025)p Fh(b)p Fi(\()p Fh(t)p Fi(\))2224 5177 y Fo(e)2268 5136 y Fi(+\()p Fg(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)2522 5112 y Fb(\000)p Fc(1)2605 5136 y Fh(z)s Fi(\))2668 5112 y Fc(2)2703 5136 y Fh(s=)p Fi(4\026)-35 b Fh(h)2867 5177 y Fj(j)p Fk( )2958 5192 y Fh(q)r(;n;m)3141 5177 y Fo(\()p Fk(r)3223 5192 y Fh(z)3263 5177 y Fk(;)17 b(!)3368 5192 y Fh(z)3407 5177 y Fo(\))p Fj(j)3473 5136 y Fh(s)3509 5177 y Fk(dz)3609 5031 y Fd(!)3675 5054 y Fi(2)p Fh(=s)3799 5177 y Fk(:)1901 5506 y Fo(35)p eop %%Page: 36 36 36 35 bop 0 -43 a Fo(W)-8 b(e)29 b(need)h(to)f(b)s(ound)g(the)h(in)m (tegral)d(in)h(this)h(expression.)44 b(W)-8 b(e)29 b(c)m(hange)h(v)-5 b(ariables)27 b(to)i Fk(y)i Fo(=)c Fj(j)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)3542 -79 y Fg(\000)p Fi(1)3636 -43 y Fk(z)t(=)p Fo(\026)-50 b Fk(h)3790 -86 y Fi(1)p Fh(=)p Fi(2)0 77 y Fo(and)30 b(use)g(the)g(estimate)f Fj(j)16 b(j)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)p Fk(y)j Fj(j)44 b(\024)h Fk(b)p Fo(\()p Fk(t)p Fo(\))p Fk(=)1651 -7 y Fj(p)p 1734 -7 56 4 v 84 x Fo(\026)-50 b Fk(h)p Fo(,)31 b(whic)m(h)f(is)f(v)-5 b(alid)27 b(when)k Fj(j)p Fk(y)t Fj(j)26 b(\024)i Fk(b)p Fo(\()p Fk(t)p Fo(\))p Fk(=)p Fo(\()p Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)3464 -7 y(p)p 3547 -7 V 84 x Fo(\026)-50 b Fk(h)p Fo(\).)43 b(This)0 198 y(yields)487 308 y Fd(Z)534 497 y Fg(j)p Fh(z)s Fg(j\025)p Fh(b)p Fi(\()p Fh(t)p Fi(\))827 425 y Fo(e)870 384 y Fi(+\()p Fg(j)p Fh(A)p Fi(\()p Fh(t)p Fi(\))p Fg(j)1124 361 y Fb(\000)p Fc(1)1208 384 y Fh(z)s Fi(\))1271 361 y Fc(2)1305 384 y Fh(s=)p Fi(4\026)-35 b Fh(h)1486 425 y Fj(j)p Fk( )1577 440 y Fh(q)r(;n;m)1760 425 y Fo(\()p Fk(r)1842 440 y Fh(z)1881 425 y Fk(;)17 b(!)1986 440 y Fh(z)2025 425 y Fo(\))p Fj(j)2091 384 y Fh(s)2144 425 y Fk(dz)327 697 y Fj(\024)487 580 y Fd(Z)534 769 y Fg(j)p Fh(y)r Fg(j\025)758 735 y Ff(b)p Fc(\()p Ff(t)p Fc(\))p 675 754 266 4 v 675 805 a Fb(k)p Ff(A)p Fc(\()p Ff(t)p Fc(\))p Fb(k)857 762 y(p)p 906 762 35 3 v 906 805 a Fc(\026)-32 b Ff(h)988 697 y Fo(e)1032 656 y Fi(+)p Fh(y)1124 633 y Fc(2)1158 656 y Fh(s=)p Fi(4)1298 697 y Fj(j)p Fk( )1389 712 y Fh(q)r(;n;m)1572 697 y Fo(\()p Fk(r)m(;)17 b(!)t Fo(\))p Fj(j)1826 656 y Fh(s)1894 697 y Fj(j)g Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)h Fo(\026)-50 b Fk(h)2374 654 y Fh(d=)p Fi(2)2501 697 y Fk(dy)t(;)1068 b Fo(\(9.9\))0 984 y(where)40 b(the)f(spherical)f(co)s(ordinates)h(\()p Fk(r)m(;)17 b(!)t Fo(\))37 b(no)m(w)j(describ)s(e)f(the)g(v)m(ector)h Fk(y)t Fo(.)61 b(Note)39 b(that)g(w)m(e)h(ha)m(v)m(e)g(used)0 1104 y(det\()p Fj(j)p Fk(A)p Fj(j)p Fo(\))28 b(=)f Fj(j)17 b Fo(det\()p Fk(A)p Fo(\))p Fj(j)p Fo(,)32 b(whic)m(h)h(follo)m(ws)e (from)h Fk(A)c Fo(=)f Fk(U)1987 1119 y Fh(A)2061 1104 y Fj(j)p Fk(A)p Fj(j)p Fo(,)32 b(where)i Fk(U)2597 1119 y Fh(A)2687 1104 y Fo(is)e(unitary)-8 b(.)146 1224 y(Since)34 b Fk(b)p Fo(\()p Fk(t)p Fo(\))p Fk(=)p Fj(k)p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(k)f Fo(has)g(a)g(strictly)f(p)s(ositiv)m(e)h (in\014m)m(um)2251 1198 y(\026)2254 1224 y Fk(b)q Fo(,)g(and)g(w)m(e)i (ultimately)30 b(c)m(ho)s(ose)k Fk(l)d Fj(')e Fk(g)t(=)3736 1140 y Fj(p)p 3818 1140 56 4 v 3818 1224 a Fo(\026)-50 b Fk(h)p Fo(,)0 1345 y(with)45 b Fk(g)k Fo(arbitrarily)42 b(small,)47 b(w)m(e)f(can)g(assume)g(the)g(in)m(tegration)d(in)i (\(9.9\))g(is)g(within)f(the)i(classically)0 1465 y(forbidden)29 b(region)g(where)i(Lemma)d(4.2)h(applies,)g(for)g(all)f(indices)h Fj(f)p Fk(q)t(;)17 b(n;)g(m)27 b Fo(:)45 b(2)p Fk(n)16 b Fo(+)g Fk(q)31 b Fo(=)d Fj(j)p Fk(j)6 b Fj(jg)28 b Fo(of)h(in)m(terest.)146 1586 y(Hence,)h(manipulations)24 b(similar)f(to)k(those)g(p)s(erformed)f(in)h(Section)f(6,)i(sho)m(w)g (that)f(\(9.9\))f(is)g(b)s(ounded)0 1706 y(ab)s(o)m(v)m(e)33 b(b)m(y)589 1909 y Fj(j)17 b Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)h Fo(\026)-50 b Fk(h)1069 1866 y Fh(d=)p Fi(2)1197 1909 y Fo(2)1246 1873 y Fh(s=)p Fi(2)p 497 1953 948 4 v 497 2045 a Fo(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))1129 2016 y Fh(s=)p Fi(2)1252 2045 y Fk(n)p Fo(!)1337 2016 y Fh(s=)p Fi(2)1504 1860 y Fd(Z)1550 2048 y Fh(S)1597 2029 y Ff(d)p Fb(\000)p Fc(1)1748 1977 y Fk(d!)e Fj(j)p Fk(Y)1965 1992 y Fh(q)r(;m)2084 1977 y Fj(j)2112 1936 y Fh(s)2182 1860 y Fd(Z)2265 1886 y Fg(1)2225 2031 y Fi(\026)2228 2048 y Fh(b=)2293 1991 y Fg(p)p 2352 1991 41 3 v 57 x Fi(\026)-35 b Fh(h)2430 1977 y Fk(dr)19 b(r)2591 1936 y Fh(d)p Fg(\000)p Fi(1+)p Fh(sq)r Fi(+2)p Fh(sn)3025 1977 y Fo(e)3068 1936 y Fg(\000)p Fh(sr)3190 1912 y Fc(2)3224 1936 y Fh(=)p Fi(4)327 2296 y Fj(\024)497 2229 y(j)e Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)g Fo(\026)-50 b Fk(h)977 2186 y Fh(d=)p Fi(2)1105 2229 y Fo(\(2)p Fk(=s)p Fo(\))1325 2193 y Fh(d)p Fi(+)p Fh(s)p Fg(j)p Fh(j)t Fg(j)1541 2229 y Fo(2)1590 2193 y Fh(s=)p Fi(2)p 497 2273 1200 4 v 623 2365 a Fo(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))1255 2336 y Fh(s=)p Fi(2)1378 2365 y Fk(n)p Fo(!)1463 2336 y Fh(s=)p Fi(2)1756 2179 y Fd(Z)1802 2368 y Fh(S)1849 2349 y Ff(d)p Fb(\000)p Fc(1)2000 2296 y Fk(d!)e Fj(j)p Fk(Y)2217 2311 y Fh(q)r(;m)2336 2296 y Fj(j)2364 2255 y Fh(s)2434 2179 y Fd(Z)2517 2206 y Fg(1)2480 2368 y Fi(0)2625 2296 y Fk(dz)h(z)2791 2255 y Fh(d)p Fg(\000)p Fi(1+)p Fh(s)p Fg(j)p Fh(j)t Fg(j)3098 2296 y Fo(e)3141 2255 y Fg(\000)p Fh(z)3232 2232 y Fc(2)327 2616 y Fj(\024)497 2549 y(j)c Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)g Fo(\026)-50 b Fk(h)977 2506 y Fh(d=)p Fi(2)1105 2549 y Fo(\(2)p Fk(=s)p Fo(\))1325 2513 y Fh(d)p Fi(+)p Fh(s)p Fg(j)p Fh(j)t Fg(j)1541 2549 y Fo(2)1590 2513 y Fh(s=)p Fi(2)p 497 2593 V 623 2685 a Fo(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))1255 2656 y Fh(s=)p Fi(2)1378 2685 y Fk(n)p Fo(!)1463 2656 y Fh(s=)p Fi(2)1756 2499 y Fd(Z)1802 2688 y Fh(S)1849 2669 y Ff(d)p Fb(\000)p Fc(1)2000 2616 y Fk(d!)e Fj(j)p Fk(Y)2217 2631 y Fh(q)r(;m)2336 2616 y Fj(j)2364 2575 y Fh(s)2417 2616 y Fo(\000)2495 2470 y Fd( )2571 2549 y Fk(d)h Fo(+)i Fk(s)p Fj(j)p Fk(j)6 b Fj(j)p 2571 2593 319 4 v 2705 2685 a Fo(2)2899 2470 y Fd(!)2981 2616 y Fk(=)p Fo(2)p Fk(:)0 2876 y Fo(This)33 b(implies)d(the)j(estimate)244 3159 y Fk(N)332 3118 y Fi(2)455 3159 y Fj(\024)625 3085 y Fo(e)669 3049 y Fg(\000)721 3032 y Fi(~)724 3049 y Fh(b=)p Fi(\026)-35 b Fh(h)850 3085 y Fj(k)p Fk(V)22 b Fj(k)1029 3049 y Fi(2)1029 3110 y Fh(p)1085 3085 y Fo(\(2)p Fk(=s)p Fo(\))1305 3049 y Fi(2\()p Fh(d=s)p Fi(+)p Fg(j)p Fh(j)t Fg(j)p Fi(\))1646 3085 y Fo(\000)17 b(\()o(\()p Fk(d)22 b Fo(+)g Fk(s)p Fj(j)p Fk(j)6 b Fj(j)p Fo(\))p Fk(=)p Fo(2\))2291 3037 y Fi(2)p Fh(=s)p 625 3136 1773 4 v 906 3239 a Fo(2)955 3210 y Fi(2)p Fh(=s)p Fg(\000)p Fi(1)1169 3239 y Fj(j)17 b Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)1578 3210 y Fi(1)p Fg(\000)p Fi(2)p Fh(=s)1792 3239 y Fo(\026)-50 b Fk(h)1847 3196 y Fh(d=)p Fi(2)p Fg(\000)p Fh(d=s)1591 3528 y Fj(\002)1880 3357 y Fd(0)1880 3506 y(@)2180 3445 y(X)1953 3633 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)2740 3390 y Fd(R)2779 3486 y Fh(S)2826 3467 y Ff(d)p Fb(\000)p Fc(1)2978 3461 y Fk(d!)19 b Fj(j)p Fk(Y)3194 3476 y Fh(q)r(;m)3314 3461 y Fj(j)3342 3424 y Fh(s)p 2585 3505 948 4 v 2585 3596 a Fo(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))3217 3568 y Fh(s=)p Fi(2)3340 3596 y Fk(n)p Fo(!)3425 3568 y Fh(s=)p Fi(2)3542 3357 y Fd(1)3542 3506 y(A)3615 3380 y Fi(2)p Fh(=s)3739 3528 y Fk(:)0 3819 y Fo(W)-8 b(e)31 b(b)s(ound)f(the)g(in)m(tegral)f(in)g(this)h(expression)h(b)m(y) g(using)f(the)h(follo)m(wing)c(crude)k(lemma.)41 b(Its)30 b(pro)s(of)g(is)f(at)0 3939 y(the)k(end)g(of)f(this)h(section.)0 4167 y Fa(Lemma)k(9.2)49 b Fe(F)-7 b(or)34 b(some)g(c)-5 b(onstants)34 b Fk(M)1542 4182 y Fi(0)1617 4167 y Fe(and)g Fk(M)1900 4182 y Fi(1)1940 4167 y Fe(,)h(we)f(have)244 4371 y Fj(j)16 b Fk(Y)345 4386 y Fh(q)r(;m)465 4371 y Fo(\()p Fk(!)t Fo(\))h Fj(j)61 b(\024)i Fk(M)946 4386 y Fi(0)1021 4371 y Fk(M)1125 4323 y Fh(q)1115 4392 y Fi(1)1163 4371 y Fk(:)2461 b Fo(\(9.10\))0 4721 y(W)-8 b(e)33 b(use)h(this)e(and)g(the)h(inequalities)e Fk(q)h Fj(\024)c(j)p Fk(j)6 b Fj(j)32 b Fo(and)1900 4599 y Fd(\022)1978 4660 y Fj(j)p Fk(j)6 b Fj(j)2000 4781 y Fk(n)2096 4599 y Fd(\023)2185 4721 y Fj(\024)28 b Fo(2)2339 4684 y Fg(j)p Fh(j)t Fg(j)2447 4721 y Fo(to)k(estimate)487 4924 y Fd(0)487 5073 y(@)803 5012 y(X)577 5200 y Fg(f)p Fh(q)r(;n;m)p Fi(:)11 b(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)1348 4956 y Fd(R)1387 5053 y Fh(S)1434 5034 y Ff(d)p Fb(\000)p Fc(1)1586 5027 y Fk(d!)19 b Fj(j)p Fk(Y)1802 5042 y Fh(q)r(;m)1922 5027 y Fj(j)1950 4991 y Fh(s)p 1193 5072 V 1193 5163 a Fo(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))1825 5134 y Fh(s=)p Fi(2)1948 5163 y Fk(n)p Fo(!)2033 5134 y Fh(s=)p Fi(2)2167 4924 y Fd(1)2167 5073 y(A)2240 4947 y Fi(2)p Fh(=s)1901 5506 y Fo(36)p eop %%Page: 37 37 37 36 bop 327 56 a Fj(\024)497 -26 y Fk(M)601 -62 y Fi(2)591 -1 y(0)658 -26 y Fj(j)p Fk(S)752 -62 y Fh(d)p Fg(\000)p Fi(1)882 -26 y Fj(j)910 -62 y Fi(2)p Fh(=s)1034 -26 y Fo(\(2)p Fk(dM)1276 -62 y Fi(2)1266 -1 y(1)1316 -26 y Fo(\))1354 -62 y Fg(j)p Fh(j)t Fg(j)1446 -26 y Fk(m)1531 -77 y Fi(2)p Fh(=s)1531 3 y Fg(j)p Fh(j)t Fg(j)p 497 33 1142 4 v 983 124 a Fk(\031)1042 96 y Fh(d=)p Fi(2)1698 -115 y Fd(0)1698 34 y(@)1787 -61 y Fg(j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)1798 -27 y Fd(X)1791 154 y Fh(n)p Fi(=0)2449 -11 y Fo(1)p 1972 33 1004 4 v 1972 124 a(\000\()p Fj(j)p Fk(j)6 b Fj(j)21 b(\000)i Fk(n)f Fo(+)g Fk(d=)p Fo(2\))2659 96 y Fh(s=)p Fi(2)2783 124 y Fk(n)p Fo(!)2868 96 y Fh(s=)p Fi(2)3002 -115 y Fd(1)3002 34 y(A)3074 -93 y Fi(2)p Fh(=s)327 446 y Fj(\024)497 364 y Fk(M)601 328 y Fi(2)591 388 y(0)658 364 y Fj(j)p Fk(S)752 328 y Fh(d)p Fg(\000)p Fi(1)882 364 y Fj(j)910 328 y Fi(2)p Fh(=s)1034 364 y Fo(\(2)p Fk(dM)1276 328 y Fi(2)1266 388 y(1)1316 364 y Fo(\))1354 328 y Fg(j)p Fh(j)t Fg(j)1446 364 y Fk(m)1531 313 y Fi(2)p Fh(=s)1531 393 y Fg(j)p Fh(j)t Fg(j)p 497 422 1142 4 v 842 514 a Fk(\031)901 485 y Fh(d=)p Fi(2)1029 514 y Fk(C)1106 485 y Fg(00)1165 514 y Fj(j)p Fk(j)6 b Fj(j)p Fo(!)1698 275 y Fd(0)1698 424 y(@)1787 329 y Fg(j)p Fh(j)t Fg(j)p Fh(=)p Fi(2)1798 363 y Fd(X)1791 544 y Fh(n)p Fi(=0)1962 325 y Fd(\022)2040 386 y Fj(j)p Fk(j)g Fj(j)2061 506 y Fk(n)2158 325 y Fd(\023)2219 337 y Fh(s=)p Fi(2)2343 275 y Fd(1)2343 424 y(A)2416 297 y Fi(2)p Fh(=s)327 819 y Fj(\024)497 737 y(j)p Fk(j)g Fj(j)599 701 y Fi(2)p Fh(=s)723 737 y Fk(M)827 701 y Fi(2)817 762 y(0)883 737 y Fj(j)p Fk(S)977 701 y Fh(d)p Fg(\000)p Fi(1)1108 737 y Fj(j)1136 701 y Fi(2)p Fh(=s)1259 737 y Fo(\(4)p Fk(dM)1501 701 y Fi(2)1491 762 y(1)1541 737 y Fo(\))1579 701 y Fg(j)p Fh(j)t Fg(j)1671 737 y Fk(m)1756 686 y Fi(2)p Fh(=s)1756 766 y Fg(j)p Fh(j)t Fg(j)p 497 796 1367 4 v 955 888 a Fk(\031)1014 859 y Fh(d=)p Fi(2)1142 888 y Fk(C)1219 859 y Fg(00)1278 888 y Fj(j)p Fk(j)g Fj(j)p Fo(!)1874 819 y Fk(:)0 1073 y Fo(Hence,)34 b(for)e(some)g(constan)m(ts)i Fk(N)1220 1088 y Fi(0)1292 1073 y Fo(and)f Fk(N)1560 1088 y Fi(1)1600 1073 y Fo(,)f(that)h(dep)s(end)g(on)g Fk(d)f Fo(and)h Fk(s)f Fo(only)-8 b(,)1294 1369 y Fk(N)71 b Fj(\024)61 b Fo(e)1624 1327 y Fg(\000)1676 1310 y Fi(~)1679 1327 y Fh(b=)p Fi(2\026)-35 b Fh(h)1951 1301 y Fj(k)p Fk(V)22 b Fj(k)2130 1316 y Fh(p)2186 1301 y Fk(N)2264 1316 y Fi(0)2320 1301 y Fk(N)2408 1250 y Fg(j)p Fh(j)t Fg(j)2398 1323 y Fi(1)p 1867 1345 702 4 v 1867 1448 a Fj(j)17 b Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)2275 1419 y Fi(1)p Fh(=p)2402 1448 y Fo(\026)-50 b Fk(h)2457 1405 y Fh(d=p)2579 1369 y Fk(:)0 1650 y Fo(By)33 b(our)g(c)m(hoice)f (of)h Fk(p)p Fo(,)1666 1742 y(1)p 1431 1786 519 4 v 1431 1878 a Fj(j)17 b Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)1839 1849 y Fi(1)p Fh(=p)2020 1809 y Fj(')60 b Fo(1)p Fk(=)p Fj(h)p Fk(t)p Fj(i)2368 1768 y Fh(d=p)0 2034 y Fo(is)32 b(in)m(tegrable.)146 2154 y(F)-8 b(or)32 b Fk(k)g Fj(\024)c Fk(l)j Fj(')d Fk(g)t(=)p Fo(\026)-50 b Fk(h)p Fo(,)33 b(with)f(su\016cien)m(tly)i(small)c Fk(g)t Fo(,)j(this)f(last)g (estimate)g(allo)m(ws)g(us)h(to)g(b)s(ound)g(the)g(cor-)0 2275 y(resp)s(onding)c(term)f(in)g Fj(k)p Fk(\030)925 2290 y Fh(l)951 2275 y Fo(\()p Fk(x;)17 b(t)p Fo(\))p Fk(\037)1222 2290 y Fi(2)1261 2275 y Fo(\()p Fk(x;)g(t)p Fo(\))p Fj(k)29 b Fo(as)g(follo)m(ws)f(\(where)i(the)f Fk(N)2541 2290 y Fh(i)2569 2275 y Fo(,)h Fk(i)e Fo(=)g(0)p Fk(;)17 b Fo(1)p Fk(;)g Fo(2)p Fk(;)g Fo(3)g Fk(:)g(:)g(:)26 b Fo(are)j(constan)m(ts\):)496 2457 y Fh(l)q Fg(\000)p Fi(1)492 2482 y Fd(X)487 2667 y Fh(k)r Fi(=0)649 2565 y Fo(\026)-50 b Fk(h)704 2522 y Fh(k)r(=)p Fi(2)851 2391 y Fd(\015)851 2441 y(\015)851 2491 y(\015)851 2540 y(\015)851 2590 y(\015)851 2640 y(\015)1004 2482 y(X)914 2670 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1247 2565 y Fk(c)1289 2580 y Fh(k)r(;j)1383 2565 y Fo(\()p Fk(t)p Fo(\))17 b Fk(\037)1572 2580 y Fi(2)1612 2565 y Fo(\()p Fk(x;)g(t)p Fo(\))g Fk(V)k Fo(\()p Fk(x)p Fo(\))c Fk(\036)2123 2580 y Fh(j)2159 2565 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))3281 2391 y Fd(\015)3280 2441 y(\015)3280 2491 y(\015)3280 2540 y(\015)3280 2590 y(\015)3280 2640 y(\015)327 2926 y Fj(\024)496 2818 y Fh(l)q Fg(\000)p Fi(1)492 2843 y Fd(X)487 3028 y Fh(k)r Fi(=0)649 2926 y Fo(\026)-50 b Fk(h)704 2884 y Fh(k)r(=)p Fi(2)834 2926 y Fj(k)p Fk(c)926 2941 y Fh(k)969 2926 y Fo(\()p Fk(t)p Fo(\))p Fj(k)1163 2755 y Fd(0)1163 2905 y(@)1342 2843 y(X)1252 3032 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1585 2926 y Fj(k)p Fk(V)22 b Fo(\()p Fk(x)p Fo(\))17 b Fk(\036)1920 2941 y Fh(j)1956 2926 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fj(k)3128 2755 y Fd(1)3127 2905 y(A)3199 2778 y Fi(1)p Fh(=)p Fi(2)327 3288 y Fj(\024)496 3180 y Fh(l)q Fg(\000)p Fi(1)492 3205 y Fd(X)487 3389 y Fh(k)r Fi(=0)649 3288 y Fo(\026)-50 b Fk(h)704 3245 y Fh(k)r(=)p Fi(2)834 3121 y Fd(s)p 917 3121 410 4 v 927 3220 a Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(k)s Fo(\)!)p 927 3264 390 4 v 1077 3356 a Fk(J)9 b Fo(!)1369 3220 y Fk(N)1457 3184 y Fh(k)1447 3245 y Fi(3)p 1369 3264 132 4 v 1394 3356 a Fk(k)s Fo(!)1542 3288 y(e)1586 3246 y Fg(\000)1638 3229 y Fi(~)1641 3246 y Fh(b=)p Fi(2\026)-35 b Fh(h)2004 3220 y Fj(k)p Fk(V)21 b Fj(k)2182 3235 y Fh(p)2238 3220 y Fk(N)2316 3235 y Fi(0)p 1829 3264 702 4 v 1829 3367 a Fj(j)c Fo(det)f Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)2237 3338 y Fi(1)p Fh(=p)2364 3367 y Fo(\026)-50 b Fk(h)2419 3324 y Fh(d=p)2573 3117 y Fd(0)2573 3266 y(@)2736 3205 y(X)2646 3393 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)2963 3288 y Fk(N)3051 3237 y Fi(2)p Fg(j)p Fh(j)t Fg(j)3041 3309 y Fi(1)3162 3117 y Fd(1)3162 3266 y(A)3235 3139 y Fi(1)p Fh(=)p Fi(2)3651 3288 y Fo(\(9.11\))327 3616 y Fj(\024)659 3549 y Fo(e)702 3513 y Fg(\000)754 3495 y Fi(~)757 3513 y Fh(b)q(=)p Fi(2\026)-35 b Fh(h)920 3549 y Fk(N)998 3564 y Fi(4)p 497 3593 V 497 3696 a Fj(j)17 b Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)906 3667 y Fi(1)p Fh(=p)1032 3696 y Fo(\026)-50 b Fk(h)1088 3653 y Fh(d=p)1266 3509 y(l)q Fg(\000)p Fi(1)1262 3533 y Fd(X)1258 3718 y Fh(k)r Fi(=0)1420 3616 y Fk(k)1474 3575 y Fh(k)r(=)p Fi(2)1604 3616 y Fo(\026)g Fk(h)1659 3574 y Fh(k)r(=)p Fi(2)1789 3616 y Fk(N)1877 3575 y Fh(k)1867 3641 y Fi(5)327 3925 y Fj(\024)659 3858 y Fo(e)702 3822 y Fg(\000)754 3804 y Fi(~)757 3822 y Fh(b)q(=)p Fi(2\026)-35 b Fh(h)920 3858 y Fk(N)998 3873 y Fi(6)p 497 3902 V 497 4004 a Fj(j)17 b Fo(det)g Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fj(j)906 3976 y Fi(1)p Fh(=p)1032 4004 y Fo(\026)-50 b Fk(h)1088 3962 y Fh(d=p)1209 3925 y Fk(:)2415 b Fo(\(9.12\))146 4190 y(It)33 b(remains)f(for)g(us)h(to)f(con)m(trol)g(in)m(tegrals)f(of)h (the)h(form)244 4418 y Fk(F)321 4377 y Fi(2)360 4418 y Fo(\()p Fk(p)p Fo(\))84 b(=)728 4301 y Fd(Z)774 4489 y Fk(I)-19 b(R)873 4442 y Ff(d)930 4418 y Fk(\037)991 4377 y Fi(2)991 4442 y(2)1031 4418 y Fo(\()p Fk(x;)17 b(t)p Fo(\))p Fj(j)p Fk(\036)1327 4433 y Fh(j)1363 4418 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fj(j)2496 4377 y Fi(2)2535 4418 y Fj(j)p Fk(D)2647 4377 y Fh(p)2686 4418 y Fk(V)k Fo(\()p Fk(a)p Fo(\()p Fk(t)p 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Fo(~)-51 b Fk(c)780 4965 y Fi(2)780 5025 y(0)838 5001 y Fo(~)g Fk(c)878 4950 y Fi(2)p Fg(j)p Fh(p)p Fg(j)878 5023 y Fi(2)1009 5001 y Fo(e)1052 4965 y Fg(\000)1104 4947 y Fi(~)1107 4965 y Fh(b)q(=)p Fi(\(2\026)-35 b Fh(h)p Fi(\))1324 5001 y Fo(\026)-50 b Fk(h)1380 4958 y Fg(j)p Fh(p)p Fg(j)p 738 5045 721 4 v 1001 5137 a Fj(h)p Fk(t)p Fj(i)1114 5108 y Fi(2)p Fh(\014)1517 4951 y Fd(Z)1564 5140 y Fg(j)p Fh(y)r Fg(j\025)1693 5122 y Fi(\026)1696 5140 y Fh(b)o(=)1760 5083 y Fg(p)p 1819 5083 41 3 v 57 x Fi(\026)-35 b Fh(h)1897 5068 y Fk(y)1949 5027 y Fi(2)p Fg(j)p Fh(p)p Fg(j)2063 5068 y Fj(j)17 b(H)2192 5083 y Fh(j)2228 5068 y Fo(\()p Fk(A)p Fo(;)g Fk(y)t Fo(\))p Fj(j)2501 5027 y Fi(2)2555 5068 y Fo(e)2598 5027 y Fh(y)2635 5004 y Fc(2)2670 5027 y Fh(=)p Fi(2)2745 5068 y Fk(=)p Fo(\(2)2881 5027 y Fg(j)p Fh(j)t Fg(j)2973 5068 y Fk(j)6 b Fo(!)17 b Fk(\031)3122 5027 y Fh(d=)p Fi(2)3232 5068 y Fo(\))g Fk(dy)t(;)1901 5506 y Fo(37)p eop %%Page: 38 38 38 37 bop 0 -43 a Fo(where)45 b(w)m(e)f(used)h(the)f(same)f(t)m(yp)s(e) i(of)e(estimates)g(as)h(ab)s(o)m(v)m(e.)77 b(W)-8 b(e)44 b(b)s(ound)g(the)g(last)f(in)m(tegral)f(in)g(this)0 77 y(expression)35 b(b)m(y)g(using)e(spherical)g(co)s(ordinates)g(and)h (noting)f(that)g(the)i(in)m(tegration)d(region)g(lies)h(within)0 198 y(the)g(classically)e(forbidden)h(region,)g(if)f Fk(g)36 b Fo(is)c(su\016cien)m(tly)h(small.)41 b(The)34 b(in)m(tegral)d(is)h(th)m(us)i(b)s(ounded)f(b)m(y)493 342 y Fd(X)260 530 y Fg(f)p Fh(q)r(;n;m)11 b Fi(:)h(2)p Fh(n)p Fi(+)p Fh(q)r Fi(=)p Fg(j)p Fh(j)t Fg(jg)954 357 y Fo(2)1003 321 y Fi(2)p Fh(n)p Fi(+)p Fh(q)r Fi(+)p Fg(j)p Fh(p)p Fg(j)p Fi(+)p Fh(d=)p Fi(2)p Fg(\000)p Fi(1)p 888 401 733 4 v 888 493 a Fk(n)p Fo(!)17 b(\000\()p Fk(q)26 b Fo(+)c Fk(n)h Fo(+)f Fk(d=)p Fo(2\))1647 425 y(\000\(2)p Fk(n)g Fo(+)g Fk(q)k Fo(+)c Fj(j)p Fk(p)p Fj(j)g Fo(+)g Fk(d=)p Fo(2\))59 b Fj(\024)i Fk(f)2797 440 y Fi(0)2853 425 y Fk(f)2912 374 y Fg(j)p Fh(p)p Fg(j)2901 446 y Fi(1)3007 425 y Fo(\()p Fj(j)p Fk(j)6 b Fj(j)22 b Fo(+)g Fj(j)p Fk(p)p Fj(j)p Fo(\)!)p Fk(=)p Fj(j)p Fk(j)6 b Fj(j)p Fo(!)p Fk(:)0 678 y Fo(So,)33 b(for)f(some)g(other)h (constan)m(ts)h(w)m(e)f(ha)m(v)m(e)1055 920 y Fk(F)14 b Fo(\()p Fk(p)p Fo(\))1257 879 y Fi(2)1356 920 y Fj(\024)1504 853 y Fk(f)1552 868 y Fi(2)1608 853 y Fk(f)1667 802 y Fg(j)p Fh(j)t Fg(j)1656 874 y Fi(3)1759 853 y Fk(f)1818 802 y Fg(j)p Fh(p)p Fg(j)1807 874 y Fi(4)1913 853 y Fo(\026)-50 b Fk(h)1968 810 y Fg(j)p Fh(p)p Fg(j)2064 853 y Fo(e)2107 816 y Fg(\000)2159 799 y Fi(~)2162 816 y Fh(b=)p Fi(\(2\026)-35 b Fh(h)q Fi(\))2379 853 y Fo(\()p Fj(j)p Fk(j)6 b Fj(j)22 b Fo(+)g Fj(j)p Fk(p)p Fj(j)p Fo(\)!)p 1504 897 1305 4 v 1986 988 a Fj(h)p Fk(t)p Fj(i)2099 960 y Fi(2)p Fh(\014)2198 988 y Fj(j)p Fk(j)6 b Fj(j)p Fo(!)2818 920 y Fk(:)146 1142 y Fo(The)34 b(corresp)s(onding)e(sum)h(in)f Fj(k)p Fk(\037)1405 1157 y Fi(2)1444 1142 y Fk(\030)1487 1157 y Fh(l)1513 1142 y Fj(k)g Fo(is)g(b)s(ounded)h(b)m(y)487 1213 y Fd(\015)487 1262 y(\015)487 1312 y(\015)487 1362 y(\015)487 1412 y(\015)487 1462 y(\015)558 1279 y Fh(l)q Fg(\000)p Fi(1)555 1304 y Fd(X)550 1488 y Fh(k)r Fi(=0)712 1387 y Fk(h)768 1346 y Fh(k)r(=)p Fi(2)1005 1304 y Fd(X)915 1492 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1248 1387 y Fk(c)1290 1402 y Fh(k)r(;j)1384 1387 y Fo(\()p Fk(t)p Fo(\))780 1725 y Fj(\002)1089 1642 y Fd(X)982 1830 y Fg(j)p Fh(p)p Fg(j\024)p Fh(l)q Fi(+1)p Fg(\000)p Fh(k)1350 1725 y Fk(\037)1411 1740 y Fi(2)1450 1725 y Fo(\()p Fk(x;)17 b(t)p Fo(\))g Fk(\036)1735 1740 y Fh(j)1772 1725 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h)o(;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))2903 1657 y Fk(D)2987 1621 y Fh(p)3026 1657 y Fk(V)22 b Fo(\()p Fk(a)p Fo(\()p Fk(t)p Fo(\)\))17 b(\()p Fk(x)22 b Fj(\000)g Fk(a)p Fo(\()p Fk(t)p Fo(\)\))3774 1621 y Fh(p)p 2903 1701 912 4 v 3320 1793 a Fk(p)p Fo(!)3841 1550 y Fd(\015)3841 1600 y(\015)3841 1650 y(\015)3841 1700 y(\015)3841 1749 y(\015)3841 1799 y(\015)327 2086 y Fj(\024)496 1978 y Fh(l)q Fg(\000)p Fi(1)492 2003 y Fd(X)487 2187 y Fh(k)r Fi(=0)649 2086 y Fk(h)705 2045 y Fh(k)r(=)p Fi(2)959 2003 y Fd(X)852 2191 y Fg(j)p Fh(p)p Fg(j\024)p Fh(l)q Fi(+1)p Fg(\000)p Fh(k)1220 2086 y Fj(k)p Fk(c)1312 2101 y Fh(k)1355 2086 y Fo(\()p Fk(t)p Fo(\))p Fj(k)1549 1915 y Fd(0)1549 2064 y(@)1728 2003 y(X)1638 2191 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)6 b Fi(+3)p Fh(k)1971 2086 y Fk(F)2048 2045 y Fi(2)2087 2086 y Fo(\()p Fk(p)p Fo(\))2229 1915 y Fd(1)2229 2064 y(A)2301 1938 y Fi(1)p Fh(=)p Fi(2)327 2447 y Fj(\024)496 2339 y Fh(l)q Fg(\000)p Fi(1)492 2364 y Fd(X)487 2548 y Fh(k)r Fi(=0)633 2447 y Fk(h)689 2406 y Fh(k)r(=)p Fi(2)802 2447 y Fk(f)861 2406 y Fh(k)850 2471 y Fi(5)913 2264 y Fd(q)p 996 2264 390 4 v 100 x Fo(\()p Fk(J)32 b Fo(+)22 b(3)p Fk(k)s Fo(\)!)p 913 2424 473 4 v 1014 2442 a Fj(p)p 1097 2442 91 4 v 84 x Fk(J)9 b Fo(!)17 b Fk(k)s Fo(!)1520 2364 y Fd(X)1412 2552 y Fg(j)p Fh(p)p Fg(j\024)p Fh(l)q 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Fo(\()p Fk(g)t Fo(\))2962 4443 y Fh(k)p 2773 4523 232 4 v 2791 4622 a Fo(\026)-50 b Fk(h)2846 4579 y Fh(k)2905 4622 y Fk(k)s Fo(!)2007 4888 y Fj(\024)84 b Fo(2)2243 4821 y(e)2286 4785 y Fg(\000)2338 4767 y Fi(~)2341 4785 y Fh(b=)p Fi(\(4\026)-35 b Fh(h)q Fi(\))p 2243 4865 299 4 v 2312 4957 a Fj(h)p Fk(t)p Fj(i)2425 4928 y Fh(\014)2584 4888 y Fo(e)2635 4847 y Fi(~)-42 b Fh(\013)p Fi(\()p Fh(g)r Fi(\))p Fh(=)p Fi(\026)-35 b Fh(h)2007 5214 y Fj(\024)2178 5146 y Fo(e)2221 5110 y Fg(\000)2273 5093 y Fi(~)2276 5110 y Fh(b=)p Fi(\(8\026)g Fh(h)q Fi(\))p 2178 5191 V 2247 5282 a Fj(h)p Fk(t)p Fj(i)2360 5253 y Fh(\014)2486 5214 y Fk(;)1138 b Fo(\(9.15\))1901 5506 y(38)p eop %%Page: 39 39 39 38 bop 0 -43 a Fo(pro)m(vided)39 b Fk(g)j Fo(is)c(small)e(enough,)41 b(since)48 b(~)-58 b Fk(\013)p Fo(\()p Fk(g)t Fo(\))37 b Fj(')i Fk(g)1870 -79 y Fi(3)p Fh(=)p Fi(2)2018 -43 y Fo(as)f Fk(g)j Fj(!)d Fo(0,)i(whic)m(h)f(is)f(exp)s(onen)m(tially)g (small)e(and)0 77 y(in)m(tegrable)c(in)f(time.)146 198 y(Finally)-8 b(,)35 b(gathering)h(estimates)g(\(9.8\),)h(\(9.12\))f (and)h(\(9.15\),)g(w)m(e)g(get)g(the)g(existence)h(of)e(p)s(ositiv)m(e) g Fk(\015)5 b Fo(,)0 318 y Fk(H)j Fo(,)32 b Fk(G)p Fo(,)h(and)f Fk(C)7 b Fo(,)33 b(suc)m(h)h(that)e Fk(g)f(<)d(G)p Fo(,)k Fk(l)r Fo(\(\026)-50 b Fk(h)q Fo(\))27 b(=)h Fk(g)t(=)p Fo(\026)-50 b Fk(h)o Fo(,)33 b(and)f(\026)-50 b Fk(h)28 b(<)g(H)40 b Fo(imply)1285 453 y Fd(Z)1368 479 y Fg(1)1331 642 y(\0001)1493 570 y Fk(dt)17 b Fj(k)p Fk(\030)1689 585 y Fh(l)1714 570 y Fo(\()p Fk(x;)g(t)p Fo(\))p Fj(k)p Fk(=)p Fo(\026)-50 b Fk(h)61 b Fj(\024)f Fk(C)24 b Fo(e)2413 529 y Fg(\000)p Fh(\015)t(=)p Fi(\026)-35 b Fh(h)2588 570 y Fk(:)0 897 y Fa(Pro)s(of)33 b(of)h(Lemma)f(9.2:)98 b Fo(Let)29 b Fk(f)38 b Fj(2)29 b(S)36 b Fo(b)s(e)30 b(the)f(function)g(that)g(is)f(giv)m(en)i(in)e(spherical)g(co)s (ordinates)h(b)m(y)1130 1198 y Fk(f)11 b Fo(\()p Fk(x)p Fo(\))60 b(=)1516 1031 y Fd(v)1516 1077 y(u)1516 1127 y(u)1516 1177 y(t)p 1604 1031 380 4 v 1770 1131 a Fo(2)p 1614 1175 361 4 v 1614 1270 a(\000\()p Fk(q)26 b Fo(+)1890 1231 y Fh(d)p 1890 1247 37 4 v 1891 1304 a Fi(2)1936 1270 y Fo(\))2033 1198 y Fk(r)2080 1157 y Fh(q)2150 1198 y Fk(e)2195 1157 y Fg(\000)p Fh(r)2284 1133 y Fc(2)2319 1157 y Fh(=)p Fi(2)2426 1198 y Fk(Y)2483 1213 y Fh(q)r(;m)2603 1198 y Fo(\()p Fk(!)t Fo(\))p Fk(:)0 1503 y Fo(F)-8 b(or)32 b(in)m(tegers)h Fk(q)e(>)d Fo(0,)k(the)h(maxim)m(um)d(absolute)j(v)-5 b(alue)32 b(of)g(this)g(function)g(is)244 1621 y Fd(v)244 1667 y(u)244 1717 y(u)244 1767 y(t)p 332 1621 380 4 v 497 1721 a Fo(2)p 342 1765 361 4 v 342 1860 a(\000\()p Fk(q)25 b Fo(+)617 1821 y Fh(d)p 617 1837 37 4 v 618 1894 a Fi(2)664 1860 y Fo(\))761 1788 y Fk(q)808 1747 y Fh(q)r(=)p Fi(2)949 1788 y Fk(e)994 1747 y Fg(\000)p Fh(q)r(=)p Fi(2)1206 1788 y Fo(max)1274 1838 y Fh(!)1437 1788 y Fj(j)17 b Fk(Y)1539 1803 y Fh(q)r(;m)1658 1788 y Fo(\()p Fk(!)t Fo(\))g Fj(j)p Fk(:)1780 b Fo(\(9.16\))146 2086 y(The)34 b(function)e Fk(f)43 b Fo(is)32 b(a)h(normalized)d (eigenfunction)i(of)g Fj(\000)p Fo(\001)23 b(+)f Fk(x)2542 2050 y Fi(2)2614 2086 y Fo(with)33 b(eigen)m(v)-5 b(alue)32 b Fk(E)h Fo(=)28 b(2)p Fk(q)e Fo(+)c Fk(d)p Fo(.)0 2207 y(Its)33 b(norm)f(in)g(the)h(Sob)s(olev)f(space)h Fj(H)1388 2222 y Fh(s)1458 2207 y Fo(for)f Fk(s)c(>)f Fo(0)32 b(satis\014es)244 2410 y Fj(k)p Fk(f)11 b Fj(k)403 2425 y Fg(H)463 2433 y Ff(s)560 2410 y Fj(\024)61 b Fk(C)768 2425 y Fi(1)807 2410 y Fo(\()p Fk(s)p Fo(\))978 2314 y Fd(\020)1044 2410 y Fj(k)p Fk(f)11 b Fj(k)38 b Fo(+)h Fj(k)p Fo(\()p Fj(\000)p Fo(\001\))1640 2369 y Fh(s=)p Fi(2)1747 2410 y Fk(f)11 b Fj(k)1873 2314 y Fd(\021)1982 2410 y Fj(\024)61 b Fk(C)2190 2425 y Fi(1)2229 2410 y Fo(\()p Fk(s)p Fo(\))2400 2314 y Fd(\020)2450 2410 y Fo(1)38 b(+)h(\(2)p Fk(q)26 b Fo(+)c Fk(d)p Fo(\))2995 2369 y Fh(s=)p Fi(2)3102 2314 y Fd(\021)3168 2410 y Fk(;)456 b Fo(\(9.17\))0 2626 y(for)32 b(some)g Fk(C)463 2641 y Fi(1)503 2626 y Fo(\()p Fk(s)p Fo(\).)146 2746 y(If)h Fk(s)27 b(>)h(d=)p Fo(2,)k(then)h(\(1)22 b(+)g Fj(j)p Fk(k)s Fj(j)1168 2710 y Fi(2)1207 2746 y Fo(\))1245 2710 y Fg(\000)p Fh(s=)p Fi(2)1439 2746 y Fo(is)33 b(in)e Fk(L)1717 2710 y Fi(2)1757 2746 y Fo(\()p Fk(I)-19 b(R)1894 2705 y Fh(d)1934 2746 y Fo(\).)44 b(So,)32 b(b)m(y)i(H\177)-49 b(older's)33 b(inequalit)m(y)-8 b(,)360 2979 y Fj(j)17 b Fk(f)11 b Fo(\()p Fk(x)p Fo(\))17 b Fj(j)59 b(\024)h Fo(\(2)p Fk(\031)t Fo(\))1020 2938 y Fg(\000)p Fh(d=)p Fi(2)1235 2879 y Fd(\015)1235 2929 y(\015)1235 2979 y(\015)1320 2945 y(b)1298 2979 y Fk(f)10 b Fo(\()p Fk(k)s Fo(\))17 b(\(1)22 b(+)g Fj(j)p Fk(k)s Fj(j)1820 2938 y Fi(2)1859 2979 y Fo(\))1897 2938 y Fh(s=)p Fi(2)2021 2879 y Fd(\015)2021 2929 y(\015)2021 2979 y(\015)2116 2879 y(\015)2116 2929 y(\015)2116 2979 y(\015)17 b Fo(\(1)k(+)h Fj(j)p Fk(k)s Fj(j)2495 2938 y Fi(2)2534 2979 y Fo(\))2572 2938 y Fg(\000)p Fh(s=)p Fi(2)2751 2879 y Fd(\015)2751 2929 y(\015)2751 2979 y(\015)60 b Fo(=)h Fk(C)3064 2994 y Fi(2)3103 2979 y Fo(\()p Fk(s)p Fo(\))32 b Fj(k)p Fk(f)11 b Fj(k)3416 2994 y Fg(H)3476 3002 y Ff(s)3513 2979 y Fk(:)0 3236 y Fo(This)33 b(and)f(\(9.17\))g(imply)f(that)h(\(9.16\))g (is)g(b)s(ounded)i(b)m(y)f Fk(C)2163 3251 y Fi(3)2202 3236 y Fo(\()p Fk(s)p Fo(\))2373 3140 y Fd(\020)2423 3236 y Fo(1)38 b(+)h(\(2)p Fk(q)26 b Fo(+)c Fk(d)p Fo(\))2968 3195 y Fh(s=)p Fi(2)3075 3140 y Fd(\021)3124 3236 y Fo(.)44 b(Th)m(us,)525 3575 y(max)593 3624 y Fh(!)756 3575 y Fj(j)16 b Fk(Y)857 3590 y Fh(q)r(;m)977 3575 y Fo(\()p Fk(!)t Fo(\))h Fj(j)58 b(\024)j Fk(C)1429 3590 y Fi(3)1468 3575 y Fo(\()p Fk(s)p Fo(\))1622 3401 y Fd(s)p 1706 3401 380 4 v 1716 3504 a Fo(\000\()p Fk(q)25 b Fo(+)1991 3464 y Fh(d)p 1991 3480 37 4 v 1992 3538 a Fi(2)2038 3504 y Fo(\))p 1716 3551 361 4 v 1871 3643 a(2)2151 3478 y Fd(\020)2201 3575 y Fo(1)38 b(+)h(\(2)p Fk(q)26 b Fo(+)c Fk(d)p Fo(\))2746 3533 y Fh(s=)p Fi(2)2853 3478 y Fd(\021)2951 3575 y Fk(q)2998 3533 y Fg(\000)p Fh(q)r(=)p Fi(2)3194 3575 y Fk(e)3239 3533 y Fh(q)r(=)p Fi(2)3348 3575 y Fk(:)0 3837 y Fo(The)34 b(lemma)c(follo)m(ws)h(from)g(this)i(b)m(y)g(an)g (application)d(of)i(Stirling's)e(form)m(ula.)p 3102 3837 67 84 v 0 4170 a Fl(10)161 b(More)53 b(General)h(Coheren)l(t)d(States)0 4448 y Fo(In)36 b(this)g(section)g(w)m(e)h(extend)h(all)c(the)i (previous)h(theorems)f(of)g(the)g(pap)s(er)g(to)g(allo)m(w)f(initial)d (conditions)0 4569 y(that)g(are)h(certain)f(in\014nite)g(linear)f(com)m (binations)g(of)h(the)h Fk(\036)2232 4584 y Fh(j)2268 4569 y Fo(.)0 4748 y Fa(Pro)s(of)46 b(of)g(Theorem)f(3.6)p Fo(:)98 b(The)41 b(strategy)f(is)g(quite)f(simple.)64 b(Let)40 b Fk(')g Fj(2)h(C)46 b Fo(ha)m(v)m(e)41 b(expansion)g Fk(')56 b Fo(=)0 4868 y Fk(c)42 4883 y Fh(j)95 4868 y Fk(\036)153 4883 y Fh(j)189 4868 y Fo(\()17 b Fk(I)-67 b(I)7 b(;)34 b(I)-67 b(I)7 b(;)34 b Fo(\026)-50 b Fk(h;)33 b Fo(0)p Fk(;)g Fo(0)p Fk(;)g(x)p Fo(\),)g(and)g(let)1352 5088 y Fk( )1415 5103 y Fi(0)1455 5088 y Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))44 b(=)g(\(\003)2048 5103 y Fh(h)2092 5088 y Fo(\()p Fk(a;)17 b(\021)t Fo(\))p Fk(')p Fo(\)\()p Fk(x)p Fo(\))1901 5506 y(39)p eop %%Page: 40 40 40 39 bop 0 -43 a Fo(b)s(e)33 b(our)f(initial)d(condition.)42 b(By)33 b(construction,)1098 168 y Fk( )1161 183 y Fi(0)1201 168 y Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))44 b(=)1730 85 y Fd(X)1688 305 y Fh(j)t Fg(2)p Fa(N)1855 258 y Ff(d)1925 168 y Fk(c)1967 183 y Fh(j)2020 168 y Fk(\036)2078 183 y Fh(j)2114 168 y Fo(\()17 b Fk(I)-67 b(I)7 b(;)33 b(I)-67 b(I)8 b(;)33 b Fo(\026)-50 b Fk(h;)34 b(a;)f(\021)t(;)g(x)p Fo(\))p Fk(:)0 495 y Fo(F)-8 b(or)32 b Fk(J)37 b(>)27 b Fo(0,)33 b(w)m(e)g(de\014ne)1126 615 y Fk( )1189 630 y Fh(J)1238 615 y Fo(\()p Fk(x;)17 b Fo(0)p Fk(;)g Fo(\026)-50 b Fk(h)p Fo(\))28 b(=)1718 532 y Fd(X)1692 720 y Fg(j)p Fh(j)t Fg(j\024)p Fh(J)1897 615 y Fk(c)1939 630 y Fh(j)1992 615 y Fk(\036)2050 630 y Fh(j)2086 615 y Fo(\()17 b Fk(I)-67 b(I)7 b(;)34 b(I)-67 b(I)7 b(;)33 b Fo(\026)-50 b Fk(h)q(;)33 b(a;)g(\021)t(;)g(x)p Fo(\))p Fk(;)0 873 y Fo(and)g(denote)g(the)g (appro)m(ximation)d(that)j(arises)f(from)g(this)g(initial)d(condition)i (b)m(y)637 1084 y Fk( )717 1094 y Fi(~)700 1111 y Fh(J)7 b Fi(\()p Fh(J)n(;)p Fi(\026)-35 b Fh(h)o Fi(\))902 1084 y Fo(\()p Fk(x;)17 b(t;)g Fo(\026)-50 b Fk(h)p Fo(\))44 b(=)1477 1001 y Fd(X)1375 1200 y Fg(j)p Fh(j)t Fg(j\024)1518 1184 y Fi(~)1503 1200 y Fh(J)5 b Fi(\()p Fh(J)n(;)p Fi(\026)-35 b Fh(h)o Fi(\))1732 1084 y Fk(c)1774 1099 y Fh(j)1810 1084 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))17 b Fk(\036)2095 1099 y Fh(j)2131 1084 y Fo(\()p Fk(A)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(B)5 b Fo(\()p Fk(t)p Fo(\))p Fk(;)17 b Fo(\026)-50 b Fk(h;)17 b(a)p Fo(\()p Fk(t)p Fo(\))p Fk(;)g(\021)t Fo(\()p Fk(t)p Fo(\))p Fk(;)g(x)p Fo(\))p Fk(:)0 1393 y Fo(W)-8 b(e)33 b(then)g(ha)m(v)m(e)486 1589 y(e)529 1548 y Fg(\000)p Fh(itH)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)889 1589 y Fk( )952 1604 y Fi(0)992 1589 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)o Fo(\))327 1792 y(=)83 b(e)529 1751 y Fg(\000)p Fh(itH)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)889 1792 y Fo(\()p Fk( )990 1807 y Fi(0)1030 1792 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)o Fo(\))22 b Fj(\000)h Fk( )1438 1807 y Fh(J)1487 1792 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)p Fo(\)\))38 b(+)h(e)1945 1751 y Fg(\000)p Fh(itH)5 b Fi(\(\026)-35 b Fh(h)q Fi(\))p Fh(=)p Fi(\026)g Fh(h)2305 1792 y Fk( )2368 1807 y Fh(J)2417 1792 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)p Fo(\))1010 b(\(10.1\))327 1996 y(=)83 b Fk( )566 2006 y Fi(~)549 2023 y Fh(J)6 b Fi(\()p Fh(J)n(;)p Fi(\026)-35 b Fh(h)p Fi(\))750 1996 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))22 b(+)g Fk(O)s Fo(\()p Fj(k)p Fo(e)1289 1955 y Fg(\000)p Fh(itH)5 b Fi(\(\026)-35 b Fh(h)p Fi(\))p Fh(=)p Fi(\026)g Fh(h)1632 1996 y Fk( )1695 2011 y Fh(J)1744 1996 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)p Fo(\))22 b Fj(\000)g Fk( )2169 2006 y Fi(~)2152 2023 y Fh(J)7 b Fi(\()p Fh(J)n(;)p Fi(\026)-35 b Fh(h)o Fi(\))2354 1996 y Fo(\()p Fk(t;)17 b Fo(\026)-50 b Fk(h)p Fo(\))p Fj(k)p Fo(\))22 b(+)g Fk(O)s Fo(\()p Fj(k)p Fk( )3001 2011 y Fi(0)3040 1996 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)o Fo(\))22 b Fj(\000)h Fk( )3448 2011 y Fh(J)3497 1996 y Fo(\(0)p Fk(;)17 b Fo(\026)-50 b Fk(h)p Fo(\)\))p Fj(k)p Fo(\))p Fk(:)0 2191 y Fo(Th)m(us,)43 b(to)38 b(mak)m(e)i(the)f (error)h(terms)f(exp)s(onen)m(tially)f(small)f(in)h(\026)-50 b Fk(h)40 b Fo(w)m(e)g(need)g(to)f(consider)h(v)-5 b(alues)39 b(of)g(the)0 2312 y(cuto\013)44 b Fk(J)52 b Fo(that)43 b(gro)m(w)h(to)f(in\014nit)m(y)g(with)g(\026)-50 b Fk(h)43 b Fo(in)g(a)g(suitable)g(w)m(a)m(y)-8 b(,)47 b(and)d(w)m(e)g(also)f (need)h(to)f(con)m(trol)g(our)0 2432 y(appro)m(ximation)30 b(as)j(a)f(function)g(of)h Fk(J)9 b Fo(.)146 2552 y(In)30 b(the)g(pro)s(ofs)e(of)h(all)e(previous)j(theorems,)g(the)g(dep)s (endence)h(of)e(the)h(appro)m(ximation)d(on)i Fk(l)i Fo(go)m(v)m(erns)0 2673 y(the)k(estimates)e(on)h(the)h(error)f(terms.) 48 b(The)35 b(\026)-50 b Fk(h)34 b Fo(dep)s(endence)i(comes)f(through)f (the)g(di\013eren)m(t)g(c)m(hoices)h(of)0 2793 y Fk(l)30 b Fj(')e Fk(g)t(=)p Fo(\026)-50 b Fk(h)29 b Fo(or)h Fk(l)g Fj(')e Fk(g)t Fo(\()p Fk(T)14 b Fo(\))p Fk(=)p Fo(\026)-50 b Fk(h)o Fo(,)30 b(with)g Fk(T)41 b Fj(')28 b Fo(ln\(1)p Fk(=)p Fo(\026)-50 b Fk(h)o Fo(\).)43 b(The)31 b(set)f Fj(C)37 b Fo(is)29 b(c)m(hosen)j(to)d(giv)m(e)h(an)g(exp)s(onen)m (tially)f(small)0 2914 y(con)m(tribution)i(as)i Fk(l)d Fj(!)d(1)33 b Fo(in)e(the)i(last)f(term)g(of)g(\(10.1\))g(with)g(the)h (c)m(hoice)244 3109 y Fk(J)k Fo(=)27 b Fk(\027)c(l)r(;)3084 b Fo(\(10.2\))0 3305 y(for)24 b(some)h Fk(\027)34 b(>)28 b Fo(0.)40 b(W)-8 b(e)26 b(need)g(only)e(sho)m(w)i(that)f(the)g(basic)g (estimates)f(in)h(the)g(pro)s(ofs)f(ab)s(o)m(v)m(e)i(are)f(unaltered)0 3425 y(b)m(y)33 b(the)g(replacemen)m(t)g(of)f Fk(J)42 b Fo(b)m(y)33 b Fk(\027)23 b(g)t(=)p Fo(\026)-50 b Fk(h)p Fo(,)32 b(for)g Fk(g)k Fo(small)30 b(enough.)146 3546 y(W)-8 b(e)38 b(can)g(do)g(this)g(b)s(ecause)h(w)m(e)f(ha)m(v)m(e)i(b)s (een)e(careful)f(to)h(mak)m(e)g(the)g Fk(J)47 b Fo(dep)s(endence)40 b(explicit)c(in)h(all)0 3666 y(the)c(k)m(ey)h(estimates,)e(suc)m(h)i (as)f(Corollary)e(5.3.)146 3786 y(In)g(the)g(con)m(tribution)e(to)i (the)f(error)h(term)f(asso)s(ciated)g(with)g Fk(\037)2479 3801 y Fi(1)2549 3786 y Fo(giv)m(en)h(b)m(y)g(\(6.8\))f(w)m(e)i(adapt)e (the)h(last)0 3907 y(step)i(b)m(y)h(using)e(the)h(estimate)254 4096 y(\()p Fk(J)9 b Fo(\()p Fk(l)r Fo(\))22 b(+)g(3)p Fk(l)r Fo(\)!)p 254 4140 474 4 v 392 4231 a Fk(J)9 b Fo(\()p Fk(l)r Fo(\)!)797 4163 y Fj(\024)61 b Fk(c)977 4178 y Fi(0)1016 4163 y Fo(\()p Fk(\027)6 b Fo(\))1173 4096 y(\(\()p Fk(\027)29 b Fo(+)22 b(3\))p Fk(l)r Fo(\))1580 4059 y Fi(\()p Fh(\027)t Fi(+3\))p Fh(l)p 1173 4140 617 4 v 1368 4231 a Fo(\()p Fk(\027)6 b(l)r Fo(\))1529 4203 y Fh(\027)t(l)1860 4163 y Fj(\024)61 b Fk(c)2040 4178 y Fi(0)2079 4163 y Fo(\()p Fk(\027)6 b Fo(\))17 b Fk(c)2268 4178 y Fi(1)2308 4163 y Fo(\()p Fk(\027)6 b Fo(\))2438 4122 y Fh(l)2480 4163 y Fk(l)2511 4122 y Fi(3)p Fh(l)2573 4163 y Fk(;)1051 b Fo(\(10.3\))0 4415 y(for)30 b(some)h(constan)m(ts)i Fk(c)863 4430 y Fi(0)902 4415 y Fo(\()p Fk(\027)6 b Fo(\))31 b(and)g Fk(c)1293 4430 y Fi(1)1333 4415 y Fo(\()p Fk(\027)6 b Fo(\).)43 b(Hence,)33 b(the)e(remainder)f(of)h(the)g(argumen)m(t)g (for)f(Lemma)g(6.1)g(is)0 4535 y(the)35 b(same,)g(with)g(up)s(dated)g (constan)m(ts.)51 b(Since)35 b(the)h(constan)m(ts)g(are)e(mo)s (di\014ed)g(in)g(a)g(time)g(indep)s(enden)m(t)0 4655 y(w)m(a)m(y)-8 b(,)34 b(the)f(long)e(time)g(estimates)i(are)f(also)g (unc)m(hanged.)146 4776 y(Consider)26 b(no)m(w)f(the)g(con)m(tribution) f(asso)s(ciated)h(with)f Fk(\037)2181 4791 y Fi(2)2246 4776 y Fo(in)g(Lemma)f(6.3.)41 b(W)-8 b(e)25 b(\014rst)g(note)g(that)g (\(10.2\))0 4896 y(implies)31 b(\(with)h(a)h(sligh)m(t)f(abuse)i(of)f (notation\))1774 4871 y(~)1751 4896 y Fk(J)9 b Fo(\()p Fk(l)r Fo(\))29 b(=)f Fk(J)9 b Fo(\()p Fk(l)r Fo(\))23 b(+)f(3)p Fk(l)j Fj(\000)e Fo(3)28 b(=)h(\()p Fk(\027)g Fo(+)22 b(3\))p Fk(l)j Fj(\000)e Fo(3)33 b(so)g(that)g(w)m(e)h(still)0 5016 y(ha)m(v)m(e)244 4991 y(~)221 5016 y Fk(J)9 b Fo(\()p Fk(l)r Fo(\))28 b Fj(')g Fk(g)t(=)p Fo(\026)-50 b Fk(h)o Fo(.)42 b(The)29 b(argumen)m(ts)g(that)f(rely)g(on)g(the)h(smallness)e (of)h Fk(g)k Fo(to)c(allo)m(w)e(us)j(to)f(use)i(of)d(Lemma)0 5137 y(\(6.2\))34 b(remain)f(in)h(force.)49 b(W)-8 b(e)35 b(th)m(us)g(arriv)m(e)g(at)f(\(6.27\).)48 b(W)-8 b(e)35 b(deal)f(with)g(it)f(b)m(y)i(using)g(\(10.3\),)f(exactly)h(as)0 5257 y(ab)s(o)m(v)m(e,)c(and)e(obtain)f(exp)s(onen)m(tial)g(deca)m(y)i (again)e(in)g(case)i Fk(l)g Fo(=)e Fk(g)t(=)p Fo(\026)-50 b Fk(h)o Fo(.)42 b(The)30 b(long)e(time)f(estimates)i(are)g(also)1901 5506 y(40)p eop %%Page: 41 41 41 40 bop 0 -43 a Fo(v)-5 b(alid)35 b(as)h(the)h(time)e(dep)s(endence)k (of)d(the)h(constan)m(ts)g(is)f(unaltered.)55 b(This)37 b(sho)m(ws)h(that)e(Theorems)h(3.1)0 77 y(and)c(3.3)f(are)g(true)h (with)g(our)f(generalized)g(initial)d(coheren)m(t)34 b(states.)146 198 y(Theorem)i(3.2)e(also)g(holds)h(for)f(these)i (initial)31 b(states)36 b(pro)m(vided)g(w)m(e)g(can)f(con)m(trol)f(the) h(sum)g(in)f(\(7.3\))0 318 y(with)f Fk(J)9 b Fo(\()p Fk(l)r Fo(\))28 b(=)g Fk(\027)6 b(l)r Fo(.)46 b(T)-8 b(o)33 b(do)g(so,)g(w)m(e)h(\014rst)g(note)f(that)g(the)g(last)f(t)m(w) m(o)i(factors)f(of)f(\(7.3\))h(can)g(b)s(e)g(b)s(ounded)h(b)m(y)0 438 y(e)43 402 y Fh(\014)86 379 y Fb(0)109 402 y Fh(J)158 438 y Fo(,)29 b(for)f(some)g Fk(\014)660 402 y Fg(0)683 438 y Fo(,)h(so)f(that)h(they)g(are)f(of)g(order)h(e)1836 402 y Fh(g)r(=)p Fi(\026)-35 b Fh(h)1952 438 y Fo(.)42 b(This)28 b(is)g(harmless)g(if)f Fk(g)32 b Fo(is)c(small)e(enough)j(b)s (ecause)0 559 y(of)j(the)h(exp)s(onen)m(tially)f(decreasing)h (prefactor.)43 b(Next,)34 b(w)m(e)f(use)h Fk(k)d Fj(\024)d Fk(l)c Fj(\000)f Fo(1)32 b(to)g(obtain)143 734 y Fd(q)p 226 734 605 4 v 105 x Fo(\()p Fk(J)f Fo(+)22 b(3)p Fk(k)s Fo(\)!)p Fk(=)p Fo(\()p Fk(J)9 b Fo(!\))60 b Fj(\024)h Fo(\()p Fk(J)31 b Fo(+)22 b(3)p Fk(k)s 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2245 y Fo(e)1867 2204 y Fh(C)1922 2180 y Fb(0)1917 2226 y Fc(2)1952 2204 y Fi(\(\()p Fh(\027)t Fi(+3\))p Fh(g)r Fi(\))2225 2180 y Fc(3)p Ff(=)p Fc(2)2324 2204 y Fh(=)p Fi(\026)g Fh(h)0 2419 y Fo(whic)m(h,)33 b(again,)e(is)i(harmless)f(for)g(su\016cien)m(tly)h(small)d Fk(g)t Fo(.)146 2540 y(T)-8 b(o)36 b(pro)m(v)m(e)h(the)g(v)-5 b(alidit)m(y)34 b(of)h(Theorem)h(3.4,)h(insert)e(\(8.2\))h(and)g (\(8.3\))f(in)g(the)i(estimates)e(ab)s(o)m(v)m(e)i(and)0 2660 y(c)m(hec)m(k)e(that)d(the)h(conclusion)f(still)e(holds.)44 b(This)32 b(is)g(straigh)m(tforw)m(ard.)146 2780 y(Finally)-8 b(,)38 b(for)g(Theorem)h(3.5)f(to)g(hold,)h(w)m(e)h(\014rst)f(m)m(ust)g (consider)f(the)h(con)m(tribution)f(asso)s(ciated)g(to)0 2901 y Fk(\037)61 2916 y Fi(1)117 2901 y Fk(\030)160 2916 y Fh(l)186 2901 y Fo(,)g(whic)m(h)g(relies)e(on)h(\(9.8\).)57 b(Here,)39 b(\(10.3\))e(applies)f(directly)-8 b(.)57 b(Next,)39 b(the)e(\014rst)h(con)m(tribution)e(from)0 3021 y Fk(\037)61 3036 y Fi(2)101 3021 y Fk(\030)144 3036 y Fh(l)197 3021 y Fo(is)27 b(\(9.11\).)41 b(It)27 b(has)h(the)g(form)f(\(7.3\))g(and)g(yields)h(exp)s(onen)m(tial)f(deca) m(y)i(in)d(the)i(same)g(w)m(a)m(y)-8 b(,)29 b(for)e Fk(g)k Fo(small)0 3141 y(enough.)44 b(It)32 b(remains)e(for)i(us)g(to)f(b)s (ound)h(\(9.13\).)43 b(With)31 b Fk(s)44 b Fj(\024)h Fk(l)23 b Fo(+)d(1)h(+)f(2)p Fk(k)47 b Fj(\024)e Fo(3)p Fk(l)r Fo(,)32 b(w)m(e)h(use)f(the)h(estimate,)489 3281 y Fd(q)p 572 3281 333 4 v 105 x Fo(\()p Fk(J)f Fo(+)22 b Fk(s)p Fo(\)!)60 b Fj(\024)g Fo(\()p Fk(\027)6 b(l)25 b Fo(+)d Fk(s)p Fo(\))1430 3345 y Fh(\027)t(l)q(=)p Fi(2)1582 3386 y Fo(\()p Fk(\027)6 b(l)25 b Fo(+)d Fk(s)p Fo(\))1910 3345 y Fh(s=)p Fi(2)2078 3386 y Fj(\024)60 b Fo(\(\()p Fk(\027)29 b Fo(+)22 b(3\))p Fk(l)r Fo(\))2622 3345 y Fh(\027)t(l)q(=)p Fi(2)2774 3386 y Fo(\(\()p Fk(\027)28 b Fo(+)22 b(3\))p Fk(g)t(=)p Fo(\026)-50 b Fk(h)o Fo(\))3303 3345 y Fh(s=)p Fi(2)0 3627 y Fo(in)24 b(\(9.14\).)40 b(The)26 b(\014rst)f(factor)g(when)h(m)m(ultiplied)21 b(b)m(y)26 b(e)1964 3574 y Fi(\026)1954 3591 y Fh(\014)s(J)2046 3627 y Fk(=)2095 3543 y Fj(p)p 2178 3543 91 4 v 84 x Fk(J)9 b Fo(!)25 b(is)f(of)h(order)g(e)2777 3591 y Fh(cg)r(=)p Fi(\026)-35 b Fh(h)2949 3627 y Fo(where)26 b Fk(c)e Fo(is)h(indep)s (enden)m(t)0 3748 y(of)k Fk(g)t Fo(.)42 b(The)31 b(\014nal)e(factor)g (allo)m(ws)f(us)j(to)e(rep)s(eat)h(the)g(argumen)m(t)g(that)f(led)g(to) h(\(9.15\).)41 b(Hence,)32 b(for)d Fk(g)k Fo(small)0 3868 y(enough,)g(w)m(e)h(get)e(an)h(exp)s(onen)m(tially)e(small)g(con)m (tribution)g(in)h(\026)-50 b Fk(h)33 b Fo(and)f(the)h(result)g(follo)m (ws.)p 3601 3868 67 84 v 0 4201 a Fl(References)0 4420 y Fo([1])49 b(Bam)m(busi,)30 b(D.,)h(Gra\016,)g(S.,)g(and)g(P)m(aul,)g (T.:)g(Long)f(Time)g(Semiclassical)e(Appro)m(ximation)h(of)h(Quan-)152 4540 y(tum)i(Flo)m(ws:)43 b(A)33 b(Pro)s(of)e(of)i(Ehrenfest)h(Time.)d (1998)h(preprin)m(t.)0 4744 y([2])49 b(Bonec)m(hi,)35 b(F.)e(and)i(De)e(Bi)m(\022)-46 b(evre,)35 b(S.:)g(Exp)s(onen)m(tial)e (mixing)f(and)i Fj(j)17 b Fo(ln)o(\(\026)-50 b Fk(h)p Fo(\))p Fj(j)34 b Fo(time)e(scales)j(in)e(quan)m(tized)152 4864 y(h)m(yp)s(erb)s(olic)f(maps)g(on)g(the)h(torus,)g(1999)f(preprin) m(t.)0 5068 y([3])49 b(Bouzouina,)35 b(A.)g(and)g(Rob)s(ert)g(D.:)f (Uniform)f(Semi-classical)f(Estimates)j(for)f(the)i(Propagation)d(of) 152 5188 y(Heisen)m(b)s(erg)g(Observ)-5 b(ables.)34 b(1999)d(preprin)m (t.)1901 5506 y(41)p eop %%Page: 42 42 42 41 bop 0 -43 a Fo([4])49 b(Com)m(b)s(escure,)e(M.:)d(The)g (e\016ciency)g(of)f(coheren)m(t)h(states)g(in)e(v)-5 b(arious)43 b(domains)f(of)g(semiclassical)152 77 y(ph)m(ysics.)34 b Fe(C.R.A.S.)g(s)n(\023)-47 b(erie)34 b(2b)k Fo(t.)f Fa(325)c Fo(pp.)g(635-648,)e(\(1997\).)0 281 y([5])49 b(Com)m(b)s(escure,)31 b(M.)e(and)g(Rob)s(ert,)g(D.:)f(Semiclassical)e (Spreading)j(of)f(Quan)m(tum)h(W)-8 b(a)m(v)m(e)30 b(P)m(ac)m(k)m(ets)h (and)152 401 y(Applications)c(near)h(Unstable)h(Fixed)f(P)m(oin)m(ts)h (of)f(the)h(Classical)e(Flo)m(w.)h Fe(Asymptotic)j(A)n(nal.)d Fa(14)p Fo(,)i(pp.)152 521 y(377{404,)h(\(1997\).)0 725 y([6])49 b(Gradsteyhn,)40 b(I.)16 b(S.)39 b(and)g(Ryzhik,)h(I.)16 b(M.:)39 b Fe(T)-7 b(able)39 b(of)h(Inte)-5 b(gr)g(als,)40 b(Series,)g(and)g(Pr)-5 b(o)g(ducts,)41 b(Fifth)f(Ed)p Fo(.)152 845 y(New)33 b(Y)-8 b(ork:)44 b(Academic)32 b(Press)i(1994.)0 1049 y([7])49 b(Hagedorn,)38 b(G.)16 b(A.:)98 b(Semiclassical)34 b(Quan)m(tum)j(Mec)m(hanics)h(I.)f(The)h (\026)-50 b Fk(h)35 b Fj(!)g Fo(0)i(Limit)d(for)j(Coheren)m(t)152 1169 y(States.)c Fe(Commun.)h(Math.)h(Phys.)d Fa(71)p Fo(,)h(77-93)e(\(1980\).)0 1372 y([8])49 b(Hagedorn,)40 b(G.)16 b(A.:)38 b(Semiclassical)e(Quan)m(tum)i(Mec)m(hanics)i(I)s(I)s (I:)e(The)h(Large)f(Order)h(Asymptotics)152 1493 y(and)32 b(More)h(General)f(States.)h Fe(A)n(nn.)i(Phys.)d Fa(135)p Fo(,)h(58{70)e(\(1981\).)0 1696 y([9])49 b(Hagedorn,)41 b(G.)16 b(A.:)40 b(Semiclassical)e(Quan)m(tum)h(Mec)m(hanics)i(IV:)f (Large)g(Order)g(Asymptotics)g(and)152 1817 y(More)30 b(General)f(States)i(in)e(More)h(than)g(One)h(Dimension.)d Fe(A)n(nn.)k(Inst.)g(H.)g(Poinc)-5 b(ar)n(\023)-47 b(e)31 b(Se)-5 b(ct.)32 b(A.)65 b Fa(42)p Fo(,)152 1937 y(363{374)31 b(\(1985\).)0 2140 y([10])49 b(Hagedorn,)40 b(G.)16 b(A.:)39 b(Raising)e(and)h(lo)m(w)m(ering)g(op)s(erators)h(for)f(semiclassical)e (w)m(a)m(v)m(e)41 b(pac)m(k)m(ets.)g Fe(A)n(nn.)152 2261 y(Phys.)32 b Fa(269)p Fo(,)h(77{104)e(\(1998\).)0 2464 y([11])49 b(Hagedorn,)32 b(G.)16 b(A.)33 b(and)g(Jo)m(y)m(e,)h(A.:)f (Semiclassical)d(Dynamics)i(with)g(Exp)s(onen)m(tially)g(Small)e(Error) 152 2584 y(Estimates.)i Fe(Commun.)i(Math.)h(Phys.)d Fa(207)p Fo(,)h(439{465)e(\(1999\).)0 2788 y([12])49 b(Hagedorn,)39 b(G.)16 b(A.)38 b(and)h(Jo)m(y)m(e,)i(A.:)d (Semiclassical)e(Dynamics)h(and)h(Exp)s(onen)m(tial)g(Asymptotics.)152 2908 y Fe(Pr)-5 b(o)g(c)g(e)g(e)g(dings)42 b(of)i(the)g(1999)f(UAB-GIT) g(International)g(Confer)-5 b(enc)g(e)42 b(on)i(Di\013er)-5 b(ential)42 b(Equations)152 3029 y(and)34 b(Mathematic)-5 b(al)34 b(Physics)f Fo(\(to)f(app)s(ear\).)0 3232 y([13])49 b(Hagedorn,)32 b(G.)16 b(A.)33 b(and)g(Jo)m(y)m(e,)h(A.:)e(In)h (preparation.)0 3435 y([14])49 b(Hepp,)41 b(K.:)e(The)h(Classical)e (Limit)e(for)i(Quan)m(tum)h(Mec)m(hanical)g(Correlation)e(F)-8 b(unctions.)39 b Fe(Com-)152 3556 y(mun.)34 b(Math.)h(Phys.)e Fa(35)p Fo(,)g(265{277)e(\(1974\).)0 3759 y([15])49 b(P)m(aul,)36 b(T.:)g(Semi-Classical)c(Metho)s(ds)37 b(with)e(Emphasis)g(on)h (Coheren)m(t)h(States)f(in)f Fe(Quasiclassic)-5 b(al)152 3880 y(Metho)g(ds)p Fo(,)40 b(J.)e(Rauc)m(h)h(and)g(B.)g(Simon)e(eds,)k (IMA)e(V)-8 b(olumes)37 b(in)h(Mathematics)g(and)h(Applications)152 4000 y Fa(95)p Fo(,)32 b(51-97,)g(Springer)g(1997.)0 4203 y([16])49 b(P)m(aul,)39 b(T.)g(and)g(Urib)s(e,)h(A.:)e(A)h (Construction)g(of)f(Quasi-Mo)s(des)g(using)g(Coheren)m(t)i(States.)g Fe(A)n(nn.)152 4324 y(I.H.P.)34 b(Se)-5 b(ct.)35 b(A.,)g(Physique)g(Th) n(\023)-47 b(eorique)38 b Fa(59)p Fo(,)33 b(357{381,)e(\(1993\).)0 4527 y([17])49 b(Reed,)44 b(M.)d(and)g(Simon,)h(B.:)g Fe(Metho)-5 b(ds)42 b(of)h(Mo)-5 b(dern)42 b(Mathematic)-5 b(al)43 b(Physics,)h(V)-7 b(ol.)42 b(II:)g(F)-7 b(ourier)152 4648 y(A)n(nalysis,)34 b(Self-A)-5 b(djointness.)31 b Fo(Academic)h(Press,)i(New)g(Y)-8 b(ork)32 b(\(1975\).)0 4851 y([18])49 b(Rob)s(ert,)h(D.:)d Fe(A)n(utour)i(de)f(l'appr)-5 b(oximation)47 b(semi-classique.)e Fo(Progress)k(in)d(Mathematics)h Fa(68)p Fo(,)152 4971 y(Birkha)s(\177)-51 b(user,)32 b(Boston)h(\(1987\).)1901 5506 y(42)p eop %%Page: 43 43 43 42 bop 0 -43 a Fo([19])49 b(Rob)s(ert,)e(D.:)d(Semi-Classical)d (Appro)m(ximation)i(in)g(Quan)m(tum)h(Mec)m(hanics.)i(A)f(Surv)m(ey)h (of)e(Old)152 77 y(and)27 b(Recen)m(t)h(Mathematical)e(Results.)h (Mathematical)e(results)j(in)e(quan)m(tum)i(mec)m(hanics)f(\(Ascona,) 152 198 y(1996\).)k Fe(Helv.)k(Phys.)g(A)-5 b(cta)78 b Fa(71)p Fo(,)32 b(44{116)g(\(1998\).)0 401 y([20])49 b(Y)-8 b(a)5 b(jima,)27 b(K.:)i(The)h(Quasi{classical)d(Limit)f(of)i (Quan)m(tum)h(Scattering)f(Theory)-8 b(.)30 b Fe(Commun.)h(Math.)152 521 y(Phys.)h Fa(69)p Fo(,)h(101{130)e(\(1979\).)0 725 y([21])49 b(Y)-8 b(a)5 b(jima,)32 b(K.:)j(The)g(Quasi{classical)d (Limit)f(of)j(Quan)m(tum)g(Scattering)g(Theory)-8 b(,)36 b(I)s(I.)e(Long)g(Range)152 845 y(Scattering.)e Fe(Duke)i(Math.)h(J.)e Fa(48)p Fo(,)g(1{21)e(\(1981\).)0 1049 y([22])49 b(Y)-8 b(a)5 b(jima,)46 b(K.:)g(Gevrey)h(F)-8 b(requency)47 b(Set)f(and)g(Semi-classical)c(Beha)m(vior)j(of)h(W)-8 b(a)m(v)m(e)46 b(P)m(ac)m(k)m(ets,)52 b(in)152 1169 y Fe(Schr\177)-50 b(odinger)29 b(Op)-5 b(er)g(ators,)31 b(The)f(Quantum)h(Me)-5 b(chanic)g(al)29 b(Many)i(Bo)-5 b(dy)31 b(Pr)-5 b(oblem)p Fo(,)28 b(Lecture)h(Notes)g(in)152 1289 y(Ph)m(ysics,)43 b Fa(403)p Fo(,)e(ed.)f(b)m(y)h(E.)e(Balslev.)g (Berlin,)h(Heidelb)s(erg,)h(New)f(Y)-8 b(ork:)57 b(Springer-V)-8 b(erlag,)40 b(1992,)152 1410 y(pp)33 b(248{264.)1901 5506 y(43)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9911260657900--