% ps file: A. Soffer and M.I. Weinstein - Nonautonomous Hamiltonians
% To appear in Journal of Statistical Physics
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11 10 bop 72 59 a Fo(The)33 b(follo)m(wing)c(h)m(yp)s(othesis)k(and)f
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%%Page: 16 16
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17 16 bop -74 59 a Fo(Pro)s(of)32 b(of)g(Prop)s(osition)f(4.1)p
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%%Page: 18 18
18 17 bop -74 59 a Fg(Prop)s(osition)36 b(4.3.)1081 308
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%%Page: 19 19
19 18 bop -74 59 a Fg(Remarks:)-74 208 y Fo(\(1\))23
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(erio)s(dic)f(p)s(erturbation,)h(is)g(made)f(explicit.)-74
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Fn(s)p Fo(\))p Fl(j)p Fn(ds)1187 1583 y Fl(\024)116 b(h)p
Fn(t)p Fl(i)1493 1542 y Fk(r)r Fi(\000)p Fm(1)1621 1583
y Fn(e)1666 1542 y Fi(\000)1731 1515 y Fh(1)p 1731 1527
31 4 v 1731 1568 a(2)1771 1542 y Fm(\000)p Fk(t)1867
1583 y Fl(\001)1917 1466 y Fe(Z)2000 1493 y Fk(M)1963
1655 y Fm(0)2096 1583 y Fn(e)2141 1542 y Fi(\000)p Fm(\000\()2277
1515 y Fh(1)p 2277 1527 V 2277 1568 a(2)2318 1542 y Fk(t)p
Fi(\000)p Fk(s)p Fm(\))2495 1583 y Fn(ds)54 b Fl(\001)109
b Fo(sup)2696 1667 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p
Fm(\000)2924 1648 y Fc(\000)p Fa(\013)3033 1583 y Fl(j)3083
1558 y Fo(~)3061 1583 y Fn(E)d Fo(\()p Fn(\034)11 b Fo(\))p
Fl(j)1187 1819 y(\024)228 b Fo(sup)1380 1903 y Fm(2\000)1459
1884 y Fc(\000)p Fa(\013)1552 1903 y Fi(\024)p Fk(\034)8
b Fi(\024)p Fk(T)1751 1819 y Fo(\()p Fl(h)p Fn(t)p Fl(i)1902
1778 y Fk(r)r Fi(\000)p Fm(1)2030 1819 y Fn(e)2075 1778
y Fi(\000)p Fm(\000)2187 1751 y Fa(t)p 2184 1763 V 2184
1804 a Fh(2)2229 1819 y Fo(\))22 b Fl(\001)g Fn(C)7 b
Fo(\000)2477 1778 y Fi(\000)p Fm(1)2593 1819 y Fl(\001)108
b Fo(sup)2643 1903 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p
Fm(\000)2871 1884 y Fc(\000)p Fa(\013)2980 1819 y Fl(j)3030
1794 y Fo(~)3008 1819 y Fn(E)d Fo(\()p Fn(\034)11 b Fo(\))p
Fl(j)1187 2055 y(\024)116 b Fn(C)7 b(e)1502 2014 y Fi(\000)p
Fm(\000)1601 1990 y Fc(\000)p Fa(\016)1823 2055 y Fo(sup)1736
2139 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fm(\000)1964
2120 y Fc(\000)p Fa(\013)2073 2055 y Fl(j)2124 2030 y
Fo(~)2101 2055 y Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p
Fl(j)p Fn(;)1362 b Fo(\(5.14\))-74 2322 y(for)32 b(some)h
Fn(\016)e(>)d Fo(0.)43 b(Therefore,)1008 2557 y(sup)896
2641 y Fm(2\000)975 2622 y Fc(\000)p Fa(\013)1068 2641
y Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)1284 2460 y Fe(\020)1333
2557 y Fl(h)p Fn(t)p Fl(i)1446 2516 y Fk(r)r Fi(\000)p
Fm(1)1607 2557 y Fn(I)1650 2572 y Fm(1)1689 2557 y Fo(\()p
Fn(t)p Fo(\))1800 2460 y Fe(\021)1910 2557 y Fl(\024)61
b Fn(C)7 b(e)2170 2516 y Fi(\000)p Fm(\000)2269 2492
y Fc(\000)p Fa(\016)2491 2557 y Fo(sup)2404 2641 y Fm(0)p
Fi(\024)p Fk(\034)h Fi(\024)p Fm(\000)2632 2622 y Fc(\000)p
Fa(\013)2741 2557 y Fl(j)2792 2532 y Fo(~)2769 2557 y
Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)721 b Fo(\(5.15\))-74
2831 y(W)-8 b(e)33 b(estimate)f Fn(I)528 2846 y Fm(2)567
2831 y Fo(\()p Fn(t)p Fo(\))h(on)f(the)h(in)m(terv)-5
b(al)32 b(2\000)1477 2794 y Fi(\000)p Fk(\013)1608 2831
y Fl(\024)d Fn(t)e Fl(\024)i Fn(T)46 b Fo(as)33 b(follo)m(ws:)511
3081 y Fl(h)p Fn(t)p Fl(i)624 3040 y Fk(r)r Fi(\000)p
Fm(1)785 3081 y Fn(I)828 3096 y Fm(2)867 3081 y Fo(\()p
Fn(t)p Fo(\))61 b Fl(\024)f(h)p Fn(t)p Fl(i)1289 3040
y Fk(r)r Fi(\000)p Fm(1)1466 2964 y Fe(Z)1549 2990 y
Fk(t)1512 3153 y Fm(\000)1556 3134 y Fc(\000)p Fa(\013)1669
3081 y Fn(e)1714 3040 y Fi(\000)p Fm(\000\()p Fk(t)p
Fi(\000)p Fk(s)p Fm(\))1986 3081 y Fl(h)p Fn(s)p Fl(i)2110
3040 y Fi(\000)p Fk(r)r Fm(+1)2325 3081 y Fn(ds)143 b
Fo(sup)2470 3165 y Fm(\000)2514 3146 y Fc(\000)p Fa(\013)2607
3165 y Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)2823 2985
y Fe(\020)2872 3081 y Fl(h)p Fn(\034)j Fl(i)3003 3040
y Fk(r)r Fi(\000)p Fm(1)3154 3056 y Fo(~)3132 3081 y
Fn(E)6 b Fo(\()p Fn(\034)11 b Fo(\))3339 2985 y Fe(\021)3725
3081 y Fo(\(5.16\))-74 3348 y(The)34 b(in)m(tegral)d(is)h(no)m(w)h(b)s
(ounded)g(ab)s(o)m(v)m(e)h(using)e(the)h(estimate)837
3593 y Fl(h)p Fn(t)p Fl(i)950 3552 y Fk(r)r Fi(\000)p
Fm(1)1127 3476 y Fe(Z)1210 3502 y Fk(t)1173 3665 y Fm(\000)1217
3646 y Fc(\000)p Fa(\013)1363 3593 y Fn(e)1408 3552 y
Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1712
3593 y Fl(h)p Fn(s)p Fl(i)1836 3552 y Fi(\000)p Fk(r)r
Fm(+1)2051 3593 y Fn(ds)59 b Fl(\024)i Fn(C)7 b Fo(\000)2483
3552 y Fi(\000)p Fm(1)2577 3593 y Fn(;)50 b(t)27 b Fl(\025)i
Fo(2\000)2932 3552 y Fi(\000)p Fk(\013)3036 3593 y Fn(:)662
b Fo(\(5.17\))72 3828 y(This)33 b(giv)m(es)944 3978 y(sup)839
4062 y Fm(2\000)918 4043 y Fc(\000)p Fa(\013)1011 4062
y Fi(\024)p Fk(t)p Fi(\024)p Fk(T)1197 3978 y Fl(h)p
Fn(t)p Fl(i)1310 3937 y Fk(r)r Fi(\000)p Fm(1)1470 3978
y Fn(I)1513 3993 y Fm(2)1553 3978 y Fo(\()p Fn(t)p Fo(\))60
b Fl(\024)g Fn(C)7 b Fo(\000)1999 3937 y Fi(\000)p Fm(1)2237
3978 y Fo(sup)2143 4062 y Fm(\000)2187 4043 y Fc(\000)p
Fa(\013)2279 4062 y Fi(\024)p Fk(\034)h Fi(\024)p Fk(T)2495
3882 y Fe(\020)2545 3978 y Fl(h)p Fn(\034)j Fl(i)2676
3937 y Fk(r)r Fi(\000)p Fm(1)2827 3953 y Fo(~)2804 3978
y Fn(E)6 b Fo(\()p Fn(\034)11 b Fo(\))3011 3882 y Fe(\021)3725
3978 y Fo(\(5.18\))72 4212 y(Assem)m(bling)26 b(the)g(estimates)g
(\(5.15\))f(and)h(\(5.18\))f(yields)h(estimate)f(\(5.4\))g(of)h(Prop)s
(osition)e(5.1.)41 b(Estimate)-74 4362 y(\(5.5\))32 b(is)g(a)h(simple)e
(consequence)k(of)d(the)h(de\014nition)f(of)g Fn(R)2091
4377 y Fk(A)2148 4362 y Fo(\()p Fn(t)p Fo(\).)-74 4659
y Fp(6.)46 b(Disp)t(ersiv)l(e)g(Estimates)h(and)d(Lo)t(cal)h(Deca)l(y)
-11 b(.)-74 4907 y Fo(In)35 b(this)f(section)h(w)m(e)g(pro)m(v)m(e)h
(the)f(lo)s(cal)d(deca)m(y)k(of)d Fn(\036)1825 4922 y
Fk(d)1900 4907 y Fo(and)i(the)f(deca)m(y)i(in)e(time)f(of)h(the)h
(remainder)e(terms)i(in)-74 5057 y(the)e(ODE)f(of)g(Section)h(4.)43
b(T)-8 b(o)33 b(this)f(end)h(w)m(e)h(will)c(rep)s(eatedly)j(use)h(the)f
(follo)m(wing:)1901 5306 y(21)p eop
%%Page: 22 22
22 21 bop -74 59 a Fg(Lemma)37 b(6.1.)49 b Fb(F)-8 b(or)32
b(an)m(y)h Fn(\021)f Fl(2)c Fo([0)p Fn(;)17 b(r)24 b
Fl(\000)f Fo(1])687 187 y Fe(\015)687 237 y(\015)687
287 y(\015)687 337 y(\015)733 195 y(Z)816 221 y Fk(t)780
383 y Fm(0)863 312 y Fn(w)933 327 y Fi(\000)991 312 y
Fn(e)1036 271 y Fi(\000)p Fk(iH)1173 280 y Fh(0)1208
271 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1380 312
y Fg(P)1457 327 y Ff(c)1497 312 y Fn(f)11 b Fo(\()p Fn(s)p
Fo(\))p Fn(ds)1775 187 y Fe(\015)1775 237 y(\015)1775
287 y(\015)1775 337 y(\015)1881 312 y Fl(\024)60 b Fn(C)7
b Fl(h)p Fn(t)p Fl(i)2208 271 y Fi(\000)p Fk(\021)2353
312 y Fo(sup)2321 390 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p
Fk(t)2547 312 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2716 271
y Fk(\021)2758 312 y Fl(k)p Fn(w)2878 327 y Fm(+)2937
312 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))560
b(\(6.1\))-74 575 y Fb(and)94 688 y Fe(\015)94 738 y(\015)94
788 y(\015)94 838 y(\015)140 696 y(Z)223 722 y Fk(t)186
884 y Fm(0)270 813 y Fn(w)340 828 y Fi(\000)398 813 y
Fn(e)443 772 y Fi(\000)p Fk(iH)580 781 y Fh(0)615 772
y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))787 813 y Fg(P)864
828 y Ff(c)904 813 y Fo(\()p Fn(H)1023 828 y Fm(0)1084
813 y Fl(\000)23 b Fn(\025)1241 828 y Fm(0)1302 813 y
Fl(\000)g Fn(\026)f Fl(\000)g Fn(i)p Fo(0\))1702 772
y Fi(\000)p Fm(1)1797 813 y Fn(f)11 b Fo(\()p Fn(s)p
Fo(\))32 b Fn(ds)2107 688 y Fe(\015)2107 738 y(\015)2107
788 y(\015)2107 838 y(\015)2213 813 y Fl(\024)60 b Fn(C)7
b Fl(h)p Fn(t)p Fl(i)2540 772 y Fi(\000)p Fk(\021)2685
813 y Fo(sup)2654 890 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p
Fk(t)2880 813 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)3049 772
y Fk(\021)3107 813 y Fl(k)p Fn(w)3227 828 y Fm(+)3286
813 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))16
b Fn(:)168 b Fo(\(6.2\))-74 1095 y(Pro)s(of.)p -74 1108
262 4 v 33 w(The)34 b(pro)s(of)f(follo)m(ws)f(from)g(the)i(assumed)g
(lo)s(cal)d(deca)m(y)k(estimates)e(on)g Fn(e)2905 1059
y Fi(\000)p Fk(iH)3042 1068 y Fh(0)3077 1059 y Fk(t)3107
1095 y Fo(;)g(see)i Fg(\(H3\))p Fo(.)45 b(Namely)-8 b(,)-74
1244 y(using)32 b(that)h Fn(r)d(>)e Fo(2,)225 1368 y
Fe(\015)225 1417 y(\015)225 1467 y(\015)225 1517 y(\015)271
1375 y(Z)354 1401 y Fk(t)317 1564 y Fm(0)400 1492 y Fn(w)470
1507 y Fi(\000)529 1492 y Fn(e)574 1451 y Fi(\000)p Fk(iH)711
1460 y Fh(0)746 1451 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p
Fm(\))918 1492 y Fg(P)995 1507 y Ff(c)1035 1492 y Fn(f)11
b Fo(\()p Fn(s)p Fo(\))32 b Fn(ds)1345 1368 y Fe(\015)1345
1417 y(\015)1345 1467 y(\015)1345 1517 y(\015)1506 1492
y Fl(\024)1699 1375 y Fe(Z)1782 1401 y Fk(t)1745 1564
y Fm(0)1828 1492 y Fl(k)p Fn(w)1948 1507 y Fi(\000)2007
1492 y Fn(e)2052 1451 y Fi(\000)p Fk(iH)2189 1460 y Fh(0)2224
1451 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2428 1492
y Fg(P)2505 1507 y Ff(c)2545 1492 y Fn(w)2615 1507 y
Fi(\000)2674 1492 y Fl(k)2724 1508 y Fi(L)p Fm(\()p Fi(H)p
Fm(\))2891 1492 y Fl(h)p Fn(s)p Fl(i)3015 1451 y Fi(\000)p
Fk(\021)3144 1492 y Fn(ds)1699 1675 y Fl(\001)48 b Fo(sup)1744
1752 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(t)1969
1675 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2138 1634 y Fk(\021)2180
1675 y Fl(k)p Fn(w)2300 1690 y Fm(+)2359 1675 y Fn(f)g
Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))1506 1937 y Fl(\024)116
b Fn(C)1793 1820 y Fe(Z)1876 1846 y Fk(t)1839 2008 y
Fm(0)1905 1937 y Fl(h)p Fn(t)22 b Fl(\000)h Fn(s)p Fl(i)2186
1896 y Fi(\000)p Fk(r)r Fm(+1)2369 1937 y Fl(h)p Fn(s)p
Fl(i)2493 1896 y Fi(\000)p Fk(\021)2621 1937 y Fn(ds)80
b Fo(sup)2767 2014 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p
Fk(t)2993 1937 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)3162 1896
y Fk(\021)3204 1937 y Fl(k)p Fn(w)3324 1952 y Fm(+)3383
1937 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))1506
2148 y Fl(\024)83 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)1856
2107 y Fi(\000)p Fk(\021)2001 2148 y Fo(sup)1970 2226
y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fk(t)2196 2148
y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2365 2107 y Fk(\021)2423
2148 y Fl(k)p Fn(w)2543 2163 y Fm(+)2602 2148 y Fn(f)g
Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))-74 2417 y(whic)m(h)32
b(pro)m(v)m(es)g(\(6.1\).)43 b(The)31 b(pro)s(of)f(of)h(\(6.2\))f(is)h
(iden)m(tical,)e(and)i(uses)i(the)e(singular)e(lo)s(cal)g(deca)m(y)j
(estimate)e(of)-74 2566 y Fg(\(H3\))p Fo(.)72 2716 y(W)-8
b(e)33 b(no)m(w)h(de\014ne)f(the)g(norms)1354 2865 y([)p
Fn(A)p Fo(])1481 2880 y Fk(\013)1530 2865 y Fo(\()p Fn(T)14
b Fo(\))60 b(=)104 b(sup)1873 2944 y Fm(0)p Fi(\024)p
Fk(\034)8 b Fi(\024)p Fk(T)2108 2865 y Fl(h)p Fn(\034)j
Fl(i)2239 2824 y Fk(\013)2289 2865 y Fl(j)p Fn(A)p Fo(\()p
Fn(\034)g Fo(\))p Fl(j)1226 b Fo(\(6.3\))-74 3090 y(and)1178
3239 y([)p Fn(\036)1263 3254 y Fk(d)1304 3239 y Fo(])1331
3254 y Fk(LD)r(;\013)1508 3239 y Fo(\()p Fn(T)14 b Fo(\))59
b(=)104 b(sup)1850 3318 y Fm(0)p Fi(\024)p Fk(\034)8
b Fi(\024)p Fk(T)2085 3239 y Fl(h)p Fn(\034)j Fl(i)2216
3198 y Fk(\013)2266 3239 y Fl(k)p Fn(w)2386 3254 y Fi(\000)2444
3239 y Fn(\036)2502 3254 y Fk(d)2543 3239 y Fo(\()p Fn(\034)g
Fo(\))p Fl(k)1051 b Fo(\(6.4\))-74 3464 y(Then)34 b(w)m(e)f(ha)m(v)m(e)
-74 3709 y Fg(Prop)s(osition)j(6.1.)49 b Fb(F)-8 b(or)32
b(an)m(y)h Fn(T)41 b(>)28 b Fo(0)p Fb(,)796 3946 y Fo([)p
Fn(\036)881 3961 y Fk(d)921 3946 y Fo(])948 3961 y Fk(LD)r(;r)r
Fi(\000)p Fm(1)1204 3946 y Fo(\()p Fn(T)14 b Fo(\))27
b Fl(\024)h Fn(C)c Fo(\()32 b Fl(k)p Fn(w)1767 3961 y
Fm(+)1826 3946 y Fn(\036)1884 3961 y Fk(d)1924 3946 y
Fo(\(0\))p Fl(k)54 b Fo(+)h Fl(jjj)p Fn(W)14 b Fl(jjj)30
b Fo([)p Fn(A)p Fo(])2715 3961 y Fk(r)r Fi(\000)p Fm(1)2843
3946 y Fo(\()p Fn(T)14 b Fo(\))32 b(\))17 b Fn(:)669
b Fo(\(6.5\))-74 4191 y(Pro)s(of.)p -74 4204 V 31 w(F)-8
b(rom)30 b(equation)h(\(4.9\))f(w)m(e)j(get,)e(using)g(the)h(assumed)g
(lo)s(cal)d(deca)m(y)k(estimate)d(for)h Fn(e)3325 4155
y Fi(\000)p Fk(iH)3462 4164 y Fh(0)3497 4155 y Fk(t)3558
4191 y Fo(and)g(\(6.1\),)281 4596 y Fl(k)o Fn(w)400 4611
y Fi(\000)459 4596 y Fn(\036)517 4611 y Fk(d)557 4596
y Fo(\()p Fn(t)p Fo(\))p Fl(k)116 b(\024)1070 4488 y
Fm(2)1028 4513 y Fe(X)1027 4695 y Fk(j)t Fm(=0)1199 4596
y Fl(k)p Fn(w)1319 4611 y Fi(\000)1377 4596 y Fn(\036)1435
4611 y Fk(j)1471 4596 y Fo(\()p Fn(t)p Fo(\))p Fl(k)834
4834 y(\024)g Fn(C)7 b Fl(h)p Fn(t)p Fl(i)1217 4793 y
Fi(\000)p Fk(r)r Fm(+1)1416 4834 y Fl(k)p Fn(w)1536 4849
y Fm(+)1595 4834 y Fn(\036)1653 4849 y Fk(d)1693 4834
y Fo(\(0\))p Fl(k)22 b Fo(+)54 b Fn(C)7 b Fl(h)p Fn(t)p
Fl(i)2210 4793 y Fi(\000)p Fk(r)r Fm(+1)2425 4834 y Fo([)p
Fn(A)p Fo(])2552 4849 y Fk(r)r Fi(\000)p Fm(1)2681 4834
y Fo(\()p Fn(t)p Fo(\))77 b(sup)2841 4911 y Fm(0)p Fi(\024)p
Fk(s)p Fi(\024)p Fk(t)3061 4834 y Fl(k)p Fn(w)3181 4849
y Fm(+)3239 4834 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn( )3530
4849 y Fm(0)3570 4834 y Fl(k)835 5048 y Fo(+)116 b Fn(C)39
b Fl(jjj)p Fn(W)14 b Fl(jjj)31 b(h)p Fn(t)p Fl(i)1554
5007 y Fi(\000)p Fk(r)r Fm(+1)1753 5048 y Fo([)p Fn(\036)1838
5063 y Fk(d)1878 5048 y Fo(])1905 5077 y Fk(LD)r(;r)r
Fi(\000)p Fm(1)2177 5048 y Fo(\()p Fn(t)p Fo(\))p Fn(:)1458
b Fo(\(6.6\))1901 5306 y(22)p eop
%%Page: 23 23
23 22 bop 72 59 a Fo(Since)28 b Fl(k)p Fn(w)442 74 y
Fm(+)501 59 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn( )792
74 y Fm(0)831 59 y Fl(k)28 b(\024)g(jjj)p Fn(W)14 b Fl(jjj)25
b(k)p Fn( )1426 74 y Fm(0)1466 59 y Fl(k)55 b Fo(=)g
Fl(jjj)p Fn(W)14 b Fl(jjj)26 b Fo(and)h Fl(jjj)p Fn(W)14
b Fl(jjj)26 b Fo(is)h(assumed)h(to)g(b)s(e)g(small,)e(m)m(ultiplying)
-74 208 y(b)s(oth)32 b(sides)i(of)e(this)g(last)g(equation)g(b)m(y)i
Fl(h)p Fn(t)p Fl(i)1524 172 y Fk(r)r Fi(\000)p Fm(1)1684
208 y Fo(and)f(taking)e(suprem)m(um)i(o)m(v)m(er)h Fn(t)28
b Fl(\024)g Fn(T)46 b Fo(yields)32 b(\(6.5\).)206 b([])72
358 y(W)-8 b(e)33 b(no)m(w)h(estimate)d Fn(E)6 b Fo(\()p
Fn(t)p Fo(\).)-74 615 y Fg(Prop)s(osition)36 b(6.2.)49
b Fb(Let)33 b Fn(T)41 b(>)27 b Fo(0)p Fb(.)44 b(F)-8
b(or)31 b(an)m(y)j Fn(\021)d Fl(2)d Fo([0)p Fn(;)17 b(r)25
b Fl(\000)d Fo(1])p Fb(:)371 864 y Fo([)p Fn(E)6 b Fo(])503
879 y Fk(\021)545 864 y Fo(\()p Fn(T)14 b Fo(\))60 b
Fl(\024)g Fn(C)983 768 y Fe(\020)1065 864 y Fl(jjj)p
Fn(W)14 b Fl(jjj)1339 823 y Fm(2)1409 864 y Fl(j)p Fn(A)p
Fo(\(0\))p Fl(j)53 b Fo(+)i Fl(jjj)p Fn(W)14 b Fl(jjj)30
b(k)p Fn(w)2271 879 y Fm(+)2330 864 y Fn(\036)2388 879
y Fk(d)2428 864 y Fo(\(0\))p Fl(k)54 b Fo(+)h Fl(jjj)p
Fn(W)14 b Fl(jjj)3062 823 y Fm(3)3131 864 y Fo([)p Fn(A)p
Fo(])3258 879 y Fk(\021)3301 864 y Fo(\()p Fn(T)g Fo(\))3480
768 y Fe(\021)3773 864 y Fo(\(6.7\))-74 1121 y(Pro)s(of.)p
-74 1134 262 4 v 72 1271 a Fn(E)6 b Fo(\()p Fn(t)p Fo(\))32
b(is)f(de\014ned)h(in)f(\(5.6\))g(and)g(\(4.19\).)42
b(F)-8 b(rom)30 b(these)j(equations)e(it)g(is)f(seen)j(that)e(w)m(e)h
(need)h(to)e(estimate)-74 1420 y(the)i(follo)m(wing)d(terms:)253
1819 y(\()p Fn(\013)353 1834 y Fm(1)393 1819 y Fo(\))83
b Fl(j)p Fn(R)616 1834 y Fm(1)p Fk(a)692 1819 y Fl(j)28
b Fo(=)f Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)1153 1719 y Fe(\014)1153
1769 y(\014)1153 1819 y(\014)1181 1722 y(\020)1231 1819
y Fn(\014)6 b( )1355 1834 y Fm(0)1394 1819 y Fn(;)17
b(e)1483 1778 y Fi(\000)p Fk(iH)1620 1787 y Fh(0)1655
1778 y Fk(t)1684 1819 y Fo(\()p Fn(H)1803 1834 y Fm(0)1865
1819 y Fl(\000)22 b Fn(\025)2021 1834 y Fm(0)2083 1819
y Fo(+)g Fn(\026)p Fo(\))2278 1778 y Fi(\000)p Fm(1)2372
1819 y Fg(P)2449 1834 y Ff(c)2489 1819 y Fn(\014)6 b( )2613
1834 y Fm(0)2652 1722 y Fe(\021)2702 1719 y(\014)2702
1769 y(\014)2702 1819 y(\014)253 2033 y Fo(\()p Fn(\013)353
2048 y Fm(2)393 2033 y Fo(\))83 b Fl(j)p Fn(R)616 2048
y Fm(1)p Fk(b)685 2033 y Fl(j)27 b Fo(=)h Fl(j)p Fn(A)p
Fo(\(0\))p Fl(j)1115 1909 y Fe(\014)1114 1959 y(\014)1114
2009 y(\014)1114 2058 y(\014)1141 1916 y(Z)1224 1943
y Fk(t)1187 2105 y Fm(0)1271 1937 y Fe(\020)1320 2033
y Fn(\014)6 b( )1444 2048 y Fm(0)1484 2033 y Fn(;)17
b(e)1573 1992 y Fi(\000)p Fk(iH)1710 2001 y Fh(0)1744
1992 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1916 2033
y Fn(e)1961 1992 y Fk(i)p Fm(\()p Fk(\026)p Fi(\000)p
Fk(\025)2150 2001 y Fh(0)2186 1992 y Fm(\))p Fk(s)2250
2033 y Fo(\()p Fn(H)2369 2048 y Fm(0)2430 2033 y Fl(\000)23
b Fn(\025)2587 2048 y Fm(0)2649 2033 y Fo(+)f Fn(\026)p
Fo(\))2844 1992 y Fi(\000)p Fm(1)2937 2033 y Fn(@)2988
2048 y Fk(s)3026 2033 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p
Fg(P)3298 2048 y Ff(c)3338 2033 y Fn(\014)6 b( )3462
2048 y Fm(0)3501 1937 y Fe(\021)3567 2033 y Fn(ds)3664
1909 y Fe(\014)3664 1959 y(\014)3664 2009 y(\014)3664
2058 y(\014)253 2273 y Fo(\()p Fn(\013)353 2288 y Fm(3)393
2273 y Fo(\))83 b Fl(j)p Fn(R)616 2288 y Fm(3)655 2273
y Fl(j)27 b Fo(=)824 2206 y(1)p 824 2250 49 4 v 824 2341
a(2)882 2273 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)1185 2173
y Fe(\014)1185 2223 y(\014)1185 2273 y(\014)o Fo(\()p
Fn(\014)6 b( )1374 2288 y Fm(0)1414 2273 y Fn(;)17 b(e)1503
2232 y Fi(\000)p Fk(iH)1640 2241 y Fh(0)1674 2232 y Fk(t)1704
2273 y Fo(\()p Fn(H)1823 2288 y Fm(0)1884 2273 y Fl(\000)23
b Fn(\025)2041 2288 y Fm(0)2102 2273 y Fl(\000)g Fn(\026)f
Fl(\000)g Fn(i)p Fo(0\))2502 2232 y Fi(\000)p Fm(1)2597
2273 y Fg(P)2674 2288 y Ff(c)2714 2273 y Fn(\014)6 b( )2838
2288 y Fm(0)2877 2273 y Fo(\))2915 2173 y Fe(\014)2915
2223 y(\014)2915 2273 y(\014)253 2506 y Fo(\()p Fn(\013)353
2521 y Fm(4)393 2506 y Fo(\))83 b Fl(j)p Fn(R)616 2521
y Fm(4)655 2506 y Fl(j)27 b Fo(=)824 2439 y(1)p 824 2483
V 824 2575 a(2)899 2382 y Fe(\014)899 2432 y(\014)899
2481 y(\014)899 2531 y(\014)927 2389 y(Z)1010 2415 y
Fk(t)973 2578 y Fm(0)1039 2506 y Fo(\()p Fn(\014)6 b( )1201
2521 y Fm(0)1241 2506 y Fn(;)17 b(e)1330 2465 y Fi(\000)p
Fk(iH)1467 2474 y Fh(0)1501 2465 y Fm(\()p Fk(t)p Fi(\000)p
Fk(s)p Fm(\))1673 2506 y Fo(\()p Fn(H)1792 2521 y Fm(0)1854
2506 y Fl(\000)22 b Fn(\025)2010 2521 y Fm(0)2072 2506
y Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2472 2465
y Fi(\000)p Fm(1)2566 2506 y Fn(e)2611 2465 y Fi(\000)p
Fk(i)p Fm(\()p Fk(\025)2758 2474 y Fh(0)2793 2465 y Fm(+)p
Fk(\026)p Fm(\))p Fk(s)2955 2506 y Fn(@)3006 2521 y Fk(s)3043
2506 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)3315 2521 y
Ff(c)3355 2506 y Fn(\014)6 b( )3479 2521 y Fm(0)3519
2506 y Fo(\))32 b Fn(ds)3686 2382 y Fe(\014)3686 2432
y(\014)3686 2481 y(\014)3686 2531 y(\014)253 2701 y Fo(\()p
Fn(\013)353 2716 y Fm(5)393 2701 y Fo(\))83 b Fl(j)p
Fo(\()p Fn(\014)6 b( )704 2716 y Fm(0)743 2701 y Fn(;)17
b(\036)845 2716 y Fm(0)883 2701 y Fo(\))p Fl(j)28 b Fo(=)f
Fl(j)p Fo(\()p Fn(\014)6 b( )1270 2716 y Fm(0)1309 2701
y Fn(;)17 b(e)1398 2660 y Fi(\000)p Fk(iH)1535 2669 y
Fh(0)1570 2660 y Fk(t)1599 2701 y Fn(\036)1657 2716 y
Fk(d)1698 2701 y Fo(\(0\)\))p Fl(j)253 2891 y Fo(\()p
Fn(\013)353 2906 y Fm(6)393 2891 y Fo(\))83 b Fl(j)p
Fo(\()p Fn(\014)6 b( )704 2906 y Fm(0)743 2891 y Fn(;)17
b(\036)845 2906 y Fm(2)883 2891 y Fo(\()p Fn(t)p Fo(\)\))p
Fl(j)28 b Fo(=)1191 2766 y Fe(\014)1191 2816 y(\014)1191
2866 y(\014)1191 2915 y(\014)1219 2773 y(Z)1302 2800
y Fk(t)1265 2962 y Fm(0)1332 2891 y Fo(\()p Fn(\014)6
b( )1494 2906 y Fm(0)1533 2891 y Fn(;)17 b(e)1622 2849
y Fi(\000)p Fk(iH)1759 2858 y Fh(0)1793 2849 y Fm(\()p
Fk(t)p Fi(\000)p Fk(s)p Fm(\))1982 2891 y Fo(cos)g Fn(\026s)32
b Fg(P)2343 2906 y Ff(c)2416 2891 y Fn(\014)6 b(\036)2535
2906 y Fk(d)2575 2891 y Fo(\()p Fn(s)p Fo(\))32 b Fn(ds)p
Fo(\))2864 2766 y Fe(\014)2864 2816 y(\014)2864 2866
y(\014)2864 2915 y(\014)2908 2891 y Fn(:)-74 3148 y Fo(The)i(estimates)
e(of)g(\()p Fn(\013)767 3163 y Fk(j)803 3148 y Fo(\))h(rep)s(eatedly)g
(use)g(Lemma)f(6.1.)43 b(Let)32 b Fn(\021)g Fl(2)c Fo([0)p
Fn(;)17 b(r)24 b Fl(\000)f Fo(1].)-74 3297 y(Estimation)31
b(of)h(\()p Fn(\013)638 3312 y Fm(1)677 3297 y Fo(\):)p
-74 3336 817 4 v 532 3696 a Fl(j)p Fn(R)634 3711 y Fm(1)p
Fk(a)710 3696 y Fl(j)83 b(\024)h(j)p Fn(A)p Fo(\(0\))p
Fl(j)1253 3600 y Fe(\020)1301 3696 y Fn(w)1371 3711 y
Fm(+)1430 3696 y Fn(\014)6 b( )1554 3711 y Fm(0)1593
3696 y Fn(;)49 b(w)1739 3711 y Fi(\000)1798 3696 y Fn(e)1843
3655 y Fi(\000)p Fk(iH)1980 3664 y Fh(0)2015 3655 y Fk(t)2044
3696 y Fo(\()p Fn(H)2163 3711 y Fm(0)2225 3696 y Fl(\000)22
b Fn(\025)2381 3711 y Fm(0)2443 3696 y Fo(+)g Fn(\026)p
Fo(\))2638 3655 y Fi(\000)p Fm(1)2732 3696 y Fg(P)2809
3711 y Ff(c)2849 3696 y Fn(w)2919 3711 y Fi(\000)3010
3696 y Fn(w)3080 3711 y Fm(+)3139 3696 y Fn(\014)6 b( )3263
3711 y Fm(0)3302 3600 y Fe(\021)821 3870 y Fl(\024)84
b Fn(C)7 b Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)31 b(jjj)p Fn(W)14
b Fl(jjj)1618 3829 y Fm(2)1688 3870 y Fl(h)p Fn(t)p Fl(i)1801
3829 y Fi(\000)p Fk(r)r Fm(+1)3773 3870 y Fo(\(6.8\))-74
4119 y(b)m(y)32 b(lo)s(cal)c(deca)m(y)33 b(estimates)d(and)h(our)g
(assumptions)f(on)h Fn(H)2112 4134 y Fm(0)2151 4119 y
Fo(,)g(since)h Fn(\025)2504 4134 y Fm(0)2561 4119 y Fl(\000)19
b Fn(\026)31 b Fo(is)f(not)h(in)f(the)h(sp)s(ectrum)g(of)g
Fn(H)3908 4134 y Fm(0)3947 4119 y Fo(.)-74 4319 y(Estimation)g(of)h(\()
p Fn(\013)638 4334 y Fm(2)677 4319 y Fo(\))p -74 4357
789 4 v 72 4468 a(F)-8 b(rom)31 b(\(5.6\))i(w)m(e)g(ha)m(v)m(e)h(that)
1211 4717 y Fl(j)p Fn(@)1290 4732 y Fk(s)1327 4717 y
Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fl(j)27 b(\024)h Fn(C)7
b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(j)p Fn(A)p Fo(\()p Fn(s)p
Fo(\))p Fl(j)22 b Fo(+)g Fl(j)p Fn(E)6 b Fo(\()p Fn(s)p
Fo(\))p Fl(j)1083 b Fo(\(6.9\))-74 4966 y(since)33 b
Fn(\032)g Fo(and)g(\000)f(are,)h(resp)s(ectiv)m(ely)-8
b(,)33 b(linear)e(and)i(quadratic)f(in)g Fl(jjj)p Fn(W)14
b Fl(jjj)p Fo(.)1901 5306 y(23)p eop
%%Page: 24 24
24 23 bop 72 59 a Fo(Applying)32 b(Lemma)f(6.1)h(to)h(\()p
Fn(\013)1220 74 y Fm(2)1259 59 y Fo(\))g(w)m(e)g(then)g(get)108
283 y Fl(j)p Fn(R)210 298 y Fm(1)p Fk(b)280 283 y Fl(j)115
b(\024)h(j)p Fn(A)p Fo(\(0\))p Fl(j)887 158 y Fe(\014)886
208 y(\014)886 258 y(\014)886 307 y(\014)914 165 y(Z)997
192 y Fk(t)960 354 y Fm(0)1043 186 y Fe(\020)1092 283
y Fn(w)1162 298 y Fm(+)1221 283 y Fn(\014)6 b( )1345
298 y Fm(0)1385 283 y Fn(;)49 b(w)1531 298 y Fi(\000)1590
283 y Fn(e)1635 241 y Fi(\000)p Fk(iH)1772 250 y Fh(0)1806
241 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1978 283
y Fo(\()p Fn(H)2097 298 y Fm(0)2159 283 y Fl(\000)22
b Fn(\025)2315 298 y Fm(0)2377 283 y Fo(+)g Fn(\026)p
Fo(\))2572 241 y Fi(\000)p Fm(1)2666 283 y Fg(P)2743
298 y Ff(c)2783 283 y Fn(w)2853 298 y Fi(\000)2944 283
y Fn(@)2995 298 y Fk(s)3032 283 y Fn(A)p Fo(\()p Fn(s)p
Fo(\))33 b Fn(w)3330 298 y Fm(+)3388 283 y Fn(\014)6
b( )3512 298 y Fm(0)3552 186 y Fe(\021)3651 283 y Fn(ds)3748
158 y Fe(\014)3748 208 y(\014)3748 258 y(\014)3748 307
y(\014)423 474 y Fl(\024)116 b Fn(C)7 b Fl(jjj)p Fn(W)14
b Fl(jjj)967 433 y Fm(2)1037 474 y Fl(h)p Fn(t)p Fl(i)1150
433 y Fi(\000)p Fk(\021)1263 474 y Fo(\()p Fl(jjj)p Fn(W)g
Fl(jjj)30 b Fo([)p Fn(A)p Fo(])1732 489 y Fk(\021)1774
474 y Fo(\()p Fn(t)p Fo(\))55 b(+)g([)p Fn(E)6 b Fo(])2203
489 y Fk(\021)2245 474 y Fo(\()p Fn(t)p Fo(\)\))16 b
Fn(:)1288 b Fo(\(6.10\))-74 683 y(Estimation)31 b(of)h(\()p
Fn(\013)638 698 y Fm(3)677 683 y Fo(\))p -74 721 789
4 v 44 w(Since)43 b Fn(\025)1081 698 y Fm(0)1151 683
y Fo(+)29 b Fn(\026)46 b Fl(2)h Fn(\033)t Fo(\()p Fn(H)1652
698 y Fm(0)1692 683 y Fo(\),)f(w)m(e)f(use)f(\(2.4\))f(together)h(with)
g(the)g(pro)s(cedure)g(used)h(to)-74 832 y(estimate)32
b(\()p Fn(\013)417 847 y Fm(2)456 832 y Fo(\).)44 b(The)33
b(result)g(is:)1287 1131 y Fl(j)p Fn(R)1389 1146 y Fm(3)1428
1131 y Fl(j)28 b(\024)g Fn(C)7 b Fl(j)p Fn(A)p Fo(\(0\))p
Fl(j)31 b(jjj)p Fn(W)14 b Fl(jjj)2225 1090 y Fm(2)2290
1131 y Fl(h)p Fn(t)p Fl(i)2403 1090 y Fi(\000)p Fk(r)r
Fm(+1)2586 1131 y Fn(:)1112 b Fo(\(6.11\))-74 1355 y(Estimation)31
b(of)h(\()p Fn(\013)638 1370 y Fm(4)677 1355 y Fo(\))p
-74 1393 V 72 1505 a(This)h(term)f(is)g(estimated)g(in)g(a)g(manner)h
(similar)c(to)j Fn(\013)2120 1520 y Fm(2)2192 1505 y
Fo(and)h Fn(\013)2444 1520 y Fm(3)2483 1505 y Fo(:)943
1803 y Fl(j)p Fn(R)1045 1818 y Fm(4)1084 1803 y Fl(j)27
b(\024)i Fn(c)p Fl(jjj)p Fn(W)14 b Fl(jjj)1561 1762 y
Fm(2)1614 1803 y Fl(fjjj)p Fn(W)g Fl(jjj)30 b Fo([)p
Fn(A)p Fo(])2095 1818 y Fk(\021)2138 1803 y Fo(\()p Fn(t)p
Fo(\))22 b(+)g([)p Fn(E)6 b Fo(])2501 1818 y Fk(\021)2543
1803 y Fo(\()p Fn(t)p Fo(\))p Fl(g)17 b(h)p Fn(t)p Fl(i)2834
1762 y Fi(\000)p Fk(\021)2930 1803 y Fn(:)768 b Fo(\(6.12\))-74
2028 y(Estimation)31 b(of)h(\()p Fn(\013)638 2043 y Fm(5)677
2028 y Fo(\):)p -74 2066 817 4 v 72 2177 a(Since,)h(b)m(y)h
(de\014nition,)d Fn(\036)1008 2192 y Fk(\013)1058 2177
y Fo(\(0\))c(=)g Fg(P)1390 2192 y Ff(c)1430 2177 y Fn(\036)1488
2192 y Fk(d)1529 2177 y Fo(\(0\))32 b(w)m(e)h(can)g(apply)f(lo)s(cal)f
(deca)m(y)j(estimates)e(for)g Fn(e)3401 2141 y Fi(\000)p
Fk(iH)3538 2150 y Fh(0)3573 2141 y Fk(t)3635 2177 y Fo(to)g(get)986
2385 y Fl(j)p Fo(\()p Fn(\014)6 b( )1176 2400 y Fm(0)1215
2385 y Fn(;)17 b(\036)1317 2400 y Fm(0)1356 2385 y Fo(\()p
Fn(t)p Fo(\)\))p Fl(j)60 b(\024)h Fn(C)7 b Fl(jjj)p Fn(W)14
b Fl(jjj)30 b(h)p Fn(t)p Fl(i)2225 2344 y Fi(\000)p Fk(r)r
Fm(+1)2435 2385 y Fl(k)p Fn(w)2555 2400 y Fm(+)2614 2385
y Fn(\036)2672 2400 y Fk(d)2712 2385 y Fo(\(0\))p Fl(k)p
Fn(:)811 b Fo(\(6.13\))-74 2632 y(Estimation)31 b(of)h(\()p
Fn(\013)638 2647 y Fm(6)677 2632 y Fo(\))p -74 2670 789
4 v 72 2781 a(Applying)g(Lemma)f(6.1)h(as)h(b)s(efore)g(w)m(e)h(get,)e
(for)g(0)c Fl(\024)g Fn(t)g Fl(\024)g Fn(T)14 b Fo(,)1017
2990 y Fl(j)p Fo(\()p Fn(\014)6 b( )1207 3005 y Fm(0)1246
2990 y Fn(;)17 b(\036)1348 3005 y Fm(2)1387 2990 y Fo(\()p
Fn(t)p Fo(\)\))p Fl(j)60 b(\024)28 b Fn(C)7 b Fl(jjj)p
Fn(W)14 b Fl(jjj)2080 2949 y Fm(2)2150 2990 y Fl(h)p
Fn(t)p Fl(i)2263 2949 y Fi(\000)p Fk(\021)2387 2990 y
Fo([)p Fn(\036)2472 3005 y Fk(d)2512 2990 y Fo(])2539
3005 y Fk(LD)r(;\021)2709 2990 y Fo(\()p Fn(T)g Fo(\))p
Fn(:)842 b Fo(\(6.14\))-74 3198 y(Using)32 b(Prop)s(osition)f(6.1)h(to)
h(estimate)e([)p Fn(\036)1477 3213 y Fk(d)1518 3198 y
Fo(])1545 3213 y Fk(LD)r(;\021)1714 3198 y Fo(\()p Fn(t)p
Fo(\))i(in)f(\(6.14\),)f(w)m(e)j(get)572 3406 y Fl(j)p
Fo(\()p Fn(\014)6 b( )762 3421 y Fm(0)801 3406 y Fn(;)17
b(\036)903 3421 y Fm(2)942 3406 y Fo(\()p Fn(t)p Fo(\)\))p
Fl(j)28 b(\024)g Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)1603
3365 y Fm(2)1672 3406 y Fl(h)p Fn(t)p Fl(i)1785 3365
y Fi(\000)p Fk(\021)1898 3406 y Fl(f)33 b(k)p Fn(w)2101
3421 y Fm(+)2159 3406 y Fn(\036)2217 3421 y Fk(d)2258
3406 y Fo(\(0\))p Fl(k)54 b Fo(+)g Fl(jjj)p Fn(W)14 b
Fl(jjj)30 b Fo([)p Fn(A)p Fo(])3048 3421 y Fk(\021)3091
3406 y Fo(\()p Fn(t)p Fo(\))i Fl(g)17 b Fn(:)397 b Fo(\(6.15\))72
3615 y(Finally)-8 b(,)33 b(com)m(bining)g(the)i(ab)s(o)m(v)m(e)h
(estimates)e(for)g(\()p Fn(\013)2028 3630 y Fk(j)2065
3615 y Fo(\))p Fn(;)51 b Fo(1)31 b Fl(\024)h Fn(j)37
b Fl(\024)32 b Fo(6,)j(w)m(e)h(can)f(no)m(w)g(b)s(ound)g([)p
Fn(E)6 b Fo(])3634 3630 y Fk(\021)3676 3615 y Fo(\()p
Fn(T)14 b Fo(\))34 b(for)-74 3764 y(an)m(y)f Fn(\021)f
Fl(2)c Fo([0)p Fn(;)17 b(r)24 b Fl(\000)f Fo(1])32 b(as)h(follo)m(ws:)
99 4122 y([)p Fn(E)6 b Fo(])231 4137 y Fk(\021)273 4122
y Fo(\()p Fn(t)p Fo(\))28 b Fl(\024)g Fn(C)610 4026 y
Fe(n)666 4122 y Fl(jjj)p Fn(W)14 b Fl(jjj)940 4081 y
Fm(2)1009 4122 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)22 b Fo(+)g
Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)1807 4137 y Fm(+)1865
4122 y Fn(\036)1923 4137 y Fk(d)1964 4122 y Fo(\(0\))p
Fl(k)21 b Fo(+)h Fl(jjj)p Fn(W)14 b Fl(jjj)2532 4081
y Fm(2)2602 4122 y Fo([)p Fn(E)6 b Fo(])2734 4137 y Fk(\021)2776
4122 y Fo(\()p Fn(T)14 b Fo(\))22 b(+)g Fl(jjj)p Fn(W)14
b Fl(jjj)3317 4081 y Fm(3)3386 4122 y Fo([)p Fn(A)p Fo(])3513
4137 y Fk(\021)3556 4122 y Fo(\()p Fn(T)g Fo(\))3703
4026 y Fe(o)3774 4122 y Fn(:)3725 4271 y Fo(\(6.16\))-74
4421 y(Since)33 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo(assumed)j(to)g(b)s
(e)f(small,)f(Prop)s(osition)g(6.2)h(follo)m(ws.)172
b([])72 4608 y(W)-8 b(e)41 b(can)g(no)m(w)g(complete)f(the)h(pro)s(ofs)
f(of)g(our)g(main)f(results,)k(Theorem)e(2.1)f(and)g(Theorem)h(2.2.)67
b(T)-8 b(o)-74 4758 y(pro)m(v)m(e)41 b(the)g(assertions)f(concerning)g
(the)g(in\014nite)f(time)g(b)s(eha)m(vior,)i(the)g(k)m(ey)g(is)f(to)f
(establish)h(lo)s(cal)d(deca)m(y)-74 4907 y(of)i Fn(\036)102
4922 y Fk(d)142 4907 y Fo(,)i(in)e(particular,)g(the)h(uniform)d(b)s
(oundedness)42 b(of)c([)p Fn(\036)2145 4922 y Fk(d)2186
4907 y Fo(])2213 4922 y Fk(LD)r(;r)r Fi(\000)p Fm(1)2468
4907 y Fo(\()p Fn(T)14 b Fo(\).)64 b(This)39 b(will)e(follo)m(w)g
(directly)i(from)-74 5057 y(Prop)s(osition)31 b(6.1)h(if)g(w)m(e)h(pro)
m(v)m(e)h(the)f(uniform)e(b)s(oundedness)k([)p Fn(A)p
Fo(])2338 5072 y Fk(r)r Fi(\000)p Fm(1)2466 5057 y Fo(\()p
Fn(T)14 b Fo(\),)32 b(or)h(equiv)-5 b(alen)m(tly)31 b([)3386
5032 y(~)3360 5057 y Fn(A)q Fo(])3461 5072 y Fk(r)r Fi(\000)p
Fm(1)3589 5057 y Fo(\()p Fn(T)14 b Fo(\).)1901 5306 y(24)p
eop
%%Page: 25 25
25 24 bop -74 59 a Fg(Prop)s(osition)36 b(6.3.)49 b Fb(There)25
b(exists)g(a)f(constan)m(t)h Fn(C)1817 74 y Fi(\003)1856
59 y Fb(,)h(dep)s(ending)e(on)g Fl(k)p Fn(\036)2604 74
y Fm(0)2643 59 y Fl(k)p Fn(;)17 b Fl(k)p Fn(w)2857 74
y Fm(+)2915 59 y Fn(\036)2973 74 y Fm(0)3012 59 y Fl(k)p
Fn(;)g Fl(jjj)p Fn(W)d Fl(jjj)p Fn(;)j Fo(\000)k Fb(and)j
Fn(r)s Fb(,)h(suc)m(h)-74 208 y(that)32 b(for)h(an)m(y)g
Fn(T)41 b(>)28 b Fo(0)1628 358 y([)p Fn(A)p Fo(])1755
373 y Fk(r)r Fi(\000)p Fm(1)1883 358 y Fo(\()p Fn(T)14
b Fo(\))28 b Fl(\024)g Fn(C)2233 373 y Fi(\003)3725 358
y Fo(\(6.17\))-74 607 y(Pro)s(of.)p -74 620 262 4 v 72
756 a(W)-8 b(e)36 b(b)s(egin)f(with)g(the)g(expansion)h(of)f
Fn(A)p Fo(\()p Fn(t)p Fo(\))g(giv)m(en)h(in)e(Prop)s(osition)g(5.1.)51
b(Multiplying)33 b(\(5.2\))i(b)m(y)h Fl(h)p Fn(t)p Fl(i)3819
720 y Fk(r)r Fi(\000)p Fm(1)3947 756 y Fo(,)-74 906 y(and)d(taking)f
(the)h(suprem)m(um)f(o)m(v)m(er)i(0)27 b Fl(\024)i Fn(t)e
Fl(\024)i Fn(T)46 b Fo(w)m(e)33 b(ha)m(v)m(e:)297 1175
y([)p Fn(A)p Fo(])424 1190 y Fk(r)r Fi(\000)p Fm(1)553
1175 y Fo(\()p Fn(T)14 b Fo(\))60 b Fl(\024)897 1029
y Fe( )963 1175 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)32 b
Fo(\000)1310 1134 y Fi(\000)p Fk(r)r Fm(+1)1547 1175
y Fo(+)159 b(sup)1678 1259 y Fm(0)p Fi(\024)p Fk(\034)8
b Fi(\024)p Fm(2\000)1941 1240 y Fc(\000)p Fa(\013)2050
1175 y Fl(j)p Fn(R)2152 1190 y Fk(A)2209 1175 y Fo(\()p
Fn(\034)j Fo(\))p Fl(j)55 b Fo(+)166 b(sup)2551 1259
y Fm(2\000)2630 1240 y Fc(\000)p Fa(\013)2723 1259 y
Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)2939 1175 y Fl(j)p
Fn(R)3041 1190 y Fk(A)3098 1175 y Fo(\()p Fn(\034)j Fo(\))p
Fl(j)3288 1029 y Fe(!)3725 1175 y Fo(\(6.18\))-74 1454
y(The)34 b(righ)m(t)d(hand)i(side)g(of)f(\(6.18\))g(is)g(estimated)g
(using)g(Prop)s(osition)f(5.1.)-74 1696 y([)p Fn(A)p
Fo(])53 1711 y Fk(r)r Fi(\000)p Fm(1)182 1696 y Fo(\()p
Fn(T)14 b Fo(\))49 b Fl(\024)504 1600 y Fe(\020)554 1696
y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)32 b Fo(\000)901 1655
y Fi(\000)p Fk(r)r Fm(+1)1138 1696 y Fo(+)55 b Fn(D)35
b Fo(\000)1446 1655 y Fi(\000)p Fk(\013r)1617 1696 y
Fo([)p Fn(E)6 b Fo(])1749 1711 y Fm(0)1788 1696 y Fo(\(2\000)1936
1655 y Fi(\000)p Fk(\013)2040 1696 y Fo(\))55 b(+)f Fn(C)2333
1711 y Fm(1)2405 1696 y Fn(e)2450 1655 y Fi(\000)p Fm(\000)2549
1632 y Fc(\000)p Fa(\016)2668 1696 y Fo([)p Fn(E)6 b
Fo(])2800 1711 y Fm(0)2840 1696 y Fo(\(2\000)2988 1655
y Fi(\000)p Fk(\013)3092 1696 y Fo(\))54 b(+)h Fn(C)39
b Fo(\000)3485 1655 y Fi(\000)p Fm(1)3580 1696 y Fo([)p
Fn(E)6 b Fo(])3712 1711 y Fk(r)r Fi(\000)p Fm(1)3840
1696 y Fo(\()p Fn(T)14 b Fo(\))3987 1600 y Fe(\021)4053
1696 y Fn(:)3725 1846 y Fo(\(6.19\))-74 1995 y(Next,)34
b(w)m(e)f(apply)g(Prop)s(osition)d(6.2)j(whic)m(h)g(yields:)181
2237 y([)p Fn(A)p Fo(])308 2252 y Fk(r)r Fi(\000)p Fm(1)436
2237 y Fo(\()p Fn(T)14 b Fo(\))116 b Fl(\024)892 2141
y Fe(\020)941 2237 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)32
b Fo(\000)1288 2196 y Fi(\000)p Fk(r)r Fm(+1)1525 2237
y Fo(+)55 b Fn(D)35 b Fo(\000)1833 2196 y Fi(\000)p Fk(\013r)2004
2237 y Fo([)p Fn(E)6 b Fo(])2136 2252 y Fm(0)2175 2237
y Fo(\(2\000)2323 2196 y Fi(\000)p Fk(\013)2428 2237
y Fo(\))54 b(+)h Fn(C)2721 2252 y Fm(1)2760 2237 y Fn(e)2805
2196 y Fi(\000)p Fm(\000)2904 2172 y Fc(\000)p Fa(\016)3023
2237 y Fo([)p Fn(E)6 b Fo(])3155 2252 y Fm(0)3194 2237
y Fo(\(2\000)3342 2196 y Fi(\000)p Fk(\013)3446 2237
y Fo(\))3484 2141 y Fe(\021)699 2420 y Fo(+)117 b Fn(C)7
b Fo(\000)1030 2378 y Fi(\000)p Fm(1)1140 2323 y Fe(\020)1190
2420 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(jjjj)p Fn(W)14 b Fl(jjj)1718
2378 y Fm(2)1809 2420 y Fo(+)54 b Fl(jjj)p Fn(W)14 b
Fl(jjj)31 b(k)p Fn(w)2364 2435 y Fm(+)2422 2420 y Fn(\036)2480
2435 y Fk(d)2520 2420 y Fo(\(0\))p Fl(k)55 b Fo(+)f Fl(jjj)p
Fn(W)14 b Fl(jjj)3154 2378 y Fm(3)3224 2420 y Fo([)p
Fn(A)p Fo(])3351 2435 y Fk(r)r Fi(\000)p Fm(1)3479 2420
y Fo(\()p Fn(T)g Fo(\))3626 2323 y Fe(\021)3692 2420
y Fn(:)72 2662 y Fo(Note)33 b(that)g(b)m(y)g(Prop)s(osition)e(6.2)h
(and)h(the)g(simple)e(b)s(ound:)880 2904 y([)p Fn(A)p
Fo(])1007 2919 y Fk(r)r Fi(\000)p Fm(1)1136 2904 y Fo(\()p
Fn(T)14 b Fo(\))60 b Fl(\024)28 b Fn(C)39 b Fl(h)p Fn(T)14
b Fl(i)1706 2862 y Fk(r)r Fi(\000)p Fm(1)1866 2904 y
Fo([)p Fn(A)p Fo(])1993 2919 y Fm(0)2033 2904 y Fo(\()p
Fn(T)g Fo(\))60 b Fl(\024)g Fn(C)40 b Fl(h)p Fn(T)14
b Fl(i)2636 2862 y Fk(r)r Fi(\000)p Fm(1)2796 2904 y
Fl(k)p Fn(\036)2904 2919 y Fm(0)2943 2904 y Fl(k)p Fn(;)705
b Fo(\(6.20\))-74 3145 y([)p Fn(E)6 b Fo(])58 3160 y
Fm(0)98 3145 y Fo(\(2\000)246 3109 y Fi(\000)p Fk(\013)350
3145 y Fo(\))32 b(is)g(b)s(ounded)i(in)e(terms)g(of)g(the)h(initial)c
(data)j(and)h Fl(jjj)p Fn(W)14 b Fl(jjj)p Fo(.)72 3295
y(By)34 b(h)m(yp)s(othesis)f Fg(\(H6\))p Fo(,)f(there)h(is)f(a)g(p)s
(ositiv)m(e)g(constan)m(t)i Fn(\022)2225 3310 y Fm(0)2265
3295 y Fo(,)e(suc)m(h)i(that)1552 3500 y Fl(jjj)p Fn(W)14
b Fl(jjj)1826 3463 y Fm(3)p 1552 3544 312 4 v 1677 3635
a Fo(\000)1934 3567 y Fl(\024)2049 3500 y(jjj)p Fn(W)g
Fl(jjj)p 2049 3544 272 4 v 2142 3635 a Fn(\022)2187 3650
y Fm(0)2330 3567 y Fn(:)1368 b Fo(\(6.21\))-74 3826 y(Therefore,)34
b(for)e Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo(su\016cien)m(tly)j(small)1628
4068 y([)p Fn(A)p Fo(])1755 4083 y Fk(r)r Fi(\000)p Fm(1)1883
4068 y Fo(\()p Fn(T)14 b Fo(\))28 b Fl(\024)g Fn(C)2233
4083 y Fi(\003)3725 4068 y Fo(\(6.22\))-74 4310 y(\(Alternativ)m(ely)-8
b(,)36 b(w)m(e)i(can)f(obtain)e(\(6.22\))h(b)m(y)h(imp)s(osing)d(that)i
Fl(jjj)p Fn(W)14 b Fl(jjj)2549 4273 y Fm(3)2586 4310
y Fo(\000)2647 4273 y Fi(\000)p Fm(1)2778 4310 y Fo(b)s(e)36
b(su\016cien)m(tly)h(small.\))53 b(Here,)-74 4459 y Fn(C)-4
4474 y Fi(\003)68 4459 y Fo(dep)s(ends)34 b(on)f Fl(k)p
Fn(\036)689 4474 y Fm(0)728 4459 y Fl(k)p Fn(;)17 b Fl(k)p
Fn(w)942 4474 y Fm(+)1000 4459 y Fn(\036)1058 4474 y
Fm(0)1097 4459 y Fl(k)p Fn(;)g Fl(jjj)p Fn(W)d Fl(jjj)p
Fn(;)j Fo(\000)29 b(and)k Fn(r)s Fo(.)72 4608 y(This)h(completes)f(the)
h(pro)s(of)e(of)h(Prop)s(osition)e(6.3)i(and)h(therewith)f(the)h
Fn(t)29 b Fl(!)g(1)j Fo(asymptotics)h(asserted)-74 4758
y(in)f(Theorems)h(2.1)f(and)h(2.2)-74 4907 y Fg(Remark:)47
b Fo(This)35 b(estimation)e(pro)s(cedure)i(di\013ers)g(from)e(one)i
(used)h(in)e([38],)h(where)h(the)f(cases)h Fn(r)26 b
Fl(\000)f Fo(1)34 b(large)-74 5057 y(and)f Fn(r)25 b
Fl(\000)d Fo(1)33 b(small)d(are)i(treated)h(di\013eren)m(tly)-8
b(.)1901 5306 y(25)p eop
%%Page: 26 26
26 25 bop 72 59 a Fo(It)33 b(remains)f(to)g(pro)m(v)m(e)i(the)f(in)m
(termediate)e(time)h(estimate)f(\(2.12\).)43 b(The)34
b(ingredien)m(ts)e(are)h(con)m(tained)g(in)-74 208 y(\(6.7\))f(and)h
(its)f(pro)s(of.)43 b(First,)31 b(b)m(y)j(\(5.3\))1198
454 y Fn(R)1272 469 y Fk(A)1329 454 y Fo(\()p Fn(t)p
Fo(\))60 b Fl(\024)28 b Fn(C)1731 337 y Fe(Z)1814 364
y Fk(t)1777 526 y Fm(0)1893 454 y Fn(e)1938 413 y Fi(\000)p
Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))2265 454
y Fl(j)o Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)50
b Fn(d\034)6 b(:)1022 b Fo(\(6.23\))-74 695 y(Let)40
b Fn(T)165 710 y Fm(0)244 695 y Fo(denote)g(an)f(arbitrary)f(\014xed)j
(p)s(ositiv)m(e)e(n)m(um)m(b)s(er.)64 b(W)-8 b(e)40 b(estimate)e
(\(6.23\))h(for)g Fn(t)g Fl(2)h Fo([0)p Fn(;)17 b(T)3526
710 y Fm(0)3565 695 y Fo(\000)3626 659 y Fi(\000)p Fm(1)3720
695 y Fo(].)64 b(W)-8 b(e)-74 844 y(b)s(ound)42 b(the)g(exp)s(onen)m
(tial)f(in)f(the)i(in)m(tegrand)f(b)m(y)i(one)e(\(explicit)f(in)m
(tegration)g(w)m(ould)h(giv)m(e)h(something)e(of)-74
994 y(order)47 b(\000)256 957 y Fi(\000)p Fm(1)350 994
y Fo(\),)j(and)d(b)s(ound)g Fl(j)p Fn(E)6 b Fo(\()p Fn(\034)11
b Fo(\))p Fl(j)46 b Fo(b)m(y)i(estimating)c(the)k(expressions)g(\()p
Fn(\013)2739 1009 y Fk(j)2775 994 y Fo(\))p Fn(;)64 b(j)57
b Fo(=)52 b(1)p Fn(;)17 b(:::;)g Fo(6)46 b(in)g(the)h(pro)s(of)-74
1143 y(of)41 b(Prop)s(osition)e(6.2.)69 b(First,)42 b(the)g(estimates)e
(of)h(Prop)s(osition)e(6.2)i(for)f Fn(\013)2727 1158
y Fm(1)2767 1143 y Fn(;)17 b(\013)2873 1158 y Fm(3)2953
1143 y Fo(and)41 b Fn(\013)3213 1158 y Fm(5)3294 1143
y Fo(are)g(useful)g(as)g(is.)-74 1292 y(In)m(tegration)32
b(of)g(the)h(b)s(ounds)g(\(6.8\),)g(\(6.11\))e(and)i(\(6.13\))f(giv)m
(es:)836 1422 y Fe(Z)919 1448 y Fk(t)882 1610 y Fm(0)998
1539 y Fn(e)1043 1498 y Fi(\000)p Fm(\000\()p Fk(t)p
Fi(\000)p Fk(\034)8 b Fm(\))1353 1539 y Fo(\()p Fn(\013)1453
1554 y Fk(j)1489 1539 y Fo(\))61 b Fl(\024)f Fn(C)40
b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2259 1554 y
Fm(+)2317 1539 y Fn(\036)p Fo(\(0\))p Fl(k)p Fn(;)49
b(j)34 b Fo(=)27 b(1)p Fn(;)17 b Fo(3)p Fn(;)g Fo(5)p
Fn(:)660 b Fo(\(6.24\))-74 1770 y(T)-8 b(o)35 b(estimate)f(the)i(con)m
(tributions)e(of)g(\()p Fn(\013)1445 1785 y Fm(2)1485
1770 y Fo(\))h(and)g(\()p Fn(\013)1850 1785 y Fm(4)1889
1770 y Fo(\),)h(\014rst)f(observ)m(e)i(that)e(b)m(y)h(\(6.9\))e(and)h
(Prop)s(osition)e(6.2)-74 1919 y(with)f Fn(\021)g Fo(=)27
b(0)1270 2069 y Fl(j)p Fn(@)1349 2084 y Fk(s)1386 2069
y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fl(j)59 b(\024)i Fn(C)39
b Fl(jjj)p Fn(W)14 b Fl(jjj)31 b(k)p Fn(w)2340 2084 y
Fm(+)2398 2069 y Fn(\036)p Fo(\(0\))p Fl(k)1094 b Fo(\(6.25\))-74
2264 y(Therefore,)34 b(using)e(lo)s(cal)e(deca)m(y)k(estimates)f(w)m(e)
g(ha)m(v)m(e:)824 2388 y Fe(Z)907 2414 y Fk(t)870 2577
y Fm(0)986 2505 y Fn(e)1031 2464 y Fi(\000)p Fm(\000\()p
Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))1341 2505 y Fo(\()p
Fn(\013)1441 2520 y Fk(j)1477 2505 y Fo(\))33 b Fn(d\034)94
b Fl(\024)116 b Fn(C)39 b(T)2094 2520 y Fm(0)2134 2505
y Fo(\000)2195 2464 y Fi(\000)p Fm(1)2322 2505 y Fl(jjj)p
Fn(W)14 b Fl(jjj)2596 2464 y Fm(3)2665 2505 y Fl(k)p
Fn(w)2785 2520 y Fm(+)2844 2505 y Fn(\036)p Fo(\(0\))p
Fl(k)1735 2679 y(\024)116 b(jjj)p Fn(W)14 b Fl(jjj)30
b(k)p Fn(w)2352 2694 y Fm(+)2411 2679 y Fn(\036)p Fo(\(0\))p
Fl(k)p Fn(;)48 b(j)34 b Fo(=)27 b(2)p Fn(;)17 b Fo(4)p
Fn(:)-74 2910 y Fo(Finally)-8 b(,)30 b(w)m(e)k(come)e(to)g(the)h(con)m
(tribution)f(of)g(\()p Fn(\013)1724 2925 y Fm(6)1763
2910 y Fo(\).)44 b(W)-8 b(e)33 b(rewrite)f(\()p Fn(\013)2470
2925 y Fm(6)2510 2910 y Fo(\))g(as)h(follo)m(ws.)430
3157 y(\()p Fn(\013)530 3172 y Fm(6)569 3157 y Fo(\))116
b(=)914 3032 y Fe(\014)914 3082 y(\014)914 3132 y(\014)914
3182 y(\014)942 3039 y(Z)1025 3066 y Fk(t)988 3228 y
Fm(0)1054 3157 y Fo(\()p Fn(\014)38 b(e)1230 3115 y Fk(iH)1312
3124 y Fh(0)1347 3115 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p
Fm(\))1519 3157 y Fg(P)1596 3172 y Ff(c)1636 3157 y Fn(\014)6
b( )1760 3172 y Fm(0)1800 3157 y Fn(;)17 b Fo(cos)g Fn(\026s)32
b(\036)2186 3172 y Fk(d)2226 3157 y Fo(\()p Fn(s)p Fo(\))g
Fn(ds)p Fo(\))2515 3032 y Fe(\014)2515 3082 y(\014)2515
3132 y(\014)2515 3182 y(\014)723 3396 y Fo(=)914 3272
y Fe(\014)914 3322 y(\014)914 3371 y(\014)914 3421 y(\014)942
3279 y(Z)1025 3305 y Fk(t)988 3468 y Fm(0)1054 3396 y
Fo(\()p Fn(w)1162 3411 y Fm(+)1221 3396 y Fn(\014)6 b(w)1352
3411 y Fm(+)1433 3396 y Fl(\001)21 b Fn(w)1552 3411 y
Fi(\000)1611 3396 y Fn(e)1656 3355 y Fk(iH)1738 3364
y Fh(0)1773 3355 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1945
3396 y Fg(P)2022 3411 y Ff(c)2062 3396 y Fn(w)2132 3411
y Fi(\000)2213 3396 y Fl(\001)h Fn(w)2333 3411 y Fm(+)2391
3396 y Fn(\014)6 b( )2515 3411 y Fm(0)2555 3396 y Fn(;)17
b Fo(cos)g Fn(\026s)32 b(w)2953 3411 y Fi(\000)3012 3396
y Fn(\036)3070 3411 y Fk(d)3110 3396 y Fo(\()p Fn(s)p
Fo(\))g Fn(ds)p Fo(\))3399 3272 y Fe(\014)3399 3322 y(\014)3399
3371 y(\014)3399 3421 y(\014)3443 3396 y Fn(:)255 b Fo(\(6.26\))-74
3642 y(Recall)24 b(that)g(b)m(y)i(\(4.9\))f Fn(\036)828
3657 y Fk(d)921 3642 y Fo(=)52 b Fn(\036)1107 3657 y
Fm(0)1178 3642 y Fo(+)32 b Fn(\036)1344 3657 y Fm(1)1415
3642 y Fo(+)g Fn(\036)1581 3657 y Fm(2)1620 3642 y Fo(,)26
b(where)h Fn(\036)2006 3657 y Fm(0)2045 3642 y Fo(\()p
Fn(t)p Fo(\))h(=)f Fn(e)2332 3606 y Fi(\000)p Fk(iH)2469
3615 y Fh(0)2504 3606 y Fk(t)2533 3642 y Fn(\036)2591
3657 y Fk(d)2632 3642 y Fo(\(0\).)40 b(Using)25 b(lo)s(cal)e(deca)m(y)j
(estimates)-74 3792 y Fg(\(H3\))p Fo(,)d(the)f(con)m(tribution)f(of)g
(the)h(term)f Fn(\036)1447 3807 y Fm(0)1487 3792 y Fo(\()p
Fn(t)p Fo(\))g(to)h(\()p Fn(\013)1828 3807 y Fm(6)1867
3792 y Fo(\))g(can)g(b)s(e)g(b)s(ounded)g(b)m(y)h Fn(C)28
b Fl(jjj)p Fn(W)14 b Fl(jjj)3101 3756 y Fm(2)3160 3792
y Fl(k)p Fn(w)3280 3807 y Fm(+)3339 3792 y Fn(\036)3397
3807 y Fk(d)3437 3792 y Fo(\(0\))p Fl(k)21 b(h)p Fn(\034)11
b Fl(i)3764 3756 y Fi(\000)p Fk(r)r Fm(+1)3947 3792 y
Fo(.)-74 3941 y(Multiplication)44 b(of)k(this)f(b)s(ound)h(b)m(y)h
Fn(e)1422 3905 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p
Fk(\034)8 b Fm(\))1747 3941 y Fo(and)48 b(in)m(tegration)f(with)g(resp)
s(ect)i(to)f Fn(t)g Fo(giv)m(es)g(the)h(b)s(ound)-74
4091 y Fn(C)39 b Fl(jjj)p Fn(W)14 b Fl(jjj)309 4054 y
Fm(2)346 4091 y Fl(k)p Fn(w)466 4106 y Fm(+)525 4091
y Fn(\036)583 4106 y Fk(d)623 4091 y Fo(\(0\))p Fl(k)p
Fo(.)43 b(T)-8 b(o)32 b(assess)i(the)e(con)m(tributions)f(from)g
Fn(\036)2337 4106 y Fm(1)2397 4091 y Fo(+)21 b Fn(\036)2552
4106 y Fm(2)2591 4091 y Fo(,)32 b(note)g(that)g(lo)s(cal)e(deca)m(y)j
(estimates)-74 4240 y Fg(\(H3\))e Fo(imply)1159 4389
y Fl(k)p Fn(w)1279 4404 y Fi(\000)1337 4389 y Fo(\()p
Fn(\036)1433 4404 y Fm(1)1494 4389 y Fo(+)22 b Fn(\036)1650
4404 y Fm(2)1690 4389 y Fo(\))p Fl(k)59 b(\024)i Fn(C)40
b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2509 4404 y
Fm(+)2567 4389 y Fn(\036)2625 4404 y Fm(0)2665 4389 y
Fl(k)p Fn(:)983 b Fo(\(6.27\))-74 4585 y(Putting)32 b(together)h(the)g
(con)m(tributions)f(from)f Fn(\036)1720 4600 y Fm(0)1792
4585 y Fo(and)i(from)e Fn(\036)2270 4600 y Fm(1)2331
4585 y Fo(+)22 b Fn(\036)2487 4600 y Fm(2)2526 4585 y
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28 27 bop -74 59 a Fg(Theorem)37 b(7.2.)49 b Fb(\(Sigal-So\013er;)30
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b Fo(\(1974\))g(315-335)-74 2964 y([10])49 b(J.S.)42
b(Ho)m(wland,)i Fj(Sc)-5 b(attering)43 b(the)-5 b(ory)44
b(for)f(Hamiltonians)g(p)-5 b(erio)g(dic)42 b(in)h(time)p
Fo(,)h(Indiana)e(Math.)g(J.)g Fg(28)127 3114 y Fo(\(1979\))31
b(471-494)-74 3346 y([11])49 b(W.)33 b(Hunzik)m(er)g(&)g(I.M.)g(Sigal,)
e Fj(Time)j(dep)-5 b(endent)34 b(sc)-5 b(attering)35
b(the)-5 b(ory)35 b(for)g(N-b)-5 b(o)g(dy)35 b(quantum)g(systems)p
Fo(,)127 3496 y(ETH-Z)s(\177)-51 b(uric)m(h)31 b(preprin)m(t)i
(\(1997\))-74 3728 y([12])49 b(W.)32 b(Hunzik)m(er,)i(I.M.)f(Sigal)e(&)
h(A.)h(So\013er,)g Fj(Minimal)h(esc)-5 b(ap)g(e)34 b(velo)-5
b(cities)p Fo(,)32 b(1998)f(preprin)m(t.)-74 3961 y([13])49
b(A.)33 b(Jensen)i(&)e(T.)h(Kato,)e Fj(Sp)-5 b(e)g(ctr)g(al)35
b(pr)-5 b(op)g(erties)35 b(of)h(Schr\177)-50 b(odinger)34
b(op)-5 b(er)g(ators)35 b(and)g(time-de)-5 b(c)g(ay)34
b(of)i(wave)127 4110 y(functions)p Fo(,)c(Duk)m(e)h(Math.)g(J.)g
Fg(46)f Fo(\(1979\),)g(583{611.)-74 4343 y([14])49 b(H.R.)d(Jauslin)e
(&)i(J.L.)g(Leb)s(o)m(witz,)j Fj(Gener)-5 b(alize)g(d)46
b(Flo)-5 b(quet)47 b(op)-5 b(er)g(ator)47 b(for)g(quasip)-5
b(erio)g(dic)g(al)5 b(ly)46 b(driven)127 4492 y(quantum)33
b(systems)p Fo(,)f(in)e(\\Mathematical)f(Ph)m(ysics)k(X",)e(K.)g(Sc)m
(hm)s(\177)-51 b(udgen)31 b(\(ed.\))h(Springer)e(\(1991\),)h(and)127
4641 y(cited)h(ref.)-74 4874 y([15])49 b(J.-L.)29 b(Journ)m(\023)-46
b(e,)31 b(A.)f(So\013er)g(&)g(C.)g(Sogge,)h Fn(L)1702
4838 y Fk(p)1769 4874 y Fl(!)d Fn(L)1963 4838 y Fk(p)1999
4814 y Fc(0)2057 4874 y Fj(estimates)k(for)g(the)h(time)f(dep)-5
b(endent)31 b(Schr\177)-50 b(odinger)127 5023 y(e)-5
b(quation)p Fo(,)32 b(Bull.)f(AMS,)i Fg(23)g Fo(\(1990\))1901
5306 y(29)p eop
%%Page: 30 30
30 29 bop -74 59 a Fo([16])49 b(C.-A.)e(Pillet)f(&)h(C.E.)i(W)-8
b(a)m(yne,)53 b Fj(Invariant)47 b(manifolds)h(for)g(a)h(class)e(of)i
(disp)-5 b(ersive,)51 b(Hamiltonian)127 208 y(p)-5 b(artial)34
b(di\013er)-5 b(ential)34 b(e)-5 b(quations)p Fo(,)32
b(J.)h(Di\013.)e(Eq.)j Fg(141)e Fo(\(1997\))g(310{326)-74
441 y([17])49 b(H.)28 b(Kitada)e(&)i(K.)g(Y)-8 b(a)5
b(jima,)27 b Fj(A)k(sc)-5 b(attering)31 b(the)-5 b(ory)31
b(for)f(time)h(dep)-5 b(endent)29 b(long)h(r)-5 b(ange)31
b(p)-5 b(otentials)p Fo(,)28 b(Duk)m(e)127 590 y(Math.)33
b(J.)f Fg(49)h Fo(\(1982\))e(341-376)-74 822 y([18])49
b(H.)32 b(Lam)m(b,)g Fj(On)h(a)i(p)-5 b(e)g(culiarity)34
b(of)g(the)g(wave-system)g(due)g(to)h(the)f(fr)-5 b(e)g(e)34
b(vibr)-5 b(ations)34 b(of)g(a)g(nucleus)g(in)g(an)127
972 y(extende)-5 b(d)34 b(me)-5 b(dium)p Fo(,)32 b(Pro)s(c.)g(London)h
(Math.)g(So)s(c.)f Fg(32)h Fo(\(1900\))f(208-211)-74
1204 y([19])49 b(L.D.)32 b(Landau)h(&)g(E.M.)h(Lifshitz,)e(Quan)m(tum)g
(Mec)m(hanics:)46 b(Non)33 b(-relativistic)d(Theory)-8
b(,)34 b(V)-8 b(olume)32 b(3)g(of)127 1354 y Fj(Course)i(of)h(The)-5
b(or)g(etic)g(al)34 b(Physics)p Fo(,)e(P)m(ergamon)g(Press,)i(Oxford)f
(1965)-74 1586 y([20])49 b(Leb)s(o)m(witz,)34 b(Jo)s(el)f(L.)h
Fj(Macr)-5 b(osc)g(opic)35 b(laws,)h(micr)-5 b(osc)g(opic)34
b(dynamics,)i(time's)f(arr)-5 b(ow)36 b(and)f(Boltzmann)-10
b('s)127 1736 y(entr)-5 b(opy)p Fo(,)32 b(Ph)m(ys.)j(A)d(194)g
(\(1993\),)g(no.)g(1-4,)g(1{27.)-74 1968 y([21])49 b(P)-8
b(.D.)31 b(Miller,)f(A.)i(So\013er)g(&)f(M.I.)i(W)-8
b(einstein,)32 b Fj(Structur)-5 b(al)35 b(instability)f(and)f
(metastability)h(of)g(br)-5 b(e)g(ather)127 2118 y(states)35
b(of)f(soliton)h(wave)f(guides)p Fo(,)e(in)g(preparation.)-74
2350 y([22])49 b(E.)24 b(Mourre,)j Fj(A)n(bsenc)-5 b(e)26
b(of)i(singular)e(c)-5 b(ontinuous)28 b(sp)-5 b(e)g(ctrum)27
b(for)g(c)-5 b(ertain)27 b(self-adjoint)f(op)-5 b(er)g(ators)p
Fo(,)26 b(Com-)127 2500 y(m)m(un.)32 b(Math.)h(Ph)m(ys.)i
Fg(78)d Fo(\(1981\))g(391)-74 2732 y([23])49 b(M.)34
b(Murata,)h Fj(R)-5 b(ate)36 b(of)g(de)-5 b(c)g(ay)36
b(of)g(lo)-5 b(c)g(al)36 b(ener)-5 b(gy)36 b(and)g(sp)-5
b(e)g(ctr)g(al)36 b(pr)-5 b(op)g(erties)35 b(of)h(el)5
b(liptic)36 b(op)-5 b(er)g(ators)p Fo(,)34 b(Jpn.)127
2881 y(J.)e(Math.)h Fg(6)g Fo(\(1980\))e(77{127)-74 3114
y([24])49 b(R.M.)28 b(Pyk)m(e)i(&)e(I.M.)h(Sigal,)e Fj(Nonline)-5
b(ar)31 b(wave)f(e)-5 b(quations:)42 b(Constr)-5 b(aints)30
b(on)h(p)-5 b(erio)g(ds)30 b(and)g(exp)-5 b(onential)127
3263 y(b)g(ounds)34 b(for)h(p)-5 b(erio)g(dic)33 b(solutions)i
Fo(,)e(Duk)m(e)g(Math.)g(J.)g Fg(88)f Fo(\(1997\))-74
3496 y([25])49 b(P)-8 b(.)37 b(P)m(erry)-8 b(,)40 b(I.M.)e(Sigal)d(&)i
(B.)g(Simon,)g Fj(Sp)-5 b(e)g(ctr)g(al)39 b(analysis)f(of)h(N-b)-5
b(o)g(dy)40 b(Schr\177)-50 b(odinger)37 b(op)-5 b(er)g(ators)p
Fo(,)38 b(Ann.)127 3645 y(Math.)33 b Fg(114)f Fo(\(1981\))g(519{567.)
-74 3878 y([26])49 b(R.)32 b(Pyk)m(e,)j(A.)d(So\013er)h(&)f(M.I.)i(W)-8
b(einstein,)32 b Fj(Stability)j(of)g(Kinks)p Fo(,)d(in)g(progress.)-74
4110 y([27])49 b(J.)25 b(Rauc)m(h,)j Fj(L)-5 b(o)g(c)g(al)28
b(de)-5 b(c)g(ay)28 b(of)g(sc)-5 b(attering)28 b(solutions)g(to)g
(Schr\177)-50 b(odinger's)27 b(e)-5 b(quation)p Fo(,)27
b(Comm.)d(Math.)i(Ph)m(ys.)127 4260 y Fg(61)32 b Fo(\(1978\),)g
(149{168.)-74 4492 y([28])49 b(M.)37 b(Reed)g(&)g(B.)f(Simon,)h(Metho)s
(ds)g(in)f(Mo)s(dern)h(Mathematical)e(Ph)m(ysics,)k(I.)e(F)-8
b(unctional)35 b(Analysis,)127 4641 y(Academic)d(Press)i(1972.)-74
4874 y([29])49 b(M.)27 b(Reed)h(&)f(B.)g(Simon,)g(Metho)s(ds)h(in)e(Mo)
s(dern)i(Mathematical)d(Ph)m(ysics,)30 b(IV.)d(Analysis)g(of)f(Op)s
(erator,)127 5023 y(Academic)32 b(Press)i(1978.)1901
5306 y(30)p eop
%%Page: 31 31
31 30 bop -74 59 a Fo([30])49 b(J.J.)33 b(Sakurai,)f(Adv)-5
b(anced)33 b(Quan)m(tum)g(Mec)m(hanics,)h(Addison)e(W)-8
b(esley)34 b(\(1967\))-74 291 y([31])49 b(T.)35 b(Sc)m(hon)m(b)s(ek,)j
Fj(De)-5 b(c)g(ay)36 b(of)h(solutions)f(of)h(Schr\177)-50
b(odinger)36 b(e)-5 b(quations)p Fo(,)35 b(Duk)m(e)h(Math.)f(J.)g(46)f
(\(1979\),)h(203{)127 441 y(213.)-74 673 y([32])49 b(I.M.)43
b(Sigal,)g Fj(Nonline)-5 b(ar)43 b(wave)g(and)h(Schr\177)-50
b(odinger)42 b(e)-5 b(quations)44 b(I.)f(Instability)h(of)g(time-p)-5
b(erio)g(dic)42 b(and)127 822 y(quasip)-5 b(erio)g(dic)33
b(solutions)p Fo(,)f(Comm)m(un.)h(Math.)f(Ph)m(ys.)j
Fg(153)e Fo(\(1993\))e(297)-74 1055 y([33])49 b(I.M.)33
b(Sigal,)e Fj(Gener)-5 b(al)35 b(char)-5 b(acteristics)34
b(of)h(nonline)-5 b(ar)34 b(dynamics)p Fo(,)e(in)g(Sp)s(ectral)g(and)h
(Scattering)f(The-)127 1204 y(ory;)43 b(Pro)s(ceedings)d(of)f(the)h(T)
-8 b(aniguc)m(hi)38 b(in)m(ternational)f(w)m(orkshop,)43
b(ed.)d(M.)g(Ik)-5 b(a)m(w)m(a,)42 b(Marcel)d(Dekk)m(er,)127
1354 y(Inc.)33 b(New)g(Y)-8 b(ork)33 b(-)f(Basel)g(-)h(Hong)f(Kong)g
(1994)-74 1586 y([34])49 b(I.M.)31 b(Sigal)d(&)i(A.-So\013er,)h
Fj(L)-5 b(o)g(c)g(al)32 b(de)-5 b(c)g(ay)32 b(and)h(velo)-5
b(city)32 b(b)-5 b(ounds)33 b(for)f(Quantum)h(pr)-5 b(op)g(agation)p
Fo(,)30 b(preprin)m(t,)127 1736 y(Princeton)i(1988)g
(\(ftp://www.math.rutgers.edu/pub/so\013er\))-74 1968
y([35])49 b(I.M.)c(Sigal)e(&)i(A.-So\013er,)j Fj(Asymptotic)f
(Completeness)e(for)h(Short)g(R)-5 b(ange)46 b(Many)g(Bo)-5
b(dy)46 b(Systems)p Fo(,)127 2118 y(Ann.)33 b(Math.)g
Fg(126)f Fo(\(1987\))g(35{108.)-74 2350 y([36])49 b(E.)h(Skibsted,)55
b Fj(Pr)-5 b(op)g(agation)50 b(estimates)g(for)g(N-b)-5
b(o)g(dy)51 b(Schr\177)-50 b(odinger)50 b(op)-5 b(er)g(ators)p
Fo(,)53 b(Comm)m(un.)c(Math.)127 2500 y(Ph)m(ys.)34 b
Fg(142)f Fo(\(1991\))e(67{98)-74 2732 y([37])49 b(A.)26
b(So\013er)h(&)f(M.I.)h(W)-8 b(einstein,)28 b Fj(Dynamic)g(the)-5
b(ory)29 b(of)g(quantum)h(r)-5 b(esonanc)g(es)28 b(and)g(p)-5
b(erturb)g(ation)30 b(the)-5 b(ory)127 2881 y(of)27 b(emb)-5
b(e)g(dde)g(d)26 b(eigenvalues)p Fo(,)f(in)f(Pro)s(ceedings)i(of)e
(Conference)j(on)d(P)m(artial)f(Di\013eren)m(tial)g(Equations)i(and)127
3031 y(Applications,)33 b(Univ)m(ersit)m(y)i(of)f(T)-8
b(oron)m(to,)35 b(June,)h(1995,)e(CRM)h(Lecture)g(Notes,)h(Eds.)g(P)-8
b(.)34 b(Greiner,)h(V.)127 3180 y(Ivrii,)c(L.Seco,)i(&)g(C.)g(Sulem)-74
3413 y([38])49 b(A.)30 b(So\013er)h(&)f(M.I.)i(W)-8 b(einstein,)30
b Fj(Time)i(dep)-5 b(endent)32 b(r)-5 b(esonanc)g(e)32
b(the)-5 b(ory)p Fo(,)31 b(to)f(app)s(ear)h(in)f(Geometric)f(and)127
3562 y(F)-8 b(unctional)30 b(Analysis.)-74 3795 y([39])49
b(A.)32 b(So\013er)g(&)g(M.I.)h(W)-8 b(einstein,)32 b
Fj(R)-5 b(esonanc)g(es,)33 b(r)-5 b(adiation)34 b(damping)f(and)h
(instability)g(in)h(Hamiltonian)127 3944 y(nonline)-5
b(ar)33 b(wave)h(e)-5 b(quations)p Fo(,)33 b(to)f(app)s(ear)g(in)g(In)m
(v)m(en)m(tiones)j(Mathematicae.)-74 4177 y([40])49 b(B.)40
b(V)-8 b(ain)m(b)s(erg)39 b Fj(Sc)-5 b(attering)42 b(of)f(waves)g(in)h
(a)f(me)-5 b(dium)41 b(dep)-5 b(ending)40 b(p)-5 b(erio)g(dic)g(al)5
b(ly)41 b(on)h(time)p Fo(,)f(Asterisque)127 4326 y Fg(210)32
b Fo(\(1992\))g(327{340)-74 4558 y([41])49 b(A.)34 b(V)-8
b(anderbau)m(whede)36 b(&)d(G.)h(Io)s(oss,)g Fj(Center)i(manifold)e
(the)-5 b(ory)36 b(in)g(in\014nite)f(dimensions)p Fo(,)e(Dynamics)127
4708 y(Rep)s(orted)f Fg(2)h Fo(\(1990\))-74 4940 y([42])49
b(J.M.G.)32 b(Vilar)f(&)h(J.M.)i(Rubi,)e(Ph)m(ys.)i(Rev.)f(Lett.)g
Fg(77)g Fo(\(1996\))e(2863)h(and)g(cited)h(ref.)1901
5306 y(31)p eop
%%Page: 32 32
32 31 bop -74 59 a Fo([43])49 b(K.)33 b(Y)-8 b(a)5 b(jima,)32
b Fj(Sc)-5 b(attering)36 b(the)-5 b(ory)36 b(for)g(Schr\177)-50
b(odinger)35 b(op)-5 b(er)g(ators)35 b(with)h(p)-5 b(otentials)35
b(p)-5 b(erio)g(dic)35 b(in)h(time)p Fo(,)e(J.)127 208
y(Math.)f(So)s(c.)f(Japan)h Fg(29)f Fo(\(1977\))g(729-743)-74
441 y([44])49 b(K.)30 b(Y)-8 b(a)5 b(jima,)29 b Fj(R)-5
b(esonanc)g(es)31 b(for)i(the)f(A)n(C-Stark)h(e\013e)-5
b(ct)p Fo(,)30 b(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Fg(78)e
Fo(\(1982\))f(331{352.)-74 673 y([45])49 b(K.)28 b(Y)-8
b(a)5 b(jima,)27 b Fj(A)32 b(multichannel)e(sc)-5 b(attering)30
b(the)-5 b(ory)32 b(for)f(some)f(time)h(dep)-5 b(endent)30
b(hamiltonians,)g(char)-5 b(ge)127 822 y(tr)g(ansfer)34
b(pr)-5 b(oblem)p Fo(,)32 b(Comm)m(un.)g(Math.)h(Ph)m(ys.)h
Fg(75)f Fo(\(1980\))e(153{178.)1901 5306 y(32)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF


