Content-Type: multipart/mixed; boundary="-------------0010041608160" This is a multi-part message in MIME format. ---------------0010041608160 Content-Type: text/plain; name="00-395.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-395.comments" AMS-Code: 81Q20 PACS-Code: 03.65.-w, 02.30.Lt Submitted to Journal of Physics A: Mathematical and General ---------------0010041608160 Content-Type: text/plain; name="00-395.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-395.keywords" semiclassical limit, optimal truncation, schrodinger equation, exponentially accurate estimates ---------------0010041608160 Content-Type: application/postscript; name="estimate_iop.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="estimate_iop.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: final_draft_iop.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 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4245 y Fk( )1461 4260 y Fj(1)1500 4245 y Fn(\()p Fk(x)p Fn(\))23 b(+)f(\026)-50 b Fk(h)1817 4175 y Ff(2)p 1817 4187 V 1817 4228 a(2)1862 4245 y Fk( )1925 4260 y Fj(2)1964 4245 y Fn(\()p Fk(x)p Fn(\))23 b(+)f(\026)-50 b Fk(h)2281 4175 y Ff(3)p 2281 4187 V 2281 4228 a(2)2326 4245 y Fk( )2389 4260 y Fj(3)2428 4245 y Fn(\()p Fk(x)p Fn(\))23 b(+)f(\026)-50 b Fk(h)2745 4175 y Ff(4)p 2745 4187 V 2745 4228 a(2)2789 4245 y Fk( )2852 4260 y Fj(4)2892 4245 y Fn(\()p Fk(x)p Fn(\))23 b(+)f Fk(:)17 b(:)g(:)322 b Fn(\(8\))0 4427 y(No)m(w)29 b(let)f(us)h(equate)h(p)s(o)m(w)m(ers)g(of)e(\026)-50 b Fk(h)1275 4357 y Ff(1)p 1275 4369 V 1275 4411 a(2)1320 4427 y Fn(.)42 b(After)29 b(replacing)e Fk(W)14 b Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))28 b(b)m(y)i(its)e(T)-8 b(a)m(ylor)28 b(series,)i(w)m(e)g(obtain,)0 4560 y(for)i Fk(n)c Fn(=)f(1)p Fk(;)17 b Fn(2)p Fk(;)g(:)g(:)g(:)o Fn(,)598 4783 y Fk(H)679 4798 y Fh(k)721 4783 y Fk( )784 4798 y Fh(n)854 4783 y Fn(+)990 4675 y Fh(n)952 4700 y Fm(X)956 4884 y Fh(l)q Fj(=1)1088 4783 y Fk(c)1130 4798 y Fh(l)q Fj(+2)1246 4783 y Fk(A)1319 4798 y Fh(k)1362 4783 y Fk(x)1417 4741 y Fh(l)q Fj(+2)1534 4783 y Fk(A)1607 4798 y Fh(k)1650 4783 y Fk( )1713 4798 y Fh(n)p Fi(\000)p Fh(l)1864 4783 y Fn(=)2007 4675 y Fh(n)1968 4700 y Fm(X)1972 4884 y Fh(l)q Fj(=1)2105 4783 y Fl(E)2158 4798 y Fh(l)2183 4783 y Fn(\()p Fk(A)2294 4798 y Fh(k)2337 4783 y Fn(\))2375 4741 y Fj(2)2414 4783 y Fk( )2477 4798 y Fh(n)p Fi(\000)p Fh(l)3581 4783 y Fn(\(9\))0 5020 y(where)34 b Fk( )345 5035 y Fj(0)412 5020 y Fn(=)28 b Fk(\036)574 5035 y Fh(k)616 5020 y Fn(.)195 5223 y(F)-8 b(or)25 b(the)h(momen)m(t)e(w)m(e)j(do)e (not)h(require)g(that)f(\011\()p Fk(x)p Fn(\))h(b)s(e)f(normalized.)40 b(Instead)26 b(w)m(e)h(c)m(ho)s(ose)f(eac)m(h)0 5356 y Fk( )63 5371 y Fh(j)125 5356 y Fn(to)e(b)s(e)h(orthogonal)e(to)h Fk(\036)1010 5371 y Fh(k)1053 5356 y Fn(.)41 b(This)24 b(do)s(es)i(not)e(a\013ect)h(the)g(eigen)m(v)-5 b(alues.)41 b(Using)25 b(the)g(selfadjoin)m(tness)p eop %%Page: 5 5 5 4 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2269 b Fn(5)0 232 y(of)32 b Fk(A)184 247 y Fh(k)227 232 y Fk(x)282 196 y Fh(l)308 232 y Fk(A)381 247 y Fh(k)424 232 y Fn(,)h(w)m(e)g(see)h(that)e(\(9\))h (requires)598 446 y Fl(E)651 461 y Fh(n)736 446 y Fn(=)895 338 y Fh(n)856 363 y Fm(X)860 548 y Fh(l)q Fj(=1)992 446 y Fk(c)1034 461 y Fh(l)q Fj(+2)1167 350 y Fm(D)1218 446 y Fk(A)1291 461 y Fh(k)1334 446 y Fk(x)1389 405 y Fh(l)q Fj(+2)1505 446 y Fk(A)1578 461 y Fh(k)1621 446 y Fk(\036)1679 461 y Fh(k)1722 446 y Fk(;)17 b( )1829 461 y Fh(n)p Fi(\000)p Fh(l)1952 350 y Fm(E)3533 446 y Fn(\(10\))598 762 y Fk( )661 777 y Fh(n)736 762 y Fn(=)27 b(\()p Fk(H)958 777 y Fh(k)1001 762 y Fn(\))1039 721 y Fi(\000)p Fj(1)1039 787 y Fh(r)1150 616 y Fm( )1254 654 y Fh(n)1215 679 y Fm(X)1219 864 y Fh(l)q Fj(=1)1352 762 y Fl(E)1405 777 y Fh(l)1430 762 y Fn(\()p Fk(A)1541 777 y Fh(k)1584 762 y Fn(\))1622 721 y Fj(2)1662 762 y Fk( )1725 777 y Fh(n)p Fi(\000)p Fh(l)1871 762 y Fl(\000)2009 654 y Fh(n)1970 679 y Fm(X)1974 864 y Fh(l)q Fj(=1)2107 762 y Fk(c)2149 777 y Fh(l)q Fj(+2)2265 762 y Fk(A)2338 777 y Fh(k)2381 762 y Fk(x)2436 721 y Fh(l)q Fj(+2)2553 762 y Fk(A)2626 777 y Fh(k)2668 762 y Fk( )2731 777 y Fh(n)p Fi(\000)p Fh(l)2855 616 y Fm(!)3533 762 y Fn(\(11\))0 995 y(where)k(\()p Fk(H)398 1010 y Fh(k)441 995 y Fn(\))479 959 y Fi(\000)p Fj(1)479 1020 y Fh(r)603 995 y Fn(is)e(the)h(in)m(v)m (erse)i(of)d(the)h(restriction)f(of)h Fk(H)2111 1010 y Fh(k)2183 995 y Fn(to)f(the)i(subspace)h(orthogonal)c(to)h Fk(\036)3529 1010 y Fh(k)3572 995 y Fn(.)42 b(If)0 1128 y(w)m(e)34 b(de\014ne)g Fk(P)489 1143 y Fh(j)t Fi(\024)p Fh(m)676 1128 y Fn(to)f(b)s(e)g(the)g(pro)5 b(jection)33 b(on)m(to)g(the)g(subspace)i(spanned)g(b)m(y)f Fl(f)17 b Fk(\036)2999 1143 y Fh(j)3079 1128 y Fn(:)45 b Fk(j)35 b Fl(\024)29 b Fk(m)17 b Fl(g)32 b Fn(then)0 1261 y(an)h(easy)h (induction)f(argumen)m(t)g(applied)f(to)h(\(11\))f(sho)m(ws)j(that)e Fk( )2447 1276 y Fh(l)2502 1261 y Fl(2)d Fn(Ran)o(\()p Fk(P)2873 1276 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(l)3115 1261 y Fn(\).)46 b(Then,)34 b(from)0 1393 y(\(10\))e(and)h(\(11\))f(w)m (e)h(immediately)d(obtain)h(the)i(follo)m(wing)d(inequalities)0 1607 y Fl(jE)81 1622 y Fh(n)127 1607 y Fl(j)83 b(\024)398 1499 y Fh(n)359 1524 y Fm(X)363 1709 y Fh(l)q Fj(=1)496 1607 y Fl(j)p Fk(c)566 1622 y Fh(l)q Fj(+2)682 1607 y Fl(j)727 1508 y Fm(\015)726 1557 y(\015)726 1607 y(\015)p Fk(A)845 1622 y Fh(k)888 1607 y Fk(x)943 1566 y Fh(l)q Fj(+2)1060 1607 y Fk(A)1133 1622 y Fh(k)1175 1607 y Fk(P)1238 1622 y Fh(j)t Fi(\024)p Fh(k)1368 1508 y Fm(\015)1368 1557 y(\015)1368 1607 y(\015)17 b Fl(k)p Fk( )1544 1622 y Fh(n)p Fi(\000)p Fh(l)1668 1607 y Fl(k)1815 b Fn(\(12\))0 1919 y Fl(k)p Fk( )113 1934 y Fh(n)160 1919 y Fl(k)28 b(\024)398 1811 y Fh(n)359 1836 y Fm(X)363 2020 y Fh(l)q Fj(=1)496 1919 y Fl(jE)577 1934 y Fh(l)602 1919 y Fl(jk)p Fk( )743 1934 y Fh(n)p Fi(\000)p Fh(l)867 1919 y Fl(k)22 b Fn(+)1075 1811 y Fh(n)1037 1836 y Fm(X)1041 2020 y Fh(l)q Fj(=1)1173 1919 y Fl(j)p Fk(c)1243 1934 y Fh(l)q Fj(+2)1359 1919 y Fl(j)1404 1819 y Fm(\015)1403 1869 y(\015)1403 1919 y(\015)q Fk(A)1523 1934 y Fh(k)1565 1919 y Fk(x)1620 1877 y Fh(l)q Fj(+2)1737 1919 y Fk(A)1810 1934 y Fh(k)1853 1919 y Fk(P)1916 1934 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3\()p Fh(n)p Fi(\000)p Fh(l)q Fj(\))2310 1819 y Fm(\015)2310 1869 y(\015)2310 1919 y(\015)17 b Fl(k)p Fk( )2486 1934 y Fh(n)p Fi(\000)p Fh(l)2610 1919 y Fl(k)873 b Fn(\(13\))0 2142 y(T)-8 b(o)33 b(obtain)e(explicit)g(b)s(ounds)i (from)f(these,)i(w)m(e)f(use)h(the)f(follo)m(wing)d(result)0 2345 y Fw(Lemma)37 b(1.)44 b Fn(F)-8 b(or)31 b Fk(m)d Fl(\025)h Fn(1)j(and)h Fk(n)27 b Fl(\025)i Fn(0,)598 2591 y Fl(k)p Fk(A)721 2606 y Fh(k)763 2591 y Fk(x)818 2550 y Fh(m)885 2591 y Fk(A)958 2606 y Fh(k)1001 2591 y Fk(P)1064 2606 y Fh(j)t Fi(\024)p Fh(n)1198 2591 y Fl(k)f(\024)g Fn(2)1440 2523 y Fg(m)p 1440 2535 55 4 v 1452 2576 a Ff(2)1508 2591 y Fn(\(2)22 b(+)g Fk(k)s Fn(\))1824 2445 y Fm(")1882 2524 y Fn(\()p Fk(n)g Fn(+)g Fk(m)h Fl(\000)g Fn(1\)!)p 1882 2568 538 4 v 1986 2659 a(\()p Fk(n)f Fn(+)g(1\)!)2429 2445 y Fm(#)2488 2441 y Ff(1)p 2488 2453 31 4 v 2488 2494 a(2)0 2880 y Fw(Pro)s(of:)80 b Fn(Since)50 b(Ran\()p Fk(xA)996 2895 y Fh(k)1039 2880 y Fk(P)1102 2895 y Fh(j)t Fi(\024)p Fh(n)1236 2880 y Fn(\))59 b Fl(\032)g Fn(Ran\()p Fk(P)1745 2895 y Fh(j)t Fi(\024)p Fh(n)p Fj(+1)1969 2880 y Fn(\))50 b(w)m(e)i(ha)m(v)m(e)g Fk(xA)2590 2895 y Fh(k)2633 2880 y Fk(P)2696 2895 y Fh(j)t Fi(\024)p Fh(n)2889 2880 y Fn(=)58 b Fk(P)3086 2895 y Fh(j)t Fi(\024)p Fh(n)p Fj(+1)3311 2880 y Fk(xA)3439 2895 y Fh(k)3482 2880 y Fk(P)3545 2895 y Fh(j)t Fi(\024)p Fh(n)3679 2880 y Fn(.)0 3013 y(Th)m(us,)598 3182 y Fl(k)p Fk(A)721 3197 y Fh(k)763 3182 y Fk(x)818 3140 y Fh(m)885 3182 y Fk(P)948 3197 y Fh(j)t Fi(\024)p Fh(n)1082 3182 y Fl(k)28 b(\024)g(k)p Fk(A)1388 3197 y Fh(k)1431 3182 y Fk(x)p Fl(k)1553 3082 y Fm(\015)1553 3132 y(\015)1553 3182 y(\015)p Fk(x)1654 3140 y Fh(m)p Fi(\000)p Fj(2)1811 3182 y Fk(P)1874 3197 y Fh(j)t Fi(\024)p Fh(n)p Fj(+1)2098 3082 y Fm(\015)2098 3132 y(\015)2098 3182 y(\015)17 b Fl(k)p Fk(xA)2339 3197 y Fh(k)2382 3182 y Fl(k)0 3350 y Fn(Let)39 b Fl(f)p Fk(\036)289 3365 y Fh(i)316 3350 y Fl(g)366 3314 y Fi(1)366 3375 y Fh(i)p Fj(=0)523 3350 y Fn(b)s(e)g(the)g(basis)f(of)g(eigenfunctions)h(of)f Fk(H)2040 3365 y Fj(0)2079 3350 y Fn(.)61 b(Then)40 b(an)m(y)f Fk(')f Fl(2)g(H)i Fn(can)f(b)s(e)f(written)h(as)0 3483 y Fk(')28 b Fn(=)195 3417 y Fm(P)283 3443 y Fi(1)283 3508 y Fh(i)p Fj(=0)418 3483 y Fk(d)469 3498 y Fh(i)497 3483 y Fk(\036)555 3498 y Fh(i)582 3483 y Fn(.)44 b(An)33 b(easy)g(calculation)e(sho)m(ws)j(that)0 3726 y Fk(A)73 3741 y Fh(k)116 3726 y Fk(x')28 b Fn(=)418 3659 y(1)p 376 3703 132 4 v 376 3721 a Fl(p)p 459 3721 49 4 v 82 x Fn(2)535 3555 y Fm(0)535 3704 y(@)607 3618 y Fh(k)r Fi(\000)p Fj(1)612 3643 y Fm(X)615 3825 y Fh(i)p Fj(=0)753 3726 y Fk(d)804 3741 y Fh(i)p Fj(+1)922 3568 y Fm(s)p 1005 3568 230 4 v 1018 3659 a Fk(i)23 b Fn(+)f(1)p 1015 3703 210 4 v 1015 3794 a Fk(k)j Fl(\000)d Fk(i)1234 3726 y(\036)1292 3741 y Fh(i)1342 3726 y Fn(+)g Fk(d)1491 3741 y Fh(k)r Fj(+1)1623 3640 y Fl(p)p 1706 3640 224 4 v 86 x Fk(k)k Fn(+)c(1)o Fk(\036)1987 3741 y Fh(k)2052 3726 y Fn(+)2218 3618 y Fi(1)2194 3643 y Fm(X)2150 3827 y Fh(i)p Fj(=)p Fh(k)r Fj(+1)2374 3726 y Fk(d)2425 3741 y Fh(i)p Fj(+1)2543 3568 y Fm(s)p 2626 3568 230 4 v 2639 3659 a Fk(i)h Fn(+)f(1)p 2636 3703 210 4 v 2636 3794 a Fk(i)h Fl(\000)f Fk(k)2855 3726 y(\036)2913 3741 y Fh(i)2941 3555 y Fm(1)2941 3704 y(A)598 4082 y Fn(+)742 4014 y(1)p 700 4058 132 4 v 700 4076 a Fl(p)p 783 4076 49 4 v 82 x Fn(2)859 3911 y Fm(0)859 4060 y(@)931 3974 y Fh(k)r Fi(\000)p Fj(1)936 3999 y Fm(X)938 4181 y Fh(i)p Fj(=1)1076 4082 y Fk(d)1127 4097 y Fh(i)p Fi(\000)p Fj(1)1246 3924 y Fm(s)p 1329 3924 230 4 v 1426 4014 a Fk(i)p 1339 4058 210 4 v 1339 4150 a(k)j Fl(\000)d Fk(i)1558 4082 y(\036)1616 4097 y Fh(i)1666 4082 y Fn(+)g Fk(d)1815 4097 y Fh(k)r Fi(\000)p Fj(1)1947 3992 y Fl(p)p 2030 3992 55 4 v 90 x Fk(k)s(\036)2142 4097 y Fh(k)2207 4082 y Fn(+)2373 3974 y Fi(1)2349 3999 y Fm(X)2305 4183 y Fh(i)p Fj(=)p Fh(k)r Fj(+1)2529 4082 y Fk(d)2580 4097 y Fh(i)p Fi(\000)p Fj(1)2698 3924 y Fm(s)p 2781 3924 230 4 v 2879 4014 a Fk(i)p 2791 4058 210 4 v 2791 4150 a(i)g Fl(\000)h Fk(k)3010 4082 y(\036)3068 4097 y Fh(i)3096 3911 y Fm(1)3096 4060 y(A)0 4319 y Fn(so)39 b(w)m(e)i(can)e(write)g Fk(A)791 4334 y Fh(k)834 4319 y Fk(x')g Fn(=)g Fk(')1171 4334 y Fj(1)1237 4319 y Fn(+)26 b Fk(')1403 4334 y Fj(2)1482 4319 y Fn(where)40 b Fk(')1834 4334 y Fj(1)p Fh(;)p Fj(2)1967 4319 y Fn(are)g(de\014ned)g(in)f(the)g(ob)m(vious)h(w)m(a)m(y)-8 b(.)64 b(Clearly)-8 b(,)0 4452 y Fl(k)p Fk(A)123 4467 y Fh(k)166 4452 y Fk(x')p Fl(k)27 b(\024)i(k)p Fk(')582 4467 y Fj(1)621 4452 y Fl(k)22 b Fn(+)g Fl(k)p Fk(')905 4467 y Fj(2)944 4452 y Fl(k)p Fn(.)43 b(No)m(w)598 4695 y Fl(k)p Fk(')712 4710 y Fj(1)751 4695 y Fl(k)801 4654 y Fj(2)868 4695 y Fn(=)983 4628 y(1)p 983 4672 49 4 v 983 4764 a(2)1058 4524 y Fm(0)1058 4674 y(@)1131 4588 y Fh(k)r Fi(\000)p Fj(1)1135 4612 y Fm(X)1138 4795 y Fh(i)p Fj(=0)1276 4695 y Fl(j)p Fk(d)1355 4710 y Fh(i)p Fj(+1)1473 4695 y Fl(j)1501 4654 y Fj(2)1553 4628 y Fk(i)23 b Fn(+)f(1)p 1550 4672 210 4 v 1550 4764 a Fk(k)j Fl(\000)e Fk(i)1791 4695 y Fn(+)f Fl(j)p Fk(d)1968 4710 y Fh(k)r Fj(+1)2100 4695 y Fl(j)2128 4654 y Fj(2)2168 4695 y Fn(\()p Fk(k)j Fn(+)d(1\))g(+)2655 4588 y Fi(1)2630 4612 y Fm(X)2587 4797 y Fh(i)p Fj(=)p Fh(k)r Fj(+1)2811 4695 y Fl(j)p Fk(d)2890 4710 y Fh(i)p Fj(+1)3008 4695 y Fl(j)3036 4654 y Fj(2)3088 4628 y Fk(i)g Fn(+)g(1)p 3085 4672 V 3085 4764 a Fk(i)g Fl(\000)h Fk(k)3304 4524 y Fm(1)3304 4674 y(A)868 5028 y Fl(\024)983 4960 y Fn(1)p 983 5004 49 4 v 983 5096 a(2)1058 4857 y Fm(0)1058 5006 y(@)1131 5028 y Fk(k)1202 4920 y Fh(k)r Fi(\000)p Fj(1)1206 4945 y Fm(X)1209 5127 y Fh(i)p Fj(=0)1347 5028 y Fl(j)p Fk(d)1426 5043 y Fh(i)p Fj(+1)1544 5028 y Fl(j)1572 4986 y Fj(2)1633 5028 y Fn(+)f(\()p Fk(k)j Fn(+)d(1\))p Fl(j)p Fk(d)2109 5043 y Fh(k)r Fj(+1)2241 5028 y Fl(j)2269 4986 y Fj(2)2330 5028 y Fn(+)g(\()p Fk(k)j Fn(+)d(2\))2812 4920 y Fi(1)2788 4945 y Fm(X)2744 5129 y Fh(i)p Fj(=)p Fh(k)r Fj(+1)2968 5028 y Fl(j)p Fk(d)3047 5043 y Fh(i)p Fj(+1)3165 5028 y Fl(j)3193 4986 y Fj(2)3232 4857 y Fm(1)3232 5006 y(A)868 5317 y Fl(\024)983 5250 y Fn(1)p 983 5294 V 983 5385 a(2)1042 5317 y(\()p Fk(k)j Fn(+)d(2\))p Fl(k)p Fk(')p Fl(k)1505 5276 y Fj(2)0 5498 y Fn(and)33 b(a)f(similar)d(argumen)m(t)k (yields)598 5693 y Fl(k)p Fk(')712 5708 y Fj(2)751 5693 y Fl(k)801 5652 y Fj(2)868 5693 y Fl(\024)983 5626 y Fn(1)p 983 5670 V 983 5762 a(2)1042 5693 y(\()p Fk(k)25 b Fn(+)d(1\))p Fl(k)p Fk(')p Fl(k)1505 5652 y Fj(2)p eop %%Page: 6 6 6 5 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2269 b Fn(6)0 232 y(Hence,)598 481 y Fl(k)p Fk(A)721 496 y Fh(k)763 481 y Fk(x)p Fl(k)28 b(\024)1001 317 y Fm(s)p 1084 317 243 4 v 1094 413 a Fk(k)e Fn(+)c(1)p 1094 457 224 4 v 1182 549 a(2)1350 481 y(+)1448 317 y Fm(s)p 1531 317 243 4 v 1541 413 a Fk(k)j Fn(+)d(2)p 1541 457 224 4 v 1628 549 a(2)1801 481 y Fl(\024)1906 376 y Fm(q)p 1989 376 348 4 v 105 x Fn(2\(2)g(+)g Fk(k)s Fn(\))0 690 y(By)33 b(taking)f(the)h(adjoin)m(t) e(of)i(previous)g(estimate)e(w)m(e)j(ha)m(v)m(e)598 872 y Fl(k)p Fk(xA)776 887 y Fh(k)819 872 y Fl(k)27 b(\024)1001 767 y Fm(q)p 1084 767 V 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\))0 1055 y(Finally)-8 b(,)30 b(w)m(e)k(mak)m(e)e(use)i(of)e(Lemma)f (5.1)h(of)g([1])h(to)f(complete)g(the)h(pro)s(of.)902 b Fd(2)195 1330 y Fn(In)26 b(\(H2\))f(w)m(e)i(assume)f(that)f Fk(V)d Fn(\()p Fk(x)p Fn(\))k(has)g(an)f(analytic)g(extension,)j(so)d (\(4\))g(has)h(a)g(non-zero)f(radius)0 1462 y(of)35 b(con)m(v)m (ergence.)56 b(Th)m(us,)38 b(there)e(is)f(a)h(constan)m(t)g Fk(c)d(>)g Fn(0)j(suc)m(h)h(that)e Fl(j)p Fk(c)2589 1477 y Fh(l)2615 1462 y Fl(j)e(\024)g Fk(c)p Fn(.)53 b(Therefore,)38 b(w)m(e)e(ma)m(y)0 1595 y(obtain)31 b(nicer)i(b)s(ounds)g(from)f(b)s (oth)g(\(12\))g(and)g(\(13\),)h(namely)-8 b(,)0 1861 y Fl(jE)81 1876 y Fh(n)127 1861 y Fl(j)83 b(\024)28 b Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))788 1753 y Fh(n)749 1778 y Fm(X)753 1962 y Fh(l)q Fj(=1)886 1861 y Fn(2)950 1792 y Fg(l)p 944 1804 31 4 v 944 1846 a Ff(2)1006 1714 y Fm(")1064 1793 y Fn(\(1)g(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 1064 1837 478 4 v 1140 1929 a(\(1)f(+)h Fk(k)s Fn(\)!)1551 1714 y Fm(#)1610 1710 y Ff(1)p 1610 1722 31 4 v 1610 1763 a(2)1671 1861 y Fl(k)p Fk( )1784 1876 y Fh(n)p Fi(\000)p Fh(l)1908 1861 y Fl(k)1575 b Fn(\(14\))0 2210 y Fl(k)p Fk( )113 2225 y Fh(n)160 2210 y Fl(k)28 b(\024)398 2102 y Fh(n)359 2127 y Fm(X)363 2311 y Fh(l)q Fj(=1)496 2210 y Fl(jE)577 2225 y Fh(l)602 2210 y Fl(jk)p Fk( )743 2225 y Fh(n)p Fi(\000)p Fh(l)867 2210 y Fl(k)22 b Fn(+)g(2)p Fk(c)p Fn(\(2)f(+)h Fk(k)s Fn(\))1482 2102 y Fh(n)1443 2127 y Fm(X)1447 2311 y Fh(l)q Fj(=1)1580 2210 y Fn(2)1644 2141 y Fg(l)p 1638 2153 V 1638 2195 a Ff(2)1699 2063 y Fm(")1758 2142 y Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(2)p Fk(l)r Fn(\)!)p 1758 2186 755 4 v 1758 2278 a(\(1)f(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!)2523 2063 y Fm(#)2581 2059 y Ff(1)p 2581 2071 31 4 v 2581 2113 a(2)2642 2210 y Fl(k)p Fk( )2755 2225 y Fh(n)p Fi(\000)p Fh(l)2879 2210 y Fl(k)604 b Fn(\(15\))0 2452 y(Finally)-8 b(,)30 b(substituting)i(\(14\))g(in)m(to)g(\(15\),)g(w)m(e)h(obtain)0 2718 y Fl(k)p Fk( )113 2733 y Fh(n)160 2718 y Fl(k)28 b(\024)g Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))788 2610 y Fh(n)749 2635 y Fm(X)753 2819 y Fh(l)q Fj(=1)935 2610 y Fh(l)886 2635 y Fm(X)888 2817 y Fh(i)p Fj(=1)1022 2718 y Fn(2)1085 2649 y Fg(i)p 1081 2661 V 1081 2703 a Ff(2)1142 2572 y Fm(")1200 2650 y Fn(\(1)g(+)g Fk(k)k Fn(+)c Fk(i)p Fn(\)!)p 1200 2694 480 4 v 1277 2786 a(\(1)g(+)g Fk(k)s Fn(\)!)1690 2572 y Fm(#)1748 2567 y Ff(1)p 1748 2579 31 4 v 1748 2621 a(2)1809 2718 y Fl(k)p Fk( )1922 2733 y Fh(l)q Fi(\000)p Fh(i)2027 2718 y Fl(kk)p Fk( )2190 2733 y Fh(n)p Fi(\000)p Fh(l)2314 2718 y Fl(k)598 3043 y Fn(+)17 b(2)p Fk(c)p Fn(\(2)k(+)h Fk(k)s Fn(\))1135 2935 y Fh(n)1096 2960 y Fm(X)1100 3145 y Fh(l)q Fj(=1)1233 3043 y Fn(2)1297 2975 y Fg(l)p 1292 2987 V 1292 3028 a Ff(2)1353 2897 y Fm(")1411 2976 y Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)h Fl(\000)f Fn(2)p Fk(l)r Fn(\)!)p 1411 3020 755 4 v 1411 3111 a(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)h Fl(\000)f Fn(3)p Fk(l)r Fn(\)!)2176 2897 y Fm(#)2235 2893 y Ff(1)p 2235 2905 31 4 v 2235 2946 a(2)2296 3043 y Fl(k)p Fk( )2409 3058 y Fh(n)p Fi(\000)p Fh(l)2533 3043 y Fl(k)950 b Fn(\(16\))195 3357 y(W)-8 b(e)96 b(pro)m(v)m(e)h(b)s (elo)m(w)e(that)g(the)h(second)h(term)e(in)g(\(16\))f(b)s(eha)m(v)m(es) k(roughly)d(lik)m(e)0 3489 y Fk(a)51 3409 y Fl(p)p 134 3409 402 4 v 80 x Fn(1)22 b(+)g Fk(k)j Fn(+)d Fk(n)17 b Fl(k)p Fk( )665 3504 y Fh(n)p Fi(\000)p Fj(1)803 3489 y Fl(k)o Fn(,)46 b(and)d(that)f(this)h(is)f(the)i(dominan)m(t)d(term)h (in)g(\(16\).)74 b(So,)46 b(w)m(e)d(exp)s(ect)i(the)0 3622 y(gro)m(wth)33 b(of)f Fl(k)p Fk( )552 3637 y Fh(n)599 3622 y Fl(k)h Fn(\(and)f(hence)i Fl(jE)1261 3637 y Fh(n)1307 3622 y Fl(j)p Fn(\))e(to)h(b)s(e)f(dominated)g(b)m(y)h(something)f(lik) m(e)g Fk(a)2972 3586 y Fh(n)3019 3538 y Fl(p)p 3102 3538 86 4 v 84 x Fk(n)p Fn(!.)0 3905 y Fw(3.)50 b(The)38 b(main)e(estimates) 0 4137 y Fn(F)-8 b(rom)30 b(\(16\))i(w)m(e)h(can)f(deriv)m(e)h(a)e(b)s (ound)i(for)e(the)h(gro)m(wth)h(of)e Fl(k)p Fk( )2316 4152 y Fh(n)2363 4137 y Fl(k)h Fn(for)g(large)e Fk(n)p Fn(.)44 b(T)-8 b(o)32 b(do)g(so,)g(w)m(e)h(ma)m(y)0 4269 y(assume)42 b(without)f(loss)g(that)g Fk(c)i Fl(\025)g Fn(1.)70 b(In)42 b(addition,)g(w)m(e)g(need)h(t)m(w)m(o)f(tec)m(hnical) f(results)h(that)f(are)0 4402 y(summarized)31 b(in)h(the)h(follo)m (wing)d(lemma.)0 4606 y Fw(Lemma)37 b(2.)43 4789 y Fn(\(i\))48 b(F)-8 b(or)32 b(eac)m(h)h Fk(k)e Fl(\025)d Fn(0)33 b(there)g(is)f(a)g (constan)m(t)i Fk(\015)1731 4804 y Fh(k)1805 4789 y Fn(so)f(that,)g (for)f(all)e Fk(m)e Fl(\025)g Fn(0,)822 4946 y Fh(m)793 4971 y Fm(X)797 5156 y Fh(l)q Fj(=0)929 4908 y Fm(")988 4987 y Fn(\(1)22 b(+)g Fk(k)j Fn(+)d Fk(m)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 988 5031 1162 4 v 1303 5122 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)2160 4908 y Fm(#)2218 4904 y Ff(1)p 2218 4916 31 4 v 2218 4957 a(2)2290 5054 y Fl(\024)28 b Fk(\015)2446 5069 y Fh(k)16 5297 y Fn(\(ii\))47 b(F)-8 b(or)32 b(all)e Fk(k)h Fl(\025)d(\000)p Fn(1)33 b(there)g(is)f(a)h(constan)m(t)g Fk(\014)38 b Fn(so)33 b(that,)g(for)f(all)e Fk(m)e Fl(\025)g Fn(0,)822 5454 y Fh(m)793 5479 y Fm(X)797 5664 y Fh(l)q Fj(=0)929 5562 y Fn(2)978 5521 y Fi(\000)1043 5494 y Ff(5)p Fg(l)p 1043 5506 51 4 v 1053 5547 a Ff(2)1124 5416 y Fm(")1183 5495 y Fn(\(1)21 b(+)h Fk(k)k Fn(+)c(3)p Fk(m)g Fl(\000)h Fn(2)p Fk(l)r Fn(\)!\(1)e(+)h Fk(k)k Fn(+)c Fk(m)g Fl(\000)h Fk(l)r Fn(\)!)p 1183 5539 1467 4 v 1259 5630 a(\(1)f(+)g Fk(k)j Fn(+)d(3)p Fk(m)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)2659 5416 y Fm(#)2717 5412 y Ff(1)p 2717 5424 31 4 v 2717 5465 a(2)2790 5562 y Fl(\024)28 b Fk(\014)p eop %%Page: 7 7 7 6 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2269 b Fn(7)0 232 y Fw(Pro)s(of:)43 b Fn(\(i\).)29 387 y Fh(m)0 412 y Fm(X)4 596 y Fh(l)q Fj(=0)137 349 y Fm(")195 428 y Fn(\(1)22 b(+)g Fk(k)j Fn(+)d Fk(m)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 195 472 1162 4 v 510 563 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)1367 349 y Fm(#)1425 345 y Ff(1)p 1425 357 31 4 v 1425 398 a(2)494 820 y Fn(=)28 b(2[\(1)21 b(+)h Fk(k)s Fn(\)!])1036 752 y Ff(1)p 1037 764 V 1037 805 a(2)1103 820 y Fn(+)g(2)1267 674 y Fm(")1376 753 y Fn(\(2)g(+)g Fk(k)s Fn(\)!)p 1325 797 429 4 v 1325 889 a(1)g(+)g Fk(k)j Fn(+)d Fk(m)1764 674 y Fm(#)1822 670 y Ff(1)p 1822 682 31 4 v 1822 723 a(2)1872 820 y Fn(+)1954 712 y Fh(m)p Fi(\000)p Fj(2)1970 737 y Fm(X)1974 922 y Fh(l)q Fj(=2)2123 674 y Fm(")2181 753 y Fn(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)h Fl(\000)f Fk(l)r Fn(\)!\(1)g(+)g Fk(k)k Fn(+)c Fk(l)r Fn(\)!)p 2181 797 1162 4 v 2496 889 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)3353 674 y Fm(#)3411 670 y Ff(1)p 3411 682 31 4 v 3411 723 a(2)493 1146 y Fl(\024)28 b Fn(2[\(1)21 b(+)h Fk(k)s Fn(\)!])1036 1077 y Ff(1)p 1037 1089 V 1037 1131 a(2)1103 1146 y Fn(+)g(2)1267 1000 y Fm(")1325 1078 y Fn(\(2)g(+)g Fk(k)s Fn(\)!)p 1325 1123 326 4 v 1377 1214 a(1)g(+)g Fk(k)1661 1000 y Fm(#)1719 995 y Ff(1)p 1719 1007 31 4 v 1719 1049 a(2)598 1461 y Fn(+)17 b(\()p Fk(m)22 b Fl(\000)g Fn(4\))75 b(max)1039 1525 y Fj(2)p Fi(\024)p Fh(l)q Fi(\024)p Fc(J)1244 1498 y Fg(m)p 1244 1510 55 4 v 1256 1551 a Ff(2)1308 1525 y Fc(K)1353 1315 y Fm(")1412 1394 y Fn(\(1)22 b(+)g Fk(k)j Fn(+)d Fk(m)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 1412 1438 1162 4 v 1727 1529 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)2584 1315 y Fm(#)2642 1311 y Ff(1)p 2642 1323 31 4 v 2642 1364 a(2)2703 1461 y Fk(;)0 1691 y Fn(where)41 b Fb(J)p Fk(\013)q Fb(K)f Fn(denotes)h(the)g (greatest)f(in)m(teger)g(less)g(than)g(or)g(equal)g(to)g Fk(\013)q Fn(.)65 b(If)40 b Fk(l)j Fl(\024)d Fb(J)3154 1651 y Fh(m)p 3154 1667 63 4 v 3168 1725 a Fj(2)3227 1691 y Fb(K)27 b Fl(\000)h Fn(1,)41 b(then)0 1823 y(2)p Fk(l)24 b Fn(+)e(1)28 b Fl(\024)g Fk(m)p Fn(.)44 b(Th)m(us,)34 b(\(1)22 b(+)g Fk(k)j Fn(+)d Fk(m)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)33 b(is)f(decreasing)h(for) f Fk(l)e Fl(\024)e Fb(J)2939 1784 y Fh(m)p 2939 1800 V 2953 1858 a Fj(2)3012 1823 y Fb(K)p Fn(.)43 b(Therefore,)29 1988 y Fh(m)0 2012 y Fm(X)4 2197 y Fh(l)q Fj(=0)137 1949 y Fm(")195 2028 y Fn(\(1)22 b(+)g Fk(k)j Fn(+)d Fk(m)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 195 2072 1162 4 v 510 2164 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)1367 1949 y Fm(#)1425 1945 y Ff(1)p 1425 1957 31 4 v 1425 1998 a(2)1497 2095 y Fl(\024)28 b Fn(2[\(1)22 b(+)g Fk(k)s Fn(\)!])2041 2027 y Ff(1)p 2041 2039 V 2041 2080 a(2)2091 2095 y Fn(+)h(2)2256 1949 y Fm(")2313 2028 y Fn(\(2)f(+)g Fk(k)s Fn(\)!)p 2313 2072 326 4 v 2365 2164 a(1)g(+)g Fk(k)2649 1949 y Fm(#)2708 1945 y Ff(1)p 2708 1957 31 4 v 2708 1998 a(2)2758 2095 y Fn(+)g([\(3)g(+)g Fk(k)s Fn(\)!])3246 2027 y Ff(1)p 3246 2039 V 3246 2080 a(2)3302 2028 y Fk(m)h Fl(\000)f Fn(4)p 3300 2072 260 4 v 3300 2164 a Fk(m)h Fn(+)f Fk(k)0 2330 y Fn(The)34 b(last)d(term)h(con)m(v)m(erges)j(as)e Fk(m)28 b Fl(!)f(1)p Fn(,)32 b(so)h(existence)h(of)e(the)h(constan)m(t) h Fk(\015)2833 2345 y Fh(k)2908 2330 y Fn(is)e(guaran)m(teed.)0 2533 y(\(ii\).)42 b(By)33 b(cancelling)d(common)i(factors,)g(w)m(e)i (ha)m(v)m(e)627 2688 y Fh(m)598 2713 y Fm(X)602 2897 y Fh(l)q Fj(=0)734 2796 y Fn(2)783 2755 y Fi(\000)848 2728 y Ff(5)p Fg(l)p 848 2740 51 4 v 858 2781 a Ff(2)980 2688 y Fh(l)937 2713 y Fm(Y)929 2894 y Fh(s)p Fj(=1)1068 2650 y Fm( )1144 2728 y Fn(1)22 b(+)g Fk(k)j Fn(+)d(3)p Fk(m)h Fl(\000)f Fn(3)p Fk(l)i Fn(+)e Fk(s)p 1144 2773 846 4 v 1193 2864 a Fn(1)g(+)g Fk(k)j Fn(+)d Fk(m)h Fl(\000)f Fk(l)j Fn(+)d Fk(s)1999 2650 y Fm(!)2075 2646 y Ff(1)p 2075 2658 31 4 v 2075 2699 a(2)0 3036 y Fn(F)-8 b(or)32 b Fk(k)f Fl(\025)d(\000)p Fn(1)k(and)h Fk(s)28 b Fl(\025)g Fn(0,)k(w)m(e)i(ha)m(v)m(e)g(0)27 b Fl(\024)h Fn(2\(1)22 b(+)g Fk(k)j Fn(+)d Fk(s)p Fn(\).)44 b(This)32 b(implies)608 3185 y(1)22 b(+)g Fk(k)j Fn(+)d(3)p Fk(m)g Fl(\000)h Fn(3)p Fk(l)h Fn(+)e Fk(s)p 608 3229 846 4 v 656 3321 a Fn(1)g(+)g Fk(k)k Fn(+)c Fk(m)g Fl(\000)h Fk(l)h Fn(+)e Fk(s)1491 3252 y Fl(\024)28 b Fn(3)0 3453 y(Therefore,)627 3602 y Fh(m)598 3627 y Fm(X)602 3812 y Fh(l)q Fj(=0)734 3710 y Fn(2)783 3669 y Fi(\000)848 3642 y Ff(5)p Fg(l)p 848 3654 51 4 v 858 3695 a Ff(2)929 3564 y Fm(")987 3643 y Fn(\(1)22 b(+)g Fk(k)k Fn(+)c(3)p Fk(m)g Fl(\000)g Fn(2)p Fk(l)r Fn(\)!\(1)g(+)g Fk(k)k Fn(+)c Fk(m)g Fl(\000)h Fk(l)r Fn(\)!)p 987 3687 1467 4 v 1064 3779 a(\(1)f(+)g Fk(k)j Fn(+)d(3)p Fk(m)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(m)p Fn(\)!)2464 3564 y Fm(#)2522 3560 y Ff(1)p 2522 3572 31 4 v 2522 3613 a(2)2595 3710 y Fl(\024)2728 3602 y Fh(m)2700 3627 y Fm(X)2704 3812 y Fh(l)q Fj(=0)2836 3710 y Fn(2)2885 3669 y Fi(\000)2950 3642 y Ff(5)p Fg(l)p 2950 3654 51 4 v 2960 3695 a Ff(2)3014 3710 y Fn(3)3078 3642 y Fg(l)p 3073 3654 31 4 v 3073 3695 a Ff(2)0 3999 y Fn(and)33 b(the)g(righ)m(t)e(hand)i(side)g(con)m(v)m(erges)i(to)d Fk(\014)h Fn(=)1781 3902 y Fm(\020)1830 3999 y Fn(1)22 b Fl(\000)2001 3899 y Fm(q)p 2084 3899 186 4 v 100 x Fn(3)p Fk(=)p Fn(2)2231 3970 y Fj(5)2270 3902 y Fm(\021)2319 3918 y Fi(\000)p Fj(1)2414 3999 y Fn(.)1191 b Fd(2)0 4273 y Fw(Theorem)37 b(1.)44 b Fn(F)-8 b(or)31 b(eac)m(h)j Fk(k)d Fl(\025)d Fn(0)k(and)h(for)f Fk(n)c Fl(\025)g Fn(1,)598 4453 y Fl(k)p Fk( )711 4468 y Fh(n)758 4453 y Fl(k)44 b(\024)28 b Fn(2)1006 4412 y Fj(3)p Fh(n)1121 4357 y Fm(h)1161 4453 y Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))1567 4357 y Fm(\020)1616 4453 y Fk(\015)1672 4412 y Fj(2)1667 4478 y Fh(k)1712 4453 y Fn([\(1)g(+)g Fk(k)s Fn(\)!])2092 4404 y Fi(\000)2157 4377 y Ff(1)p 2157 4389 31 4 v 2157 4431 a(2)2224 4453 y Fn(+)g Fk(\014)2383 4357 y Fm(\021)o(i)2471 4372 y Fh(n)2551 4453 y Fn([\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])3120 4377 y Ff(1)p 3120 4389 V 3120 4431 a(2)3181 4453 y Fk(:)0 4727 y Fw(Pro)s(of:)43 b Fn(Fix)32 b Fk(k)f Fl(\025)d Fn(0.)43 b(F)-8 b(or)32 b Fk(n)c Fn(=)f(1)33 b(the)g(inequalit)m(y)e(\(16\))h(b)s(ecomes)598 4907 y Fl(k)p Fk( )711 4922 y Fj(1)750 4907 y Fl(k)c(\024)g Fn(2)982 4866 y Fj(2)1021 4907 y Fk(c)p Fn(\(2)22 b(+)g Fk(k)s Fn(\)2)1421 4839 y Ff(1)p 1421 4851 V 1421 4892 a(2)1465 4907 y Fn(\(2)g(+)g Fk(k)s Fn(\))1774 4839 y Ff(1)p 1774 4851 V 1774 4892 a(2)1819 4907 y Fk(:)0 5087 y Fn(F)-8 b(rom)31 b(the)i(pro)s(of)f(of)g(Lemma)f(2)h(it)g(follo)m(ws) f(that)i Fk(\015)1905 5051 y Fj(2)1900 5111 y Fh(k)1944 5087 y Fn([\(1)22 b(+)g Fk(k)s Fn(\)!])2324 5050 y Fi(\000)2389 5023 y Ff(1)p 2389 5035 V 2389 5076 a(2)2456 5087 y Fn(+)g Fk(\014)33 b Fl(\025)28 b Fn(1.)43 b(Th)m(us,)598 5266 y(2)647 5225 y Fj(2)686 5266 y Fk(c)p Fn(\(2)22 b(+)g Fk(k)s Fn(\)2)1086 5198 y Ff(1)p 1086 5210 V 1086 5251 a(2)1147 5266 y Fn(\(2)g(+)g Fk(k)s Fn(\))1456 5198 y Ff(1)p 1456 5210 V 1456 5251 a(2)1528 5266 y Fl(\024)28 b Fn(2)1682 5225 y Fj(3)1721 5266 y Fn(2)p Fk(c)p Fn(\(2)22 b(+)g Fk(k)s Fn(\)\()p Fk(\015)2205 5225 y Fj(2)2200 5291 y Fh(k)2244 5266 y Fn([\(1)g(+)g Fk(k)s Fn(\)!])2624 5225 y Fi(\000)2689 5198 y Ff(1)p 2689 5210 V 2689 5251 a(2)2756 5266 y Fn(+)g Fk(\014)6 b Fn(\))17 b([\(2)k(+)h Fk(k)s Fn(\)!])3359 5198 y Ff(1)p 3359 5210 V 3359 5251 a(2)3404 5266 y Fk(;)0 5446 y Fn(so)33 b(the)g(assertion)f(is)g(true)h (for)f Fk(n)c Fn(=)g(1)k(and)h(for)f(eac)m(h)h Fk(k)e Fl(\025)d Fn(0.)195 5579 y(No)m(w)67 b(assume)g(the)g(inequalit)m(y)e (is)h(satis\014ed)g(for)g(all)e Fk(i)86 b(<)f(n)p Fn(.)145 b(Let)66 b Fk(b)86 b Fn(=)f(2)p Fk(c)p Fn(\(2)44 b(+)0 5712 y Fk(k)s Fn(\))109 5616 y Fm(\020)158 5712 y Fk(\015)214 5676 y Fj(2)209 5737 y Fh(k)270 5712 y Fn([\(1)22 b(+)g Fk(k)s Fn(\)!])650 5663 y Fi(\000)715 5636 y Ff(1)p 715 5648 V 715 5690 a(2)782 5712 y Fn(+)g Fk(\014)941 5616 y Fm(\021)990 5712 y Fn(.)71 b(As)42 b(w)m(e)g(ha)m(v)m(e)h(p)s(oin)m (ted)e(out)h(ab)s(o)m(v)m(e,)i(w)m(e)f(ma)m(y)e(assume)i(that)e Fk(c)i Fl(\025)g Fn(1.)p eop %%Page: 8 8 8 7 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2269 b Fn(8)0 232 y(W)-8 b(e)34 b(already)f(kno)m(w)i(that)e Fk(\015)1038 196 y Fj(2)1033 257 y Fh(k)1078 232 y Fn([\(1)22 b(+)h Fk(k)s Fn(\)!])1459 196 y Fi(\000)1524 169 y Ff(1)p 1524 181 31 4 v 1524 222 a(2)1592 232 y Fn(+)f Fk(\014)35 b Fl(\025)30 b Fn(1,)k(so)g(w)m(e) g(conclude)g(that)g Fk(b)c Fl(\025)g Fn(1.)46 b(F)-8 b(rom)32 b(\(16\))h(it)0 365 y(follo)m(ws)e(that)0 606 y Fl(k)p Fk( )113 621 y Fh(n)160 606 y Fl(k)d(\024)g Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))788 498 y Fh(n)749 523 y Fm(X)753 707 y Fh(l)q Fj(=1)935 498 y Fh(l)886 523 y Fm(X)888 705 y Fh(i)p Fj(=1)1022 606 y Fn(2)1085 538 y Fg(i)p 1081 550 V 1081 591 a Ff(2)1142 460 y Fm(")1200 538 y Fn(\(1)g(+)g Fk(k)k Fn(+)c Fk(i)p Fn(\)!)p 1200 583 480 4 v 1277 674 a(\(1)g(+)g Fk(k)s Fn(\)!)1690 460 y Fm(#)1748 456 y Ff(1)p 1748 468 31 4 v 1748 509 a(2)1809 606 y Fn(2)1858 565 y Fj(3\()p Fh(n)p Fi(\000)p Fh(i)p Fj(\))2074 606 y Fk(b)2115 565 y Fh(n)p Fi(\000)p Fh(i)2242 606 y Fn([\(1)6 b(+)g Fk(k)j Fn(+)d Fk(n)g Fl(\000)g Fk(l)r Fn(\)!\(1)g(+)g Fk(k)j Fn(+)d Fk(l)i Fl(\000)e Fk(i)p Fn(\)!])3401 538 y Ff(1)p 3396 550 V 3396 591 a(2)365 931 y Fn(+)39 b(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))924 823 y Fh(n)886 848 y Fm(X)890 1033 y Fh(l)q Fj(=1)1022 931 y Fn(2)1086 863 y Fg(l)p 1081 875 V 1081 916 a Ff(2)1142 785 y Fm(")1201 864 y Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(2)p Fk(l)r Fn(\)!)p 1201 908 755 4 v 1201 1000 a(\(1)f(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!)1966 785 y Fm(#)2024 781 y Ff(1)p 2024 793 31 4 v 2024 834 a(2)2085 931 y Fn(2)2134 890 y Fj(3\()p Fh(n)p Fi(\000)p Fh(l)q Fj(\))2348 931 y Fk(b)2389 890 y Fh(n)p Fi(\000)p Fh(l)2513 931 y Fn([\(1)f(+)g Fk(k)j Fn(+)d Fk(n)g Fl(\000)h Fk(l)r Fn(\)!])3234 863 y Ff(1)p 3234 875 V 3234 916 a(2)238 1257 y Fl(\024)28 b Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\)2)781 1215 y Fj(3)p Fh(n)864 1257 y Fk(b)905 1215 y Fh(n)p Fi(\000)p Fj(1)1097 1149 y Fh(n)1059 1174 y Fm(X)1063 1358 y Fh(l)q Fj(=1)1244 1149 y Fh(l)1195 1174 y Fm(X)1198 1356 y Fh(i)p Fj(=1)1332 1257 y Fn(2)1381 1215 y Fi(\000)1446 1188 y Ff(5)p Fg(i)p 1445 1200 53 4 v 1457 1242 a Ff(2)1529 1110 y Fm(")1587 1189 y Fn(\(1)g(+)g Fk(k)j Fn(+)d Fk(i)p Fn(\)!)p 1587 1233 480 4 v 1664 1325 a(\(1)g(+)g Fk(k)s Fn(\)!)2077 1110 y Fm(#)2135 1106 y Ff(1)p 2135 1118 31 4 v 2135 1159 a(2)2196 1257 y Fn([\(1)6 b(+)g Fk(k)j Fn(+)d Fk(n)g Fl(\000)g Fk(l)r Fn(\)!\(1)g(+)g Fk(k)j Fn(+)d Fk(l)i Fl(\000)e Fk(i)p Fn(\)!])3355 1188 y Ff(1)p 3351 1200 V 3351 1242 a(2)365 1582 y Fn(+)39 b(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\)2)918 1541 y Fj(3)p Fh(n)1000 1582 y Fk(b)1041 1541 y Fh(n)p Fi(\000)p Fj(1)1234 1474 y Fh(n)1195 1499 y Fm(X)1199 1683 y Fh(l)q Fj(=1)1332 1582 y Fn(2)1381 1541 y Fi(\000)1446 1514 y Ff(5)p Fg(l)p 1446 1526 51 4 v 1456 1567 a Ff(2)1527 1436 y Fm(")1585 1515 y Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(2)p Fk(l)r Fn(\)!)p 1585 1559 755 4 v 1585 1650 a(\(1)f(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!)2350 1436 y Fm(#)2408 1432 y Ff(1)p 2408 1444 31 4 v 2408 1485 a(2)2470 1582 y Fn([\(1)f(+)g Fk(k)j Fn(+)d Fk(n)g Fl(\000)h Fk(l)r Fn(\)!])3191 1506 y Ff(1)p 3191 1518 V 3191 1560 a(2)238 1907 y Fl(\024)28 b Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\)2)781 1866 y Fj(3)p Fh(n)864 1907 y Fk(b)905 1866 y Fh(n)p Fi(\000)p Fj(1)1059 1761 y Fm(")1117 1840 y Fn(\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!)p 1117 1884 505 4 v 1206 1976 a(\(1)g(+)g Fk(k)s Fn(\)!)1631 1761 y Fm(#)1690 1757 y Ff(1)p 1690 1769 31 4 v 1690 1810 a(2)1790 1799 y Fh(n)1751 1824 y Fm(X)1755 2009 y Fh(l)q Fj(=1)1888 1761 y Fm(")1946 1840 y Fn(\(1)g(+)g Fk(k)j Fn(+)d Fk(n)g Fl(\000)h Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)p 1946 1884 1135 4 v 2261 1976 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!)3091 1761 y Fm(#)3149 1757 y Ff(1)p 3149 1769 31 4 v 3149 1810 a(2)365 2233 y Fl(\002)514 2125 y Fh(l)465 2150 y Fm(X)467 2332 y Fh(i)p Fj(=1)601 2087 y Fm(")660 2165 y Fn(\(1)f(+)h Fk(k)k Fn(+)c Fk(l)i Fl(\000)f Fk(i)p Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(i)p Fn(\)!)p 660 2210 1112 4 v 977 2301 a(\(1)g(+)g Fk(k)j Fn(+)d Fk(l)r Fn(\)!)1781 2087 y Fm(#)1840 2082 y Ff(1)p 1840 2094 31 4 v 1840 2136 a(2)1906 2233 y Fn(+)39 b(2)p Fk(c)p Fn(\(2)22 b(+)g Fk(k)s Fn(\)2)2460 2192 y Fj(3)p Fh(n)2542 2233 y Fk(b)2583 2192 y Fh(n)p Fi(\000)p Fj(1)2737 2233 y Fn([\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])3305 2157 y Ff(1)p 3305 2169 V 3305 2210 a(2)365 2556 y Fl(\002)503 2448 y Fh(n)465 2473 y Fm(X)469 2657 y Fh(l)q Fj(=1)601 2556 y Fn(2)650 2515 y Fi(\000)715 2488 y Ff(5)p Fg(l)p 715 2500 51 4 v 725 2541 a Ff(2)796 2410 y Fm(")854 2489 y Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)h Fl(\000)f Fn(2)p Fk(l)r Fn(\)!\(1)g(+)g Fk(k)j Fn(+)d Fk(n)h Fl(\000)f Fk(l)r Fn(\)!)p 854 2533 1413 4 v 931 2624 a(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)g Fl(\000)h Fn(3)p Fk(l)r Fn(\)!\(1)f(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!)2277 2410 y Fm(#)2335 2406 y Ff(1)p 2335 2418 31 4 v 2335 2459 a(2)0 2793 y Fn(W)-8 b(e)33 b(no)m(w)g(use)h (Lemma)d(2)h(to)g(obtain)598 2976 y Fl(k)p Fk( )711 2991 y Fh(n)758 2976 y Fl(k)27 b(\024)941 2879 y Fm(h)980 2976 y Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))1386 2879 y Fm(\020)1436 2976 y Fk(\015)1492 2935 y Fj(2)1487 3000 y Fh(k)1548 2976 y Fn([\(1)g(+)g Fk(k)s Fn(\)!])1928 2927 y Fi(\000)1993 2900 y Ff(1)p 1993 2912 V 1993 2953 a(2)2059 2976 y Fn(+)g Fk(\014)2218 2879 y Fm(\021i)2323 2976 y Fn(2)2372 2935 y Fj(3)p Fh(n)2454 2976 y Fk(b)2495 2935 y Fh(n)p Fi(\000)p Fj(1)2650 2976 y Fn([\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])3218 2900 y Ff(1)p 3218 2912 V 3218 2953 a(2)835 3166 y Fn(=)28 b(2)988 3125 y Fj(3)p Fh(n)1070 3166 y Fk(b)1111 3125 y Fh(n)1175 3166 y Fn([\(1)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])1744 3091 y Ff(1)p 1744 3103 V 1744 3144 a(2)3632 3349 y Fd(2)195 3553 y Fn(The)41 b(next)f(step)h(is)e(to)h(b)s(ound)f(the)i(error)e (made)g(b)m(y)i(taking)e(a)g(\014nite)h(n)m(um)m(b)s(er)g(of)f(terms)h (in)0 3686 y(b)s(oth)c(series)g(\(8\))f(and)h(\(6\).)53 b(W)-8 b(e)36 b(use)h(the)g(follo)m(wing)c(approac)m(h,)k(whic)m(h)f (follo)m(ws)f(the)h(ideas)g(of)f([3].)0 3818 y(F)-8 b(or)32 b Fk(N)38 b Fl(\025)28 b Fn(1)k(de\014ne)598 4052 y Fk(E)670 4067 y Fh(N)765 4052 y Fn(=)27 b Fk(e)913 4067 y Fh(k)978 4052 y Fn(+)1076 3944 y Fh(N)7 b Fi(\000)p Fj(1)1093 3969 y Fm(X)1087 4150 y Fh(n)p Fj(=1)1246 4052 y Fn(\026)-50 b Fk(h)1311 3982 y Fg(n)p 1311 3994 39 4 v 1315 4035 a Ff(2)1364 4052 y Fl(E)1417 4067 y Fh(n)1779 4052 y Fn(\011)1855 4067 y Fh(N)1922 4052 y Fn(\()p Fk(x)p Fn(\))28 b(=)g Fk(\036)2243 4067 y Fh(k)2285 4052 y Fn(\()p Fk(x)p Fn(\))22 b(+)2536 3944 y Fh(N)7 b Fi(\000)p Fj(1)2553 3969 y Fm(X)2546 4150 y Fh(n)p Fj(=1)2706 4052 y Fn(\026)-50 b Fk(h)2771 3982 y Fg(n)p 2771 3994 V 2775 4035 a Ff(2)2824 4052 y Fk( )2887 4067 y Fh(n)2934 4052 y Fn(\()p Fk(x)p Fn(\))468 b(\(17\))0 4285 y(These)34 b(are)f(the)g(truncation)f(at)g (order)h Fk(N)43 b Fn(of)32 b(the)h(R-S)f(series.)44 b(W)-8 b(e)33 b(de\014ne)598 4468 y Fk(\030)641 4483 y Fh(N)708 4468 y Fn(\()p Fk(x)p Fn(\))28 b(=)f Fk(A)1043 4483 y Fh(k)1086 4468 y Fn(\()p Fk(H)1205 4483 y Fj(0)1266 4468 y Fn(+)22 b Fk(W)14 b Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))22 b Fl(\000)h Fk(E)1894 4483 y Fh(N)1961 4468 y Fn(\))p Fk(A)2072 4483 y Fh(k)2115 4468 y Fn(\011)2191 4483 y Fh(N)2258 4468 y Fn(\()p Fk(x)p Fn(\))867 4701 y(=)987 4530 y Fm(2)987 4679 y(4)1042 4701 y Fk(H)1123 4716 y Fh(k)1188 4701 y Fn(+)f Fk(A)1359 4716 y Fh(k)1402 4701 y Fk(W)14 b Fn(\(\026)-50 b Fk(h)o Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1810 4716 y Fh(k)1875 4701 y Fl(\000)1975 4593 y Fh(N)7 b Fi(\000)p Fj(1)1992 4618 y Fm(X)1990 4800 y Fh(j)t Fj(=1)2145 4701 y Fn(\026)-50 b Fk(h)2211 4626 y Fg(j)p 2210 4643 31 4 v 2210 4684 a Ff(2)2254 4701 y Fl(E)2307 4716 y Fh(j)2343 4701 y Fn(\()p Fk(A)2454 4716 y Fh(k)2497 4701 y Fn(\))2535 4659 y Fj(2)2575 4530 y Fm(3)2575 4679 y(5)2646 4593 y Fh(N)7 b Fi(\000)p Fj(1)2663 4618 y Fm(X)2647 4799 y Fh(m)p Fj(=0)2816 4701 y Fn(\026)-50 b Fk(h)2881 4630 y Fg(m)p 2881 4642 55 4 v 2893 4684 a Ff(2)2950 4701 y Fk( )3013 4716 y Fh(m)3080 4701 y Fn(\()p Fk(x)p Fn(\))322 b(\(18\))0 4952 y(This)63 b(function)g(is)g (the)g(error)g(in)g(the)h(time-indep)s(enden)m(t)e(Sc)m(hr\177)-49 b(odinger)64 b(equation)f(due)g(to)0 5085 y(truncation,)34 b(at)g(order)h Fk(N)10 b Fn(,)35 b(of)f(b)s(oth)g(\(8\))g(and)h(\(6\).) 48 b(There)36 b(is)e(plethora)f(of)h(cancellations)f(in)h(\(18\),)0 5218 y(that)e(are)h(describ)s(ed)g(in)f(the)h(App)s(endix.)44 b(After)33 b(making)e(these)j(cancellations,)d(w)m(e)j(ha)m(v)m(e)0 5457 y Fk(\030)43 5472 y Fh(N)110 5457 y Fn(\()p Fk(x)p Fn(\))28 b(=)372 5349 y Fh(N)7 b Fi(\000)p Fj(1)389 5374 y Fm(X)383 5555 y Fh(n)p Fj(=0)542 5457 y Fn(\026)-50 b Fk(h)607 5387 y Fg(n)p 607 5399 39 4 v 611 5440 a Ff(2)660 5457 y Fk(A)733 5472 y Fh(k)776 5457 y Fk(W)882 5416 y Fj([)p Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)p Fj(])1176 5457 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1479 5472 y Fh(k)1521 5457 y Fk( )1584 5472 y Fh(n)1632 5457 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1885 5349 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)1919 5374 y Fm(X)1899 5558 y Fh(n)p Fj(=)p Fh(N)2090 5457 y Fn(\026)-50 b Fk(h)2155 5387 y Fg(n)p 2155 5399 V 2159 5440 a Ff(2)2311 5349 y Fh(N)7 b Fi(\000)p Fj(1)2328 5374 y Fm(X)2224 5559 y Fh(l)q Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)2568 5457 y Fl(E)2621 5472 y Fh(l)2647 5457 y Fn(\()p Fk(A)2758 5472 y Fh(k)2801 5457 y Fn(\))2839 5416 y Fj(2)2878 5457 y Fk( )2941 5472 y Fh(n)p Fi(\000)p Fh(l)3065 5457 y Fn(\()p Fk(x)p Fn(\))p eop %%Page: 9 9 9 8 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2269 b Fn(9)0 239 y(where)47 b Fk(W)401 203 y Fj([)p Fh(j)t Fj(])476 239 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))50 b(=)g Fk(V)21 b Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))32 b Fl(\000)1330 172 y Fm(P)1418 192 y Fh(j)1418 264 y(l)q Fj(=0)1550 239 y Fn(\026)-50 b Fk(h)1616 167 y Fg(l)p Fe(\000)p Ff(2)p 1616 181 99 4 v 1650 222 a(2)1728 239 y Fk(c)1770 254 y Fh(l)1796 239 y Fk(x)1851 203 y Fh(l)1923 239 y Fn(is)45 b(the)h(T)-8 b(a)m(ylor)45 b(series)i(error)e(\(at)g(order)h Fk(j)6 b Fn(\))46 b(of)0 372 y Fk(V)22 b Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\).)43 b(The)34 b(T)-8 b(a)m(ylor)32 b(theorem)g(sa)m(ys)i(that)598 608 y Fk(W)704 567 y Fj([)p Fh(j)t Fj(])779 608 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))28 b(=)f(\026)-50 b Fk(h)1206 534 y Fg(j)s Fe(\000)p Ff(1)p 1206 550 107 4 v 1244 591 a(2)1336 541 y Fk(V)1415 504 y Fj([)p Fh(j)t Fj(+1])1581 541 y Fn(\()p Fk(\021)t Fn(\))p 1336 585 373 4 v 1363 676 a(\()p Fk(j)29 b Fn(+)22 b(1\)!)1718 608 y Fk(x)1773 567 y Fh(j)t Fj(+1)0 834 y Fn(for)32 b(some)g Fk(\021)g Fl(2)c Fn(\()p Fl(\000j)p Fk(x)p Fl(j)p Fk(;)17 b Fl(j)p Fk(x)p Fl(j)p Fn(\).)43 b(F)-8 b(rom)31 b(this,)h(w)m(e)i(obtain)0 1079 y Fk(\030)43 1094 y Fh(N)110 1079 y Fn(\()p Fk(x)p Fn(\))28 b(=)f(\026)-50 b Fk(h)437 1009 y Fg(N)p 437 1021 54 4 v 449 1062 a Ff(2)522 971 y Fh(N)7 b Fi(\000)p Fj(1)538 996 y Fm(X)532 1177 y Fh(n)p Fj(=0)692 1079 y Fk(A)765 1094 y Fh(k)817 1011 y Fk(V)896 975 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1190 1011 y Fn(\()p Fk(\021)1276 1026 y Fh(n)1323 1011 y Fn(\))p 817 1056 544 4 v 819 1147 a(\()p Fk(N)33 b Fn(+)22 b(2)g Fl(\000)g Fk(n)p Fn(\)!)1371 1079 y Fk(x)1426 1038 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)1682 1079 y Fk(A)1755 1094 y Fh(k)1797 1079 y Fk( )1860 1094 y Fh(n)1908 1079 y Fn(\()p Fk(x)p Fn(\))598 1380 y Fl(\000)708 1272 y Fj(2)p Fh(N)g Fi(\000)p Fj(2)743 1297 y Fm(X)722 1481 y Fh(n)p Fj(=)p Fh(N)914 1380 y Fn(\026)-50 b Fk(h)979 1310 y Fg(n)p 979 1322 39 4 v 983 1363 a Ff(2)1135 1272 y Fh(N)7 b Fi(\000)p Fj(1)1152 1297 y Fm(X)1048 1482 y Fh(l)q Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)1392 1380 y Fl(E)1445 1395 y Fh(l)1471 1380 y Fn(\()p Fk(A)1582 1395 y Fh(k)1624 1380 y Fn(\))1662 1339 y Fj(2)1702 1380 y Fk( )1765 1395 y Fh(n)p Fi(\000)p Fh(l)1889 1380 y Fn(\()p Fk(x)p Fn(\))1513 b(\(19\))0 1630 y(with)32 b Fk(\021)270 1645 y Fh(n)345 1630 y Fl(2)c Fn(\()p Fl(\000j)p Fk(x)p Fl(j)p Fk(;)17 b Fl(j)p Fk(x)p Fl(j)p Fn(\))p Fk(;)50 b(n)27 b Fn(=)h(0)p Fk(;)17 b(:)g(:)g(:)f(;)h(N)32 b Fl(\000)23 b Fn(1.)195 1762 y(Our)28 b(main)f(result)h(relies)f(on)h (an)g(estimate)f(of)h Fl(k)p Fn(\()p Fk(H)20 b Fl(\000)13 b Fk(E)2249 1777 y Fh(N)2317 1762 y Fn(\))p Fk(A)2428 1777 y Fh(k)2471 1762 y Fn(\011)2547 1777 y Fh(N)2614 1762 y Fn(\()p Fk(x)p Fn(\))p Fl(k)p Fn(,)29 b(so)f(w)m(e)h(need)h(an)e (upp)s(er)0 1895 y(b)s(ound)j(of)g(the)g Fk(L)641 1859 y Fj(2)681 1895 y Fn(-norm)f(of)h Fk(A)1150 1854 y Fi(\000)p Fj(1)1150 1920 y Fh(k)1244 1895 y Fk(\030)1287 1910 y Fh(N)1354 1895 y Fn(\()p Fk(x)p Fn(\).)43 b(Note)31 b(that)g Fk(A)2072 1854 y Fi(\000)p Fj(1)2072 1920 y Fh(k)2198 1895 y Fn(is)f(an)i(un)m(b)s(ounded)g(op)s(erator,)f(but)g(that)0 2028 y Fk(\030)43 2043 y Fh(N)110 2028 y Fn(\()p Fk(x)p Fn(\))i(is)f(clearly)f(in)h(its)g(domain.)0 2232 y Fw(Prop)s(osition)k (1.)43 b Fn(F)-8 b(or)32 b(eac)m(h)h Fk(k)e Fl(\025)d Fn(0,)33 b(there)g(exist)g(p)s(ositiv)m(e)f(constan)m(ts)i Fk(A)p Fn(,)e Fk(B)38 b Fn(and)33 b Fk(N)3211 2247 y Fj(0)3283 2232 y Fn(suc)m(h)h(that)598 2371 y Fm(\015)598 2421 y(\015)598 2471 y(\015)p Fk(A)717 2430 y Fi(\000)p Fj(1)717 2496 y Fh(k)811 2471 y Fk(\030)854 2486 y Fh(N)921 2471 y Fn(\()p Fk(x)p Fn(\))1052 2371 y Fm(\015)1052 2421 y(\015)1052 2471 y(\015)28 b Fl(\024)1263 2363 y Fj(2)p Fh(N)1252 2388 y Fm(X)1231 2572 y Fh(n)p Fj(=)p Fh(N)1409 2471 y Fk(AB)1561 2430 y Fh(n)1608 2471 y Fn(\026)-50 b Fk(h)1673 2401 y Fg(n)p 1673 2413 V 1677 2454 a Ff(2)1742 2471 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])2311 2395 y Ff(1)p 2311 2407 31 4 v 2311 2449 a(2)0 2707 y Fn(whenev)m(er)35 b Fk(N)j Fl(\025)28 b Fk(N)727 2722 y Fj(0)800 2707 y Fn(and)k(\026)-50 b Fk(h)28 b Fl(\024)g Fn(1.)0 2911 y Fw(Pro)s(of:)70 b Fn(W)-8 b(e)46 b(separately)g(pro)m(v) m(e)h(estimates)f(inside)f(and)h(outside)f(a)h(closed)g(in)m(terv)-5 b(al)44 b([)p Fl(\000)p Fk(R)q(;)17 b(R)q Fn(],)0 3044 y(where)35 b Fk(R)c Fn(=)494 2963 y Fl(p)p 577 2963 530 4 v 81 x Fn(2)p Fk(k)25 b Fn(+)d(6)p Fk(N)32 b Fn(+)23 b(3)o(.)48 b(Let)34 b Fk(\037)1418 3059 y Fh(R)1510 3044 y Fn(b)s(e)g(the)g(c)m(haracteristic)g(function)f(de\014ned)j(b)m(y)e Fk(\037)3332 3059 y Fh(R)3390 3044 y Fn(\()p Fk(x)p Fn(\))c(=)g(1)0 3177 y(if)h Fk(x)e Fl(2)f Fn([)p Fl(\000)p Fk(R)q(;)17 b(R)q Fn(],)33 b(and)f Fk(\037)902 3192 y Fh(R)960 3177 y Fn(\()p Fk(x)p Fn(\))c(=)g(0)k(otherwise.)44 b(F)-8 b(rom)31 b(\(19\))h(it)g(follo)m(ws)f(that)0 3322 y Fm(\015)0 3371 y(\015)0 3421 y(\015)p Fk(A)119 3380 y Fi(\000)p Fj(1)119 3446 y Fh(k)214 3421 y Fk(\030)257 3436 y Fh(N)323 3421 y Fn(\()p Fk(x)p Fn(\))454 3322 y Fm(\015)454 3371 y(\015)454 3421 y(\015)45 b Fl(\024)28 b Fn(\026)-50 b Fk(h)715 3351 y Fg(N)p 715 3363 54 4 v 727 3404 a Ff(2)800 3313 y Fh(N)7 b Fi(\000)p Fj(1)816 3338 y Fm(X)810 3519 y Fh(n)p Fj(=0)969 3272 y Fm(\015)969 3322 y(\015)969 3371 y(\015)969 3421 y(\015)969 3471 y(\015)1026 3354 y Fk(V)1104 3318 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1398 3354 y Fn(\()p Fk(\021)1484 3369 y Fh(n)1531 3354 y Fn(\))p 1026 3398 544 4 v 1028 3490 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)g Fk(n)p Fn(\)!)1579 3421 y Fk(x)1634 3380 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)1890 3421 y Fk(A)1963 3436 y Fh(k)2006 3421 y Fk( )2069 3436 y Fh(n)2116 3421 y Fn(\()p Fk(x)p Fn(\))2247 3272 y Fm(\015)2247 3322 y(\015)2247 3371 y(\015)2247 3421 y(\015)2247 3471 y(\015)672 3723 y Fn(+)787 3615 y Fj(2)p Fh(N)g Fi(\000)p Fj(2)821 3639 y Fm(X)801 3823 y Fh(n)p Fj(=)p Fh(N)992 3723 y Fn(\026)-50 b Fk(h)1057 3652 y Fg(n)p 1057 3664 39 4 v 1061 3706 a Ff(2)1214 3615 y Fh(N)7 b Fi(\000)p Fj(1)1230 3639 y Fm(X)1126 3824 y Fh(l)q Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)1471 3723 y Fl(j)o(E)1551 3738 y Fh(l)1577 3723 y Fl(j)16 b(k)p Fk( )1734 3738 y Fh(n)p Fi(\000)p Fh(l)1858 3723 y Fn(\()p Fk(x)p Fn(\))p Fl(k)545 4034 y(\024)28 b Fn(\026)-50 b Fk(h)715 3964 y Fg(N)p 715 3976 54 4 v 727 4017 a Ff(2)800 3926 y Fh(N)7 b Fi(\000)p Fj(1)816 3951 y Fm(X)810 4132 y Fh(n)p Fj(=0)969 3885 y Fm(\015)969 3934 y(\015)969 3984 y(\015)969 4034 y(\015)969 4084 y(\015)1026 3967 y Fk(V)1104 3930 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1398 3967 y Fn(\()p Fk(\021)1484 3982 y Fh(n)1531 3967 y Fn(\))p 1026 4011 544 4 v 1028 4102 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)g Fk(n)p Fn(\)!)1579 4034 y Fk(x)1634 3993 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)1907 4034 y Fn(\()o(1)22 b Fl(\000)h Fk(\037)2176 4049 y Fh(R)2234 4034 y Fn(\()p Fk(x)p Fn(\)\))17 b Fk(A)2493 4049 y Fh(k)2535 4034 y Fk( )2598 4049 y Fh(n)2646 4034 y Fn(\()p Fk(x)p Fn(\))2777 3885 y Fm(\015)2777 3934 y(\015)2777 3984 y(\015)2777 4034 y(\015)2777 4084 y(\015)672 4335 y Fn(+)39 b(\026)-50 b Fk(h)852 4265 y Fg(N)p 852 4277 54 4 v 864 4318 a Ff(2)936 4227 y Fh(N)7 b Fi(\000)p Fj(1)953 4252 y Fm(X)947 4433 y Fh(n)p Fj(=0)1106 4186 y Fm(\015)1106 4236 y(\015)1106 4285 y(\015)1106 4335 y(\015)1106 4385 y(\015)1162 4268 y Fk(V)1241 4232 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1535 4268 y Fn(\()p Fk(\021)1621 4283 y Fh(n)1668 4268 y Fn(\))p 1162 4312 544 4 v 1164 4404 a(\()p Fk(N)33 b Fn(+)22 b(2)g Fl(\000)g Fk(n)p Fn(\)!)1716 4335 y Fk(x)1771 4294 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)2027 4335 y Fk(\037)2088 4350 y Fh(R)2146 4335 y Fn(\()p Fk(x)p Fn(\))p Fk(A)2350 4350 y Fh(k)2393 4335 y Fk( )2456 4350 y Fh(n)2503 4335 y Fn(\()p Fk(x)p Fn(\))2634 4186 y Fm(\015)2634 4236 y(\015)2634 4285 y(\015)2634 4335 y(\015)2634 4385 y(\015)672 4637 y Fn(+)787 4529 y Fj(2)p Fh(N)g Fi(\000)p Fj(2)821 4554 y Fm(X)801 4737 y Fh(n)p Fj(=)p Fh(N)992 4637 y Fn(\026)-50 b Fk(h)1057 4567 y Fg(n)p 1057 4579 39 4 v 1061 4620 a Ff(2)1214 4529 y Fh(N)7 b Fi(\000)p Fj(1)1230 4554 y Fm(X)1126 4738 y Fh(l)q Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)1471 4637 y Fl(j)o(E)1551 4652 y Fh(l)1577 4637 y Fl(j)16 b(k)p Fk( )1734 4652 y Fh(n)p Fi(\000)p Fh(l)1858 4637 y Fn(\()p Fk(x)p Fn(\))p Fl(k)1494 b Fn(\(20\))0 4915 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(used)g(that)e(\026)-50 b Fk(h)28 b Fl(\024)g Fn(1)33 b(implies)d(\026)-50 b Fk(h)1750 4845 y Fg(n)p 1750 4857 V 1754 4899 a Ff(2)1830 4915 y Fl(\024)28 b Fn(\026)-50 b Fk(h)2000 4845 y Fg(N)p 2000 4857 54 4 v 2012 4899 a Ff(2)2100 4915 y Fn(for)32 b Fk(n)c Fl(\025)h Fk(N)10 b Fn(.)195 5048 y(T)-8 b(o)30 b(estimate)f(the)i(\014rst)f(term)g(of)f(\(20\))h(w)m(e)h(need)g(to)e (con)m(trol)h(the)g(T)-8 b(a)m(ylor)30 b(series)g(error)g(outside)0 5181 y([)p Fl(\000)p Fk(R)q(;)17 b(R)q Fn(].)51 b(An)36 b(easy)g(argumen)m(t)f(using)f(the)i(Cauc)m(h)m(y)h(in)m(tegral)c(form) m(ula)g(\(and)i(the)h(h)m(yp)s(othesis)g(on)0 5314 y Fk(V)22 b Fn(;)32 b(see)i([2,)e(3]\))h(sho)m(ws)h(that)e(for)g Fk(\021)g Fl(2)c Fn(\()p Fl(\000j)p Fk(x)p Fl(j)p Fk(;)17 b Fl(j)p Fk(x)p Fl(j)p Fn(\))32 b(and)h(for)f(all)e Fk(n)e Fl(\025)g Fn(0,)0 5583 y Fk(\016)47 5542 y Fh(n)104 5397 y Fm(\014)104 5447 y(\014)104 5497 y(\014)p Fk(V)210 5460 y Fj([)p Fh(n)p Fj(])296 5497 y Fn(\()p Fk(\021)t Fn(\))424 5397 y Fm(\014)424 5447 y(\014)424 5497 y(\014)p 104 5560 348 4 v 235 5651 a Fk(n)p Fn(!)489 5583 y Fl(\024)g Fk(M)g Fn(exp)881 5487 y Fm(h)921 5583 y Fk(\034)11 b Fn(\()p Fl(j)p Fk(x)p Fl(j)22 b Fn(+)g Fk(\016)t Fn(\))1328 5542 y Fj(2)1367 5487 y Fm(i)1434 5583 y Fl(\024)28 b Fk(M)g Fn(exp)q(\(2)p Fk(\034)11 b(\016)1997 5542 y Fj(2)2036 5583 y Fn(\))17 b(exp)q(\(2)p Fk(\034)11 b(x)2435 5542 y Fj(2)2475 5583 y Fn(\))28 b(=:)f Fk(K)d Fn(exp)q(\(2)p Fk(\034)11 b(x)3122 5542 y Fj(2)3162 5583 y Fn(\))333 b(\(21\))p eop %%Page: 10 10 10 9 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(10)0 232 y(Later)27 b(w)m(e)h(will)d(assume)i(that)g Fk(\016)32 b Fl(\024)c Fn(1.)41 b(Also,)28 b(w)m(e)g(recall)d(that)i(the)h(eigenfunctions)e (of)h(the)g(harmonic)0 365 y(oscillator)34 b Fk(H)503 380 y Fj(0)580 365 y Fn(are)j Fk(\036)805 380 y Fh(k)847 365 y Fn(\()p Fk(x)p Fn(\))f(=)f Fk(\031)1184 329 y Fi(\000)1249 301 y Ff(1)p 1249 313 31 4 v 1249 355 a(4)1293 365 y Fn(\(2)1380 329 y Fh(k)1422 365 y Fk(k)s Fn(!\))1541 329 y Fi(\000)1606 301 y Ff(1)p 1606 313 V 1606 355 a(2)1668 365 y Fn(exp)q(\()p Fl(\000)1942 326 y Fj(1)p 1942 342 36 4 v 1942 400 a(2)1987 365 y Fk(x)2042 329 y Fj(2)2082 365 y Fn(\))p Fk(H)2201 380 y Fh(n)2248 365 y Fn(\()p Fk(x)p Fn(\),)j(where)h Fk(H)2812 380 y Fh(k)2854 365 y Fn(\()p Fk(x)p Fn(\))e(are)h(the)f(Hermite)0 498 y(p)s(olynomials.) 195 631 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)d(estimate)0 835 y Fw(Lemma)37 b(3.)44 b Fn(F)-8 b(or)31 b Fk(n)d Fn(=)g(0)p Fk(;)17 b Fn(1)p Fk(;)g(:)g(:)g(:)e(;)i(N)32 b Fl(\000)23 b Fn(1)32 b(and)h(for)f Fk(R)d Fn(=)2088 754 y Fl(p)p 2171 754 530 4 v 81 x Fn(2)p Fk(k)c Fn(+)d(6)p Fk(N)33 b Fn(+)22 b(3)o(,)0 916 y Fm(\015)0 966 y(\015)0 1016 y(\015)0 1066 y(\015)0 1115 y(\015)46 1066 y Fk(\016)93 1024 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)358 998 y Fk(V)437 962 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])731 998 y Fn(\()p Fk(\021)817 1013 y Fh(n)864 998 y Fn(\))p 358 1042 544 4 v 360 1134 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)912 1066 y Fk(x)967 1024 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)1239 1066 y Fn([1)22 b Fl(\000)h Fk(\037)1498 1081 y Fh(R)1556 1066 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(P)1794 1081 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2147 916 y Fm(\015)2147 966 y(\015)2147 1016 y(\015)2147 1066 y(\015)2147 1115 y(\015)493 1366 y Fl(\024)631 1298 y Fk(K)7 b Fn(2)780 1233 y Ff(2+)p Fg(k)q Ff(+3)p Fg(n)p 779 1247 227 4 v 877 1287 a Ff(2)1020 1298 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)d Fk(N)33 b Fn(+)22 b(2)p Fk(n)p Fn(\)!])1846 1234 y Ff(1)p 1846 1246 31 4 v 1846 1287 a(2)p 608 1342 1306 4 v 608 1452 a Fk(\031)677 1388 y Ff(1)p 677 1400 31 4 v 677 1441 a(4)721 1452 y Fn(\(1)g Fl(\000)h Fn(4)p Fk(\034)11 b Fn(\))1080 1387 y Ff(3+)p Fg(k)q Ff(+)p Fg(N)5 b Ff(+2)p Fg(n)p 1080 1401 327 4 v 1228 1441 a Ff(2)1420 1452 y Fn([\()p Fk(k)26 b Fn(+)c(3)p Fk(n)p Fn(\)!])1869 1388 y Ff(1)p 1869 1400 31 4 v 1869 1441 a(2)0 1681 y Fw(Pro)s(of)37 b(of)h(Lemma)f(3:)43 b Fn(Consider)33 b(an)m(y)h Fk(\036)1614 1696 y Fh(i)1642 1681 y Fn(\()p Fk(x)p Fn(\))e(with)h Fk(i)28 b Fn(=)f(0)p Fk(;)17 b Fn(1)p Fk(;)g(:)g(:)g(:)e(;)i(k)25 b Fn(+)d(3)p Fk(n)g Fn(+)g(1.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)0 1784 y Fm(\015)0 1834 y(\015)0 1884 y(\015)0 1934 y(\015)0 1984 y(\015)46 1934 y Fk(\016)93 1893 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)358 1866 y Fk(V)437 1830 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])731 1866 y Fn(\()p Fk(\021)817 1881 y Fh(n)864 1866 y Fn(\))p 358 1910 544 4 v 360 2002 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)912 1934 y Fk(x)967 1893 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)1239 1934 y Fn([1)22 b Fl(\000)h Fk(\037)1498 1949 y Fh(R)1556 1934 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(\036)1789 1949 y Fh(i)1816 1934 y Fn(\()p Fk(x)p Fn(\))1947 1784 y Fm(\015)1947 1834 y(\015)1947 1884 y(\015)1947 1934 y(\015)1947 1984 y(\015)1994 1807 y Fj(2)493 2184 y Fl(\024)28 b Fk(K)688 2143 y Fj(2)744 2067 y Fm(Z)790 2255 y Fh(I)-18 b(R)p Fi(n)p Fj([)p Fi(\000)p Fh(R;R)p Fj(])1133 2184 y Fk(e)1178 2143 y Fj(4)p Fh(\034)8 b(x)1292 2119 y Ff(2)1331 2184 y Fk(x)1386 2143 y Fj(2\()p Fh(N)f Fj(+1)p Fi(\000)p Fh(n)p Fj(\))1748 2184 y Fl(j)p Fk(\036)1834 2199 y Fh(i)1862 2184 y Fn(\()p Fk(x)p Fn(\))p Fl(j)2021 2135 y Fj(2)2077 2184 y Fk(dx)493 2451 y Fl(\024)644 2384 y Fn(2)p Fk(K)783 2348 y Fj(2)p 608 2428 251 4 v 608 2534 a Fk(\031)677 2470 y Ff(1)p 677 2482 31 4 v 677 2523 a(2)721 2534 y Fn(2)770 2505 y Fh(i)798 2534 y Fk(i)p Fn(!)885 2334 y Fm(Z)968 2360 y Fi(1)931 2523 y Fh(R)1043 2451 y Fk(e)1088 2410 y Fi(\000)p Fj(\(1)p Fi(\000)p Fj(4)p Fh(\034)h Fj(\))p Fh(x)1401 2387 y Ff(2)1424 2451 y Fk(x)1479 2410 y Fj(2\()p Fh(N)f Fj(+1)p Fi(\000)p Fh(n)p Fj(\))1841 2451 y Fl(j)p Fk(H)1950 2466 y Fh(i)1978 2451 y Fn(\()p Fk(x)p Fn(\))p Fl(j)2137 2403 y Fj(2)2193 2451 y Fk(dx)0 2679 y Fn(Because)28 b Fk(R)h Fl(\025)575 2600 y(p)p 658 2600 251 4 v 79 x Fn(2)p Fk(i)22 b Fn(+)g(1)k(w)m(e)h (ma)m(y)f(mak)m(e)h(use)g(of)f(Lemma)e(3.1)i(of)g([2])g(to)g(estimate)f (the)i(last)f(in)m(tegral.)0 2812 y(This)33 b(lemma)d(states)k(that)e Fl(j)p Fk(H)1135 2827 y Fh(i)1163 2812 y Fn(\()p Fk(x)p Fn(\))p Fl(j)27 b(\024)h Fn(2)1503 2776 y Fh(i)1531 2812 y Fl(j)p Fk(x)p Fl(j)1642 2776 y Fh(i)1703 2812 y Fn(whenev)m(er)35 b Fk(x)28 b Fl(\025)2319 2733 y(p)p 2402 2733 V 79 x Fn(2)p Fk(i)22 b Fn(+)g(1.)44 b(Th)m(us,)0 2915 y Fm(\015)0 2965 y(\015)0 3015 y(\015)0 3065 y(\015)0 3114 y(\015)46 3065 y Fk(\016)93 3024 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)358 2997 y Fk(V)437 2961 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])731 2997 y Fn(\()p Fk(\021)817 3012 y Fh(n)864 2997 y Fn(\))p 358 3041 544 4 v 360 3133 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)912 3065 y Fk(x)967 3024 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)1239 3065 y Fn([1)22 b Fl(\000)h Fk(\037)1498 3080 y Fh(R)1556 3065 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(\036)1789 3080 y Fh(i)1816 3065 y Fn(\()p Fk(x)p Fn(\))1947 2915 y Fm(\015)1947 2965 y(\015)1947 3015 y(\015)1947 3065 y(\015)1947 3114 y(\015)1994 2938 y Fj(2)493 3345 y Fl(\024)608 3278 y Fn(2)657 3241 y Fh(i)p Fj(+1)775 3278 y Fk(K)865 3241 y Fj(2)p 608 3322 297 4 v 669 3428 a Fk(\031)738 3364 y Ff(1)p 738 3376 31 4 v 738 3417 a(2)783 3428 y Fk(i)p Fn(!)931 3228 y Fm(Z)1014 3254 y Fi(1)977 3416 y Fh(R)1105 3345 y Fk(e)1150 3304 y Fi(\000)p Fj(\(1)p Fi(\000)p Fj(4)p Fh(\034)8 b Fj(\))p Fh(x)1463 3280 y Ff(2)1503 3345 y Fk(x)1558 3304 y Fj(2\()p Fh(N)f Fj(+1)p Fi(\000)p Fh(n)p Fj(+)p Fh(i)p Fj(\))1983 3345 y Fk(dx)493 3608 y Fl(\024)608 3541 y Fn(2)657 3505 y Fh(i)p Fj(+1)775 3541 y Fk(K)865 3505 y Fj(2)p 608 3585 297 4 v 669 3691 a Fk(\031)738 3627 y Ff(1)p 738 3639 31 4 v 738 3680 a(2)783 3691 y Fk(i)p Fn(!)931 3491 y Fm(Z)1014 3518 y Fi(1)977 3680 y Fj(0)1105 3608 y Fk(e)1150 3567 y Fi(\000)p Fj(\(1)p Fi(\000)p Fj(4)p Fh(\034)h Fj(\))p Fh(x)1463 3544 y Ff(2)1503 3608 y Fk(x)1558 3567 y Fj(2\()p Fh(N)f Fj(+1)p Fi(\000)p Fh(n)p Fj(+)p Fh(i)p Fj(\))1983 3608 y Fk(dx)494 3910 y Fn(=)608 3824 y(2)657 3788 y Fh(i)685 3824 y Fk(K)775 3788 y Fj(2)814 3824 y Fn(\000)892 3728 y Fm(\020)941 3824 y Fk(N)33 b Fn(+)22 b(1)g Fl(\000)h Fk(n)f Fn(+)g Fk(i)g Fn(+)1662 3785 y Fj(1)p 1662 3801 36 4 v 1662 3858 a(2)1708 3728 y Fm(\021)p 608 3887 1150 4 v 702 3993 a Fk(\031)771 3929 y Ff(1)p 770 3941 31 4 v 770 3982 a(2)815 3993 y Fk(i)p Fn(!\(1)g Fl(\000)h Fn(4)p Fk(\034)11 b Fn(\))1224 3956 y Fh(N)c Fj(+1)p Fi(\000)p Fh(n)p Fj(+)p Fh(i)p Fj(+)1619 3929 y Ff(1)p 1619 3941 V 1619 3982 a(2)493 4199 y Fl(\024)638 4131 y Fn(2)687 4095 y Fh(i)715 4131 y Fk(K)805 4095 y Fj(2)845 4131 y Fn(\()p Fk(N)33 b Fn(+)22 b(1)f Fl(\000)i Fk(n)f Fn(+)g Fk(i)p Fn(\)!)p 608 4175 962 4 v 608 4281 a Fk(\031)677 4217 y Ff(1)p 677 4229 31 4 v 677 4271 a(2)721 4281 y Fk(i)p Fn(!\(1)g Fl(\000)h Fn(4)p Fk(\034)11 b Fn(\))1130 4245 y Fh(N)c Fj(+1)p Fi(\000)p Fh(n)p Fj(+)p Fh(i)p Fj(+)1525 4217 y Ff(1)p 1525 4229 V 1525 4271 a(2)0 4439 y Fn(T)-8 b(o)30 b(obtain)e(last)h(inequalit)m(y)g(here,)i(w)m(e)f(ha)m (v)m(e)h(used)g(the)f(fact)g(that)f(\000\()p Fk(x)p Fn(\))h(is)f(an)h (increasing)f(function)0 4572 y(when)34 b Fk(x)28 b Fl(\025)g Fn(2)k(.)44 b(Using)32 b(the)h(Sc)m(h)m(w)m(arz)i(inequalit)m(y)-8 b(,)31 b(w)m(e)j(then)f(conclude)g(that)0 4653 y Fm(\015)0 4703 y(\015)0 4753 y(\015)0 4803 y(\015)0 4853 y(\015)46 4803 y Fk(\016)93 4762 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)358 4735 y Fk(V)437 4699 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])731 4735 y Fn(\()p Fk(\021)817 4750 y Fh(n)864 4735 y Fn(\))p 358 4780 544 4 v 360 4871 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)912 4803 y Fk(x)967 4762 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)1239 4803 y Fn([1)22 b Fl(\000)h Fk(\037)1498 4818 y Fh(R)1556 4803 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(P)1794 4818 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2147 4653 y Fm(\015)2147 4703 y(\015)2147 4753 y(\015)2147 4803 y(\015)2147 4853 y(\015)493 5131 y Fl(\024)614 4985 y Fm(")663 5023 y Fj(1+)p Fh(k)r Fj(+3)p Fh(n)734 5048 y Fm(X)737 5230 y Fh(i)p Fj(=0)951 5064 y Fn(2)1000 5027 y Fh(i)1028 5064 y Fk(K)1118 5027 y Fj(2)1158 5064 y Fn(\()p Fk(N)32 b Fn(+)22 b(1)g Fl(\000)h Fk(n)f Fn(+)g Fk(i)p Fn(\)!)p 951 5108 901 4 v 973 5214 a Fk(\031)1042 5150 y Ff(1)p 1042 5162 31 4 v 1042 5203 a(2)1086 5214 y Fk(i)p Fn(!\(1)h Fl(\000)f Fn(4)p Fk(\034)11 b Fn(\))1495 5185 y Fh(N)c Fi(\000)p Fh(n)p Fj(+)p Fh(i)p Fj(+1)1861 4985 y Fm(#)1920 4968 y Ff(1)p 1920 4980 V 1920 5021 a(2)0 5366 y Fn(Note)33 b(that)f(the)h(terms)f(on)h(the)g(righ)m(t)e(hand)i(side)f(are)h (increasing)f(in)f Fk(i)p Fn(.)44 b(The)34 b(sum)e(is)g(b)s(ounded)h(b) m(y)0 5499 y(the)g(n)m(um)m(b)s(er)g(of)f(terms)g(times)f(the)i(\(2)22 b(+)f Fk(k)k Fn(+)d(3)p Fk(n)p Fn(\))32 b(times)f(the)i(largest)f (term.)43 b(The)33 b(lemma)d(follo)m(ws)0 5632 y(b)s(ecause)k(\(2)22 b(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\))p Fk(=)p Fn(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\))27 b Fl(\024)i Fn(2.)1961 b Fd(2)p eop %%Page: 11 11 11 10 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(11)195 240 y(F)-8 b(rom)46 b(the)h(pro)s(of)f(of)h(Lemma)f(1)h(w)m(e)h(kno)m(w)g(that)f Fl(k)p Fk(xA)2333 255 y Fh(k)2376 240 y Fl(k)52 b(\024)2608 139 y Fm(q)p 2691 139 348 4 v 101 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\).)87 b(Also,)50 b(w)m(e)e(are)0 372 y(assuming)32 b(that)g Fk(\016)g Fl(\024)c Fn(1.)43 b(Th)m(us,)34 b(the)f(\014rst)g (term)f(of)h(\(20\))f(satis\014es)0 546 y(1)49 510 y Fj(st)141 546 y Fn(term)27 b Fl(\024)h Fn(\026)-50 b Fk(h)539 476 y Fg(N)p 539 488 54 4 v 551 529 a Ff(2)607 546 y Fk(\016)654 505 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)496 755 y Fl(\002)596 647 y Fh(N)g Fi(\000)p Fj(1)613 672 y Fm(X)606 853 y Fh(n)p Fj(=0)766 605 y Fm(\015)766 655 y(\015)766 705 y(\015)766 755 y(\015)766 804 y(\015)812 755 y Fk(\016)859 713 y Fh(N)g Fj(+2)p Fi(\000)p Fh(n)1124 687 y Fk(V)1202 651 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1497 687 y Fn(\()p Fk(\021)1583 702 y Fh(n)1630 687 y Fn(\))p 1124 731 544 4 v 1126 823 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)1678 755 y Fk(x)1733 713 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)2005 755 y Fn([1)f Fl(\000)g Fk(\037)2231 770 y Fh(R)2288 755 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(P)2526 770 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2879 605 y Fm(\015)2879 655 y(\015)2879 705 y(\015)2879 755 y(\015)2879 804 y(\015)2925 755 y Fl(k)p Fk(xA)3103 770 y Fh(k)3146 755 y Fl(k)g(k)o Fk( )3325 770 y Fh(n)3373 755 y Fl(k)369 1006 y(\024)474 901 y Fm(q)p 557 901 348 4 v 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\)\026)-50 b Fk(h)970 936 y Fg(N)p 970 948 54 4 v 981 989 a Ff(2)1037 1006 y Fk(\016)1084 965 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)496 1242 y Fl(\002)596 1134 y Fh(N)g Fi(\000)p Fj(1)613 1159 y Fm(X)606 1340 y Fh(n)p Fj(=0)766 1092 y Fm(\015)766 1142 y(\015)766 1192 y(\015)766 1242 y(\015)766 1292 y(\015)812 1242 y Fk(\016)859 1201 y Fh(N)g Fj(+2)p Fi(\000)p Fh(n)1124 1174 y Fk(V)1202 1138 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1497 1174 y Fn(\()p Fk(\021)1583 1189 y Fh(n)1630 1174 y Fn(\))p 1124 1219 544 4 v 1126 1310 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)1678 1242 y Fk(x)1733 1201 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)2005 1242 y Fn([1)22 b Fl(\000)g Fk(\037)2263 1257 y Fh(R)2321 1242 y Fn(\()p Fk(x)p Fn(\)])17 b Fk(P)2559 1257 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2912 1092 y Fm(\015)2912 1142 y(\015)2912 1192 y(\015)2912 1242 y(\015)2912 1292 y(\015)2975 1242 y Fl(k)p Fk( )3088 1257 y Fh(n)3135 1242 y Fl(k)0 1472 y Fn(W)-8 b(e)25 b(use)h(Lemma)d(3,)j(the)f (estimate)f(of)g Fl(k)p Fk( )1522 1487 y Fh(n)1570 1472 y Fn(\()p Fk(x)p Fn(\))p Fl(k)g Fn(of)h(Theorem)g(1,)h(and)f(arrange)f (factorials)f(to)h(obtain)0 1658 y(1)49 1622 y Fj(st)141 1658 y Fn(term)j Fl(\024)474 1553 y Fm(q)p 557 1553 348 4 v 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\)2)964 1588 y Ff(2+)p Fg(k)p 963 1602 111 4 v 1004 1643 a Ff(2)1089 1658 y Fn(\026)-50 b Fk(h)1154 1588 y Fg(N)p 1154 1600 54 4 v 1165 1641 a Ff(2)1221 1658 y Fk(\016)1268 1617 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1481 1658 y Fk(K)g(\031)1630 1617 y Fi(\000)1695 1590 y Ff(1)p 1694 1602 31 4 v 1694 1643 a(4)1739 1658 y Fn(\(1)22 b Fl(\000)h Fn(4)p Fk(\034)11 b Fn(\))2088 1617 y Fi(\000)2153 1588 y Fg(k)q Ff(+3)p Fg(N)5 b Fe(\000)p Ff(1)p 2153 1602 243 4 v 2259 1643 a(2)2410 1658 y Fk(b)2451 1617 y Fh(N)i Fi(\000)p Fj(1)496 1915 y Fl(\002)91 b Fn(max)596 1974 y Fj(0)p Fi(\024)p Fh(n)p Fi(\024)p Fh(N)-5 b Fi(\000)-12 b Fj(1)930 1769 y Fm(")988 1847 y Fn(\(1)22 b(+)g Fk(k)k Fn(+)c(3)p Fk(n)p Fn(\)\(2)f(+)h Fk(k)k Fn(+)c(3)p Fk(n)p Fn(\))p 988 1892 1053 4 v 1314 1983 a(2)g(+)g Fk(k)j Fn(+)d Fk(n)2050 1769 y Fm(#)2109 1765 y Ff(1)p 2109 1777 31 4 v 2109 1818 a(2)496 2230 y Fl(\002)596 2122 y Fh(N)7 b Fi(\000)p Fj(1)613 2147 y Fm(X)606 2328 y Fh(n)p Fj(=0)766 2230 y Fn(2)825 2162 y Ff(9)p Fg(n)p 824 2174 69 4 v 844 2215 a Ff(2)907 2084 y Fm(")966 2163 y Fn(\(2)22 b(+)g Fk(k)j Fn(+)d Fk(N)32 b Fn(+)22 b(2)p Fk(n)p Fn(\)!\(2)g(+)g Fk(k)k Fn(+)c Fk(n)p Fn(\)!)p 966 2207 1267 4 v 1322 2298 a(\(2)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\)!)2242 2084 y Fm(#)2300 2080 y Ff(1)p 2300 2092 31 4 v 2300 2133 a(2)0 2455 y Fn(Because)34 b(2)22 b(+)g Fk(k)j Fn(+)d(3)p Fk(n)28 b Fl(\024)g Fn(6)22 b(+)g(3)p Fk(k)j Fn(+)d(3)p Fk(n)33 b Fn(for)f(all)e Fk(k)h Fl(\025)d Fn(0)k(and)h Fk(n)28 b Fl(\025)g Fn(0,)k(w)m(e)i(ha)m(v)m(e)678 2706 y(max)598 2765 y Fj(0)p Fi(\024)p Fh(n)p Fi(\024)p Fh(N)7 b Fi(\000)p Fj(1)955 2560 y Fm(")1014 2639 y Fn(\(1)22 b(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\)\(2)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\))p 1014 2683 1053 4 v 1339 2774 a(2)g(+)g Fk(k)j Fn(+)d Fk(n)2076 2560 y Fm(#)2134 2556 y Ff(1)p 2134 2568 31 4 v 2134 2609 a(2)2207 2706 y Fl(\024)28 b Fn(3)2371 2638 y Ff(1)p 2370 2650 V 2370 2691 a(2)2415 2706 y Fn(\()p Fk(k)d Fn(+)d(3)p Fk(N)33 b Fl(\000)22 b Fn(2\))2983 2638 y Ff(1)p 2983 2650 V 2983 2691 a(2)0 2924 y Fn(W)-8 b(e)39 b(c)m(hange)g(the)g(summation)d(index)j(to)f Fk(l)i Fn(=)d Fk(N)g Fl(\000)26 b Fk(n)39 b Fn(and)f(then)h(use)h(the)f (second)g(inequalit)m(y)f(of)0 3057 y(Lemma)31 b(2)h(to)h(conclude)0 3231 y(1)49 3195 y Fj(st)141 3231 y Fn(term)27 b Fl(\024)474 3126 y Fm(q)p 557 3126 348 4 v 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\)2)964 3161 y Ff(2+)p Fg(k)q Ff(+9)p Fg(N)p 963 3175 242 4 v 1069 3216 a Ff(2)1219 3231 y Fn(3)1278 3163 y Ff(1)p 1278 3175 31 4 v 1278 3216 a(2)1322 3231 y Fn(\026)-50 b Fk(h)1388 3161 y Fg(N)p 1388 3173 54 4 v 1399 3214 a Ff(2)1455 3231 y Fk(\016)1502 3190 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1714 3231 y Fk(K)g(\031)1863 3190 y Fi(\000)1928 3163 y Ff(1)p 1928 3175 31 4 v 1928 3216 a(4)1973 3231 y Fn(\(1)22 b Fl(\000)g Fn(4)p Fk(\034)11 b Fn(\))2321 3190 y Fi(\000)2386 3161 y Fg(k)q Ff(+3)p Fg(N)5 b Fe(\000)p Ff(2)p 2386 3175 243 4 v 2493 3216 a(2)2643 3231 y Fk(b)2684 3190 y Fh(N)i Fi(\000)p Fj(1)496 3429 y Fl(\002)23 b Fn(\()p Fk(k)i Fn(+)d(3)p Fk(N)32 b Fl(\000)23 b Fn(2\))1164 3361 y Ff(1)p 1164 3373 31 4 v 1164 3414 a(2)1208 3429 y Fn([\(2)f(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\)!])1806 3361 y Ff(1)p 1807 3373 V 1807 3414 a(2)496 3693 y Fl(\002)624 3585 y Fh(N)596 3610 y Fm(X)600 3795 y Fh(l)q Fj(=1)732 3693 y Fn(2)781 3652 y Fi(\000)846 3625 y Ff(5)p Fg(l)p 846 3637 51 4 v 856 3678 a Ff(2)927 3547 y Fm(")986 3626 y Fn(\(2)21 b(+)h Fk(k)k Fn(+)c(3)p Fk(N)32 b Fl(\000)23 b Fn(2)p Fk(l)r Fn(\)!\(2)f(+)g Fk(k)j Fn(+)d Fk(N)33 b Fl(\000)22 b Fk(l)r Fn(\)!)p 986 3670 1473 4 v 1062 3762 a(\(2)g(+)g Fk(k)j Fn(+)d(3)p Fk(N)33 b Fl(\000)22 b Fn(3)p Fk(l)r Fn(\)!\(2)g(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\)!)2468 3547 y Fm(#)2527 3543 y Ff(1)p 2527 3555 31 4 v 2527 3596 a(2)369 3972 y Fl(\024)474 3867 y Fm(q)p 557 3867 315 4 v 105 x Fn(3\(2)c(+)g Fk(k)s Fn(\))o(2)931 3902 y Ff(3+)p Fg(k)q Ff(+9)p Fg(N)p 930 3916 242 4 v 1036 3956 a Ff(2)1186 3972 y Fn(\026)-50 b Fk(h)1251 3901 y Fg(N)p 1251 3913 54 4 v 1263 3955 a Ff(2)1319 3972 y Fk(\016)1366 3930 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1578 3972 y Fk(K)g Fn(\(1)f Fl(\000)g Fn(4)p Fk(\034)11 b Fn(\))1984 3930 y Fi(\000)2049 3902 y Fg(k)q Ff(+3)p Fg(N)p 2048 3916 165 4 v 2115 3956 a Ff(2)2227 3972 y Fk(b)2268 3930 y Fh(N)c Fi(\000)p Fj(1)2426 3972 y Fk(\014)496 4170 y Fl(\002)23 b Fn(\()p Fk(k)i Fn(+)d(3)p Fk(N)32 b Fl(\000)23 b Fn(2\))1164 4102 y Ff(1)p 1164 4114 31 4 v 1164 4155 a(2)1208 4170 y Fn([\(2)f(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\)!])1806 4102 y Ff(1)p 1807 4114 V 1807 4155 a(2)3533 4170 y Fn(\(22\))195 4344 y(W)-8 b(e)23 b(no)m(w)g(estimate)f(the)h (second)h(term)e(in)g(\(20\).)39 b(A)23 b(argumen)m(t)f(in)m(v)m (olving)f(the)i(Cauc)m(h)m(y)i(in)m(tegral)0 4477 y(form)m(ula,)31 b(similar)e(to)j(the)h(one)g(men)m(tioned)f(ab)s(o)m(v)m(e,)i(sho)m(ws) g(that)163 4731 y(sup)0 4814 y Fh(n)p Fi(\025)p Fj(0)p Fh(;)11 b(x)p Fi(2)p Fj([)p Fi(\000)p Fh(R;R)p Fj(])489 4731 y Fk(\016)536 4690 y Fh(n)592 4545 y Fm(\014)592 4595 y(\014)592 4645 y(\014)p Fk(V)699 4609 y Fj([)p Fh(n)p Fj(])785 4645 y Fn(\()p Fk(x)p Fn(\))916 4545 y Fm(\014)916 4595 y(\014)916 4645 y(\014)p 592 4708 352 4 v 725 4800 a Fk(n)p Fn(!)981 4731 y Fl(\024)61 b Fk(M)27 b Fn(exp)1406 4635 y Fm(h)1445 4731 y Fk(\034)11 b Fn(\()1536 4646 y Fl(p)p 1620 4646 530 4 v 1620 4731 a Fn(2)p Fk(k)25 b Fn(+)d(6)p Fk(N)32 b Fn(+)22 b(3\))g(+)g Fk(\016)t Fn(\))2392 4690 y Fj(2)2431 4635 y Fm(i)2498 4731 y Fl(\024)28 b Fk(D)20 b Fn(exp)q(\(12)p Fk(\034)11 b(N)f Fn(\))0 4967 y(where)34 b Fk(D)c Fn(:=)e Fk(M)f Fn(exp)q(\(2)p Fk(\034)11 b Fn(\(2)p Fk(k)25 b Fn(+)d Fk(\016)1242 4931 y Fj(2)1304 4967 y Fn(+)g(3\)\).)43 b(Th)m(us,)34 b(the)f(second)h(term)e(in)g(\(20\))g(satis\014es)0 5203 y(2)49 5167 y Fj(nd)164 5203 y Fn(term)27 b Fl(\024)h Fn(\026)-50 b Fk(h)562 5133 y Fg(N)p 562 5145 54 4 v 574 5186 a Ff(2)630 5203 y Fk(\016)677 5162 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)906 5095 y Fh(N)g Fi(\000)p Fj(1)922 5120 y Fm(X)916 5301 y Fh(n)p Fj(=0)1076 5054 y Fm(\015)1076 5104 y(\015)1076 5153 y(\015)1076 5203 y(\015)1076 5253 y(\015)1122 5203 y Fk(\016)1169 5162 y Fh(N)g Fj(+2)p Fi(\000)p Fh(n)1434 5136 y Fk(V)1512 5100 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1807 5136 y Fn(\()p Fk(\021)1893 5151 y Fh(n)1940 5136 y Fn(\))p 1434 5180 544 4 v 1436 5271 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)1988 5203 y Fk(x)2043 5162 y Fh(N)7 b Fj(+2)p Fi(\000)p Fh(n)2298 5203 y Fk(\037)2359 5218 y Fh(R)2417 5203 y Fn(\()p Fk(x)p Fn(\))p Fk(A)2621 5218 y Fh(k)2664 5203 y Fk(P)2727 5218 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)2990 5054 y Fm(\015)2990 5104 y(\015)2990 5153 y(\015)2990 5203 y(\015)2990 5253 y(\015)3053 5203 y Fl(k)p Fk( )3166 5218 y Fh(n)3213 5203 y Fl(k)392 5447 y(\024)28 b Fn(\026)-50 b Fk(h)562 5377 y Fg(N)p 562 5389 54 4 v 574 5430 a Ff(2)630 5447 y Fk(\016)677 5406 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)519 5655 y Fl(\002)619 5547 y Fh(N)g Fi(\000)p Fj(1)636 5572 y Fm(X)629 5753 y Fh(n)p Fj(=0)789 5506 y Fm(\015)789 5556 y(\015)789 5605 y(\015)789 5655 y(\015)789 5705 y(\015)835 5655 y Fk(\016)882 5614 y Fh(N)g Fj(+2)p Fi(\000)p Fh(n)1147 5588 y Fk(V)1226 5552 y Fj([)p Fh(N)g Fj(+2)p Fi(\000)p Fh(n)p Fj(])1520 5588 y Fn(\()p Fk(\021)1606 5603 y Fh(n)1653 5588 y Fn(\))p 1147 5632 544 4 v 1149 5724 a(\()p Fk(N)32 b Fn(+)22 b(2)g Fl(\000)h Fk(n)p Fn(\)!)1701 5655 y Fk(\037)1762 5670 y Fh(R)1820 5655 y Fn(\()p Fk(x)p Fn(\))p Fk(x)2006 5614 y Fh(N)7 b Fj(+1)p Fi(\000)p Fh(n)2261 5655 y Fk(P)2324 5670 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2677 5506 y Fm(\015)2677 5556 y(\015)2677 5605 y(\015)2677 5655 y(\015)2677 5705 y(\015)2740 5655 y Fl(k)p Fk(xA)2918 5670 y Fh(k)2961 5655 y Fl(k)17 b(k)p Fk( )3141 5670 y Fh(n)3188 5655 y Fl(k)p eop %%Page: 12 12 12 11 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(12)392 294 y Fl(\024)497 189 y Fm(q)p 580 189 348 4 v 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\)\026)-50 b Fk(h)993 224 y Fg(N)p 993 236 54 4 v 1005 278 a Ff(2)1061 294 y Fk(\016)1108 253 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1320 294 y Fk(D)s(e)1449 253 y Fj(12)p Fh(\034)h(N)1642 187 y(N)f Fi(\000)p Fj(1)1659 211 y Fm(X)1652 393 y Fh(n)p Fj(=0)1812 195 y Fm(\015)1812 245 y(\015)1812 294 y(\015)p Fk(x)1913 253 y Fh(N)g Fj(+1)p Fi(\000)p Fh(n)2169 294 y Fk(P)2232 309 y Fh(j)t Fi(\024)p Fh(k)r Fj(+3)p Fh(n)p Fj(+1)2585 195 y Fm(\015)2585 245 y(\015)2585 294 y(\015)16 b Fl(k)p Fk( )2760 309 y Fh(n)2808 294 y Fl(k)0 518 y Fn(So)51 b(using)g(our)f(b)s(ound)i(from)d(Theorem)j (1,)j(Lemma)50 b(5.1)g(of)h([1],)k(and)d(switc)m(hing)e(indices)h(b)m (y)0 651 y Fk(n)28 b Fn(=)f Fk(N)33 b Fl(\000)23 b Fk(l)r Fn(,)33 b(w)m(e)g(obtain)0 819 y(2)49 782 y Fj(nd)164 819 y Fn(term)27 b Fl(\024)497 714 y Fm(q)p 580 714 348 4 v 105 x Fn(2\(2)22 b(+)g Fk(k)s Fn(\)2)987 749 y Fg(N)5 b Ff(+1)p 987 763 131 4 v 1037 804 a(2)1131 819 y Fk(D)s(e)1260 778 y Fj(12)p Fh(\034)j(N)1437 819 y Fn(\026)-50 b Fk(h)1502 749 y Fg(N)p 1502 761 54 4 v 1514 802 a Ff(2)1570 819 y Fk(\016)1617 778 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1829 819 y Fk(b)1870 778 y Fh(N)g Fi(\000)p Fj(1)519 1075 y Fl(\002)619 967 y Fh(N)g Fi(\000)p Fj(1)636 992 y Fm(X)629 1174 y Fh(n)p Fj(=0)789 1075 y Fn(2)848 1007 y Ff(5)p Fg(n)p 848 1019 69 4 v 867 1060 a Ff(2)947 929 y Fm(")1005 1008 y Fn(\(2)22 b(+)g Fk(k)k Fn(+)c Fk(N)32 b Fn(+)22 b(2)p Fk(n)p Fn(\)!\(1)g(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!)p 1005 1052 1267 4 v 1362 1144 a(\(1)g(+)g Fk(k)j Fn(+)d(3)p Fk(n)p Fn(\)!)2282 929 y Fm(#)2340 925 y Ff(1)p 2340 937 31 4 v 2340 978 a(2)392 1327 y Fl(\024)497 1222 y Fm(q)p 580 1222 348 4 v 105 x Fn(2\(2)g(+)g Fk(k)s Fn(\)2)987 1257 y Ff(6)p Fg(N)5 b Ff(+1)p 987 1271 162 4 v 1052 1312 a(2)1162 1327 y Fk(D)s(e)1291 1286 y Fj(12)p Fh(\034)j(N)1467 1327 y Fn(\026)-50 b Fk(h)1533 1257 y Fg(N)p 1533 1269 54 4 v 1544 1310 a Ff(2)1600 1327 y Fk(\016)1647 1286 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1859 1327 y Fk(b)1900 1286 y Fh(N)g Fi(\000)p Fj(1)519 1583 y Fl(\002)23 b Fn([\(2)f(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\)!])1217 1515 y Ff(1)p 1218 1527 31 4 v 1218 1568 a(2)1359 1583 y Fn(max)1279 1643 y Fj(0)p Fi(\024)p Fh(n)p Fi(\024)p Fh(N)d Fi(\000)p Fj(1)1637 1437 y Fm( )1712 1516 y Fn(2)22 b(+)g Fk(k)k Fn(+)c(3)p Fk(n)p 1712 1560 451 4 v 1737 1652 a Fn(2)g(+)g Fk(k)j Fn(+)d Fk(n)2172 1437 y Fm(!)2248 1433 y Ff(1)p 2248 1445 31 4 v 2248 1486 a(2)519 1899 y Fl(\002)647 1791 y Fh(N)619 1816 y Fm(X)623 2000 y Fh(l)q Fj(=1)755 1899 y Fn(2)804 1858 y Fi(\000)869 1830 y Ff(5)p Fg(l)p 869 1842 51 4 v 879 1884 a Ff(2)950 1753 y Fm(")1009 1831 y Fn(\(2)g(+)g Fk(k)j Fn(+)d(3)p Fk(N)32 b Fl(\000)23 b Fn(2)p Fk(l)r Fn(\)!\(2)f(+)g Fk(k)j Fn(+)d Fk(N)33 b Fl(\000)22 b Fk(l)r Fn(\)!)p 1009 1875 1473 4 v 1085 1967 a(\(2)g(+)g Fk(k)j Fn(+)d(3)p Fk(N)33 b Fl(\000)22 b Fn(3)p Fk(l)r Fn(\)!\(2)g(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\)!)2492 1753 y Fm(#)2550 1748 y Ff(1)p 2550 1760 31 4 v 2550 1802 a(2)392 2153 y Fl(\024)497 2048 y Fm(q)p 580 2048 348 4 v 105 x Fn(3\(2)22 b(+)g Fk(k)s Fn(\)2)977 2112 y Fj(3)p Fh(N)7 b Fj(+1)1169 2153 y Fk(D)s(e)1298 2112 y Fj(12)p Fh(\034)h(N)1475 2153 y Fk(\014)e Fn(\026)-50 b Fk(h)1601 2083 y Fg(N)p 1601 2095 54 4 v 1612 2136 a Ff(2)1668 2153 y Fk(\016)1715 2112 y Fi(\000)p Fh(N)7 b Fi(\000)p Fj(2)1928 2153 y Fk(b)1969 2112 y Fh(N)g Fi(\000)p Fj(1)2127 2153 y Fn([\(2)21 b(+)h Fk(k)k Fn(+)c Fk(N)10 b Fn(\)!])2725 2085 y Ff(1)p 2725 2097 31 4 v 2725 2138 a(2)3533 2153 y Fn(\(23\))195 2321 y(Finally)-8 b(,)26 b(to)i(estimate)f(the)h(third)g(term)f(of)h(\(20\))f(w)m(e)i (need)g(the)f(inequalit)m(y)f(\(14\),)h(the)h(estimate)0 2453 y(of)j(Theorem)h(1)f(and)h(the)g(\014rst)g(part)f(of)g(Lemma)g(2)0 2677 y(3)49 2641 y Fj(rd)152 2677 y Fn(term)27 b Fl(\024)502 2569 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)536 2594 y Fm(X)516 2778 y Fh(n)p Fj(=)p Fh(N)707 2677 y Fn(\026)-50 b Fk(h)772 2607 y Fg(n)p 772 2619 39 4 v 776 2660 a Ff(2)841 2569 y Fh(N)7 b Fi(\000)p Fj(1)858 2594 y Fm(X)862 2779 y Fh(l)q Fj(=1)1011 2677 y Fl(jE)1092 2692 y Fh(l)1118 2677 y Fl(jk)p Fk( )1259 2692 y Fh(n)p Fi(\000)p Fh(l)1382 2677 y Fl(k)380 3003 y(\024)29 b Fn(2)p Fk(c)p Fn(\(2)21 b(+)h Fk(k)s Fn(\))892 2895 y Fj(2)p Fh(N)7 b 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Fj(1)2010 1246 y Fl(\024)2148 1146 y Fm(\015)2148 1196 y(\015)2148 1246 y(\015)p Fn(\()p Fk(H)29 b Fl(\000)23 b Fk(E)2514 1261 y Fh(N)2581 1246 y Fn(\(\026)-50 b Fk(h)p Fn(\)\))2750 1197 y Fi(\000)p Fj(1)2845 1146 y Fm(\015)2845 1196 y(\015)2845 1246 y(\015)16 b Fk(:)0 1492 y Fn(Because)42 b Fk(H)49 b Fn(is)40 b(selfadjoin)m(t,)i Fl(k)p Fn(\()p Fk(H)34 b Fl(\000)29 b Fk(E)6 b Fn(\))1549 1456 y Fi(\000)p Fj(1)1643 1492 y Fl(k)41 b Fn(=)h(dist)p Fl(f)p Fk(E)6 b(;)17 b(\033)t Fn(\()p Fk(H)8 b Fn(\))p Fl(g)2456 1456 y Fi(\000)p Fj(1)2589 1492 y Fn(b)m(y)42 b(the)f(Sp)s(ectral)g(Theorem.)0 1625 y(Th)m(us,)598 1864 y(dist)16 b Fl(f)p Fk(E)894 1879 y Fh(N)961 1864 y Fn(\(\026)-50 b Fk(h)p Fn(\))p Fk(;)17 b(\033)t Fn(\()p Fk(H)8 b Fn(\))p Fl(g)59 b(\024)1606 1756 y Fj(2)p Fh(N)1595 1781 y Fm(X)1575 1965 y Fh(n)p Fj(=)p Fh(N)1752 1864 y Fk(AB)1904 1823 y Fh(n)1951 1864 y Fn(\026)-50 b Fk(h)2016 1794 y Fg(n)p 2016 1806 39 4 v 2020 1847 a Ff(2)2085 1864 y Fn([\(2)22 b(+)g Fk(k)k Fn(+)c Fk(n)p 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b Fk(h)p Fn(\))39 b(is)f(asymptotic)f(to) h Fk(E)1964 3062 y Fh(k)2007 3047 y Fn(\(\026)-50 b Fk(h)p Fn(\).)61 b(So,)40 b(w)m(e)g(ma)m(y)e(apply)g(Lemma)f(4)h(\(with)0 3180 y Fk(\014)e Fn(=)29 b(\026)-50 b Fk(h)262 3110 y Ff(1)p 262 3122 31 4 v 262 3163 a(2)306 3180 y Fn(\))34 b(to)f(see)j(that)d(there)i(exists)g(an)e(\026)-50 b Fk(h)1584 3195 y Fj(3)1653 3180 y Fk(>)30 b Fn(0,)k(so)g(that)g(for)f (eac)m(h)i(\026)-50 b Fk(h)30 b Fl(\024)h Fn(\026)-50 b Fk(h)2822 3195 y Fj(3)2861 3180 y Fn(,)34 b(w)m(e)h(ha)m(v)m(e)h(an)d Fk(N)3508 3195 y Fj(3)3548 3180 y Fn(\(\026)-50 b Fk(h)p Fn(\),)0 3313 y(suc)m(h)34 b(that)598 3496 y Fl(j)o Fk(E)697 3511 y Fh(k)740 3496 y Fn(\(\026)-50 b Fk(h)p Fn(\))22 b Fl(\000)h Fk(E)1065 3511 y Fh(N)1133 3496 y Fn(\(\026)-50 b Fk(h)p Fn(\))p Fl(j)27 b(\024)1434 3428 y Fn(1)p 1434 3472 49 4 v 1434 3564 a(4)3533 3496 y(\(30\))0 3697 y(for)32 b(all)f Fk(N)363 3712 y Fj(0)430 3697 y Fl(\024)d Fk(N)38 b Fl(\024)28 b Fk(N)834 3712 y Fj(3)874 3697 y Fn(\(\026)-50 b 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y Fn(\(\026)-50 b Fk(h)p Fn(\))p Fl(g)p Fn(.)84 b(This)0 4328 y(completes)32 b(the)h(pro)s(of.)2764 b Fd(2)0 4602 y Fw(Remark:)56 b Fn(It)40 b(is)f(clear)f(from)g (inequalit)m(y)h(\(26\))f(that)h(indeed)h(w)m(e)g(ha)m(v)m(e)h Fl(j)p Fk(E)2857 4617 y Fh(N)2951 4602 y Fl(\000)27 b Fk(E)3127 4617 y Fh(k)3170 4602 y Fl(j)38 b(\024)3363 4563 y Fj(1)p 3363 4579 36 4 v 3363 4637 a(4)3447 4602 y Fn(for)h(all)0 4735 y Fk(N)78 4750 y Fj(0)159 4735 y Fl(\024)i Fk(N)51 b Fl(\024)42 b Fk(N)603 4750 y Fj(1)642 4735 y Fn(\(\026)-50 b Fk(h)p Fn(\).)67 b(So)41 b(without)e(harm)h(w)m (e)h(can)g(extend)h Fk(N)2385 4750 y Fh(k)2428 4735 y Fn(\(\026)-50 b Fk(h)p Fn(\))40 b(c)m(hosen)i(in)d(the)i(pro)s(of)f(to) g(b)s(e)0 4868 y Fk(N)78 4883 y Fh(k)121 4868 y Fn(\(\026)-50 b Fk(h)p Fn(\))27 b(=)h Fk(N)461 4883 y Fj(1)501 4868 y Fn(\(\026)-50 b Fk(h)p Fn(\).)0 5143 y Fw(Theorem)43 b(3.)59 b Fn(F)-8 b(or)37 b(eac)m(h)i(0)d Fk(<)h(g)j(<)c(B)1508 5106 y Fi(\000)p Fj(2)1603 5143 y Fn(,)j(there)f(exists)h(0)d Fk(<)h Fn(\026)-50 b Fk(h)2451 5158 y Fh(g)2491 5143 y Fn(,)39 b(suc)m(h)g(that)f(for)f(eac)m(h)i(\026)-50 b Fk(h)37 b Fl(\024)g Fn(\026)-50 b Fk(h)3639 5158 y Fh(g)3679 5143 y Fn(,)0 5275 y(there)33 b(is)f Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\),)33 b(so)f(that)598 5394 y Fm(\014)598 5444 y(\014)598 5494 y(\014)o Fk(E)697 5509 y Fh(N)7 b Fj(\(\026)-35 b Fh(h)q Fj(\))860 5494 y Fn(\(\026)-50 b Fk(h)p Fn(\))23 b Fl(\000)f Fk(E)1185 5509 y Fh(k)1228 5494 y Fn(\(\026)-50 b Fk(h)p Fn(\))1359 5394 y Fm(\014)1359 5444 y(\014)1359 5494 y(\014)60 b Fl(\024)h Fn(\003)17 b(exp)1835 5373 y Fm(\022)1896 5494 y Fl(\000)2000 5427 y Fn(\000)p 2000 5471 61 4 v 2003 5562 a(\026)-50 b Fk(h)2071 5373 y Fm(\023)3533 5494 y Fn(\(31\))0 5696 y(for)32 b(some)g(\000)c Fk(>)g Fn(0)k(and)h(\003)27 b Fk(>)h Fn(0.)p eop %%Page: 15 15 15 14 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(15)0 232 y Fw(Pro)s(of:)47 b Fn(W)-8 b(e)35 b(mimic)d(a)i(pro)s(of)f(giv)m(en)i(in)f([2].)49 b(Fix)34 b(0)c Fk(<)h(g)j(<)d(B)2336 196 y Fi(\000)p Fj(2)2431 232 y Fn(.)49 b(Then)36 b(0)30 b Fk(<)h(B)3029 196 y Fj(2)3069 232 y Fk(g)j(<)d Fn(1,)k(so)f(there)0 365 y(is)e Fk(!)f(>)d Fn(0,)k(suc)m(h)i(that)e Fk(B)912 329 y Fj(2)952 365 y Fk(g)f Fn(=)c(exp)r(\()p Fl(\000)p Fk(!)t Fn(\).)43 b(Consider)33 b(the)g(function)598 598 y Fk(f)11 b Fn(\(\026)-50 b Fk(h)p Fn(\))27 b(=)h Fk(Ag)20 b Fn(exp)1224 452 y Fm(")1273 598 y Fl(\000)1360 530 y Fk(!)t Fn(\(1)h(+)h Fk(k)s Fn(\))p 1360 575 364 4 v 1517 666 a(4)1733 452 y Fm(#)1798 598 y Fn(\026)-50 b Fk(h)1854 555 y Fi(\000)1919 526 y Ff(4+)p Fg(k)p 1918 540 111 4 v 1959 581 a Ff(2)2060 598 y Fn(exp)2226 477 y Fm(\022)2287 598 y Fl(\000)2374 530 y Fk(!)t(g)p 2374 575 115 4 v 2379 666 a Fn(2\026)g Fk(h)2499 477 y Fm(\023)2577 598 y Fk(:)0 830 y Fn(W)-8 b(e)33 b(kno)m(w)h(that)e Fk(f)11 b Fn(\(\026)-50 b Fk(h)p Fn(\))27 b Fk(>)h Fn(0)k(on)h(\(0)p Fk(;)17 b Fl(1)p Fn(\),)31 b(and)i Fk(f)11 b 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Fk(h)28 b Fl(\024)356 1632 y Fn(^)353 1658 y(\026)-50 b Fk(h)408 1673 y Fh(g)471 1658 y Fn(de\014ne)24 b Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\))23 b(b)m(y)g(the)h(iden)m(tit) m(y)e(2)r(+)r Fk(k)6 b Fn(+)r(2)p Fk(N)k Fn(\(\026)-50 b Fk(h)q Fn(\))27 b(=)h Fb(J)2336 1615 y Fh(g)p 2334 1635 41 4 v 2334 1692 a Fj(\026)-35 b Fh(h)2385 1658 y Fb(K)p Fn(.)40 b(Then)24 b(clearly)e Fk(N)10 b Fn(\(\026)-50 b Fk(h)p Fn(\))28 b Fl(\025)g Fk(N)3469 1673 y Fj(0)3509 1658 y Fn(.)40 b(On)0 1791 y(the)30 b(other)g(hand,)h(since)g(w)m(e)f (can)h(assume)f Fk(B)j Fl(\025)28 b Fn(1)i(and)g(2)16 b(+)h Fk(k)i Fn(+)e Fk(n)28 b Fl(\024)g Fk(g)t(=)p Fn(\026)-50 b Fk(h)29 b Fn(for)g Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\))28 b Fl(\024)g Fk(n)g Fl(\024)g Fn(2)p Fk(N)10 b Fn(\(\026)-50 b Fk(h)p Fn(\))0 1923 y(w)m(e)34 b(ha)m(v)m(e)31 2040 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)q Fj(\))68 2074 y Fm(X)0 2262 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))273 2157 y Fk(AB)425 2116 y Fh(n)472 2157 y Fn(\026)-50 b Fk(h)537 2087 y Fg(n)p 537 2099 39 4 v 541 2140 a Ff(2)606 2157 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)e Fk(n)p Fn(\)!])1175 2081 y Ff(1)p 1175 2093 31 4 v 1175 2134 a(2)1264 2157 y Fl(\024)1417 2040 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1454 2074 y Fm(X)1385 2262 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1658 2157 y Fk(AB)1810 2116 y Fh(n)1858 2157 y Fn(\026)-50 b Fk(h)1923 2087 y Fg(n)p 1923 2099 39 4 v 1927 2140 a Ff(2)1975 2157 y Fn(\(2)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\))2462 2087 y Ff(2+)p Fg(k)q Ff(+)p Fg(n)p 2462 2101 196 4 v 2545 2142 a Ff(2)1264 2495 y Fl(\024)28 b Fk(A)p Fn(\026)-50 b Fk(h)1497 2453 y Fi(\000)1562 2424 y Ff(2+)p Fg(k)p 1562 2438 111 4 v 1602 2479 a Ff(2)1735 2378 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1772 2412 y Fm(X)1704 2601 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1977 2399 y Fm(h)2016 2495 y Fk(B)2095 2454 y Fj(2)2134 2495 y Fn(\026)-50 b Fk(h)p Fn(\(2)22 b(+)g Fk(k)k Fn(+)c Fk(n)p Fn(\))2667 2399 y Fm(i)2716 2394 y Ff(2+)p Fg(k)q Ff(+)p Fg(n)p 2716 2408 196 4 v 2799 2448 a Ff(2)1264 2834 y Fl(\024)28 b Fk(A)p Fn(\026)-50 b Fk(h)1497 2791 y Fi(\000)1562 2763 y Ff(2+)p Fg(k)p 1562 2777 111 4 v 1602 2817 a Ff(2)1735 2717 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1772 2751 y Fm(X)1704 2939 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1977 2738 y Fm(\020)2026 2834 y Fk(B)2105 2793 y Fj(2)2145 2834 y Fk(g)2196 2738 y Fm(\021)2255 2732 y Ff(2+)p Fg(k)q Ff(+)p Fg(n)p 2255 2746 196 4 v 2337 2787 a Ff(2)0 3101 y Fn(No)m(w)33 b(use)h(that)e Fk(B)681 3065 y Fj(2)721 3101 y Fk(g)f Fn(=)c(exp)q(\()p Fl(\000)p Fk(!)t Fn(\))h Fk(<)f Fn(1)32 b(and)h(the)g(fact)f Fk(a)2082 3065 y Fh(n)2157 3101 y Fl(\025)c Fk(a)2313 3065 y Fh(n)p Fj(+1)2483 3101 y Fn(if)k Fk(a)c(<)f Fn(1)32 b(to)h(obtain)31 3242 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)q Fj(\))68 3276 y Fm(X)0 3465 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))273 3359 y Fk(AB)425 3318 y Fh(n)472 3359 y Fn(\026)-50 b Fk(h)537 3289 y Fg(n)p 537 3301 39 4 v 541 3343 a Ff(2)606 3359 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)e Fk(n)p Fn(\)!])1175 3284 y Ff(1)p 1175 3296 31 4 v 1175 3337 a(2)493 3698 y Fl(\024)28 b Fk(A)p Fn(\026)-50 b Fk(h)726 3655 y Fi(\000)791 3627 y Ff(2+)p Fg(k)p 791 3641 111 4 v 831 3681 a Ff(2)964 3581 y Fj(2)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)p Fj(\))1001 3615 y Fm(X)933 3803 y Fh(n)p Fj(=)p Fh(N)7 b Fj(\(\026)-35 b Fh(h)o Fj(\))1206 3698 y Fn(exp)1371 3577 y Fm(\032)1433 3698 y Fl(\000)1520 3631 y Fk(!)p 1520 3675 65 4 v 1528 3766 a Fn(2)1612 3698 y([2)22 b(+)g Fk(k)j Fn(+)d Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\)])2229 3577 y Fm(\033)494 3986 y Fn(=)28 b Fk(A)p Fn(\026)-50 b Fk(h)726 3943 y Fi(\000)791 3914 y Ff(2+)p Fg(k)p 791 3928 111 4 v 831 3969 a Ff(2)916 3986 y Fk(e)961 3945 y Fi(\000)1026 3917 y Fg(!)p 1026 3929 41 4 v 1031 3971 a Ff(4)1077 3945 y Fj(\(2+)p Fh(k)r Fj(\))1281 3986 y Fn([1)22 b(+)g Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\)])16 b(exp)1906 3865 y Fm(\032)1968 3986 y Fl(\000)2055 3918 y Fk(!)p 2055 3963 65 4 v 2063 4054 a Fn(4)2146 3986 y([)q(2)21 b(+)h Fk(k)k Fn(+)c(2)p Fk(N)10 b Fn(\(\026)-50 b Fk(h)p Fn(\)])2812 3865 y Fm(\033)493 4218 y Fl(\024)28 b Fk(A)p Fn(\026)-50 b Fk(h)726 4175 y Fi(\000)791 4147 y Ff(2+)p Fg(k)p 791 4161 111 4 v 831 4201 a Ff(2)916 4218 y Fk(e)961 4177 y Fi(\000)1026 4150 y Fg(!)p 1026 4162 41 4 v 1031 4203 a Ff(4)1077 4177 y Fj(\(2+)p Fh(k)r Fj(\))1281 4218 y Fn([2)22 b(+)g Fk(k)j Fn(+)d(2)p Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\)])16 b(exp)2129 4097 y Fm(\024)2173 4218 y Fl(\000)2260 4151 y Fk(!)p 2260 4195 65 4 v 2268 4287 a Fn(4)2351 4097 y Fm(\022)2425 4151 y Fk(g)p 2422 4195 57 4 v 2422 4287 a(h)2510 4218 y Fl(\000)23 b Fn(1)2659 4097 y Fm(\023\025)493 4451 y Fl(\024)28 b Fk(Ag)t(e)767 4410 y Fi(\000)832 4382 y Fg(!)p 831 4394 41 4 v 836 4436 a Ff(4)882 4410 y Fj(\(1+)p Fh(k)r Fj(\))1070 4451 y Fn(\026)-50 b Fk(h)1125 4408 y Fi(\000)1190 4379 y Ff(4+)p Fg(k)p 1190 4393 111 4 v 1230 4434 a Ff(2)1331 4451 y Fn(exp)1497 4329 y Fm(\022)1558 4451 y Fl(\000)1645 4383 y Fk(!)t(g)p 1645 4427 115 4 v 1650 4519 a Fn(4\026)g Fk(h)1770 4329 y Fm(\023)3533 4451 y Fn(\(32\))493 4683 y Fl(\024)28 b Fk(Ag)t(e)767 4642 y Fi(\000)832 4615 y Fg(!)p 831 4627 41 4 v 836 4668 a Ff(4)882 4642 y Fj(\(1+)p Fh(k)r Fj(\))1070 4683 y Fn(\026)-50 b Fk(h)1125 4623 y Fi(\000)1190 4594 y Ff(4+)p Fg(k)p 1190 4608 111 4 v 1230 4649 a Ff(2)1125 4705 y Fj(4)1331 4683 y Fn(exp)1497 4562 y Fm(\022)1558 4683 y Fl(\000)1660 4616 y Fk(!)t(g)p 1645 4660 144 4 v 1645 4751 a Fn(4\026)g Fk(h)1749 4766 y Fj(4)1799 4562 y Fm(\023)493 4931 y Fl(\024)608 4864 y Fn(1)p 608 4908 49 4 v 608 5000 a(4)666 4931 y Fk(:)0 5153 y Fn(Th)m(us,)34 b Fk(N)10 b Fn(\(\026)-50 b Fk(h)q Fn(\))28 b Fl(\024)g Fk(N)705 5168 y Fh(k)748 5153 y Fn(\(\026)-50 b Fk(h)p Fn(\).)44 b(Therefore,)33 b(Prop)s(osition)e(2)i(holds)f(for)g(\026)-50 b Fk(h)28 b(<)2619 5127 y Fn(^)2616 5153 y(\026)-50 b Fk(h)2671 5168 y Fh(g)2711 5153 y Fn(,)33 b(whic)m(h)g(along)e(with)i (\(32\))0 5286 y(implies)598 5388 y Fm(\014)598 5438 y(\014)598 5488 y(\014)o Fk(E)697 5504 y Fh(N)7 b Fj(\(\026)-35 b Fh(h)q Fj(\))860 5488 y Fn(\(\026)-50 b Fk(h)p Fn(\))23 b Fl(\000)f Fk(E)1185 5503 y Fh(k)1228 5488 y Fn(\(\026)-50 b Fk(h)p Fn(\))1359 5388 y Fm(\014)1359 5438 y(\014)1359 5488 y(\014)60 b Fl(\024)h Fk(Ag)t(e)1754 5447 y Fi(\000)1819 5420 y Fg(!)p 1818 5432 41 4 v 1823 5473 a Ff(4)1869 5447 y Fj(\(1+)p Fh(k)r Fj(\))2056 5488 y Fn(\026)-50 b Fk(h)2112 5445 y Fi(\000)2177 5417 y Ff(4+)p Fg(k)p 2176 5431 111 4 v 2217 5471 a Ff(2)2318 5488 y Fn(exp)2484 5367 y Fm(\022)2545 5488 y Fl(\000)2632 5421 y Fk(!)t(g)p 2632 5465 115 4 v 2637 5556 a Fn(4\026)g Fk(h)2757 5367 y Fm(\023)2835 5488 y Fk(;)p eop %%Page: 16 16 16 15 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(16)0 232 y(for)32 b(all)f(\026)-50 b Fk(h)27 b Fl(\024)476 206 y Fn(^)472 232 y(\026)-50 b Fk(h)528 247 y Fh(g)568 232 y Fn(.)43 b(Finally)-8 b(,)30 b(de\014ne)598 432 y(\026)-50 b Fk(h)653 447 y Fh(g)721 432 y Fn(=)27 b(max)1022 311 y Fm(\032)1085 432 y Fn(\026)-50 b Fk(h)27 b Fl(\024)1276 405 y Fn(^)1273 432 y(\026)-50 b Fk(h)1328 447 y Fh(g)1395 432 y Fn(:)28 b(\026)-50 b Fk(h)1505 389 y Fi(\000)1570 360 y Ff(4+)p Fg(k)p 1570 374 111 4 v 1611 415 a Ff(2)1712 432 y Fn(exp)1878 311 y Fm(\022)1939 432 y Fl(\000)2026 364 y Fk(!)t(g)p 2026 409 115 4 v 2031 500 a Fn(8\026)g Fk(h)2151 311 y Fm(\023)2239 432 y Fl(\024)29 b Fn(1)2394 311 y Fm(\033)0 628 y Fn(Then)34 b(\(31\))e(is)g(true)h(for)f(all)e(\026)-50 b Fk(h)28 b Fl(\024)g Fn(\026)-50 b Fk(h)1292 643 y Fh(g)1365 628 y Fn(with)32 b(\000)c(:=)f Fk(!)t(g)t(=)p Fn(8)k(and)i(\003)27 b(:=)h Fk(Ag)20 b Fn(exp)d(\()p Fl(\000)p Fk(!)t Fn(\(1)22 b(+)g Fk(k)s Fn(\))p Fk(=)p Fn(4\).)218 b Fd(2)0 903 y Fw(Prop)s(osition)35 b(3.)43 b Fn(Let)33 b Fk(h)966 918 y Fh(k)1041 903 y Fn(and)f Fk(N)1308 918 y Fh(k)1351 903 y Fn(\(\026)-50 b Fk(h)p Fn(\))32 b(b)s(e)g(de\014ned)i(as)e(in)g (Prop)s(osition)e(2.)43 b(Let)3046 878 y(~)3033 903 y(\011)3109 918 y Fh(N)3176 903 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))32 b(b)s(e)h(the)0 1036 y(v)m(ector)38 b(obtained)e(b)m(y)i(normalizing)33 b(the)k(v)m(ector)h(obtained)f(b)m (y)g(our)g(construction.)56 b(Let)3371 1010 y(~)3357 1036 y(\011)3433 1051 y Fh(k)3476 1036 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))0 1168 y(b)s(e)33 b(the)g(normalized)d (exact)k(eigen)m(v)m(ector)g(of)e Fk(H)8 b Fn(.)43 b(Then)598 1297 y Fm(\015)598 1346 y(\015)598 1396 y(\015)657 1371 y Fn(~)644 1396 y(\011)720 1411 y Fh(N)787 1396 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))22 b Fl(\000)1152 1371 y Fn(~)1139 1396 y(\011)1215 1411 y Fh(k)1257 1396 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))1487 1297 y Fm(\015)1487 1346 y(\015)1487 1396 y(\015)61 b Fl(\024)g Fn(8)1828 1288 y Fj(2)p Fh(N)1817 1313 y Fm(X)1797 1497 y Fh(n)p Fj(=)p Fh(N)1991 1396 y Fk(A)17 b(B)2160 1355 y Fh(n)2223 1396 y Fn(\026)-50 b Fk(h)2288 1326 y Fg(n)p 2288 1338 39 4 v 2292 1379 a Ff(2)2374 1396 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])2943 1321 y Ff(1)p 2943 1333 31 4 v 2943 1374 a(2)3004 1396 y Fk(;)0 1627 y Fn(for)32 b(all)f(\026)-50 b Fk(h)27 b Fl(\024)h Fn(\026)-50 b Fk(h)528 1642 y Fh(k)603 1627 y Fn(and)32 b Fk(N)870 1642 y Fj(0)938 1627 y Fl(\024)c Fk(N)38 b Fl(\024)28 b Fk(N)1342 1642 y Fh(k)1385 1627 y Fn(\(\026)-50 b Fk(h)p Fn(\).)0 1831 y Fw(Pro)s(of:)43 b Fn(Let)33 b(us)g(note)g(that)f (\(27\))g(means)598 1965 y Fm(\015)598 2014 y(\015)598 2064 y(\015)p Fn(\()p Fk(H)d Fl(\000)23 b Fk(E)964 2079 y Fh(N)1032 2064 y Fn(\(\026)-50 b Fk(h)p Fn(\)\))1231 2039 y(~)1217 2064 y(\011)1293 2079 y Fh(N)1360 2064 y Fn(\(\026)g Fk(h)p Fn(;)17 b Fk(x)p Fn(\))1590 1965 y Fm(\015)1590 2014 y(\015)1590 2064 y(\015)28 b Fl(\024)1800 1956 y Fj(2)p Fh(N)1790 1981 y Fm(X)1769 2165 y Fh(n)p Fj(=)p Fh(N)1947 2064 y Fk(AB)2099 2023 y Fh(n)2146 2064 y Fn(\026)-50 b Fk(h)2211 1994 y Fg(n)p 2211 2006 39 4 v 2215 2047 a Ff(2)2280 2064 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])2849 1989 y Ff(1)p 2849 2001 31 4 v 2849 2042 a(2)3533 2064 y Fn(\(33\))0 2311 y(W)-8 b(e)40 b(can)g(write)631 2286 y(~)617 2311 y(\011)693 2326 y Fh(N)761 2311 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))40 b(=)f Fk(w)1216 2326 y Fh(N)1297 2286 y Fn(~)1283 2311 y(\011)1359 2326 y Fh(k)1402 2311 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))27 b(+)g(\012)1832 2326 y Fh(N)1899 2311 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\),)42 b(where)f(\012)2557 2326 y Fh(N)2664 2311 y Fn(is)f(orthogonal)d(to)3404 2286 y(~)3391 2311 y(\011)3467 2326 y Fh(k)3549 2311 y Fn(and)0 2444 y Fl(j)p Fk(w)98 2459 y Fh(N)165 2444 y Fl(j)193 2408 y Fj(2)248 2444 y Fn(+)16 b Fl(k)p Fn(\012)460 2459 y Fh(N)528 2444 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fl(k)808 2408 y Fj(2)875 2444 y Fn(=)27 b(1.)42 b(Since)30 b(these)h(functions)f (are)g(de\014ned)h(up)f(to)f(a)g(global)f(phase,)j(w)m(e)g(can)0 2577 y(assume)i(that)f(indeed)h(0)28 b Fk(<)f(w)1107 2592 y Fh(N)1202 2577 y Fl(\024)h Fn(1.)43 b(F)-8 b(rom)31 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y Fh(n)1738 5653 y Fn(\026)-50 b Fk(h)1803 5583 y Fg(n)p 1803 5595 39 4 v 1807 5636 a Ff(2)1889 5653 y Fn([\(2)22 b(+)g Fk(k)j Fn(+)d Fk(n)p Fn(\)!])2458 5577 y Ff(1)p 2458 5589 31 4 v 2458 5630 a(2)2519 5653 y Fk(;)987 b Fn(\(38\))p eop %%Page: 17 17 17 16 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(17)0 232 y(for)32 b(all)f(\026)-50 b Fk(h)27 b Fl(\024)h Fn(\026)-50 b Fk(h)528 247 y Fh(k)603 232 y Fn(and)32 b Fk(N)870 247 y Fj(0)938 232 y Fl(\024)c Fk(N)38 b Fl(\024)28 b Fk(N)1342 247 y Fh(k)1385 232 y Fn(\(\026)-50 b Fk(h)p Fn(\).)43 b(No)m(w)34 b(com)m(bine)e(\(34\))g(and)g(\(38\).)845 b Fd(2)195 436 y Fn(The)37 b(next)g(result)f(follo)m(ws)e(from)h(Prop)s (osition)f(3)i(in)f(the)h(same)g(w)m(a)m(y)h(that)f(Theorem)g(3)g(do)s (es)0 569 y(from)31 b(Prop)s(osition)g(2.)0 773 y Fw(Theorem)37 b(4.)44 b Fn(If)32 b Fk(g)t Fn(,)g(\026)-50 b Fk(h)868 788 y Fh(g)908 773 y Fn(,)33 b Fk(N)10 b Fn(\(\026)-50 b Fk(h)p Fn(\),)33 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Fh(k)2297 3064 y Fk( )2360 3079 y Fh(m)2427 3064 y Fn(\()p Fk(x)p Fn(\))395 3379 y(+)493 3271 y Fh(N)7 b Fi(\000)p Fj(1)509 3296 y Fm(X)493 3477 y Fh(m)p Fj(=0)662 3379 y Fn(\026)-50 b Fk(h)727 3309 y Fg(m)p 727 3321 55 4 v 739 3362 a Ff(2)796 3379 y Fk(A)869 3394 y Fh(k)912 3379 y Fk(W)1018 3338 y Fj([)p Fh(N)7 b Fj(+2])1214 3379 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1517 3394 y Fh(k)1560 3379 y Fk( )1623 3394 y Fh(m)1690 3379 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1943 3271 y Fh(N)7 b Fi(\000)p Fj(1)1960 3296 y Fm(X)1943 3477 y Fh(m)p Fj(=0)2113 3271 y Fh(N)g Fi(\000)p Fj(1)2130 3296 y Fm(X)2128 3478 y Fh(j)t Fj(=1)2283 3379 y Fn(\026)-50 b Fk(h)2348 3305 y Fg(j)s Ff(+)p Fg(m)p 2348 3321 130 4 v 2397 3362 a Ff(2)2491 3379 y Fl(E)2544 3394 y Fh(j)2581 3379 y Fn(\()p Fk(A)2692 3394 y Fh(k)2734 3379 y Fn(\))2772 3338 y Fj(2)2812 3379 y Fk( )2875 3394 y Fh(m)2942 3379 y Fn(\()p Fk(x)p Fn(\))269 3694 y(=)389 3586 y Fh(N)7 b Fi(\000)p Fj(1)406 3611 y Fm(X)399 3792 y Fh(n)p Fj(=1)559 3694 y Fn(\026)-50 b Fk(h)624 3624 y Fg(n)p 624 3636 39 4 v 628 3677 a Ff(2)676 3694 y Fk(H)757 3709 y Fh(k)800 3694 y Fk( )863 3709 y Fh(n)910 3694 y Fn(\()p Fk(x)p Fn(\))23 b(+)1162 3586 y Fh(N)7 b Fi(\000)p Fj(1)1178 3611 y Fm(X)1172 3792 y Fh(n)p Fj(=1)1332 3694 y Fn(\026)-50 b Fk(h)1397 3624 y Fg(n)p 1397 3636 V 1401 3677 a Ff(2)1506 3586 y Fh(n)1467 3611 y Fm(X)1466 3794 y Fh(j)t Fj(=1)1605 3694 y Fk(c)1647 3709 y Fh(j)t Fj(+2)1774 3694 y Fk(A)1847 3709 y Fh(k)1889 3694 y Fk(x)1944 3653 y Fh(j)t Fj(+2)2071 3694 y Fk(A)2144 3709 y Fh(k)2187 3694 y Fk( )2250 3709 y Fh(n)p Fi(\000)p Fh(j)2385 3694 y Fn(\()p Fk(x)p Fn(\))395 4010 y(+)493 3902 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(1)527 3927 y Fm(X)506 4110 y Fh(n)p Fj(=)p Fh(N)698 4010 y Fn(\026)-50 b Fk(h)763 3940 y Fg(n)p 763 3952 V 767 3993 a Ff(2)969 3902 y Fh(N)941 3927 y Fm(X)832 4110 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)7 b Fj(+1)1187 4010 y Fk(c)1229 4025 y Fh(j)t Fj(+2)1355 4010 y Fk(A)1428 4025 y Fh(k)1471 4010 y Fk(x)1526 3968 y Fh(j)t Fj(+2)1653 4010 y Fk(A)1726 4025 y Fh(k)1769 4010 y Fk( )1832 4025 y Fh(n)p Fi(\000)p Fh(j)1967 4010 y Fn(\()p Fk(x)p Fn(\))395 4326 y(+)493 4218 y Fh(N)g Fi(\000)p Fj(1)509 4243 y Fm(X)493 4425 y Fh(m)p Fj(=0)662 4326 y Fn(\026)-50 b Fk(h)727 4256 y Fg(m)p 727 4268 55 4 v 739 4310 a Ff(2)796 4326 y Fk(A)869 4341 y Fh(k)912 4326 y Fk(W)1018 4285 y Fj([)p Fh(N)7 b Fj(+2])1214 4326 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1517 4341 y Fh(k)1560 4326 y Fk( )1623 4341 y Fh(m)1690 4326 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1943 4218 y Fh(N)7 b Fi(\000)p Fj(1)1960 4243 y Fm(X)1953 4425 y Fh(n)p Fj(=1)2113 4326 y Fn(\026)-50 b Fk(h)2178 4256 y Fg(n)p 2178 4268 39 4 v 2182 4310 a Ff(2)2287 4218 y Fh(n)2248 4243 y Fm(X)2247 4426 y Fh(j)t Fj(=1)2386 4326 y Fl(E)2439 4341 y Fh(j)2475 4326 y Fn(\()p Fk(A)2586 4341 y Fh(k)2629 4326 y Fn(\))2667 4285 y Fj(2)2706 4326 y Fk( )2769 4341 y Fh(n)p Fi(\000)p Fh(j)2904 4326 y Fn(\()p Fk(x)p Fn(\))395 4642 y Fl(\000)494 4534 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)528 4559 y Fm(X)508 4742 y Fh(n)p Fj(=)p Fh(N)699 4642 y Fn(\026)-50 b Fk(h)764 4572 y Fg(n)p 764 4584 V 768 4625 a Ff(2)926 4534 y Fh(N)7 b Fi(\000)p Fj(1)943 4559 y Fm(X)833 4742 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)1188 4642 y Fl(E)1241 4657 y Fh(j)1277 4642 y Fn(\()p Fk(A)1388 4657 y Fh(k)1431 4642 y Fn(\))1469 4601 y Fj(2)1508 4642 y Fk( )1571 4657 y Fh(n)p Fi(\000)p Fh(j)1706 4642 y Fn(\()p Fk(x)p Fn(\))0 4896 y(The)41 b(\014rst,)j(second)d(and)g (\014fth)g(terms)f(of)g(last)g(equation)g(cancel)h(b)s(ecause)h(of)e (\(9\).)67 b(In)41 b(the)g(third)0 5029 y(term)32 b(de\014ne)i Fk(m)28 b Fn(=)f Fk(n)c Fl(\000)f Fk(j)39 b Fn(and)33 b(then)g Fk(p)27 b Fn(=)h Fk(n)22 b Fl(\000)h Fk(N)10 b Fn(.)44 b(This)33 b(yields)0 5268 y Fk(\030)43 5283 y Fh(N)110 5268 y Fn(\()p Fk(x)p Fn(\))28 b(=)389 5160 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(1)423 5185 y Fm(X)403 5369 y Fh(n)p Fj(=)p Fh(N)594 5268 y Fn(\026)-50 b Fk(h)659 5198 y Fg(n)p 659 5210 V 663 5252 a Ff(2)791 5160 y Fh(N)7 b Fi(\000)p Fj(1)807 5185 y Fm(X)728 5369 y Fh(m)p Fj(=)p Fh(n)p Fi(\000)p Fh(N)1023 5268 y Fk(c)1065 5283 y Fh(n)p Fi(\000)p Fh(m)p Fj(+2)1319 5268 y Fk(A)1392 5283 y Fh(k)1435 5268 y Fk(x)1490 5227 y Fh(n)p Fi(\000)p Fh(m)p Fj(+2)1745 5268 y Fk(A)1818 5283 y Fh(k)1861 5268 y Fk( )1924 5283 y Fh(m)1991 5268 y Fn(\()p Fk(x)p Fn(\))395 5579 y(+)493 5471 y Fh(N)g Fi(\000)p Fj(1)509 5496 y Fm(X)493 5677 y Fh(m)p Fj(=0)662 5579 y Fn(\026)-50 b Fk(h)727 5509 y Fg(m)p 727 5521 55 4 v 739 5562 a Ff(2)796 5579 y Fk(A)869 5594 y Fh(k)912 5579 y Fk(W)1018 5538 y Fj([)p Fh(N)7 b Fj(+2])1214 5579 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1517 5594 y Fh(k)1560 5579 y Fk( )1623 5594 y Fh(m)1690 5579 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1943 5471 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)1977 5496 y Fm(X)1957 5680 y Fh(n)p Fj(=)p Fh(N)2148 5579 y Fn(\026)-50 b Fk(h)2213 5509 y Fg(n)p 2213 5521 39 4 v 2217 5562 a Ff(2)2375 5471 y Fh(N)7 b Fi(\000)p Fj(1)2392 5496 y Fm(X)2282 5680 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)2637 5579 y Fl(E)2690 5594 y Fh(j)2726 5579 y Fn(\()p Fk(A)2837 5594 y Fh(k)2880 5579 y Fn(\))2918 5538 y Fj(2)2957 5579 y Fk( )3020 5594 y Fh(n)p Fi(\000)p Fh(j)3155 5579 y Fn(\()p Fk(x)p Fn(\))p eop %%Page: 18 18 18 17 bop 0 17 a Fo(Exp)-5 b(onential)5 b(ly)34 b(ac)-5 b(cur)g(ate)35 b(estimates)2221 b Fn(18)269 294 y(=)389 187 y Fh(N)7 b Fi(\000)p Fj(1)406 211 y Fm(X)403 393 y Fh(p)p Fj(=0)559 187 y Fh(N)g Fi(\000)p Fj(1)576 211 y Fm(X)559 386 y Fh(m)p Fj(=)p Fh(p)729 294 y Fn(\026)-50 b Fk(h)794 220 y Fg(p)p Ff(+)p Fg(N)p 794 237 133 4 v 845 278 a Ff(2)940 294 y Fk(c)982 309 y Fh(p)p Fj(+)p Fh(N)7 b Fi(\000)p Fh(m)p Fj(+2)1347 294 y Fk(A)1420 309 y Fh(k)1463 294 y Fk(x)1518 253 y Fh(p)p Fj(+)p Fh(N)g Fi(\000)p Fh(m)p Fj(+2)1883 294 y Fk(A)1956 309 y Fh(k)1999 294 y Fk( )2062 309 y Fh(m)2129 294 y Fn(\()p Fk(x)p Fn(\))395 609 y(+)493 501 y Fh(N)g Fi(\000)p Fj(1)509 526 y Fm(X)493 707 y Fh(m)p Fj(=0)662 609 y Fn(\026)-50 b Fk(h)727 539 y Fg(m)p 727 551 55 4 v 739 592 a Ff(2)796 609 y Fk(A)869 624 y Fh(k)912 609 y Fk(W)1018 568 y Fj([)p Fh(N)7 b Fj(+2])1214 609 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1517 624 y Fh(k)1560 609 y Fk( )1623 624 y Fh(m)1690 609 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1943 501 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)1977 526 y Fm(X)1957 709 y Fh(n)p Fj(=)p Fh(N)2148 609 y Fn(\026)-50 b Fk(h)2213 539 y Fg(n)p 2213 551 39 4 v 2217 592 a Ff(2)2375 501 y Fh(N)7 b Fi(\000)p Fj(1)2392 526 y Fm(X)2282 709 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)2637 609 y Fl(E)2690 624 y Fh(j)2726 609 y Fn(\()p Fk(A)2837 624 y Fh(k)2880 609 y Fn(\))2918 568 y Fj(2)2957 609 y Fk( )3020 624 y Fh(n)p Fi(\000)p Fh(j)3155 609 y Fn(\()p Fk(x)p Fn(\))269 925 y(=)389 817 y Fh(N)g Fi(\000)p Fj(1)406 842 y Fm(X)389 1024 y Fh(m)p Fj(=0)559 925 y Fn(\026)-50 b Fk(h)624 855 y Fg(m)p 624 867 55 4 v 636 909 a Ff(2)741 817 y Fh(m)712 842 y Fm(X)709 1024 y Fh(p)p Fj(=0)851 925 y Fn(\026)g Fk(h)916 851 y Fg(p)p Ff(+)p Fg(N)5 b Fe(\000)p Fg(m)p 916 868 235 4 v 1019 909 a Ff(2)1165 925 y Fk(c)1207 940 y Fh(p)p Fj(+)p Fh(N)i Fi(\000)p Fh(m)p Fj(+2)1572 925 y Fk(A)1645 940 y Fh(k)1688 925 y Fk(x)1743 884 y Fh(p)p Fj(+)p Fh(N)g Fi(\000)p Fh(m)p Fj(+2)2109 925 y Fk(A)2182 940 y Fh(k)2224 925 y Fk( )2287 940 y Fh(m)2354 925 y Fn(\()p Fk(x)p Fn(\))395 1240 y(+)493 1132 y Fh(N)g Fi(\000)p Fj(1)509 1157 y Fm(X)493 1338 y Fh(m)p Fj(=0)662 1240 y Fn(\026)-50 b Fk(h)727 1170 y Fg(m)p 727 1182 55 4 v 739 1223 a Ff(2)796 1240 y Fk(A)869 1255 y Fh(k)912 1240 y Fk(W)1018 1198 y Fj([)p Fh(N)7 b Fj(+2])1214 1240 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1517 1255 y Fh(k)1560 1240 y Fk( )1623 1255 y Fh(m)1690 1240 y Fn(\()p Fk(x)p Fn(\))22 b Fl(\000)1943 1132 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)1977 1157 y Fm(X)1957 1340 y Fh(n)p Fj(=)p Fh(N)2148 1240 y Fn(\026)-50 b Fk(h)2213 1170 y Fg(n)p 2213 1182 39 4 v 2217 1223 a Ff(2)2375 1132 y Fh(N)7 b Fi(\000)p Fj(1)2392 1157 y Fm(X)2282 1340 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)2637 1240 y Fl(E)2690 1255 y Fh(j)2726 1240 y Fn(\()p Fk(A)2837 1255 y Fh(k)2880 1240 y Fn(\))2918 1198 y Fj(2)2957 1240 y Fk( )3020 1255 y Fh(n)p Fi(\000)p Fh(j)3155 1240 y Fn(\()p Fk(x)p Fn(\))269 1556 y(=)389 1448 y Fh(N)g Fi(\000)p Fj(1)406 1473 y Fm(X)389 1655 y Fh(m)p Fj(=0)559 1556 y Fn(\026)-50 b Fk(h)624 1486 y Fg(m)p 624 1498 55 4 v 636 1540 a Ff(2)693 1556 y Fk(A)766 1571 y Fh(k)825 1410 y Fm(")873 1448 y Fh(m)p Fj(+2)890 1473 y Fm(X)893 1656 y Fh(i)p Fj(=2)1043 1556 y Fn(\026)g Fk(h)1108 1485 y Fg(i)p Ff(+)p Fg(N)5 b Fe(\000)p Fg(m)p Fe(\000)p Ff(2)p 1108 1499 304 4 v 1244 1540 a(2)1425 1556 y Fk(c)1467 1571 y Fh(i)p Fj(+)p Fh(N)i Fi(\000)p Fh(m)1730 1556 y Fk(x)1785 1515 y Fh(i)p Fj(+)p Fh(N)g Fi(\000)p Fh(m)2071 1556 y Fn(+)22 b Fk(W)2275 1515 y Fj([)p Fh(N)7 b Fj(+2])2472 1556 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))2702 1410 y Fm(#)2767 1556 y Fk(A)2840 1571 y Fh(k)2883 1556 y Fk( )2946 1571 y Fh(m)3013 1556 y Fn(\()p Fk(x)p Fn(\))395 1859 y Fl(\000)494 1751 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)528 1776 y Fm(X)508 1960 y Fh(n)p Fj(=)p Fh(N)699 1859 y Fn(\026)-50 b Fk(h)764 1789 y Fg(n)p 764 1801 39 4 v 768 1842 a Ff(2)926 1751 y Fh(N)7 b Fi(\000)p Fj(1)943 1776 y Fm(X)833 1960 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)1188 1859 y Fl(E)1241 1874 y Fh(j)1277 1859 y Fn(\()p Fk(A)1388 1874 y Fh(k)1431 1859 y Fn(\))1469 1818 y Fj(2)1508 1859 y Fk( )1571 1874 y Fh(n)p Fi(\000)p Fh(j)1706 1859 y Fn(\()p Fk(x)p Fn(\))0 2146 y(Finally)-8 b(,)30 b(note)j(that)f(\026)-50 b Fk(h)844 2072 y Fg(j)s Fe(\000)p Ff(2)p 844 2088 107 4 v 882 2129 a(2)965 2146 y Fk(c)1007 2161 y Fh(j)1044 2146 y Fk(x)1099 2110 y Fh(j)1158 2146 y Fn(+)22 b Fk(W)1362 2110 y Fj([)p Fh(j)t Fj(+1])1527 2146 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))28 b(=)f Fk(W)1994 2110 y Fj([)p Fh(j)t Fj(])2070 2146 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\).)43 b(It)33 b(then)g(follo)m(ws)f(that)0 2394 y Fk(\030)43 2409 y Fh(N)110 2394 y Fn(\()p Fk(x)p Fn(\))c(=)372 2286 y Fh(N)7 b Fi(\000)p Fj(1)389 2311 y Fm(X)373 2492 y Fh(m)p Fj(=0)542 2394 y Fn(\026)-50 b Fk(h)607 2324 y Fg(m)p 607 2336 55 4 v 619 2377 a Ff(2)676 2394 y Fk(A)749 2409 y Fh(k)792 2394 y Fk(W)898 2352 y Fj([)p Fh(N)7 b Fi(\000)p Fh(m)p Fj(+1])1211 2394 y Fn(\(\026)-50 b Fk(h)p Fn(;)17 b Fk(x)p Fn(\))p Fk(A)1514 2409 y Fh(k)1557 2394 y Fk( )1620 2409 y Fh(m)1687 2394 y Fn(\()p Fk(x)p Fn(\))23 b Fl(\000)1940 2286 y Fj(2)p Fh(N)7 b Fi(\000)p Fj(2)1975 2311 y Fm(X)1954 2494 y Fh(n)p Fj(=)p Fh(N)2145 2394 y Fn(\026)-50 b Fk(h)2210 2324 y Fg(n)p 2210 2336 39 4 v 2214 2377 a Ff(2)2372 2286 y Fh(N)7 b Fi(\000)p Fj(1)2389 2311 y Fm(X)2280 2494 y Fh(j)t Fj(=)p Fh(n)p Fi(\000)p Fh(N)g Fj(+1)2634 2394 y Fl(E)2687 2409 y Fh(j)2723 2394 y Fn(\()p Fk(A)2834 2409 y Fh(k)2877 2394 y Fn(\))2915 2352 y Fj(2)2955 2394 y Fk( )3018 2409 y Fh(m)p Fi(\000)p Fh(j)3172 2394 y Fn(\()p Fk(x)p Fn(\))0 2743 y Fw(References)0 2950 y Fv([1])41 b(Com)n(b)r(es)27 b(J)h(M,)f(Duclos)h(P)f(and)g (Seiler)h(R)f(1983)f Fa(J.)k(F)-6 b(unct.)29 b(A)n(nal.)f Ft(52)f Fv(257{301)0 3058 y([2])41 b(Hagedorn)26 b(G)i(A)g(and)f(Jo)n (y)n(e)f(A)i(1999)e Fa(Commun.)k(Math.)h(Phys.)e Ft(207)e Fv(439{465)0 3166 y([3])41 b(Hagedorn)36 b(G)i(A)h(and)e(Jo)n(y)n(e)f (A)i(2000)e(Exp)r(onen)n(tially)h(Accurate)g(Semiclassical)g(Dynamics:) 57 b(Propagation,)212 3274 y(Lo)r(calization,)38 b(Ehrenfest)e(Times,)i (Scattering)e(and)g(More)g(General)g(States,)i(to)f(app)r(ear)e(in)i Fa(A)n(nn.)g(Henri)212 3381 y(Poinc)l(ar)n(\023)-40 b(e.)0 3489 y Fv([4])41 b(Hel\013er)28 b(B)f(and)h(Sj\177)-42 b(ostrand)27 b(J)g(1984)f Fa(Comm.)31 b(PDE)d Ft(9)f Fv(337{408)0 3597 y([5])41 b(Ho)n(wland)27 b(J)g(S)h(1997)e Fa(J.)k(Phys.)h(A:)f(Math.)h(Gen.)d Ft(30)f Fv(2069{2076)0 3705 y([6])41 b(Simon)28 b(B)f(1983)f Fa(A)n(nn.)j(Inst.)h(Henri)f (Poinc)l(ar)n(\023)-40 b(e)31 b Fv(A)d Ft(38)g Fv(295{307)0 3813 y([7])41 b(Simon)28 b(B)f(1984)f Fa(A)n(nn.)j(of)i(Math.)g(\(2\))d Ft(120)f Fv(89{118)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0010041608160--