Content-Type: multipart/mixed; boundary="-------------0309031122606" This is a multi-part message in MIME format. ---------------0309031122606 Content-Type: text/plain; name="03-401.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-401.keywords" adiabatic approximation, exponential asymptotics ---------------0309031122606 Content-Type: application/postscript; name="hagjoy8.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hagjoy8.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: hagjoy8.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips hagjoy8.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.03:1714 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 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3502 y Fg(2)359 3487 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))63 b(=)f Fk(\037)832 3502 y Fg(2)871 3487 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))57 b(+)1272 3400 y Fj(p)p 1355 3400 49 4 v 87 x Fm(2)35 b Fk(e)1484 3446 y Fh(\000)p Fi(E)15 b(\016)r(=\017)1753 3419 y Fm(1)p 1753 3464 V 1753 3555 a(2)1863 3313 y Fb(8)1863 3387 y(<)1863 3537 y(:)1953 3487 y Fm(erf)2081 3316 y Fb(0)2081 3465 y(@)2170 3323 y(s)p 2253 3323 189 4 v 2308 3419 a Fk(E)p 2263 3464 169 4 v 2263 3555 a Fm(2)i Fk(\016)j(\017)2476 3487 y(t)2528 3316 y Fb(1)2528 3465 y(A)2640 3487 y Fm(+)38 b(1)2820 3313 y Fb(9)2820 3387 y(=)2820 3537 y(;)2945 3487 y Fk(\037)3006 3502 y Fg(1)3045 3487 y Fm(\()p Fk(\017;)c(t)p Fm(\))57 b(+)g Fk(O)s Fm(\()p Fk(e)3607 3446 y Fh(\000)p Fi(E)15 b(\016)r(=\017)3847 3487 y Fk(\017)3886 3446 y Fi(\026)3933 3487 y Fm(\))p Fk(:)0 3914 y Fa(Remarks)0 4035 y(0.)44 b Fm(Recall)31 b(that)h(the)h(function)f(erf)h(is)f (de\014ned)i(b)m(y)244 4277 y(erf\()p Fk(x)p Fm(\))60 b(=)739 4209 y(2)p 692 4253 142 4 v 692 4274 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y(that)i(the)h(sum)g(equals)g(8)p Fk(=)p Fm(3)f(when)h Fk(n)28 b Fm(=)g(3)k(and)h Fk(n)28 b Fm(=)f(4.)146 1706 y(F)-8 b(or)30 b Fk(n)e Fj(\025)g Fm(5,)j(w)m(e)h(separate)g(the)f (\014rst)g(t)m(w)m(o)g(terms)g(and)g(last)f(t)m(w)m(o)h(terms)g(to)f (see)i(that)f(the)g(sum)g(equals)244 1976 y(2)38 b(+)461 1908 y(2)p 456 1952 59 4 v 456 2044 a Fk(n)563 1976 y Fm(+)678 1868 y Fi(n)p Fh(\000)p Fg(2)684 1893 y Fb(X)683 2075 y Fi(j)t Fg(=2)854 1908 y Fm(\()p Fk(n)22 b Fj(\000)h Fk(j)6 b Fm(\)!)32 b Fk(j)6 b Fm(!)p 854 1952 435 4 v 1028 2044 a Fk(n)p Fm(!)1298 1976 y Fk(:)0 2257 y Fm(The)42 b(largest)f(terms)g(in)g(the)h(sum)f(o)m(v)m(er)h Fk(j)47 b Fm(in)41 b(this)g(expression)i(come)e(from)f Fk(j)48 b Fm(=)43 b(2)e(and)g Fk(j)49 b Fm(=)42 b Fk(n)29 b Fj(\000)f Fm(2.)0 2416 y(Those)35 b(terms)f(equal)988 2349 y(2)p 831 2393 363 4 v 831 2484 a Fk(n)p Fm(\()p Fk(n)23 b Fj(\000)f Fm(1\))1204 2416 y(,)35 b(and)f(there)h(are)f(\()p Fk(n)23 b Fj(\000)h Fm(3\))33 b(terms.)48 b(Th)m(us,)37 b(the)d(left)f(hand)i(side)f(of)f(\(3.7\))0 2587 y(is)f(b)s(ounded)h(b) m(y)244 2839 y(2)38 b(+)461 2772 y(2)p 456 2816 59 4 v 456 2908 a Fk(n)563 2839 y Fm(+)692 2772 y(2\()p Fk(n)22 b Fj(\000)h Fm(3\))p 687 2816 363 4 v 687 2908 a Fk(n)p Fm(\()p Fk(n)g Fj(\000)g Fm(1\))1121 2839 y Fj(\024)60 b Fm(2)39 b(+)1470 2772 y(2)p 1470 2816 49 4 v 1470 2908 a(5)1568 2839 y(+)1692 2772 y(1)p 1692 2816 V 1692 2908 a(5)1811 2839 y Fk(<)1957 2772 y Fm(8)p 1957 2816 V 1957 2908 a(3)2016 2839 y Fk(:)0 3120 y Fm(This)26 b(last)f(step)i(relies)e (on)g(the)h(observ)-5 b(ation)26 b(that)1856 3074 y Fg(2\()p Fi(n)p Fh(\000)p Fg(3\))p 1852 3097 231 4 v 1852 3155 a Fi(n)p Fg(\()p Fi(n)p Fh(\000)p Fg(1\))2119 3120 y Fm(tak)m(es)h(the)f(v)-5 b(alue)25 b(1)p Fk(=)p Fm(5)g(when)i Fk(n)h Fm(=)f(5)f(and)g Fk(n)i Fm(=)f(6,)0 3263 y(and)33 b(that)f(it)g(is)g(decreasing)h(for)f Fk(n)c Fj(\025)g Fm(6.)p 1582 3279 84 100 v 146 3501 a(F)-8 b(or)35 b(an)m(y)i Fk(y)t Fm(\()p Fk(t)p Fm(\))49 b(=)850 3435 y Fb(P)938 3522 y Fi(j)1007 3501 y Fk(y)1055 3516 y Fi(j)1108 3501 y Fk(e)1153 3516 y Fi(j)1190 3501 y Fm(\()p Fk(t)p Fm(\),)37 b(with)e Fj(f)p Fk(y)1688 3516 y Fi(j)1724 3501 y Fj(g)e(2)h Fk(l)1938 3465 y Fg(1)1978 3501 y Fm(,)i(w)m(e)h(de\014ne)h Fj(k)p Fk(y)t Fj(k)48 b Fm(=)2800 3435 y Fb(P)2888 3522 y Fi(j)2957 3501 y Fj(j)p Fk(y)3033 3516 y Fi(j)3069 3501 y Fj(j)p Fm(.)53 b(W)-8 b(e)37 b(note)f(that)f(for)0 3621 y Fk(t)40 b Fj(2)h Fk(I)-24 b(R)p Fm(,)42 b Fj(j)p Fk(y)t Fm(\()p Fk(t)p Fm(\))p Fj(j)55 b(\024)i(k)p Fk(y)t Fj(k)p Fm(.)64 b(Since)40 b Fk(G)1334 3636 y Fi(n)1420 3621 y Fm(is)g(obtained)f(from)g Fk(g)2219 3636 y Fi(n)2305 3621 y Fm(b)m(y)i(dropping)e(comp)s(onen)m(ts)h(in)f(the)h Fk(e)3752 3636 y Fi(j)3789 3621 y Fm(\()p Fk(t)p Fm(\))0 3742 y(basis,)c(w)m(e)g(note)f(that)g Fj(k)p Fk(G)976 3757 y Fi(n)1023 3742 y Fj(k)48 b(\024)h(k)p Fk(g)1344 3757 y Fi(n)1390 3742 y Fj(k)p Fm(.)i(Th)m(us,)37 b(the)f(follo)m(wing) c(lemma)h(implies)f(\(3.1\))j(and)g(\(3.2\))g(since)10 3823 y Fi(d)p 10 3839 62 4 v 10 3897 a(dt)98 3862 y Fm(\(1)22 b Fj(\006)h Fk(it)p Fm(\))413 3826 y Fh(\000)p Fi(j)549 3862 y Fm(=)44 b Fj(\007)17 b Fk(i)g(j)23 b Fm(\(1)e Fj(\006)i Fk(it)p Fm(\))1190 3826 y Fh(\000)p Fi(j)t Fh(\000)p Fg(1)1372 3862 y Fm(.)0 4101 y Fa(Lemma)37 b(3.3)49 b Fj(k)p Fk(g)686 4116 y Fi(n)733 4101 y Fj(k)44 b(\024)g Fm(\()p Fk(n)23 b Fj(\000)f Fm(1\)!)p Fc(.)0 4447 y Fa(Pro)s(of)97 b Fm(W)-8 b(e)33 b(pro)m(v)m(e)h(that)f(the)g (sequence)i Fk(a)1635 4462 y Fi(n)1726 4447 y Fm(=)44 b Fj(k)p Fk(g)1943 4462 y Fi(n)1990 4447 y Fj(k)p Fk(=)p Fm(\()p Fk(n)22 b Fj(\000)g Fm(1\)!)33 b(is)f(b)s(ounded)h(ab)s(o)m(v)m (e)h(b)m(y)f(1.)146 4568 y(By)g(\(2.6\),)g(Lemmas)e(3.1)h(and)h(3.2,)f (w)m(e)i(see)g(that)e Fk(n)44 b Fj(\025)h Fm(2)33 b(implies)244 4831 y Fk(a)295 4846 y Fi(n)p Fg(+1)492 4831 y Fj(\024)61 b Fk(a)681 4846 y Fi(n)767 4831 y Fm(+)971 4763 y(4)17 b Fk(a)1088 4727 y Fg(2)1088 4787 y Fi(n)p Fh(\000)p Fg(1)p 892 4808 412 4 v 892 4899 a Fm(3)p Fk(n)p Fm(\()p Fk(n)22 b Fj(\000)h Fm(1\))1313 4831 y Fk(:)2359 b Fm(\(3.8\))0 5090 y(F)-8 b(rom)37 b(\(2.4\))h(and)h(\(2.5\))f(w)m(e)i(ha)m(v)m(e)g Fk(a)1368 5105 y Fg(1)1463 5090 y Fm(=)54 b Fk(a)1644 5105 y Fg(2)1739 5090 y Fm(=)g(1)p Fk(=)p Fm(2.)62 b(By)39 b(explicit)e(computation,)i(w)m(e)h(observ)m(e)g(that)0 5210 y Fk(a)51 5225 y Fg(3)118 5210 y Fm(=)232 5171 y Fg(17)p 232 5187 71 4 v 232 5245 a(32)357 5210 y Fj(\024)k Fm(1)22 b Fj(\000)659 5171 y Fg(4)p 659 5187 36 4 v 659 5245 a(9)704 5210 y Fm(,)33 b(and)f Fk(a)1004 5225 y Fg(4)1072 5210 y Fm(=)1185 5171 y Fg(197)p 1185 5187 106 4 v 1185 5245 a(384)1345 5210 y Fj(\024)45 b Fm(1)22 b Fj(\000)1665 5171 y Fg(4)p 1648 5187 71 4 v 1648 5245 a(12)1728 5210 y Fm(.)1901 5506 y(10)p eop %%Page: 11 11 11 10 bop 146 -43 a Fm(The)40 b(lemma)d(no)m(w)j(follo)m(ws)e(b)m(y)h (induction)g(\(starting)f(at)g Fk(n)h Fm(=)g(4\))f(and)i(the)f(follo)m (wing)d(statemen)m(t:)0 77 y(If)d Fk(n)27 b Fj(\025)i Fm(4)p Fk(;)64 b(a)480 92 y Fi(n)p Fh(\000)p Fg(1)662 77 y Fj(\024)45 b Fm(1)38 b Fj(\000)1092 38 y Fg(4)p 998 54 224 4 v 998 112 a(3\()p Fi(n)p Fh(\000)p Fg(1\))1263 77 y Fm(and)33 b Fk(a)1504 92 y Fi(n)1595 77 y Fj(\024)45 b Fm(1)39 b Fj(\000)1952 38 y Fg(4)p 1931 54 79 4 v 1931 112 a(3)p Fi(n)2019 77 y Fm(,)33 b(then)g Fk(a)2352 92 y Fi(n)p Fg(+1)2533 77 y Fj(\024)45 b Fm(1)39 b Fj(\000)2963 38 y Fg(4)p 2869 54 224 4 v 2869 112 a(3\()p Fi(n)p Fg(+1\))3102 77 y Fm(.)146 211 y(T)-8 b(o)33 b(pro)m(v)m(e)h(this)e(statemen)m(t,)h (w)m(e)h(use)f(\(3.8\))f(to)h(see)g(that)g(for)f Fk(n)c Fj(\025)g Fm(4)244 493 y Fk(a)295 508 y Fi(n)p Fg(+1)515 493 y Fj(\024)84 b Fm(1)38 b Fj(\000)919 426 y Fm(4)p 889 470 108 4 v 889 562 a(3)p Fk(n)1045 493 y Fm(+)1351 426 y(4)p 1170 470 412 4 v 1170 562 a(3)p Fk(n)p Fm(\()p Fk(n)22 b Fj(\000)h Fm(1\))1625 347 y Fb( )1707 493 y Fm(1)f Fj(\000)2040 426 y Fm(4)p 1887 470 354 4 v 1887 562 a(3\()p Fk(n)g Fj(\000)h Fm(1\))2267 347 y Fb(!)2333 370 y Fg(2)516 814 y Fm(=)84 b(1)38 b Fj(\000)1041 747 y Fm(4)p 889 791 352 4 v 889 882 a(3\()p Fk(n)23 b Fm(+)f(1\))1290 814 y Fj(\000)1434 747 y Fm(8\(3)p Fk(n)1628 711 y Fg(2)1690 747 y Fm(+)g(10)p Fk(n)g Fj(\000)g Fm(29\))p 1416 791 803 4 v 1416 882 a(27\()p Fk(n)g Fm(+)g(1\))p Fk(n)p Fm(\()p Fk(n)g Fj(\000)h Fm(1\))2180 854 y Fg(3)515 1114 y Fj(\024)84 b Fm(1)38 b Fj(\000)1041 1046 y Fm(4)p 889 1090 352 4 v 889 1182 a(3\()p Fk(n)23 b Fm(+)f(1\))1251 1114 y Fk(:)p 1376 1130 84 100 v 146 1485 a Fm(Lemma)31 b(3.1)g(is)h(no)m(w)h(pro)m(v)m(en)g(b)m(y)g(the)f(commen)m(ts)g(b)s (efore)g(Lemma)f(3.3)g(and)h(the)h(follo)m(wing)c(lemma.)0 1724 y Fa(Lemma)37 b(3.4)49 b Fc(Ther)-5 b(e)34 b(exists)h Fk(C)7 b Fc(,)35 b(such)f(that)h Fj(k)p Fk(h)1804 1739 y Fi(n)1851 1724 y Fj(k)44 b(\024)h Fk(C)24 b Fm(\()p Fk(n)e Fj(\000)h Fm(2\)!)33 b(log\()p Fk(n)22 b Fj(\000)h Fm(2\))34 b Fc(and)0 1844 y Fj(k)p Fk(h)106 1808 y Fh(0)106 1869 y Fi(n)153 1844 y Fj(k)44 b(\024)h Fk(C)23 b Fm(\()p Fk(n)g Fj(\000)f Fm(1\)!)33 b(log\()p Fk(n)23 b Fj(\000)f Fm(2\))p Fc(.)0 2191 y Fa(Pro)s(of)97 b Fm(The)29 b(\014rst)e(estimate) g(implies)e(the)i(second)i(since)f Fk(h)2234 2206 y Fi(n)2308 2191 y Fm(is)f(in)g(the)g(span)h(of)f Fk(e)3046 2206 y Fi(j)3083 2191 y Fm(\()p Fk(t)p Fm(\))g(with)g Fk(j)34 b Fj(\024)28 b Fm(2)p Fk(n)11 b Fj(\000)g Fm(2.)146 2311 y(De\014ne)37 b Fk(b)493 2326 y Fi(n)590 2311 y Fm(=)50 b Fj(k)p Fk(h)822 2326 y Fi(n)868 2311 y Fj(k)p Fk(=)p Fm(\(\()p Fk(n)25 b Fj(\000)f Fm(2\)!\).)53 b(Since)36 b Fk(h)1773 2326 y Fi(n)1856 2311 y Fm(is)f(obtained)h(from)e Fk(g)2642 2326 y Fi(n)2725 2311 y Fm(b)m(y)i(dropping)f(comp)s(onen)m (ts)i(in)0 2432 y(the)c Fk(e)213 2447 y Fi(j)250 2432 y Fm(\()p Fk(t)p Fm(\))f(basis,)h(w)m(e)h(ha)m(v)m(e)g Fj(k)p Fk(h)1135 2447 y Fi(n)1181 2432 y Fj(k)45 b(\024)f(k)p Fk(g)1494 2447 y Fi(n)1541 2432 y Fj(k)p Fm(.)f(Th)m(us,)34 b(b)m(y)g(Lemma)d(3.3,)h Fk(b)2644 2447 y Fi(n)2736 2432 y Fj(\024)45 b Fk(n)22 b Fj(\000)h Fm(1.)146 2552 y(W)-8 b(e)33 b(rewrite)g(\(2.6\),)f(using)g Fk(g)1207 2567 y Fi(n)1298 2552 y Fm(=)44 b Fk(G)1495 2567 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b(e)26 b(then)g(drop)f(the)h Fk(e)802 3148 y Fg(2)p Fi(n)p Fg(+1)974 3133 y Fm(\()p Fk(t)p Fm(\))g(and)f Fk(e)1338 3148 y Fg(2)p Fi(n)p Fg(+2)1511 3133 y Fm(\()p Fk(t)p Fm(\))g(comp)s(onen)m(ts)h(of)f(this)g (expression)h(to)f(obtain)f(an)i(expression)0 3303 y(for)h Fk(h)200 3318 y Fi(n)p Fg(+1)337 3303 y Fm(.)42 b(This)27 b(in)m(v)m(olv)m(es)h(dropping)f(the)g(en)m(tire)h(term)e Fk(i)2109 3235 y(dG)2237 3250 y Fi(n)p 2109 3280 175 4 v 2154 3371 a Fk(dt)2294 3303 y Fm(,)i(as)g(w)m(ell)e(as)h(parts)h (of)f(other)g(terms.)42 b(Since)0 3449 y(the)33 b(norm)f(decreases)j (when)e(w)m(e)h(drop)e(comp)s(onen)m(ts,)i(w)m(e)f(see)h(that)119 3734 y Fj(k)p Fk(h)225 3749 y Fi(n)p Fg(+1)362 3734 y Fj(k)60 b(\024)610 3563 y Fb(0)610 3712 y(@)699 3584 y(\015)699 3634 y(\015)699 3684 y(\015)699 3734 y(\015)699 3784 y(\015)755 3666 y Fk(dh)862 3681 y Fi(n)p 755 3710 154 4 v 789 3802 a Fk(dt)919 3584 y Fb(\015)919 3634 y(\015)919 3684 y(\015)919 3734 y(\015)919 3784 y(\015)1004 3734 y Fm(+)1119 3559 y 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b(this)g(implies)244 5047 y Fk(b)285 5062 y Fi(n)p Fg(+1)483 5047 y Fj(\024)60 b Fk(b)661 5062 y Fi(n)748 5047 y Fm(+)913 4979 y(4)p Fk(=)p Fm(3)p 872 5023 229 4 v 872 5115 a Fk(n)22 b Fj(\000)h Fm(1)1150 5047 y(+)1315 4979 y(5)p Fk(=)p Fm(3)p 1274 5023 V 1274 5115 a Fk(n)g Fj(\000)f Fm(1)1552 5047 y(+)1717 4979 y(1)p Fk(=)p Fm(3)p 1676 5023 V 1676 5115 a Fk(n)h Fj(\000)f Fm(1)1975 5047 y(=)60 b Fk(b)2152 5062 y Fi(n)2222 5047 y Fm(+)2385 4979 y(10)p Fk(=)p Fm(3)p 2330 5023 305 4 v 2330 5115 a(\()p Fk(n)22 b Fj(\000)h Fm(1\))2644 5047 y Fk(;)1901 5506 y Fm(11)p eop %%Page: 12 12 12 11 bop 0 -43 a Fm(for)32 b Fk(n)c Fj(\025)g Fm(2.)43 b(Th)m(us,)244 227 y Fk(b)285 242 y Fi(n)393 227 y Fm(=)529 119 y Fi(n)p Fh(\000)p Fg(1)535 144 y Fb(X)531 328 y Fi(k)r Fg(=2)695 227 y Fm(\()16 b Fk(b)790 242 y Fi(k)r Fg(+1)962 227 y Fj(\000)40 b Fk(b)1120 242 y Fi(k)1179 227 y Fm(\))60 b(=)1414 119 y Fi(n)p Fh(\000)p Fg(1)1420 144 y Fb(X)1416 328 y Fi(k)r Fg(=2)1604 159 y Fm(10)p Fk(=)p Fm(3)p 1590 203 225 4 v 1590 295 a Fk(k)25 b Fj(\000)e Fm(1)1884 227 y Fj(\024)2032 159 y Fm(10)p 2032 203 98 4 v 2056 295 a(3)2173 227 y(\()16 b(1)39 b(+)f(log)q(\()p Fk(n)22 b Fj(\000)g Fm(2\))17 b(\))f Fk(:)0 511 y Fm(This)33 b(implies)d(the)j(lemma)d(and)j(completes)f(the)h(pro)s(of)f(of)g(Prop) s(osition)f(3.1.)p 2984 527 84 100 v 146 749 a(The)43 b(functions)f Fk(e)831 764 y Fi(j)868 749 y Fm(\()p Fk(t)p Fm(\))g(are)g(in)f Fk(L)1382 713 y Fg(1)1422 749 y Fm(\()p Fk(I)-24 b(R)p Fm(\))42 b(for)g Fk(j)49 b Fj(\025)44 b Fm(3.)72 b(The)43 b(functions)f Fk(e)2835 764 y Fg(1)2874 749 y Fm(\()p Fk(t)p Fm(\))g(and)g Fk(e)3271 764 y Fg(2)3311 749 y Fm(\()p Fk(t)p Fm(\))g(are)g(not)g(in)0 870 y Fk(L)66 833 y Fg(1)106 870 y Fm(\()p Fk(I)-24 b(R)p Fm(\),)41 b(but)e Fk(e)573 885 y Fg(1)613 870 y Fm(\()p Fk(t)p Fm(\))26 b(+)h Fk(e)898 885 y Fg(2)937 870 y Fm(\()p Fk(t)p Fm(\))39 b(is.)63 b(The)39 b(follo)m(wing)e(lemma)f(facilitates) h(getting)h Fk(L)3040 833 y Fg(1)3119 870 y Fm(information)d(ab)s(out)0 990 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b Fm(.)44 b(The)32 b(function)e Fk(i)17 b(g)1706 1779 y Fh(0)1702 1840 y Fi(N)7 b Fh(\000)p Fg(1)1890 1816 y Fm(con)m(tains)31 b(no)g Fk(e)2450 1831 y Fg(1)2489 1816 y Fm(\()p Fk(t)p Fm(\))g(or)g Fk(e)2794 1831 y Fg(2)2834 1816 y Fm(\()p Fk(t)p Fm(\))f(comp)s(onen)m(t.)43 b(Since)32 b(the)0 1936 y Fk(g)47 1951 y Fi(n)131 1936 y Fm(are)k(b)s(ounded,)j Fk(f)820 1870 y Fb(P)908 1896 y Fi(N)7 b Fh(\000)p Fg(2)908 1961 y Fi(j)t Fg(=1)1098 1936 y Fk(g)1145 1951 y Fi(j)1198 1936 y Fk(g)1245 1951 y Fi(N)g Fh(\000)p Fi(j)t Fh(\000)p Fg(1)1527 1936 y Fm(is)36 b(in)g Fk(L)1813 1900 y Fg(1)1853 1936 y Fm(\()p Fk(I)-24 b(R)q Fm(\))36 b(since)i Fk(f)47 b Fm(is.)56 b(The)38 b(only)f(w)m(a)m(y)h(this)e(function)h(can)0 2056 y(b)s(e)e(in)g Fk(L)318 2020 y Fg(1)392 2056 y Fm(is)g(if)f(it's)g Fk(e)795 2071 y Fg(1)835 2056 y Fm(\()p Fk(t)p Fm(\))h(and)g Fk(e)1218 2071 y Fg(2)1258 2056 y Fm(\()p Fk(t)p Fm(\))g(comp)s(onen)m (ts)h(are)f(equal.)50 b(This)36 b(implies)c(the)k(result)f(for)f Fk(g)3613 2071 y Fi(N)3680 2056 y Fm(,)i(and)0 2177 y(the)d(induction)f (can)g(pro)s(ceed.)146 2297 y(The)f(second)h(statemen)m(t)f(follo)m(ws) e(from)g(the)h(\014rst)h(b)s(ecause)h(the)e(absolute)g(v)-5 b(alues)30 b(of)g(\()p Fk(e)3405 2312 y Fg(1)3445 2297 y Fm(\()p Fk(t)p Fm(\))17 b(+)g Fk(e)3711 2312 y Fg(2)3751 2297 y Fm(\()p Fk(t)p Fm(\)\))0 2417 y(and)33 b Fk(e)235 2432 y Fi(j)271 2417 y Fm(\()p Fk(t)p Fm(\))g(for)f Fk(j)i Fj(\025)28 b Fm(3)k(are)h(dominated)e(b)m(y)j(\(1)21 b(+)h Fk(t)1846 2381 y Fg(2)1886 2417 y Fm(\))1924 2381 y Fh(\000)p Fg(1)2051 2417 y Fm(whic)m(h)33 b(has)g(in)m(tegral)e Fk(\031)t Fm(.)146 2538 y(The)f(third)e(statemen)m(t)h(follo)m(ws)e (since)i Fk(h)1639 2553 y Fi(n)1715 2538 y Fm(is)f(obtained)g(from)f Fk(g)2479 2553 y Fi(n)2554 2538 y Fm(b)m(y)j(remo)m(v)-5 b(al)27 b(of)h(the)h Fk(n)3374 2502 y Fg(th)3473 2538 y Fm(order)g(p)s(ole)0 2658 y(terms.)p 407 2675 V 146 2897 a(W)-8 b(e)30 b(no)m(w)h(examine)e Fk(G)968 2912 y Fi(n)1044 2897 y Fm(more)g(closely)-8 b(.)42 b(W)-8 b(e)30 b(note)g(that)f(highest)h(order)f(p)s(ole)g(terms)h(in)e Fk(g)3430 2912 y Fi(n)3507 2897 y Fm(at)h Fk(t)f Fm(=)f Fj(\006)p Fk(i)0 3017 y Fm(satisfy)33 b(the)g(recurrence)h(relation)244 3302 y Fk(G)321 3261 y Fh(\006)321 3327 y Fi(n)p Fg(+1)458 3302 y Fm(\()p Fk(t)p Fm(\))60 b(=)765 3131 y Fb(0)765 3281 y(@)871 3302 y Fk(i)931 3235 y(dG)1059 3199 y Fh(\006)1059 3259 y Fi(n)p 931 3279 187 4 v 981 3371 a Fk(dt)1128 3302 y Fm(\()p Fk(t)p Fm(\))54 b Fj(\006)1457 3235 y Fm(1)p Fk(=)p Fm(4)p 1436 3279 191 4 v 1436 3371 a Fk(t)22 b Fj(\007)h Fk(i)1685 3194 y Fi(n)p Fh(\000)p Fg(1)1691 3219 y Fb(X)1690 3401 y Fi(j)t Fg(=1)1867 3302 y Fk(G)1944 3261 y Fh(\006)1944 3327 y Fi(j)2003 3302 y Fm(\()p Fk(t)p Fm(\))33 b Fk(G)2224 3261 y Fh(\006)2224 3327 y Fi(n)p Fh(\000)p Fi(j)2358 3302 y Fm(\()p Fk(t)p Fm(\))2486 3131 y Fb(1)2485 3281 y(A)2575 3302 y Fk(;)0 3651 y Fm(with)f Fk(G)299 3610 y Fh(\006)299 3675 y Fg(1)358 3651 y Fm(\()p Fk(t)p Fm(\))60 b(=)28 b Fj(\006)753 3583 y Fm(1)p 753 3628 49 4 v 753 3719 a(4)925 3583 y(1)p 854 3628 191 4 v 854 3719 a Fk(t)23 b Fj(\007)f Fk(i)1087 3651 y Fm(and)33 b Fk(G)1354 3610 y Fh(\006)1354 3675 y Fg(2)1413 3651 y Fm(\()p Fk(t)p Fm(\))60 b(=)1730 3583 y Fj(\007)17 b Fk(i)p 1730 3628 128 4 v 1769 3719 a Fm(4)2038 3583 y(1)p 1910 3628 306 4 v 1910 3719 a(\()p Fk(t)22 b Fj(\007)h Fk(i)p Fm(\))2176 3690 y Fg(2)2225 3651 y Fm(.)44 b(F)-8 b(rom)31 b(this)h(it)g(follo)m(ws)f(that)244 3935 y Fk(G)321 3950 y Fi(n)368 3935 y Fm(\()p Fk(t)p Fm(\))60 b(=)g Fk(i)17 b(\015)776 3950 y Fi(n)856 3839 y Fb(\020)922 3935 y Fk(e)967 3950 y Fg(2)p Fi(n)p Fh(\000)p Fg(1)1140 3935 y Fm(\()p Fk(t)p Fm(\))38 b(+)h(\()p Fj(\000)p Fm(1\))1606 3894 y Fi(n)p Fh(\000)p Fg(1)1760 3935 y Fk(e)1805 3950 y Fg(2)p Fi(n)1887 3935 y Fm(\()p Fk(t)p Fm(\))2015 3839 y Fb(\021)2081 3935 y Fk(;)1591 b Fm(\(3.9\))0 4138 y(where)34 b Fk(\015)333 4153 y Fi(n)412 4138 y Fm(satis\014es)f(the)g(real)f(n)m (umerical)f(recurrence)j(relation)244 4408 y Fk(\015)295 4423 y Fi(n)p Fg(+1)492 4408 y Fm(=)60 b Fk(n)33 b(\015)770 4423 y Fi(n)871 4408 y Fj(\000)1013 4341 y Fm(1)p 1013 4385 49 4 v 1013 4476 a(4)1121 4300 y Fi(n)p Fh(\000)p Fg(1)1127 4325 y Fb(X)1126 4507 y Fi(j)t Fg(=1)1303 4408 y Fk(\015)1354 4423 y Fi(j)1423 4408 y Fk(\015)1474 4423 y Fi(n)p Fh(\000)p Fi(j)1607 4408 y Fk(;)2017 b Fm(\(3.10\))0 4695 y(with)32 b Fk(\015)273 4710 y Fg(1)357 4695 y Fm(=)44 b Fk(\015)528 4710 y Fg(2)611 4695 y Fm(=)g(1)p Fk(=)p Fm(4.)146 4815 y(By)33 b(Lemma)f(3.1,)g(the)h(quan)m(tit)m(y)g Fk(\014)1445 4830 y Fi(n)1536 4815 y Fm(=)45 b Fk(\015)1708 4830 y Fi(n)1754 4815 y Fk(=)p Fm(\()p Fk(n)22 b Fj(\000)h Fm(1\)!)32 b(is)g(b)s(ounded.)44 b(It)33 b(satis\014es)244 5090 y Fk(\014)299 5105 y Fi(n)p Fg(+1)496 5090 y Fm(=)60 b Fk(\014)687 5105 y Fi(n)789 5090 y Fj(\000)931 5023 y Fm(1)p 931 5067 V 931 5159 a(4)1039 4982 y Fi(n)p Fh(\000)p Fg(1)1046 5007 y Fb(X)1044 5190 y Fi(j)t Fg(=1)1231 5023 y Fm(\(\()p Fk(j)28 b Fj(\000)23 b Fm(1\)!\)\(\()p Fk(n)f Fj(\000)g Fk(j)28 b Fj(\000)23 b Fm(1\)!\))p 1231 5067 971 4 v 1674 5159 a Fk(n)p Fm(!)2244 5090 y Fk(\014)2299 5105 y Fi(j)2368 5090 y Fk(\014)2423 5105 y Fi(n)p Fh(\000)p Fi(j)2558 5090 y Fk(;)1066 b Fm(\(3.11\))1901 5506 y(12)p eop %%Page: 13 13 13 12 bop 0 -43 a Fm(with)41 b Fk(\014)286 -28 y Fg(1)384 -43 y Fm(=)59 b Fk(\014)574 -28 y Fg(2)672 -43 y Fm(=)g(1)p Fk(=)p Fm(4.)68 b(F)-8 b(rom)40 b(this)h(relation)e(and)i(Lemma)f(3.2,) j(it)d(follo)m(ws)g(that)h Fk(\014)3339 -28 y Fi(n)3427 -43 y Fm(has)h(a)e(limit)0 77 y Fk(\014)61 41 y Fh(\003)135 77 y Fm(as)34 b Fk(n)h Fm(tends)h(to)e(in\014nit)m(y)-8 b(.)49 b(T)-8 b(o)34 b(see)i(this,)f(supp)s(ose)g(that)g(the)g (sequence)i Fk(\014)2843 92 y Fi(m)2944 77 y Fm(is)d(p)s(ositiv)m(e)g (and)h(strictly)0 198 y(decreasing)e(for)f Fk(m)c Fj(\024)g Fk(n)p Fm(.)44 b(Then,)244 467 y(0)60 b Fk(<)489 359 y Fi(n)p Fh(\000)p Fg(1)495 384 y Fb(X)494 567 y Fi(j)t Fg(=1)681 400 y Fm(\(\()p Fk(j)28 b Fj(\000)23 b Fm(1\)!\)\(\()p Fk(n)f Fj(\000)g Fk(j)28 b Fj(\000)23 b Fm(1\)!\))p 681 444 971 4 v 1124 536 a Fk(n)p Fm(!)1694 467 y Fk(\014)1749 482 y Fi(j)1818 467 y Fk(\014)1873 482 y Fi(n)p Fh(\000)p Fi(j)2068 467 y Fk(<)2214 400 y Fm(8)p 2214 444 49 4 v 2214 536 a(3)2446 400 y Fk(\014)2507 364 y Fg(2)2501 425 y(1)p 2315 444 363 4 v 2315 536 a Fk(n)p Fm(\()p Fk(n)g Fj(\000)f Fm(1\))2688 467 y Fk(:)936 b Fm(\(3.12\))0 754 y(Th)m(us,)34 b(iterating)d(and)i(using)f Fk(\014)1170 769 y Fg(1)1237 754 y Fm(=)c Fk(\014)1396 769 y Fg(2)1463 754 y Fm(=)f(1)p Fk(=)p Fm(4,)32 b(w)m(e)i(ha)m(v)m(e)244 1016 y Fk(\014)299 1031 y Fi(n)p Fg(+1)519 1016 y Fk(>)83 b(\014)733 1031 y Fi(n)835 1016 y Fj(\000)977 949 y Fm(2)p 977 993 49 4 v 977 1085 a(3)1209 949 y Fk(\014)1270 913 y Fg(2)1264 974 y(1)p 1078 993 363 4 v 1078 1085 a Fk(n)p Fm(\()p Fk(n)23 b Fj(\000)f Fm(1\))1511 1016 y Fk(>)60 b(\014)1702 1031 y Fg(2)1796 1016 y Fj(\000)1938 949 y Fm(2)p Fk(\014)2048 913 y Fg(2)2042 974 y(1)p 1938 993 149 4 v 1988 1085 a Fm(3)2186 908 y Fi(n)2148 933 y Fb(X)2146 1116 y Fi(j)t 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Fj(\000)f Fm(1)1419 1938 y Fj(\000)1720 1870 y Fm(1)p 1545 1914 398 4 v 1545 2006 a Fk(n)g Fm(+)g Fk(p)h Fj(\000)f Fm(1)1970 1792 y Fb(!)2052 1938 y Fk(;)1572 b Fm(\(3.14\))0 2198 y(so)33 b(that,)244 2401 y Fk(\014)299 2416 y Fi(n)406 2401 y Fm(=)60 b Fk(\014)603 2360 y Fh(\003)675 2401 y Fm(\(1)54 b(+)h Fk(O)s Fm(\()p Fk(n)1121 2360 y Fh(\000)p Fg(1)1214 2401 y Fm(\)\))p Fk(:)2334 b Fm(\(3.15\))0 2767 y Fa(Remark)97 b Fm(Later,)33 b(w)m(e)g(will)e(see)i(that)g Fk(\014)1532 2726 y Fh(\003)1598 2767 y Fm(=)1783 2700 y(1)p 1712 2744 191 4 v 1712 2844 a Fk(\031)1771 2762 y Fj(p)p 1854 2762 49 4 v 82 x Fm(2)1912 2767 y(.)0 3116 y Fl(4)161 b(Optimal)55 b(T)-13 b(runcation)0 3335 y Fm(W)-8 b(e)33 b(b)s(egin)f(this)g(section)h(b)m(y)g(studying) 244 3582 y Fk(\020)287 3597 y Fi(n)333 3582 y Fm(\()p Fk(\017;)h(t)p Fm(\))60 b(=)g Fk(i)17 b(\017)856 3515 y(@)5 b( )p 856 3559 124 4 v 872 3650 a(@)g(t)1029 3582 y Fj(\000)39 b Fk(H)8 b Fm(\()p Fk(t)p Fm(\))17 b Fk( )t(;)2243 b Fm(\(4.1\))0 3886 y(where)38 b Fk( )j Fm(is)c(giv)m(en)g(b)m(y)h (\(2.2\))e(with)h Fk(g)t Fm(\()p Fk(\017;)c(t)p Fm(\))51 b(=)1836 3778 y Fi(n)1797 3803 y Fb(X)1796 3986 y Fi(j)t Fg(=1)1952 3886 y Fk(g)1999 3901 y Fi(j)2035 3886 y Fm(\()p Fk(t)p Fm(\))17 b Fk(\017)2202 3845 y Fi(j)2239 3886 y Fm(.)56 b(W)-8 b(e)38 b(ultimately)c(c)m(ho)s(ose)k Fk(n)d Fm(=)g([)-18 b([1)p Fk(=\017)p Fm(])g(])27 b Fj(\000)e Fm(1,)0 4090 y(where)34 b([)-18 b([)p Fk(k)s Fm(])g(])34 b(denotes)f(the)g(greatest)h(in)m(teger)e(less)h(than)g(or)f(equal)g (to)g Fk(k)s Fm(.)146 4245 y(By)h(explicit)f(calculation,)e Fk(\020)1207 4260 y Fi(n)1253 4245 y Fm(\()p Fk(\017;)k(t)p Fm(\))f(equals)g Fk(e)1838 4209 y Fi(it=)p Fg(\(2)p Fi(\017)p Fg(\))2078 4245 y Fk(e)2123 4133 y Fb(R)2179 4154 y Ff(t)2162 4230 y Fe(0)2218 4196 y Fg(\()p Fi(f)7 b Fg(\()p Fi(s)p Fg(\))2398 4138 y Fb(P)2486 4158 y Ff(n)2486 4225 y(j)s Fe(=1)2619 4196 y Fi(g)2653 4206 y Ff(j)2685 4196 y Fg(\()p Fi(s)p Fg(\))12 b Fi(\017)2813 4173 y Ff(j)2845 4196 y Fg(\))g 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b(\014rst)h(term)g(in)g(\(4.2\))f(satis\014es)244 5131 y Fk(\017)283 5090 y Fi(n)p Fg(+1)437 5131 y Fj(k)p Fk(G)564 5090 y Fh(0)564 5156 y Fi(n)611 5131 y Fj(k)62 b Fm(=)h(2)17 b Fk(\014)983 5146 y Fi(n)1046 5131 y Fk(\017)1085 5090 y Fi(n)p Fg(+1)1239 5131 y Fm(\()p Fk(n)p Fm(!\))p Fk(:)2272 b Fm(\(4.3\))1901 5506 y(13)p eop %%Page: 14 14 14 13 bop 0 -43 a Fc(When)35 b Fk(n)g Fc(is)f(chosen)g(to)h(b)-5 b(e)35 b Fk(n)44 b Fm(=)g([)-18 b([1)p Fk(=\017)p Fm(])g(])40 b Fj(\000)f Fm(1)p Fc(,)34 b(the)h(norm)g(of)f(the)h(r)-5 b(emaining)34 b(terms)g(in)h(\(4.2\))f(satis\014es)244 73 y Fb(\015)244 123 y(\015)244 173 y(\015)17 b Fk(\020)350 188 y Fi(n)396 173 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))39 b Fj(\000)g Fk(i)17 b(\017)851 132 y Fi(n)p Fg(+1)1005 173 y Fk(G)1082 132 y Fh(0)1082 197 y Fi(n)1145 73 y Fb(\015)1145 123 y(\015)1145 173 y(\015)63 b Fj(\024)g Fk(C)24 b(\017)1527 132 y Fi(n)p Fg(+1)1681 173 y Fm(\(\()p Fk(n)e Fj(\000)h Fm(1\)!\))33 b(log\()p Fk(n)22 b Fj(\000)h Fm(2\))p Fk(:)1119 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y(1)p 1636 1679 V 1636 1771 a(2)1727 1703 y(\()p Fk(n)23 b Fj(\000)f Fm(2\)!)60 b(=)2265 1635 y(4)p 2265 1679 V 2265 1771 a(3)2357 1703 y Fk(\017)2396 1662 y Fi(n)p Fg(+1)2566 1703 y Fm(\()p Fk(n)22 b Fj(\000)h Fm(2\)!)p Fk(:)774 b Fm(\(4.6\))0 1988 y(W)-8 b(e)33 b(no)m(w)g(pro)m(v)m(e)h(that)244 2098 y Fb(\015)244 2148 y(\015)244 2198 y(\015)244 2248 y(\015)244 2298 y(\015)244 2348 y(\015)336 2165 y Fg(2)p Fi(n)p Fg(+1)360 2190 y Fb(X)307 2374 y Fi(k)r Fg(=)p Fi(n)p Fg(+2)566 2273 y Fk(\017)605 2232 y Fi(k)665 2273 y Fk(f)876 2165 y Fi(n)837 2190 y Fb(X)740 2374 y Fi(j)t Fg(=)p Fi(k)r Fh(\000)p Fi(n)p Fh(\000)p Fg(1)1087 2273 y Fk(g)1134 2288 y Fi(j)1187 2273 y Fk(g)1234 2288 y Fi(k)r Fh(\000)p Fi(j)t Fh(\000)p Fg(1)1470 2098 y Fb(\015)1470 2148 y(\015)1470 2198 y(\015)1470 2248 y(\015)1470 2298 y(\015)1470 2348 y(\015)1577 2273 y Fj(\024)60 b Fk(C)1791 2232 y Fh(00)1850 2273 y Fk(\017)1889 2232 y Fi(n)p Fg(+1)2043 2273 y Fm(\()p Fk(n)23 b Fj(\000)f Fm(1\)!)1324 b(\(4.7\))0 2563 y(for)32 b(some)g Fk(C)470 2527 y Fh(00)545 2563 y Fm(when)i Fk(n)45 b Fm(=)f([)-18 b([1)p Fk(=\017)p Fm(])g(])40 b Fj(\000)f Fm(1.)146 2684 y(W)-8 b(e)33 b(b)s(egin)f(the)h(pro)s(of)f(of)g (\(4.7\))g(with)g(a)g(tec)m(hnical)h(lemma:)0 2997 y Fa(Lemma)k(4.2)49 b Fc(F)-7 b(or)25 b(p)-5 b(ositive)26 b(inte)-5 b(gers)26 b Fk(l)k Fj(\024)e Fk(m=)p Fm(2)p Fc(,)1935 2889 y Fi(m)p Fh(\000)p Fi(l)1945 2914 y Fb(X)1949 3099 y Fi(p)p Fg(=)p Fi(l)2101 2997 y Fm(\()p Fk(p)p Fm(!\))17 b(\(\()p Fk(m)s Fj(\000)s Fk(p)p Fm(\)!\))54 b Fj(\024)71 b Fm(\()p Fk(m)s Fj(\000)s Fm(2)p Fk(l)5 b Fm(+)s(1\))17 b(\()p Fk(l)r Fm(!\))g(\(\()p Fk(m)s Fj(\000)s Fk(l)r Fm(\)!\))p Fc(.)0 3306 y Fa(Pro)s(of)486 3451 y Fi(m)p Fh(\000)p Fi(l)495 3476 y Fb(X)499 3661 y Fi(p)p Fg(=)p Fi(l)657 3559 y Fm(\()p Fk(p)p Fm(!\))g(\(\()p Fk(m)22 b Fj(\000)h Fk(p)p Fm(\)!\))327 3831 y(=)83 b(\()p Fk(l)r Fm(!\)\(\()p Fk(m)22 b Fj(\000)h Fk(l)r Fm(\)!\))39 b(+)g(\(\()p Fk(l)24 b Fm(+)e(1\)!\)\(\()p Fk(m)g Fj(\000)h Fk(l)h Fj(\000)f 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Fg(=)p Fi(k)r Fh(\000)p Fi(n)p Fh(\000)p Fg(1)2702 4753 y Fk(g)2749 4768 y Fi(j)2802 4753 y Fk(g)2849 4768 y Fi(k)r Fh(\000)p Fi(j)t Fh(\000)p Fg(1)3098 4753 y Fm(in)29 b(\(4.2\))g(is)g(b)s(ounded)0 5046 y(b)m(y)k Fk(\017)174 5005 y Fi(k)244 4979 y Fm(1)p 244 5023 49 4 v 244 5115 a(2)336 4938 y Fi(m)p Fh(\000)p Fi(l)346 4963 y Fb(X)350 5148 y Fi(p)p Fg(=)p Fi(l)508 5046 y Fk(p)p Fm(!)17 b(\(\()p Fk(m)22 b Fj(\000)h Fk(p)p Fm(\)!\))60 b Fj(\024)h Fk(\017)1273 5005 y Fi(k)1332 5046 y Fm(\(2)p Fk(n)23 b Fj(\000)f Fk(k)j Fm(+)d(2\))17 b(\(\()p Fk(k)25 b Fj(\000)d Fk(n)h Fj(\000)f Fm(2\)!\))17 b(\(\()p Fk(n)22 b Fj(\000)h Fm(1\)!\))p Fk(=)p Fm(2.)1901 5506 y(14)p eop %%Page: 15 15 15 14 bop 146 -43 a Fm(T)-8 b(o)33 b(pro)m(v)m(e)h(\(4.7\),)e(w)m(e)i (no)m(w)f(sum)f(this)g(quan)m(tit)m(y)i(o)m(v)m(er)f Fk(k)s Fm(:)515 83 y Fg(2)p Fi(n)p Fg(+1)539 108 y Fb(X)486 292 y Fi(k)r Fg(=)p Fi(n)p Fg(+2)745 191 y Fk(\017)784 149 y Fi(k)844 191 y Fm(\(2)p Fk(n)22 b Fj(\000)h Fk(k)i Fm(+)d(2\))17 b(\(\()p Fk(k)24 b Fj(\000)f Fk(n)f Fj(\000)h Fm(2\)!\))17 b(\(\()p Fk(n)22 b Fj(\000)g Fm(1\)!\))p Fk(=)p Fm(2)327 541 y(=)83 b Fk(\017)525 500 y Fi(n)p Fg(+1)679 541 y Fm(\(\()p Fk(n)22 b Fj(\000)h Fm(1\)!\))1149 433 y Fg(2)p Fi(n)p Fg(+1)1173 458 y Fb(X)1120 643 y Fi(k)r Fg(=)p Fi(n)p Fg(+2)1389 474 y Fm(2)p Fk(n)f Fj(\000)h Fk(k)i Fm(+)d(2)p 1389 518 452 4 v 1591 609 a(2)1868 541 y Fk(\017)1907 500 y Fi(k)r Fh(\000)p Fi(n)p Fh(\000)p Fg(1)2154 541 y Fm(\(\()p Fk(k)j Fj(\000)e Fk(n)f Fj(\000)h Fm(2\)!\))327 891 y(=)83 b Fk(\017)525 850 y Fi(n)p Fg(+1)679 891 y Fm(\(\()p Fk(n)22 b Fj(\000)h Fm(1\)!\))1130 784 y Fi(n)p Fh(\000)p Fg(1)1136 808 y Fb(X)1120 990 y Fi(m)p Fg(=0)1316 824 y Fk(n)f Fj(\000)h Fk(m)p 1316 868 266 4 v 1424 960 a Fm(2)1607 891 y Fk(\017)1646 850 y Fi(m)p Fg(+1)1820 891 y Fm(\()p Fk(m)p Fm(!\))p Fk(:)1664 b Fm(\(4.8\))146 1123 y(W)-8 b(e)33 b(no)m(w)h(\014x)f Fk(n)44 b Fm(=)g([)-18 b([1)p Fk(=\017)p Fm(])g(])40 b Fj(\000)g Fm(1)32 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Fj(\024)59 b Fk(C)23 b Fm(\()p Fk(j)6 b(=n)p Fm(\))1709 2221 y Fi(j)1762 2257 y Fk(e)1807 2221 y Fh(\000)p Fi(j)1915 2185 y Fj(p)p 1998 2185 46 4 v 72 x Fk(j)47 b Fm(for)40 b(some)h Fk(C)48 b(>)41 b Fm(0.)68 b(So,)43 b(the)e(quan)m(tit)m(y)g(on)g(the)0 2378 y(righ)m(t)32 b(hand)h(side)f(here)i(is)e(b)s(ounded.)44 b(This)33 b(and)f(\(4.8\))g(imply)f(\(4.7\),)h(whic)m(h)h(pro)m(v)m(es) i(the)e(lemma.)p 3755 2394 84 100 v 146 2616 a(W)-8 b(e)33 b(note)g(that)f(our)h(c)m(hoice)g(of)f Fk(n)c Fm(=)f([)-18 b([1)p Fk(=\017)p Fm(])g(])24 b Fj(\000)e Fm(1)33 b(implies)d(that)i (there)h(exists)h(a)e Fk(C)j(>)27 b Fm(0)32 b(suc)m(h)j(that)244 2691 y Fb(\014)244 2741 y(\014)244 2791 y(\014)244 2841 y(\014)244 2890 y(\014)244 2940 y(\014)288 2748 y(Z)444 2758 y Fi(n)406 2782 y Fb(X)404 2965 y Fi(j)t Fg(=1)560 2865 y Fk(g)607 2880 y Fi(j)643 2865 y Fm(\()p Fk(t)p Fm(\))17 b Fk(\017)810 2824 y Fi(j)864 2865 y Fk(dt)967 2691 y Fb(\014)966 2741 y(\014)966 2791 y(\014)966 2841 y(\014)966 2890 y(\014)966 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Fb(Z)327 2186 y Fi(t)290 2348 y Fh(\0001)573 2209 y Fk(e)618 2173 y Fi(is=\017)p 463 2254 391 4 v 463 2345 a Fm(\(1)22 b(+)g Fk(is)p Fm(\))787 2316 y Fi(m)880 2277 y Fk(ds)62 b Fm(=)1178 2151 y Fb(r)p 1261 2151 171 4 v 1316 2209 a Fk(\031)p 1271 2254 151 4 v 1271 2345 a Fm(2)17 b Fk(m)1464 2156 y Fb(\032)1543 2277 y Fm(erf)1671 2156 y Fb(\022)1749 2151 y(r)p 1832 2151 106 4 v 1842 2209 a Fk(m)p 1842 2254 86 4 v 1860 2345 a Fm(2)1972 2277 y Fk(t)2024 2156 y Fb(\023)2124 2277 y Fm(+)38 b(1)2304 2156 y Fb(\033)2423 2277 y Fm(+)57 b Fk(O)s Fm(\()p Fk(m)2757 2236 y Fh(\000)p Fi(\015)2856 2277 y Fm(\))p Fk(;)778 b Fm(\(5.1\))0 2520 y Fc(and)244 2638 y Fb(Z)327 2664 y Fi(t)290 2826 y Fh(\0001)574 2687 y Fk(e)619 2651 y Fi(is=\017)p 463 2732 393 4 v 463 2823 a Fm(\(1)22 b Fj(\000)g Fk(is)p Fm(\))788 2794 y Fi(m)882 2755 y Fk(ds)62 b Fm(=)g Fk(O)s Fm(\()p Fk(m)1380 2714 y Fh(\000)p Fi(\015)1479 2755 y Fm(\))p Fk(:)2155 b Fm(\(5.2\))0 3020 y Fa(Pro)s(of)97 b Fm(W)-8 b(e)38 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y(error)k(in)g(eac)m(h)i (of)e(the)h(in)m(tegrals,)e(if)h(w)m(e)h(drop)g(the)g(con)m(tributions) f(from)f Fj(j)p Fk(s)p Fj(j)c(\025)h Fm(1.)146 3969 y(Next,)e(let)c Fk(\016)31 b(>)d Fm(0)22 b(b)s(e)h(small)d(and)j Fk(a)k Fm(=)h Fk(m)1599 3933 y Fi(\016)1637 3969 y Fk(=)1686 3898 y Fj(p)p 1769 3898 86 4 v 71 x Fk(m)p Fm(.)40 b(If)23 b Fk(a)28 b Fj(\024)g(j)p Fk(s)p Fj(j)f(\024)h Fm(1,)c(then)f(\(1)r(+)r Fk(s)2952 3933 y Fg(2)2991 3969 y Fm(\))3029 3933 y Fh(\000)p Fi(m=)p Fg(2)3265 3969 y Fj(\024)45 b Fm(\(1)r(+)r Fk(a)3605 3933 y Fg(2)3643 3969 y Fm(\))3681 3933 y Fh(\000)p Fi(m=)p Fg(2)3873 3969 y Fm(.)0 4089 y(Th)m(us)244 4187 y Fb(Z)327 4213 y Fg(1)290 4376 y Fi(a)383 4304 y Fm(\(1)22 b(+)g Fk(s)636 4263 y Fg(2)675 4304 y Fm(\))713 4263 y Fh(\000)p Fi(m=)p Fg(2)922 4304 y Fk(ds)60 b Fj(\024)g Fm(\(1)22 b Fj(\000)h Fk(a)p Fm(\))17 b(\(1)k(+)h Fk(a)1788 4263 y Fg(2)1828 4304 y Fm(\))1866 4263 y Fh(\000)p Fi(m=)p Fg(2)2118 4304 y Fj(\024)61 b Fm(\(1)22 b(+)g Fk(a)2514 4263 y Fg(2)2553 4304 y Fm(\))2591 4263 y Fh(\000)p Fi(m=)p Fg(2)2783 4304 y Fk(:)889 b Fm(\(5.3\))0 4531 y(No)m(w,)33 b(if)f Fk(\016)k Fm(is)c(small)e(enough,)k(\(5.3\))e(is)g(of)g(the)h (order)g(of)244 4761 y Fk(e)289 4720 y Fh(\000)354 4693 y Ff(m)p 354 4705 55 4 v 366 4746 a Fe(2)430 4720 y Fg(ln\(1+)p Fi(a)643 4697 y Fe(2)679 4720 y Fg(\))738 4761 y Fm(=)27 b Fk(e)886 4720 y Fh(\000)951 4693 y Ff(m)1005 4672 y Fe(2)p Ff(\016)p 952 4705 120 4 v 996 4746 a Fe(2)1080 4720 y Fg(+)p Fi(O)r Fg(\(1)p Fi(=m)1350 4697 y Fe(1)p Fd(\000)p Fe(4)p Ff(\016)1494 4720 y Fg(\))1554 4761 y Fm(=)g Fk(O)s Fm(\()p Fk(e)1818 4720 y Fh(\000)1883 4693 y Ff(m)1937 4672 y Fe(2)p Ff(\016)p 1882 4705 V 1927 4746 a Fe(2)2015 4761 y Fm(\))h Fk(<<)f(O)s Fm(\()p Fk(m)2461 4720 y Fh(\000)p Fg(1)2555 4761 y Fm(\))p Fk(:)1079 b Fm(\(5.4\))146 4954 y(Therefore,)34 b(if)e(w)m(e)h(let)f Fk(A)1062 4969 y Fi(m)1156 4954 y Fm(=)c Fj(f)p Fk(s)f Fm(:)45 b Fj(j)p Fk(s)p Fj(j)27 b(\024)h Fk(a)g Fm(=)f Fk(m)1956 4918 y Fi(\016)1995 4954 y Fk(=)2044 4883 y Fj(p)p 2126 4883 86 4 v 2126 4954 a Fk(m)q Fj(g)p Fm(,)32 b(then)244 5097 y Fb(Z)327 5123 y Fi(t)290 5285 y Fh(\0001)574 5146 y Fk(e)619 5110 y Fi(is=\017)p 463 5191 393 4 v 463 5282 a Fm(\(1)22 b Fj(\006)g Fk(is)p Fm(\))788 5253 y Fi(m)882 5214 y Fk(ds)59 b Fm(=)1174 5097 y Fb(Z)1257 5123 y Fi(t)1221 5285 y Fh(\000)p Fi(a)1471 5146 y Fk(e)1516 5110 y Fi(is=\017)p 1360 5191 V 1360 5282 a Fm(\(1)22 b Fj(\006)h Fk(is)p Fm(\))1686 5253 y Fi(m)1795 5214 y Fk(\037)1856 5229 y Fi(A)1909 5237 y Ff(m)1972 5214 y Fm(\()p Fk(s)p Fm(\))54 b(+)h Fk(O)s Fm(\()p Fk(m)2480 5173 y Fh(\000)p Fg(1)2574 5214 y Fm(\))p Fk(:)1060 b Fm(\(5.5\))1901 5506 y(16)p eop %%Page: 17 17 17 16 bop 146 -43 a Fm(W)-8 b(e)33 b(no)m(w)h(concen)m(trate)f(on)g (\(5.1\).)244 227 y(\(1)22 b(+)g Fk(is)p Fm(\))568 186 y Fh(\000)p Fi(m)750 227 y Fm(=)60 b(\(1)21 b(+)h Fk(s)1138 186 y Fg(2)1178 227 y Fm(\))1216 186 y Fh(\000)p Fi(m=)p Fg(2)1457 106 y Fb(\022)1545 160 y Fm(1)g Fj(\000)g Fk(is)p 1545 204 250 4 v 1546 296 a Fm(1)f(+)h Fk(is)1821 106 y Fb(\023)1882 121 y Fi(m=)p Fg(2)2036 227 y Fk(:)0 462 y Fm(W)-8 b(e)33 b(separately)g(examine)f(the)h(logs)f(of)g(the)h (factors)f(on)h(the)g(righ)m(t)e(hand)i(side)g(of)f(this)g(equation.) 146 582 y(First,)244 802 y(log)386 705 y Fb(\020)436 802 y Fm(\(1)22 b(+)g Fk(s)689 760 y Fg(2)728 802 y Fm(\))766 760 y Fh(\000)p Fi(m=)p Fg(2)958 705 y Fb(\021)1068 802 y Fm(=)60 b Fj(\000)1291 734 y Fk(m)p 1291 778 86 4 v 1310 870 a Fm(2)1436 802 y(log\(1)22 b(+)g Fk(s)1815 760 y Fg(2)1854 802 y Fm(\))p Fk(:)0 1029 y Fm(On)33 b(the)g(supp)s(ort)g(of)f Fk(\037)864 1044 y Fi(A)917 1052 y Ff(m)979 1029 y Fm(,)h(this)f(equals)244 1252 y Fj(\000)331 1184 y Fk(m)p 331 1228 V 350 1320 a Fm(2)476 1155 y Fb(\020)542 1252 y Fk(s)588 1210 y Fg(2)666 1252 y Fj(\000)39 b Fk(s)828 1210 y Fg(4)868 1252 y Fk(=)p Fm(2)f(+)h Fk(:)17 b(:)g(:)1250 1155 y Fb(\021)1359 1252 y Fm(=)61 b Fj(\000)17 b Fk(m)g(s)1738 1210 y Fg(2)1777 1252 y Fk(=)p Fm(2)38 b(+)h Fk(O)s Fm(\()p Fk(m)2229 1210 y Fh(\000)p Fg(\(1)p Fh(\000)p Fg(4)p Fi(\016)r Fg(\))2501 1252 y Fm(\))p Fk(:)0 1479 y Fm(Exp)s(onen)m(tiating,)32 b(w)m(e)i(ha)m(v)m(e)244 1680 y(\(1)22 b(+)g Fk(s)497 1639 y Fg(2)536 1680 y Fm(\))574 1639 y Fh(\000)p Fi(m=)p Fg(2)826 1680 y Fm(=)60 b Fk(e)1007 1639 y Fh(\000)12 b Fi(m)g(s)1181 1615 y Fe(2)1216 1639 y Fi(=)p Fg(2)1345 1680 y Fm(+)55 b Fk(O)s Fm(\()p Fk(m)1677 1639 y Fh(\000)p Fg(\(1)p Fh(\000)p Fg(4)p Fi(\016)r Fg(\))1949 1680 y Fm(\))98 b(on)32 b(the)h(supp)s(ort)g(of)f Fk(\037)2921 1695 y Fi(A)2974 1703 y Ff(m)3037 1680 y Fm(.)635 b(\(5.6\))146 1882 y(Second,)244 2147 y(log)386 2001 y Fb( )469 2025 y(\022)556 2079 y Fm(1)22 b Fj(\000)h Fk(is)p 556 2123 250 4 v 557 2215 a Fm(1)f(+)g Fk(is)833 2025 y Fb(\023)894 2040 y Fi(m=)p Fg(2)1048 2001 y Fb(!)1174 2147 y Fm(=)60 b Fj(\000)33 b Fk(i)f(m)50 b Fm(arctan\()p Fk(s)p Fm(\))27 b(=)h Fj(\000)33 b Fk(i)g(m)g Fm(\()p Fk(s)22 b Fm(+)g Fk(O)s Fm(\()p Fk(s)2804 2106 y Fg(3)2842 2147 y Fm(\)\))p Fk(:)0 2405 y Fm(On)33 b(the)g(supp)s(ort)g(of)f Fk(\037)864 2420 y Fi(A)917 2428 y Ff(m)979 2405 y Fm(,)h(this)f(equals)244 2606 y Fj(\000)h Fk(i)g(m)g(s)54 b Fm(+)h Fk(O)s Fm(\()p Fk(m)970 2565 y Fh(\000)p Fg(\(1)p Fi(=)p Fg(2)p Fh(\000)p Fg(3)p Fi(\016)r Fg(\))1313 2606 y Fm(\))p Fk(:)0 2808 y Fm(Exp)s(onen)m(tiating,)32 b(w)m(e)i(ha)m(v)m(e)244 2952 y Fb(\022)332 3006 y Fm(1)21 b Fj(\000)i Fk(is)p 332 3050 V 333 3141 a Fm(1)e(+)h Fk(is)608 2952 y Fb(\023)669 2967 y Fi(m=)p Fg(2)866 3073 y Fm(=)60 b Fk(e)1047 3032 y Fh(\000)12 b Fi(i)g(m)g(s)1316 3073 y Fm(+)54 b Fk(O)s Fm(\()p Fk(m)1647 3032 y Fh(\000)p Fg(\(1)p Fi(=)p Fg(2)p Fh(\000)p Fg(3)p Fi(\016)r Fg(\))1990 3073 y Fm(\))98 b(on)32 b(the)h(supp)s(ort)g(of)f Fk(\037)2962 3088 y Fi(A)3015 3096 y Ff(m)3078 3073 y Fm(.)594 b(\(5.7\))146 3313 y(F)-8 b(or)32 b Fk(m)c Fm(=)g([)-18 b([1)p Fk(=\017)p Fm(])g(],)33 b(w)m(e)h(ha)m(v)m(e)244 3527 y Fk(e)289 3486 y Fi(is=\017)474 3527 y Fm(=)60 b Fk(e)655 3486 y Fi(i)12 b(m)g(s)834 3527 y Fk(e)879 3486 y Fi(is)p Fg(\(1)p Fi(=\017)p Fh(\000)p Fi(m)p Fg(\))1272 3527 y Fm(=)60 b Fk(e)1453 3486 y Fi(i)12 b(m)g(s)1649 3431 y Fb(\020)1715 3527 y Fm(1)39 b(+)f Fk(O)s Fm(\()p Fk(m)2118 3486 y Fh(\000)p Fg(\(1)p Fi(=)p Fg(2)p Fh(\000)p Fi(\016)r Fg(\))2426 3527 y Fm(\))2481 3431 y Fb(\021)2591 3527 y Fm(=)60 b Fk(e)2772 3486 y Fi(i)12 b(m)g(s)2973 3527 y Fm(+)55 b Fk(O)s Fm(\()p Fk(m)3305 3486 y Fh(\000)p Fg(\(1)p Fi(=)p Fg(2)p Fh(\000)p Fi(\016)r Fg(\))3613 3527 y Fm(\))p Fk(;)21 b Fm(\(5.8\))0 3729 y(on)32 b(the)h(supp)s(ort)g (of)g Fk(\037)837 3744 y Fi(A)890 3752 y Ff(m)952 3729 y Fm(.)146 3849 y(The)38 b(supp)s(ort)f(of)f(the)h(in)m(tegrand)f(in)g (\(5.5\))g(has)h(length)f Fk(O)s Fm(\()p Fk(m)2479 3813 y Fh(\000)p Fg(\(1)p Fi(=)p Fg(2)p Fh(\000)p Fi(\016)r Fg(\))2787 3849 y Fm(\).)55 b(Using)36 b(this,)h(\(5.6\),)g(\(5.7\),)0 3970 y(and)c(\(5.8\))f(in)g(\(5.5\))g(w)m(e)h(see)h(that)e(for)g Fk(\016)37 b Fm(small)30 b(enough,)244 4120 y Fb(Z)327 4146 y Fi(t)290 4309 y Fh(\0001)573 4170 y Fk(e)618 4134 y Fi(is=\017)p 463 4214 391 4 v 463 4306 a Fm(\(1)22 b(+)g Fk(is)p Fm(\))787 4277 y Fi(m)880 4237 y Fk(ds)83 b Fm(=)1218 4120 y Fb(Z)1302 4146 y Fi(t)1265 4309 y Fh(\000)p Fi(a)1515 4170 y Fk(e)1560 4134 y Fi(is=\017)p 1404 4214 V 1404 4306 a Fm(\(1)22 b(+)g Fk(is)p Fm(\))1728 4277 y Fi(m)1837 4237 y Fk(\037)1898 4252 y Fi(A)1951 4260 y Ff(m)2014 4237 y Fm(\()p Fk(s)p Fm(\))17 b Fk(ds)54 b Fm(+)g Fk(O)s Fm(\()p Fk(m)2635 4196 y Fh(\000)p Fg(1)2729 4237 y Fm(\))1060 4537 y(=)1218 4420 y Fb(Z)1302 4446 y Fi(t)1265 4608 y Fh(\000)p Fi(m)1382 4589 y Fd(\000)p Fe(\(1)p Ff(=)p Fe(2)p Fd(\000)p Ff(\016)r Fe(\))1690 4537 y Fk(e)1735 4496 y Fh(\000)12 b Fi(m)g(s)1909 4472 y Fe(2)1943 4496 y Fi(=)p Fg(2)2050 4537 y Fk(\037)2111 4552 y Fi(A)2164 4560 y Ff(m)2227 4537 y Fm(\()p Fk(s)p Fm(\))17 b Fk(ds)54 b Fm(+)g Fk(O)s Fm(\()p Fk(m)2848 4496 y Fh(\000)p Fg(\(1)p Fh(\000)p Fg(4)p Fi(\016)r Fg(\))3121 4537 y Fm(\))p Fk(:)0 4774 y Fm(By)33 b(simple)e(estimates)h (on)h(the)g(tail)d(of)i(the)h(Gaussian,)f(this)h(equals)244 4897 y Fb(Z)327 4924 y Fi(t)290 5086 y Fh(\0001)453 5015 y Fk(e)498 4973 y Fh(\000)12 b Fi(m)g(s)672 4950 y Fe(2)706 4973 y Fi(=)p Fg(2)797 5015 y Fk(ds)55 b Fm(+)f Fk(O)s Fm(\()p Fk(m)1280 4973 y Fh(\000)p Fg(\(1)p Fh(\000)p Fg(4)p Fi(\016)r Fg(\))1553 5015 y Fm(\))p Fk(;)0 5257 y Fm(whic)m(h)33 b(implies)d(\(5.1\).)1901 5506 y(17)p eop %%Page: 18 18 18 17 bop 146 -43 a Fm(W)-8 b(e)35 b(no)m(w)g(turn)f(to)g(\(5.2\).)48 b(The)35 b(analysis)f(is)g(v)m(ery)h(similar,)d(except)k(that)e(w)m(e)h (use)g(the)g(conjugate)f(of)0 77 y(\(5.7\).)43 b(This)33 b(leads)f(us)h(to)244 180 y Fb(Z)327 207 y Fi(t)290 369 y Fh(\0001)574 230 y Fk(e)619 194 y Fi(is=\017)p 463 274 393 4 v 463 366 a Fm(\(1)22 b Fj(\000)g Fk(is)p Fm(\))788 337 y Fi(m)882 298 y Fk(ds)82 b Fm(=)1220 180 y Fb(Z)1303 207 y Fi(t)1266 369 y Fh(\000)p Fi(a)1517 230 y Fk(e)1562 194 y Fi(is=\017)p 1406 274 V 1406 366 a Fm(\(1)22 b Fj(\000)g Fk(is)p Fm(\))1731 337 y Fi(m)1841 298 y Fk(\037)1902 313 y Fi(A)1955 321 y Ff(m)2017 298 y Fm(\()p Fk(s)p Fm(\))17 b Fk(ds)54 b Fm(+)h Fk(O)s Fm(\()p Fk(m)2639 256 y Fh(\000)p Fg(1)2732 298 y Fm(\))1061 597 y(=)1220 480 y Fb(Z)1303 506 y Fi(t)1266 669 y Fh(\000)p Fi(m)1383 649 y Fd(\000)p Fe(\(1)p Ff(=)p Fe(2)p Fd(\000)p Ff(\016)r Fe(\))1692 597 y Fk(e)1737 556 y Fh(\000)11 b Fi(m)i(s)1911 533 y Fe(2)1945 556 y Fi(=)p Fg(2)2052 597 y Fk(e)2097 556 y Fg(2)f Fi(i)g(m)g(s)2291 597 y Fk(\037)2352 612 y Fi(A)2405 620 y Ff(m)2468 597 y Fm(\()p Fk(s)p Fm(\))17 b Fk(ds)54 b Fm(+)g Fk(O)s Fm(\()p Fk(m)3089 556 y Fh(\000)p Fg(\(1)p Fh(\000)p Fg(4)p Fi(\016)r Fg(\))3362 597 y Fm(\))p Fk(:)272 b Fm(\(5.9\))0 787 y(In)m(tegrating)32 b(b)m(y)h(parts,)g(w)m(e)h(ha)m(v)m(e)244 863 y Fb(Z)360 980 y Fk(e)405 939 y Fh(\000)12 b Fi(m)g(s)579 915 y Fe(2)613 939 y Fi(=)p Fg(2)721 980 y Fk(e)766 939 y Fg(2)g Fi(i)g(m)g(s)976 980 y Fk(ds)60 b Fm(=)1355 913 y(1)p 1279 957 201 4 v 1279 1048 a(2)17 b Fk(i)g(m)1522 980 y(e)1567 939 y Fh(\000)12 b Fi(m)g(s)1741 915 y Fe(2)1775 939 y Fi(=)p Fg(2)1883 980 y Fk(e)1928 939 y Fg(2)g Fi(i)g(m)g(s)2176 980 y Fm(+)2333 913 y(1)p 2317 957 83 4 v 2317 1048 a(2)p Fk(i)2458 863 y Fb(Z)2574 980 y Fk(s)17 b(e)2682 939 y Fh(\000)12 b Fi(m)g(s)2856 915 y Fe(2)2890 939 y Fi(=)p Fg(2)2997 980 y Fk(e)3042 939 y Fg(2)h Fi(i)e(m)h(s)3253 980 y Fk(ds:)0 1222 y Fm(Since)254 1105 y Fb(Z)337 1131 y Fh(1)300 1294 y(\0001)462 1222 y Fj(j)p Fk(s)p Fj(j)17 b Fk(e)626 1181 y Fh(\000)11 b Fi(m)h(s)799 1158 y Fe(2)834 1181 y Fi(=)p Fg(2)925 1222 y Fk(ds)44 b Fm(=)1214 1155 y(2)p 1196 1199 86 4 v 1196 1290 a Fk(m)1291 1222 y Fm(,)32 b(the)g(in)m(tegral)d(on)j(the)g(righ)m(t)e(hand)i(side)f(of)g(\(5.9\)) g(is)g Fk(O)s Fm(\(1)p Fk(=m)p Fm(\))f(for)h(all)0 1392 y Fk(t)p Fm(,)i(and)g(\(5.2\))f(follo)m(ws)f(for)h Fk(\016)k Fm(small)31 b(enough.)p 1750 1408 84 100 v 146 1630 a(In)i(the)g(follo) m(wing)d(section,)j(w)m(e)h(will)c(also)h(need)j(the)f(follo)m(wing)d (trivial)g(estimate)i(for)g(an)m(y)h Fk(m)28 b 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b(erf)33 b(\()1079 5152 y Fb(q)p 1162 5152 129 4 v 105 x Fm([)-18 b([)p Fk(x)p Fm(])g(])r Fk(\034)11 b Fm(\))39 b(+)f Fk(O)s Fm(\(1)p Fk(=x)p Fm(\))p Fk(;)146 b Fm(uniformly)31 b(in)80 b Fk(\034)6 b(:)1002 b Fm(\(6.5\))1901 5506 y(19)p eop %%Page: 20 20 20 19 bop 0 -43 a Fm(Finally)-8 b(,)28 b(the)i(remainder)g(term)f(in)g (\(6.3\))h(is)f(of)h(order)g Fk(O)s Fm(\()2117 -115 y Fj(p)p 2199 -115 40 4 v 2199 -43 a Fk(\017)17 b(e)2300 -79 y Fh(\000)p Fg(1)p Fi(=\017)2459 -43 y Fm(\))30 b(so)g(that)g(the)g (quan)m(tit)m(y)h(\(6.1\))f(equals)244 223 y Fj(\000)j Fk(\014)415 182 y Fh(\003)486 223 y Fk(\031)k(e)623 182 y Fh(\000)p Fg(1)p Fi(=\017)830 49 y Fb(8)830 124 y(<)830 273 y(:)921 223 y Fm(erf)1048 52 y Fb(0)1048 202 y(@)1138 62 y(s)p 1221 62 125 4 v 1259 156 a Fm(1)p 1231 200 105 4 v 1231 292 a(2)17 b Fk(\017)1378 223 y(t)1430 52 y Fb(1)1430 202 y(A)1541 223 y Fm(+)39 b(1)54 b(+)h Fk(O)s Fm(\()p Fk(\017)2045 182 y Fi(\026)2091 223 y Fm(\))2146 49 y Fb(9)2145 124 y(=)2145 273 y(;)2236 223 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3410 986 a Fk(n)3702 918 y Fm(from)0 1081 y(Lemma)32 b(4.1)h(and)g(a)h(factor)e(of)1171 1002 y Fj(p)p 1254 1002 228 4 v 79 x Fk(n)23 b Fm(+)f(1)33 b(that)g(app)s(ears)h(in)e(the)i(denominator)e(of)h(\(5.1\).)45 b(Therefore,)35 b(b)m(y)0 1201 y(estimates)f(similar)c(to)k(those)g(ab) s(o)m(v)m(e,)i(\(6.2\))d(is)h(of)f(order)h Fk(O)s Fm(\()p Fk(e)2298 1165 y Fh(\000)p Fg(1)p Fi(=\017)2472 1130 y Fj(p)p 2555 1130 40 4 v 71 x Fk(\017)g Fm(ln)o(\(1)p Fk(=\017)p Fm(\)\),)g(as)g Fk(\017)d Fj(!)e Fm(0.)48 b(Hence,)35 b(for)0 1321 y(an)m(y)e(0)28 b Fk(<)f(\026)h(<)f Fm(1)p Fk(=)p Fm(2,)244 1588 y(\011)320 1603 y Fg(1)359 1588 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))83 b(=)g Fk( )875 1603 y Fg(1)915 1588 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))55 b Fj(\000)g Fk(\014)1357 1547 y Fh(\003)1428 1588 y Fk(\031)37 b(e)1565 1547 y Fh(\000)p Fg(1)p Fi(=\017)1772 1414 y Fb(8)1772 1488 y(<)1772 1638 y(:)1863 1588 y Fm(erf)1990 1417 y Fb(0)1990 1566 y(@)2080 1426 y(s)p 2163 1426 125 4 v 2201 1521 a Fm(1)p 2173 1565 105 4 v 2173 1656 a(2)17 b Fk(\017)2320 1588 y(t)2372 1417 y Fb(1)2372 1566 y(A)2483 1588 y Fm(+)39 b(1)54 b(+)h Fk(O)s Fm(\()p Fk(\017)2987 1547 y Fi(\026)3033 1588 y Fm(\))3088 1414 y Fb(9)3087 1488 y(=)3087 1638 y(;)3210 1588 y Fk( )3273 1603 y Fg(2)3313 1588 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))653 1837 y(+)83 b Fk(O)s Fm(\()p Fk(e)973 1796 y Fh(\000)p Fg(1)p Fi(=\017)1147 1760 y Fj(p)p 1230 1760 40 4 v 77 x Fk(\017)34 b Fm(ln)o(\(1)p Fk(=\017)p Fm(\)\))p Fk(;)2037 b Fm(\(6.7\))0 2016 y(uniformly)30 b(for)i Fk(t)c Fj(2)g Fk(I)-24 b(R)q Fm(.)146 2137 y(A)28 b(similar)c(analysis)j(of)g(the)i(solution)d(\011)1634 2152 y Fg(2)1673 2137 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))27 b(of)g(\(1.1\))h(that)f(coincides)g(with)h Fk( )3140 2152 y Fg(2)3180 2137 y Fm(\()p Fk(\017;)33 b(t)p Fm(\))28 b(as)g Fk(t)g Fj(!)f(\0001)0 2257 y Fm(leads)32 b(to)h(the)g(estimate) 244 2499 y(\011)320 2514 y Fg(2)359 2499 y Fm(\()p Fk(\017;)h(t)p Fm(\))83 b(=)g Fk( )875 2514 y Fg(2)915 2499 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))55 b(+)f Fk(\014)1355 2458 y Fh(\003)1427 2499 y Fk(\031)36 b(e)1563 2458 y Fh(\000)p Fg(1)p Fi(=\017)1771 2324 y Fb(8)1771 2399 y(<)1771 2548 y(:)1861 2499 y Fm(erf)1989 2328 y Fb(0)1989 2477 y(@)2078 2337 y(s)p 2161 2337 125 4 v 2199 2431 a Fm(1)p 2171 2475 105 4 v 2171 2567 a(2)17 b Fk(\017)2318 2499 y(t)2370 2328 y Fb(1)2370 2477 y(A)2481 2499 y Fm(+)39 b(1)54 b(+)h Fk(O)s Fm(\()p Fk(\017)2985 2458 y Fi(\026)3031 2499 y Fm(\))3086 2324 y Fb(9)3086 2399 y(=)3086 2548 y(;)3209 2499 y Fk( )3272 2514 y Fg(1)3311 2499 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))653 2747 y(+)83 b Fk(O)s Fm(\()p Fk(e)973 2706 y Fh(\000)p Fg(1)p Fi(=\017)1147 2671 y Fj(p)p 1230 2671 40 4 v 76 x Fk(\017)34 b Fm(ln)o(\(1)p Fk(=\017)p Fm(\)\))p Fk(;)2037 b Fm(\(6.8\))0 2927 y(uniformly)30 b(for)i Fk(t)c Fj(2)g Fk(I)-24 b(R)q Fm(.)146 3165 y(F)-8 b(or)39 b(an)m(y)g(0)g Fk(<)f(\026)g(<)h Fm(1)p Fk(=)p Fm(2,)h(the)f(follo)m(wing)e(estimates)h(follo)m(w)g(directly)g(from)g (\(6.7\),)i(\(6.8\))e(and)i(the)0 3286 y(unitarit)m(y)31 b(of)i Fk(U)584 3301 y Fi(\017)616 3286 y Fm(\()p Fk(t;)17 b(r)s Fm(\):)719 3465 y(\011)795 3480 y Fi(j)831 3465 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))83 b(=)g Fk( )1347 3480 y Fi(j)1384 3465 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))54 b(+)h Fk(O)s Fm(\()p Fk(e)1941 3424 y Fh(\000)p Fg(1)p Fi(=\017)2099 3465 y Fm(\))p Fk(;)632 3669 y Fj(k)p Fk( )745 3684 y Fi(j)782 3669 y Fm(\()p Fk(\017;)33 b(t)p Fm(\))p Fj(k)83 b Fm(=)g(1)55 b(+)f Fk(O)s Fm(\()p Fk(e)1679 3628 y Fh(\000)p Fg(1)p Fi(=\017)1853 3669 y Fk(\017)1892 3628 y Fi(\026)1939 3669 y Fm(\))p Fk(;)212 b Fm(and)244 3872 y Fj(h)17 b Fk( )363 3887 y Fg(1)402 3872 y Fm(\()p Fk(\017;)34 b(t)p Fm(\))p Fk(;)f( )736 3887 y Fg(2)776 3872 y Fm(\()p Fk(\017;)h(t)p Fm(\))17 b Fj(i)82 b Fm(=)h Fk(O)s Fm(\()p Fk(e)1445 3831 y Fh(\000)p Fg(2)p Fi(=\017)1619 3872 y Fk(\017)1658 3831 y Fi(\026)1705 3872 y Fm(\))0 4052 y(for)32 b Fk(j)i Fm(=)27 b(1)p Fk(;)17 b Fm(2.)43 b(Th)m(us,)34 b(b)m(y)g(\(4.10\))d(and)i(\(4.11\),)f(w)m(e)i(ha)m(v)m (e)279 4268 y(lim)244 4332 y Fh(j)p Fi(t)p Fh(j!1)466 4268 y Fj(jh)p Fm(\010)603 4283 y Fi(j)640 4268 y Fm(\()p Fk(t)p Fm(\))p Fk(;)f( )874 4283 y Fi(j)911 4268 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))p Fj(ij)60 b Fm(=)1368 4144 y Fb(\014)1368 4194 y(\014)1368 4244 y(\014)1368 4293 y(\014)1412 4268 y Fk(e)1457 4217 y Fg(\()p Fh(\000)p Fg(1\))1601 4194 y Ff(j)s Fe(+1)1724 4154 y Fb(R)1779 4175 y Fd(1)1763 4251 y(\0001)1888 4217 y Fi(f)7 b Fg(\()p Fi(s)p Fg(\))p Fi(g)r Fg(\()p Fi(\017;)12 b Fg(\()p Fh(\000)p Fg(1\))2284 4194 y Ff(j)s Fe(+1)2394 4217 y Fi(s)p Fg(\))g Fi(ds)2556 4144 y Fb(\014)2556 4194 y(\014)2556 4244 y(\014)2556 4293 y(\014)2644 4268 y Fm(=)60 b(1)54 b(+)h Fk(O)s Fm(\()p Fk(e)3175 4227 y Fh(\000)p Fg(1)p Fi(=\017)3349 4268 y Fk(\017)3388 4227 y Fi(\026)3435 4268 y Fm(\))p Fk(:)146 4633 y Fm(W)-8 b(e)30 b(pro)m(v)m(e)g(that)f(the)g(v)-5 b(alue)29 b(of)f Fk(\014)1359 4596 y Fh(\003)1427 4633 y Fm(is)h(1)p Fk(=)p Fm(\()p Fk(\031)1717 4551 y Fj(p)p 1799 4551 49 4 v 1799 4633 a Fm(2\))g(b)m(y)h(comparing)d(the)i (asymptotics)g(ab)s(o)m(v)m(e)h(as)f Fk(t)f Fj(!)f(1)0 4753 y Fm(with)32 b(the)h(transition)e(amplitude)g(\(1.9\).)146 4873 y(This)i(pro)m(v)m(es)h(Theorem)f(1.1)f(with)244 5053 y Fk(\037)305 5068 y Fi(j)342 5053 y Fm(\()p Fk(\017;)17 b(t)p Fm(\))27 b(=)h Fk( )730 5068 y Fi(j)767 5053 y Fm(\()p Fk(\017;)17 b(t=\016)t Fm(\))2642 b(\(6.9\))0 5232 y(when)32 b(w)m(e)g(return)g(to)f(the)g(original)d(notation)i Fk(\017)e Fj(7!)f Fk(\017=)p Fm(\()p Fk(E)c(\016)t Fm(\))31 b(and)g(original)d(time)i(v)-5 b(ariable)29 b Fk(t)f Fj(7!)f Fk(t=\016)t Fm(.)p 3817 5249 84 100 v 1901 5506 a(20)p eop %%Page: 21 21 21 20 bop 0 -43 a Fl(References)0 176 y Fm([1])49 b(J.E.)29 b(Avron)h(and)e(A.)i(Elgart,)e(An)h(Adiabatic)f(Theorem)h(without)f(a)h (Gap)f(Condition,)g(in)g Fc(Op)-5 b(er)g(ator)152 296 y(the)g(ory)38 b(advanc)-5 b(es)37 b(and)h(Applic)-5 b(ations,)38 b(vol)g(108)p Fm(.)e(J.)g(Dittric)m(h,)g(P)-8 b(.)37 b(Exner,)h(and)f(M.)g(T)-8 b(ater)36 b(editors,)152 417 y(Birkhauser,)d(1999.)0 620 y([2])49 b(M.V.)37 b(Berry)-8 b(,)38 b(Uniform)c(asymptotic)i(smo)s(othing)e(of)i(Stok)m(es's)j (discon)m(tin)m(uities,)d Fc(Pr)-5 b(o)g(c.)38 b(R)-5 b(oy.)39 b(So)-5 b(c.)152 740 y(L)g(ond.)34 b(A)f Fa(422)p Fm(,)g(7-21,)e(\(1989\).)0 944 y([3])49 b(M.V.)39 b(Berry)-8 b(,)42 b(Histories)c(of)g(adiabatic)f(quan)m(tum)i(transitions,)h Fc(Pr)-5 b(o)g(c.)40 b(R)-5 b(oy.)41 b(So)-5 b(c.)40 b(L)-5 b(ond.)40 b(A)g Fa(429)p Fm(,)152 1064 y(61-72,)31 b(\(1990\).)0 1268 y([4])49 b(M.V.)38 b(Berry)-8 b(,)39 b(Geometric)d(amplitude)f(factors)i(in)g(adiabatic)e(quan)m(tum)j (transitions,)f Fc(Pr)-5 b(o)g(c.)39 b(R)-5 b(oy.)152 1388 y(So)g(c.)34 b(L)-5 b(ond.)34 b(A)f Fa(430)p Fm(,)g(405-411,)e (\(1990\).)0 1591 y([5])49 b(M.V.)29 b(Berry)h(and)f(R.)g(Lim,)f(Univ)m (ersal)h(transition)f(prefactors)h(deriv)m(ed)h(b)m(y)g(sup)s (eradiabatic)e(renor-)152 1712 y(malization,)h Fc(J.)35 b(Phys.)g(A)d Fa(26)p Fm(,)h(4737-4747,)d(\(1993\).)0 1915 y([6])49 b(M.)33 b(Born)f(and)h(V.)g(F)-8 b(o)s(c)m(k,)32 b(Bew)m(eis)i(des)g(Adiabatensatzes,)f Fc(Z.)i(Phys.)d Fa(51)p Fm(,)h(165-180)e(\(1928\).)0 2119 y([7])49 b(O.)33 b(Costin,)h(L.)g(Dupaigne,)f(and,)i(M.)f(D.)f(Krusk)-5 b(al,)34 b(Borel)f(Summation)e(of)i(Adiabatic)g(In)m(v)-5 b(arian)m(ts,)152 2239 y(Preprin)m(t)33 b(in)e(preparation.)0 2442 y([8])49 b(O.)22 b(Costin)g(and)g(M.)h(Krusk)-5 b(al,)24 b(On)e(optimal)d(truncation)j(of)g(div)m(ergen)m(t)h(series)f (solutions)g(of)f(nonlinear)152 2563 y(di\013eren)m(tial)30 b(systems;)35 b(Berry)e(smo)s(othing,)e Fc(Pr)-5 b(o)g(c.)34 b(R)-5 b(oy.)35 b(So)-5 b(c.)35 b(L)-5 b(ond.)34 b(A)f Fa(455)p Fm(,)g(1931-1956,)d(\(1999\).)0 2766 y([9])49 b(A.M.)29 b(Dykhne,)h(Adiabatic)d(p)s(erturbation)g(of)h(discrete)h(sp) s(ectrum)f(states,)j Fc(Sov.)f(Phys.)h(JETP.)d Fa(14)p Fm(,)152 2887 y(941-943)i(\(1962\).)0 3090 y([10])49 b(G.)29 b(A.)h(Hagedorn,)h(Pro)s(of)e(of)h(the)h(Landau{Zener)f(F)-8 b(orm)m(ula)28 b(in)h(an)h(Adiabatic)f(Limit)e(with)j(Small)152 3210 y(Eigen)m(v)-5 b(alue)32 b(Gaps,)g Fc(Commun.)i(Math.)h(Phys.)e Fa(136)p Fm(,)g(433-449)d(\(1991\).)0 3414 y([11])49 b(G.)24 b(A.)g(Hagedorn)h(and)g(A.)g(Jo)m(y)m(e,)i(Elemen)m(tary)e(Exp) s(onen)m(tial)f(Error)g(Estimates)g(for)g(the)h(Adiabatic)152 3534 y(Appro)m(ximation,)31 b Fc(J.)j(Math.)i(A)n(nal.)e(Appl.)e Fa(267)p Fm(,)h(235-246)e(\(2002\).)0 3737 y([12])49 b(C.)41 b(Ho)m(wls,)j(The)e(Borel-Laplace)d(approac)m(h)j(to)f(h)m(yp)s (erasymptotics,)j(\(R.I.M.S.,)h(Univ)m(ersit)m(y)c(of)152 3858 y(Ky)m(oto,)32 b(Japan,)h(Conference)h(Pro)s(ceedings,)g(ed.)f(Y.) g(T)-8 b(ak)m(ei\))32 b Fa(968)p Fm(,)h(31-48)e(\(1997\))0 4061 y([13])49 b(J.-T.)42 b(Hw)m(ang)h(and)f(P)-8 b(.)43 b(P)m(ec)m(h)m(uk)-5 b(as,)47 b(The)d(adiabatic)c(theorem)i(in)g(the)g (complex)g(plane)g(and)h(the)152 4182 y(semi-classical)h(calculation)h (of)i(non-adiabatic)e(transition)g(amplitudes,)50 b Fc(J.)e(Chem.)g (Phys.)f Fa(67)p Fm(,)152 4302 y(4640-4653)30 b(\(1977\).)0 4505 y([14])49 b(V.)35 b(Jaksic)g(and)g(J.)g(Segert,)h(Exp)s(onen)m (tial)e(approac)m(h)h(to)g(the)g(adiabatic)e(limit)e(and)k(the)g (Landau-)152 4626 y(Zener)e(form)m(ula,)d Fc(R)-5 b(ev.)35 b(Math.)g(Phys.)e Fa(4)p Fm(,)f(529-574)f(\(1992\).)0 4829 y([15])49 b(A.)38 b(Jo)m(y)m(e,)j(Non-trivial)35 b(prefactors)j(in)f(Adiabatic)g(T)-8 b(ransition)37 b(Probabilities)e (Induced)40 b(b)m(y)f(High)152 4950 y(Order)33 b(Complex)f (Degeneracies,)h Fc(J.)i(Phys.)g(A)e Fa(26)p Fm(,)f(6517-6540,)f (\(1993\).)0 5153 y([16])49 b(A.)32 b(Jo)m(y)m(e,)i(Pro)s(of)e(of)g (the)h(Landau-Zener)g(form)m(ula,)e Fc(Asymptotic)k(A)n(nalysis)d Fa(9)p Fm(,)g(209-258,)f(\(1994\).)1901 5506 y(21)p eop %%Page: 22 22 22 21 bop 0 -43 a Fm([17])49 b(A.)35 b(Jo)m(y)m(e,)i(Exp)s(onen)m(tial) e(Asymptotics)h(in)e(a)h(Singular)f(Limit)e(for)j(n-Lev)m(el)h (Scattering)f(Systems,)152 77 y Fc(SIAM)g(J.)g(Math.)g(A)n(nal.)d Fa(28)p Fm(,)g(669-703)f(\(1997\).)0 278 y([18])49 b(A.)30 b(Jo)m(y)m(e,)j(H.)d(Kunz,)i(and)e(C.-E.)i(P\014ster,)g(Exp)s(onen)m (tial)e(deca)m(y)i(and)e(geometric)g(asp)s(ect)h(of)f(transi-)152 398 y(tion)h(probabilities)f(in)i(the)h(adiabatic)e(limit,)e Fc(A)n(nn.)34 b(Phys.)e Fa(208)p Fm(,)h(299-332)e(\(1991\).)0 599 y([19])49 b(A.)35 b(Jo)m(y)m(e)i(and)e(C.-E.)h(P\014ster,)i(Exp)s (onen)m(tially)c(small)f(adiabatic)h(in)m(v)-5 b(arian)m(t)34 b(for)h(the)h(Sc)m(hr\177)-49 b(odinger)152 720 y(equation,)32 b Fc(Commun.Math.Phys.)g Fa(140)p Fm(,)h(15-41)e(\(1991\).)0 921 y([20])49 b(A.)27 b(Jo)m(y)m(e)h(and)g(C.-E.)f(P\014ster,)j(Sup)s (eradiabatic)c(ev)m(olution)g(and)h(adiabatic)e(transition)h (probabilit)m(y)152 1041 y(b)s(et)m(w)m(een)34 b(t)m(w)m(o)f (non-degenerate)f(lev)m(els)h(isolated)d(in)i(the)g(sp)s(ectrum,)h Fc(J.)h(Math.)h(Phys.)d Fa(34)p Fm(,)g(454-479)152 1161 y(\(1993\).)0 1362 y([21])49 b(A.)f(Jo)m(y)m(e)h(and)f(C.-E.)h (P\014ster,)k(Semiclassical)45 b(Asymptotics)j(Bey)m(ond)i(all)c (Orders)i(for)g(Simple)152 1483 y(Scattering)32 b(Systems)i(,)e Fc(IAM)j(J.)g(Math.)g(A)n(nal.)d Fa(26)p Fm(,)h(944-977,)e(\(1995\).)0 1683 y([22])49 b(A.)33 b(Jo)m(y)m(e)i(and)f(C.-E.)g(P\014ster,)i(Exp)s (onen)m(tial)d(Estimates)g(in)g(Adiabatic)f(Quan)m(tum)h(Ev)m(olution,) h(in)152 1804 y Fc(XIIth)k(International)g(Congr)-5 b(ess)38 b(of)g(Mathematic)-5 b(al)39 b(Physics,)g(ICMP)g('97)p Fm(,)e(p.)h(309-315,)e(De)h(Wit,)152 1924 y(Brac)m(k)m(en,)d(Gould)d (and)i(P)m(earce)h(Eds,)g(In)m(ternational)d(Press,)j(Cam)m(bridge,)e (1999.)0 2125 y([23])49 b(A.)24 b(Jo)m(y)m(e)i(and)f(C.-E.)h (P\014ster,)h(Complex)e(WKB)f(metho)s(d)h(for)f(3-lev)m(el)g (scattering)g(systems,)k Fc(Asymp.)152 2245 y(A)n(nal.)k Fa(23)p Fm(,)g(91-109,)f(\(2000\).)0 2446 y([24])49 b(L.D.)36 b(Landau,)h Fc(Col)5 b(le)-5 b(cte)g(d)38 b(Pap)-5 b(ers)38 b(of)g(L.D.L)-5 b(andau)p Fm(,)36 b(P)m(ergamon)h(Press,)i(Oxford,)e (London,)h(Ed-)152 2567 y(in)m(burgh,)32 b(New)i(Y)-8 b(ork,)32 b(P)m(aris,)h(F)-8 b(rankfurt,)32 b(1965.)0 2768 y([25])49 b(R.)31 b(Lim)f(and)h(M.V.)h(Berry)-8 b(,)33 b(Sup)s(eradiabatic)d(trac)m(king)h(of)g(quan)m(tum)h(ev)m (olution,)f Fc(J.)j(Phys.)g(A)e Fa(24)p Fm(,)152 2888 y(3255-3264)e(\(1991\).)0 3089 y([26])49 b(A.)34 b(Martinez,)g(Precise) h(exp)s(onen)m(tial)e(estimates)h(in)f(adiabatic)f(theory)-8 b(,)35 b Fc(J.)h(Math.)g(Phys.)e Fa(35)g Fm(\(8\),)152 3209 y(3889-3915)c(\(1994\).)0 3410 y([27])49 b(J.)42 b(B.)h(McLeo)s(d,)j(Smo)s(othing)41 b(of)h(Stok)m(es)j(discon)m(tin)m (uities,)f Fc(Pr)-5 b(o)g(c.)44 b(R)-5 b(oy.)45 b(So)-5 b(c.)43 b(L)-5 b(ondon)42 b Fa(A)49 b(437)p Fm(,)152 3531 y(343-354,)31 b(\(1992\).)0 3731 y([28])49 b(G.)38 b(Nenciu,)i(On)e(the)h(adiabatic)e(theorem)h(of)g(quan)m(tum)h(mec)m (hanics,)h Fc(J.)h(Phys.)f(A)f Fa(13)f Fm(L15-L18)152 3852 y(\(1980\).)0 4053 y([29])49 b(G.)35 b(Nenciu,)j(Adiabatic)d (theorems)h(and)h(sp)s(ectral)f(concen)m(tration,)h Fc(Commun.)g(Math.) i(Phys.)d Fa(82)152 4173 y Fm(121-135)30 b(\(1981\).)0 4374 y([30])49 b(G.)32 b(Nenciu,)i(Asymptotic)f(in)m(v)-5 b(arian)m(t)32 b(subspaces,)37 b(adiabatic)31 b(theorems)j(and)f(blo)s (c)m(k)g(diagonalisa-)152 4494 y(tion,)c(in)f Fc(R)-5 b(e)g(c)g(ent)31 b(developments)f(in)i(quantum)f(me)-5 b(chanics)30 b(\(Poiana)h(Br)-5 b(a)n(\030)-42 b(sov,)30 b(1989\))p Fm(,)f Fc(Math.)j(Phys.)152 4615 y(Stud.)p Fm(,)h Fa(12)p Fm(,)f(133-149,)f(Klu)m(w)m(er)i(Acad.)g(Publ.,)g (Dordrec)m(h)m(t,)g(1991.)0 4816 y([31])49 b(G.)27 b(Nenciu,)j(Linear)d (adiabatic)f(theory)-8 b(.)29 b(Exp)s(onen)m(tial)e(estimates,)i Fc(Commun.)h(Math.)h(Phys.)d Fa(152)p Fm(,)152 4936 y(479-496)i (\(1993\).)0 5137 y([32])49 b(A.)e(Olde)h(Daalh)m(uis,)i(Hyp)s (erasymptotics)e(and)g(the)g(Stok)m(es')h(phenomenon,)j Fc(Pr)-5 b(o)g(c.)49 b(R)-5 b(oy.)49 b(So)-5 b(c.)152 5257 y(Edinbur)g(gh)32 b Fa(123)37 b(A)p Fm(,)32 b(731-743,)f (\(1993\).)1901 5506 y(22)p eop %%Page: 23 23 23 22 bop 0 -43 a Fm([33])49 b(J.)e(Sj\177)-49 b(ostrand,)51 b(Remarque)d(sur)f(des)i(pro)5 b(jecteurs)49 b(adiabatiques)d(du)i(p)s (oin)m(t)e(de)i(vue)g(pseudo)s(d-)152 77 y(i\013)m(\023)-46 b(eren)m(tiel,)31 b Fc(C.)j(R.)h(A)-5 b(c)g(ad.)35 b(Sci.)f(Paris)f Fa(317)f Fm(S)m(\023)-46 b(er.)33 b(I)g Fa(22)p Fm(,)g(217-220)d (\(1993\).)0 281 y([34])49 b(C.)28 b(Zener:)42 b(Non-adiabatic)26 b(crossing)j(of)f(energy)h(lev)m(els,)h Fc(Pr)-5 b(o)g(c.)30 b(R)-5 b(oy.)32 b(So)-5 b(c.)30 b(L)-5 b(ondon)27 b Fa(137)p Fm(,)j(696-702)152 401 y(\(1932\).)1901 5506 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0309031122606--