Content-Type: multipart/mixed; boundary="-------------0107131040326" This is a multi-part message in MIME format. ---------------0107131040326 Content-Type: text/plain; name="01-272.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-272.keywords" Scattering Theory, Schr\"odinger-type Equations, Almost Invariant Subspaces ---------------0107131040326 Content-Type: application/postscript; name="Paper.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Paper.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Paper.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o Paper.ps Paper.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.07.13:1742 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 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b(and)f(\()p Fn(H)7 b Fo(2\))29 b(whose)515 623 y Fn(L)572 593 y Fl(2)608 623 y Fo(-norms)21 b(are)f(uniformly)h(b)r(ounded)h(with)f(resp)r(ect)g (to)g Fn(x)j Fm(2)f Fj(R)2493 593 y Fk(n)2538 623 y Fo(,)g (automatically)d(satis\014es)515 722 y(\()p Fn(H)7 b Fo(3\))697 734 y Fl(0)770 722 y Fo(\(or)35 b(\()p Fn(H)7 b Fo(3)p Fn(A)p Fo(\))1156 734 y Fl(0)1193 722 y Fo(\))36 b(with)g Fn(V)1506 734 y Fi(1)1614 722 y Fo(=)g(0.)60 b(In)36 b(the)g(follo)n(wing,)h(w)n(e)e(denote)h(b)n(y)f(\005)3078 734 y Fl(0)3116 722 y Fo(\()p Fn(x)p Fo(\))i(the)515 822 y(sp)r(ectral)j(pro)5 b(jection)40 b(of)h Fn(V)19 b Fo(\()p Fn(x)p Fo(\))42 b(corresp)r(onding)d(to)i Fn(\033)2285 834 y Fl(0)2322 822 y Fo(\()p Fn(x)p Fo(\).)78 b(\005)2596 834 y Fl(0)2634 822 y Fo(\()p Fn(x)p Fo(\))42 b(is)f(a)f(smo)r(oth)h (\(or)515 922 y(analytic)35 b(on)g(\000)1014 934 y Fk(a)1054 922 y Fo(,)i(if)f(\(H1A\))g(holds\))g(function)g(of)f Fn(x)h Fo(with)g(v)-5 b(alue)35 b(in)h Fm(L)p Fo(\()p Fm(H)q Fo(;)14 b Fm(D)r Fo(\).)62 b(Let)35 b(us)515 1021 y(denote)h(b)n(y)h(\005)978 1033 y Fl(0)1052 1021 y Fo(the)g(b)r (ounded)g(op)r(erator)e(from)h Fn(L)2158 991 y Fl(2)2195 1021 y Fo(\()p Fj(R)2299 991 y Fk(n)2344 1021 y Fo(;)14 b Fm(H)q Fo(\))37 b(to)g Fn(L)2689 991 y Fl(2)2725 1021 y Fo(\()p Fj(R)2829 991 y Fk(n)2874 1021 y Fo(;)14 b Fm(D)r Fo(\))38 b(acting)e(as)515 1121 y(m)n(ultiplication)g(b)n(y)g (\005)1242 1133 y Fl(0)1279 1121 y Fo(\()p Fn(x)p Fo(\).)63 b(Asso)r(ciated)36 b(to)f Fn(V)19 b Fo(\()p Fn(x)p Fo(\))37 b(let)g(us)f(consider)f(the)h(Sc)n(hr\177)-42 b(odinger)515 1220 y(Hamiltonian)1532 1320 y Fn(P)12 b Fo(\()p Fn(h)p Fo(\))24 b(=)e Fm(\000)p Fn(h)1933 1286 y Fl(2)1970 1320 y Fo(\001)2039 1332 y Fk(x)2099 1320 y Fo(+)c Fn(V)h Fo(\()p Fn(x)p Fo(\))848 b(\(2.4\))515 1457 y(with)37 b(domain)g Fn(H)1094 1426 y Fl(2)1131 1457 y Fo(\()p Fj(R)1235 1426 y Fk(n)1280 1457 y Fo(;)14 b Fm(D)r Fo(\))38 b(and)f(v)-5 b(alues)37 b(in)h Fn(L)2045 1426 y Fl(2)2081 1457 y Fo(\()p Fj(R)2185 1426 y Fk(n)2230 1457 y Fo(;)14 b Fm(H)q Fo(\).)66 b(In)38 b(the)f(follo)n(wing,)i(giv)n(en)d(a)515 1556 y(b)r(ounded)28 b(op)r(erator)e Fn(A)i Fo(on)f Fn(L)1451 1526 y Fl(2)1488 1556 y Fo(\()p Fj(R)1592 1526 y Fk(n)1637 1556 y Fo(;)14 b Fm(H)q Fo(\),)28 b(w)n(e)f(denote)g(b)n(y)h Fm(k)p Fn(A)p Fm(k)2479 1568 y Fk(\016)2542 1556 y Fo(the)g(w)n(eigh)n (ted)f(norm)1180 1717 y Fm(k)p Fn(A)p Fm(k)1326 1729 y Fk(\016)1385 1717 y Fo(=)50 b(Sup)1473 1791 y Fk(t)p Fi(2)p Fl([0)p Fk(;)p Fl(1])1680 1717 y Fm(kh)p Fn(x)p Fm(i)1833 1683 y Fk(t\016)1895 1717 y Fn(A)p Fm(h)p Fn(x)p Fm(i)2068 1683 y Fl(\(1)p Fi(\000)p Fk(t)p Fl(\))p Fk(\016)2268 1717 y Fm(k)2310 1732 y Fi(L)p Fl(\()p Fk(L)2428 1716 y Fe(2)2459 1732 y Fl(\()p Fg(R)2541 1716 y Ff(n)2582 1732 y Fl(;)p Fi(H)p Fl(\)\))515 1946 y Fo(and)31 b(simply)g(b)n(y)h Fm(k)p Fn(A)p Fm(k)e Fo(the)i(one)f(corresp)r(onding)f(to)h Fn(\016)h Fo(=)d(0.)48 b(Our)31 b(main)g(results)g(are)g(the)515 2045 y(follo)n(wing)515 2194 y Fj(Theorem)f(2.1)41 b Fd(Assume)26 b Fo(\()p Fn(H)7 b Fo(1\))p Fn(;)14 b Fo(\()p Fn(H)7 b Fo(2\))27 b Fd(and)h Fo(\()p Fn(H)7 b Fo(3\))2160 2206 y Fk(\016)2223 2194 y Fd(hold)29 b(true)e(for)h(some)g Fn(\016)e Fm(\025)c Fo(0)p Fd(.)38 b(Then,)515 2294 y(for)46 b(any)g Fo(\010)52 b Fm(2)g Fn(C)1122 2264 y Fi(1)1116 2315 y Fl(0)1193 2294 y Fo(\()p Fj(R)p Fo(\))p Fd(,)e(ther)l(e)c (exists)f(a)h(pr)l(oje)l(ction)h Fo(\005\()p Fn(h)p Fo(\))g Fd(\(dep)l(ending)g(on)e Fo(\010)p Fd(\))h(on)515 2394 y Fn(L)572 2363 y Fl(2)608 2394 y Fo(\()p Fj(R)712 2363 y Fk(n)758 2394 y Fo(;)14 b Fm(H)q Fo(\))29 b Fd(such)h(that)1527 2555 y Fm(k)p Fo(\005)18 b Fm(\000)g Fo(\005)1794 2567 y Fl(0)1831 2555 y Fm(k)1873 2567 y Fk(\016)1933 2555 y Fo(=)k Fm(O)r Fo(\()p Fn(h)p Fo(\))1008 b(\(2.5\))515 2716 y Fd(and,)30 b(for)h(any)f Fn(\037)23 b Fm(2)h Fn(C)1212 2686 y Fi(1)1206 2736 y Fl(0)1282 2716 y Fo(\()p Fj(R)p Fo(\))30 b Fd(such)g(that)g Fn(\037)p Fo(\010)23 b(=)f Fn(\037)p Fd(,)30 b(we)g(have)1204 2877 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[\005)p Fn(;)i(P)e Fo(])p Fm(k)1679 2889 y Fk(\016)1734 2877 y Fo(+)18 b Fm(k)p Fo([\005)p Fn(;)c(P)e Fo(])p Fn(\037)p Fo(\()p Fn(P)g Fo(\))p Fm(k)2292 2889 y Fk(\016)2351 2877 y Fo(=)22 b Fm(O)r Fo(\()p Fn(h)2586 2843 y Fi(1)2657 2877 y Fo(\))519 b(\(2.6\))515 3055 y Fd(uniformly)29 b(with)f(r)l(esp)l(e)l(ct)f(to)h Fn(h)23 b(>)g Fo(0)k Fd(smal)t(l)i(enough.)38 b(Mor)l(e)l(over,)30 b Fo(\005\()p Fn(h)p Fo(\))24 b(=)f(\005)2935 3067 y Fl(0)2986 3055 y Fo(+)3065 2976 y Fc(X)3067 3154 y Fk(j)s Fi(\025)p Fl(1)3199 3055 y Fo(\005)3261 3067 y Fk(j)3296 3055 y Fn(h)3344 3020 y Fk(j)515 3261 y Fd(mo)l(dulo)j Fm(O)r Fo(\()p Fn(h)945 3231 y Fi(1)1016 3261 y Fo(\))g Fd(and,)h(for)f(any)g Fn(j)i Fm(\025)23 b Fo(1)p Fd(,)j Fo(\005)1845 3273 y Fk(j)1903 3261 y Fo(=)d(\010\()p Fn(P)12 b Fo(\))2190 3240 y(^)2180 3261 y(\005)2242 3273 y Fk(j)2287 3261 y Fo(+)d(\(1)g Fm(\000)g Fo(\010\()p Fn(P)j Fo(\)\))2749 3240 y(^)2739 3261 y(\005)2801 3273 y Fk(j)2837 3261 y Fo(\010\()p Fn(P)g Fo(\))26 b Fd(wher)l(e)3292 3240 y Fo(^)3282 3261 y(\005)3344 3273 y Fk(j)515 3360 y Fd(is)34 b(a)g(di\013er)l(ential)h(op)l(er)l(ator)f(of)h(or)l(der)f Fn(j)39 b Fd(with)34 b(smo)l(oth)g(c)l(o)l(e\016cients)h(de)l(c)l (aying)g(as)f Fm(h)p Fn(x)p Fm(i)3290 3330 y Fi(\000)p Fk(\016)515 3460 y Fd(uniformly)c(on)g Fj(R)1084 3430 y Fk(n)1129 3460 y Fd(.)515 3621 y Fj(Theorem)g(2.2)41 b Fd(Assume)d Fo(\()p Fn(H)7 b Fo(1)p Fn(A)p Fo(\))p Fn(;)14 b Fo(\()p Fn(H)7 b Fo(2\))38 b Fd(and)i Fo(\()p Fn(H)7 b Fo(3)p Fn(A)p Fo(\))2319 3633 y Fk(\016)2394 3621 y Fd(hold)40 b(true)e(for)i(some)f Fn(\016)k Fm(\025)c Fo(0)p Fd(.)515 3721 y(Then,)30 b(for)h(any)f Fo(\010)23 b Fm(2)g Fn(C)1274 3691 y Fi(1)1268 3741 y Fl(0)1345 3721 y Fo(\()p Fj(R)p Fo(\))29 b Fd(and)h(for)h(any)f Fn(B)d(>)c Fo(0)29 b Fd(su\016ciently)g(lar)l(ge,)i(ther)l(e)f(exists)f (an)515 3820 y(ortho)l(gonal)36 b(pr)l(oje)l(ction)f Fo(\005\()p Fn(h)p Fo(\))g Fd(on)g Fn(L)1705 3790 y Fl(2)1741 3820 y Fo(\()p Fj(R)1845 3790 y Fk(n)1890 3820 y Fo(;)14 b Fm(H)q Fo(\))35 b Fd(\(dep)l(ending)h(on)e Fo(\010)g Fd(and)h Fn(B)t Fd(\))f(and)h Fn(c)d(>)e Fo(0)515 3920 y Fd(such)g(that)1527 4081 y Fm(k)p Fo(\005)18 b Fm(\000)g Fo(\005)1794 4093 y Fl(0)1831 4081 y Fm(k)1873 4093 y Fk(\016)1933 4081 y Fo(=)k Fm(O)r Fo(\()p Fn(h)p Fo(\))1008 b(\(2.7\))515 4242 y Fd(and)1165 4342 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[\005)p Fn(;)i(P)e Fo(])p Fm(k)1640 4354 y Fk(\016)1694 4342 y Fo(+)18 b Fm(k)p Fo([\005)p Fn(;)c(P)e Fo(])p Fn(\037)p Fo(\()p Fn(P)g Fo(\))p Fm(k)2252 4354 y Fk(\016)2311 4342 y Fo(=)23 b Fm(O)r Fo(\()p Fn(e)2538 4308 y Fi(\000)p Fk(c=h)2697 4342 y Fo(\))479 b(\(2.8\))515 4571 y Fd(uniformly)22 b(with)g(r)l(esp)l(e)l(ct)f(to)g Fn(h)i(>)g Fo(0)d Fd(smal)t(l)i(enough.)37 b(Mor)l(e)l(over,)25 b Fo(\005\()p Fn(h)p Fo(\))e(=)g(\005)2888 4583 y Fl(0)2925 4571 y Fo(+)2990 4460 y Fl([1)p Fk(=B)s(h)p Fl(])3028 4492 y Fc(X)3031 4669 y Fk(j)s Fl(=1)3200 4571 y Fo(\005)3262 4583 y Fk(j)3298 4571 y Fn(h)3346 4536 y Fk(j)515 4775 y Fd(mo)l(dulo)29 b Fm(O)r Fo(\()p Fn(e)939 4745 y Fi(\000)p Fk(c=h)1098 4775 y Fo(\))f Fd(and,)i(for)f(any)g Fn(j)f Fm(\025)23 b Fo(1)p Fd(,)28 b Fo(\005)1940 4787 y Fk(j)1999 4775 y Fo(=)22 b(\010\()p Fn(P)12 b Fo(\))2285 4754 y(^)2275 4775 y(\005)2337 4787 y Fk(j)2388 4775 y Fo(+)j(\(1)h Fm(\000)f Fo(\010\()p Fn(P)d Fo(\)\))2869 4754 y(^)2859 4775 y(\005)2921 4787 y Fk(j)2957 4775 y Fo(\010\()p Fn(P)g Fo(\))28 b Fd(wher)l(e)525 4865 y Fo(^)515 4886 y(\005)577 4898 y Fk(j)638 4886 y Fd(is)e(a)g(di\013er)l(ential)h(op)l (er)l(ator)f(of)h(or)l(der)f Fn(j)31 b Fd(with)26 b(c)l(o)l (e\016cients)g(analytic)h(on)f Fo(\000)2973 4898 y Fk(a)3009 4882 y Fb(0)3035 4886 y Fn(;)14 b Fo(0)23 b Fn(<)f(a)3268 4856 y Fi(0)3314 4886 y Fn(<)515 4985 y(a)p Fd(,)30 b(de)l(c)l(aying)h (as)f Fm(h)p Fn(x)p Fm(i)1164 4955 y Fi(\000)p Fk(\016)1283 4985 y Fd(uniformly)h(on)f Fo(\000)1833 4997 y Fk(a)1869 4981 y Fb(0)1895 4985 y Fd(.)1926 5255 y Fo(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fp(3)134 b(Pro)t(of)45 b(of)h(the)f(main)g(results) 515 705 y Fo(The)20 b(k)n(ey)g(p)r(oin)n(t)h(in)g(the)g(pro)r(of)e(of)i (the)g(t)n(w)n(o)f(Theorems)f(is)h(\(as)h(in)f([NeSo,)h(Sj]\))g(the)g (construc-)515 821 y(tion)33 b(of)f(a)h(formal)f(sym)n(b)r(ol)37 b(^)-47 b Fn(\031)t Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(h)p Fo(\))32 b(=)f(\005)1934 833 y Fl(0)1972 821 y Fo(\()p Fn(x)p Fo(\))23 b(+)2192 742 y Fc(X)2194 920 y Fk(j)s Fi(\025)p Fl(1)2331 821 y Fo(^)-47 b Fn(\031)2373 833 y Fk(j)2408 821 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))p Fn(h)2644 787 y Fk(j)2712 821 y Fm(2)32 b Fn(C)2864 787 y Fi(1)2935 821 y Fo(\()p Fj(R)3039 787 y Fl(2)p Fk(n)3117 821 y Fo(;)14 b Fm(L)p Fo(\()p Fm(H)q Fo(\)\))515 1016 y(suc)n(h)34 b(that,)i(denoting)e(b)n(y)g Fn(p)p Fo(\()p Fn(x;)14 b(\030)t Fo(\))34 b(=)g Fn(\030)1793 986 y Fl(2)1853 1016 y Fo(+)23 b Fn(V)18 b Fo(\()p Fn(x)p Fo(\))36 b(the)e(sym)n(b)r (ol)g(of)g Fn(P)12 b Fo(\()p Fn(h)p Fo(\),)36 b(satisfy)e(\(at)g(a)515 1116 y(formal)27 b(series)f(lev)n(el\))1746 1287 y(^)-46 b Fn(\031)26 b Fo(=)h(^)-46 b Fn(\031)1953 1252 y Fi(\003)2014 1287 y Fo(=)27 b(^)-46 b Fn(\031)s Fo(#)t(^)g Fn(\031)940 b Fo(\(3.1\))1746 1411 y(^)-46 b Fn(\031)s Fo(#)p Fn(p)18 b Fm(\000)h Fn(p)p Fo(#)t(^)-46 b Fn(\031)26 b Fo(=)c(0)890 b(\(3.2\))515 1582 y(\(here)30 b(and)h(in)g(the)g(follo)n(wing,)f(#)h (denotes)f(the)h(W)-7 b(eyl)31 b(comp)r(osition)f(of)h(sym)n(b)r (ols\).)46 b(Our)515 1682 y(\014rst)27 b(lemma)h(is)f(the)h(follo)n (wing)515 1839 y Fj(Lemma)h(3.1)41 b Fd(Assume)32 b Fo(\()p Fn(H)7 b Fo(1\))p Fn(;)14 b Fo(\()p Fn(H)7 b Fo(2\))32 b Fd(and)i Fo(\()p Fn(H)7 b Fo(3\))2110 1851 y Fk(\016)2179 1839 y Fd(hold)34 b(true)f(for)g(some)h Fn(\016)e Fm(\025)c Fo(0)p Fd(.)48 b(Then,)515 1939 y(ther)l(e)30 b(exist)f(functions)1500 2050 y Fo(^)-46 b Fn(\031)1543 2062 y Fk(j)1578 2050 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))24 b(=)1900 1971 y Fc(X)1878 2153 y Fi(j)p Fk(\013)p Fi(j\024)p Fk(j)2057 2050 y Fn(a)2101 2062 y Fk(\013;j)2199 2050 y Fo(\()p Fn(x)p Fo(\))p Fn(\030)2350 2016 y Fk(\013)3208 2050 y Fo(\(3.3\))515 2290 y Fd(p)l(olynomial)31 b(of)e(de)l(gr)l(e)l(e)g Fn(j)k Fd(with)d(r)l(esp)l(e)l(ct)e(to)g Fn(\030)t Fd(,)i(with)f(smo)l (oth)g(c)l(o)l(e\016cient)g Fn(a)2875 2302 y Fk(\013;j)2973 2290 y Fo(\()p Fn(x)p Fo(\))24 b Fm(2)f(L)p Fo(\()p Fm(H)q Fo(\))515 2389 y Fd(such)30 b(that)f(the)h(formal)h(symb)l(ol)1322 2572 y Fo(^)-47 b Fn(\031)t Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(h)p Fo(\))24 b(=)e(\005)1814 2584 y Fl(0)1852 2572 y Fo(\()p Fn(x)p Fo(\))d(+)2065 2494 y Fc(X)2067 2672 y Fk(j)s Fi(\025)p Fl(1)2203 2572 y Fo(^)-46 b Fn(\031)2246 2584 y Fk(j)2281 2572 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))p Fn(h)2517 2538 y Fk(j)2553 2572 y Fn(;)632 b Fo(\(3.4\))515 2829 y Fd(satis\014es,)30 b(at)g(a)g(formal)h(series)g(level,)1251 3000 y Fo(^)-46 b Fn(\031)1297 2966 y Fi(\003)1358 3000 y Fo(=)27 b(^)-46 b Fn(\031)s(;)104 b Fo(^)-47 b Fn(\031)s Fo(#)t(^)h Fn(\031)27 b Fo(=)g(^)-46 b Fn(\031)s(;)103 b Fo(^)-46 b Fn(\031)s Fo(#)p Fn(p)18 b Fm(\000)g Fn(p)p Fo(#)t(^)-46 b Fn(\031)27 b Fo(=)22 b(0)561 b(\(3.5\))515 3171 y Fd(and,)30 b(in)g(addition,)i(one)e(has)1365 3342 y Fn(a)1409 3354 y Fk(\013;j)1507 3342 y Fo(\()p Fn(x)p Fo(\))24 b(=)f Fm(O)r Fo(\()p Fm(h)p Fn(x)p Fm(i)1941 3308 y Fi(\000)p Fk(\016)2031 3342 y Fo(\))p Fn(;)99 b(f)9 b(or)25 b(j)j Fm(\025)23 b Fo(1)679 b(\(3.6\))515 3514 y Fd(uniformly)33 b(on)f Fj(R)1089 3484 y Fk(n)1134 3514 y Fd(.)46 b(Mor)l(e)l(over,)35 b(if)e Fo(\()p Fn(H)7 b Fo(1)p Fn(A)p Fo(\))32 b Fd(and)h Fo(\()p Fn(H)7 b Fo(3)p Fn(A)p Fo(\))2366 3526 y Fk(\016)2434 3514 y Fd(hold)34 b(for)f(some)g Fn(\016)d Fm(\025)d Fo(0)p Fd(,)33 b(then)515 3613 y Fn(a)559 3625 y Fk(\013;j)656 3613 y Fo(\()p Fn(x)p Fo(\))27 b Fd(ar)l(e)e(analytic)i(on)e Fo(\000)1406 3625 y Fk(a)1446 3613 y Fd(,)h(and)g(\(3.6\))g(holds)h(uniformly)f(on)f Fo(\000)2605 3583 y Fi(0)2605 3634 y Fk(a)2645 3613 y Fd(,)i(for)f(any)g Fo(0)c Fn(<)h(a)3176 3583 y Fi(0)3222 3613 y Fn(<)g(a)p Fd(.)515 3770 y(Pr)l(o)l(of:)49 b Fo(The)32 b(construction)g(of)h(a)g(sym)n(b)r(ol)j(^)-46 b Fn(\031)36 b Fo(satisfying)c(\(3.1\))h(and)f(\(3.2\))h(can)f(b)r(e)h(done)515 3870 y(follo)n(wing)26 b(exactly)h(the)h(same)e(sc)n(heme)h(used)h(in)f (the)h(pro)r(of)f(of)g(Lemma)g(3.2)g(of)g([NeSo])h(in)515 3969 y(the)23 b(case)g(of)g(Klein-Gordon)f(matrix-v)-5 b(alued)22 b(op)r(erators.)34 b(F)-7 b(or)23 b(the)g(sak)n(e)f(of)i (completeness)515 4069 y(w)n(e)30 b(recall)g(here)g(the)h(main)f (argumen)n(ts.)45 b(Let)30 b Fn(\015)5 b Fo(\()p Fn(x)p Fo(\))31 b(b)r(e)g(the)g(con)n(tin)n(uous)f(lo)r(op)g(that)h(sur-)515 4169 y(rounds)d Fn(\033)836 4181 y Fl(0)874 4169 y Fo(\()p Fn(x)p Fo(\))j(and)e(lea)n(v)n(es)f Fn(\033)1466 4181 y Fl(1)1504 4169 y Fo(\()p Fn(x)p Fo(\))i(in)g(its)g(exterior.)41 b(In)30 b(the)g(case)e Fn(\016)33 b Fo(is)c(strictly)g(p)r(ositiv)n(e,) 515 4268 y(w)n(e)i(can)h(also)e(supp)r(ose)i(that)g Fn(\015)5 b Fo(\()p Fn(x)p Fo(\))33 b(admits)e(a)h(limit)g(lo)r(op)g(as)f Fm(j)p Fn(x)p Fm(j)g(!)f Fo(+)p Fm(1)p Fo(.)49 b(Let)32 b(us)g(con-)515 4368 y(sider)27 b(\012)d(=:)f Fm(f)p Fo(\()p Fn(x;)14 b(\030)t(;)g(z)t Fo(\))23 b Fm(2)h Fj(R)1393 4338 y Fl(2)p Fk(n)1490 4368 y Fm(\002)18 b Fj(C)24 b Fo(;)14 b Fn(z)22 b Fm(\000)c Fn(\030)1887 4338 y Fl(2)1948 4368 y Fm(2)24 b Fn(\015)5 b Fo(\()p Fn(x)p Fo(\))p Fm(g)p Fo(.)38 b(F)-7 b(or)27 b(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(z)t Fo(\))24 b Fm(2)g Fo(\012,)k Fn(p)p Fo(\()p Fn(x;)14 b(\030)t Fo(\))20 b Fm(\000)e Fn(z)31 b Fo(is)515 4468 y(in)n(v)n(ertible)23 b(and)g Fn(q)1072 4480 y Fl(0)1109 4468 y Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(z)t Fo(\))24 b(=)e(\()p Fn(p)p Fo(\()p Fn(x;)14 b(\030)t Fo(\))c Fm(\000)g Fn(z)t Fo(\))1910 4437 y Fi(\000)p Fl(1)2024 4468 y Fo(is)23 b(smo)r(oth)g(and)h(b)r(ounded.)36 b(Let)23 b(us)h(de\014ne)515 4567 y(the)k(sym)n(b)r(ol)1417 4679 y Fn(r)r Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(h;)g(z)t Fo(\))23 b(=)1920 4600 y Fc(X)1922 4778 y Fk(j)s Fi(\025)p Fl(1)2054 4679 y Fn(r)2091 4691 y Fk(j)2126 4679 y Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(z)t Fo(\))p Fn(h)2442 4644 y Fk(j)3208 4679 y Fo(\(3.7\))515 4907 y(b)n(y)1180 5006 y(\()p Fn(p)p Fo(\()p Fn(x;)g(\030)t Fo(\))19 b Fm(\000)f Fn(z)t Fo(\))c(#)p Fn(q)1739 5018 y Fl(0)1776 5006 y Fo(\()p Fn(x;)g(\030)t Fo(;)g Fn(z)t Fo(\))24 b(=)e(1)c Fm(\000)g Fn(r)r Fo(\()p Fn(x;)c(\030)t Fo(;)g Fn(h;)g(z)t Fo(\))p Fn(:)495 b Fo(\(3.8\))1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fo(and)27 b Fn(q)s Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(h;)g(z)t Fo(\))28 b(b)n(y)1573 639 y Fn(q)f Fo(=)22 b Fn(q)1761 651 y Fl(0)1817 639 y Fo(+)c Fn(q)1937 651 y Fl(0)1974 639 y Fo(#)2057 560 y Fc(X)2059 739 y Fk(j)s Fi(\025)p Fl(1)2191 639 y Fn(r)2230 605 y Fl(#)p Fk(j)3208 639 y Fo(\(3.9\))515 875 y(One)27 b(can)g(easily)g(pro)n(v)n(e)f(\(see)h(e.g.)36 b(the)28 b(App)r(endix)h(in)e([NeSo]\),)h(that)g(for)f(all)g Fn(j)h Fo(=)23 b(1)p Fn(;)14 b Fo(2)p Fn(;)g(:::;)515 975 y Fo(and)27 b(for)g(an)n(y)g(m)n(ultindex)h Fn(\013;)14 b(\014)32 b Fo(there)c(exist)f Fn(K)6 b Fo(\()p Fn(\013;)14 b(\014)t(;)g(j)5 b Fo(\))24 b Fn(>)f Fo(0)k(suc)n(h)g(that,)1020 1152 y(Sup)946 1225 y Fl(\()p Fk(x;\030)r Fl(;)p Fk(z)r Fl(\))p Fi(2)p Fl(\012)1232 1152 y Fm(h)p Fn(x)p Fm(i)1343 1118 y Fk(\016)1381 1152 y Fm(h)p Fn(\030)t Fm(i)1485 1118 y Fi(\000j)p Fk(\013)p Fi(j\000)p Fk(j)1707 1152 y Fm(k)p Fn(@)1798 1118 y Fk(\013)1793 1172 y(\030)1844 1152 y Fn(@)1893 1118 y Fk(\014)1888 1172 y(x)1938 1152 y Fn(q)1975 1164 y Fk(j)2010 1152 y Fo(\()p Fn(x;)14 b(\030)t(;)g(z)t Fo(\))p Fm(k)2320 1167 y Fi(L)p Fl(\()p Fi(H)p Fl(\))2502 1152 y Fm(\024)22 b Fn(K)6 b Fo(\()p Fn(\013;)14 b(\014)t(;)g(j)5 b Fo(\))220 b(\(3.10\))515 1396 y(If)38 b(\()p Fn(H)7 b Fo(1)p Fn(A)p Fo(\))38 b(and)g(\()p Fn(H)7 b Fo(3)p Fn(A)p Fo(\))1306 1408 y Fk(\016)1380 1396 y Fo(hold)38 b(for)f(some)g Fn(\016)43 b Fm(\025)d Fo(0,)g(one)d(can)g(tak)n(e)h(\(see)f(e.g.)67 b([BoKr)o(]\))515 1512 y Fn(K)6 b Fo(\()p Fn(\013;)14 b(\014)t(;)g(j)5 b Fo(\))23 b(=)g Fn(K)1061 1469 y Fk(j)s Fl(+)p Fi(j)p Fk(\013)p Fi(j)p Fl(+)p Fi(j)p Fk(\014)s Fi(j)p Fl(+1)1055 1534 y(1)1444 1512 y Fn(\013)p Fo(!)p Fn(\014)t Fo(!)p Fn(j)5 b Fo(!)22 b(for)e(some)g(constan)n(t)h Fn(K)2399 1524 y Fl(1)2458 1512 y Fn(>)i Fo(0)d(indep)r(enden)n(t)i(of)f(m)n (ulti-)515 1612 y(index)26 b Fn(\013;)14 b(\014)32 b Fo(and)26 b(of)h Fn(j)5 b Fo(.)37 b(Moreo)n(v)n(er,)24 b(one)i(can)g(c)n(hec)n(k)g(that,)h(for)f Fn(j)i Fm(\025)23 b Fo(1,)j Fn(q)2777 1624 y Fk(j)2812 1612 y Fo(\()p Fn(x;)14 b(\030)t(;)g(z)t Fo(\))27 b(is)g(giv)n(en)515 1711 y(b)n(y)g(\(see)h (e.g.[Ba)n(]\))808 1972 y Fn(q)845 1984 y Fk(j)903 1972 y Fo(=)22 b Fn(q)1027 1984 y Fl(0)1118 1865 y(2)p Fk(j)1090 1893 y Fc(X)1079 2069 y Fk(m)p Fl(=1)1404 1893 y Fc(X)1235 2087 y Fl(\(\()p Fk(\013)1330 2095 y Ff(i)1357 2087 y Fl(\))p Fk(;)p Fl(\()p Fk(\014)1467 2095 y Ff(i)1492 2087 y Fl(\)\))p Fi(2)p Fk(E)1641 2059 y Ff(j)1638 2099 y(m)1707 1972 y Fn(C)1766 1984 y Fk(m)1829 1972 y Fo(\(\()p Fn(\013)p Fo(\))1978 1984 y Fk(i)2007 1972 y Fn(;)14 b Fo(\()p Fn(\014)t Fo(\))2159 1984 y Fk(i)2187 1972 y Fo(\))2258 1868 y Fk(m)2234 1893 y Fc(Y)2233 2070 y Fk(i)p Fl(=1)2355 1880 y Fc(\020)2405 1972 y Fo(\()p Fn(@)2486 1938 y Fk(\013)2529 1946 y Ff(i)2481 1993 y Fk(x)2559 1972 y Fn(@)2608 1932 y Fk(\014)2646 1940 y Ff(i)2603 1997 y Fk(\030)2676 1972 y Fn(p)p Fo(\))p Fn(q)2787 1984 y Fl(0)2824 1880 y Fc(\021)3167 1972 y Fo(\(3.11\))515 2333 y(where)38 b Fn(E)832 2299 y Fk(j)827 2354 y(m)933 2333 y Fo(=)k Fm(f)p Fo(\(\()p Fn(\013)1199 2345 y Fk(i)1226 2333 y Fo(\))p Fn(;)14 b Fo(\()p Fn(\014)1374 2345 y Fk(i)1403 2333 y Fo(\)\))42 b Fm(2)h Fo(\()p Fj(Z)1697 2299 y Fk(n)1697 2354 y Fl(+)1753 2333 y Fo(\))1785 2291 y Fk(m)1874 2333 y Fm(\002)26 b Fo(\()p Fj(Z)2055 2299 y Fk(n)2055 2354 y Fl(+)2110 2333 y Fo(\))2143 2291 y Fk(m)2248 2333 y Fo(;)14 b Fm(j)2352 2229 y Fk(m)2322 2254 y Fc(X)2328 2431 y Fk(i)p Fl(=1)2456 2333 y Fn(\013)2509 2345 y Fk(i)2536 2333 y Fm(j)43 b Fo(=)e Fm(j)2776 2229 y Fk(m)2745 2254 y Fc(X)2751 2431 y Fk(i)p Fl(=1)2879 2333 y Fn(\014)2926 2345 y Fk(i)2954 2333 y Fm(j)h Fo(=)g Fn(j)5 b Fm(g)38 b Fo(and)515 2510 y Fn(C)574 2522 y Fk(m)637 2510 y Fo(\(\()p Fn(\013)p Fo(\))786 2522 y Fk(i)815 2510 y Fn(;)14 b Fo(\()p Fn(\014)t Fo(\))967 2522 y Fk(i)996 2510 y Fo(\))27 b(are)g(univ)n(ersal)f(constan)n(ts.)36 b(Let)28 b(us)f(set)1341 2687 y(\000\()p Fn(x;)14 b(\030)t Fo(\))24 b(=)e Fm(f)p Fn(z)k Fm(2)d Fj(C)g Fo(;)37 b Fn(z)22 b Fm(\000)c Fn(\030)2213 2653 y Fl(2)2274 2687 y Fm(2)23 b Fn(\015)5 b Fo(\()p Fn(x)p Fo(\))p Fm(g)515 2864 y Fo(and)27 b(let)h(us)g(de\014ne)1333 2964 y(^)-46 b Fn(\031)s Fo(\()p Fn(x;)14 b(\030)t Fo(;)g Fn(h)p Fo(\))24 b(=)f(\005)1826 2976 y Fl(0)1863 2964 y Fo(\()p Fn(x)p Fo(\))d(+)2077 2885 y Fc(X)2079 3063 y Fk(j)s Fi(\025)p Fl(1)2215 2964 y Fo(^)-46 b Fn(\031)2258 2976 y Fk(j)2293 2964 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))p Fn(h)2529 2929 y Fk(j)515 3195 y Fo(where)845 3394 y(\005)907 3406 y Fl(0)944 3394 y Fo(\()p Fn(x)p Fo(\))84 b(=)1328 3338 y Fn(i)p 1297 3375 92 4 v 1297 3451 a Fo(2)p Fn(\031)1412 3281 y Fc(I)1458 3469 y Fl(\000\()p Fk(x;\030)r Fl(\))1659 3394 y Fn(q)1696 3406 y Fk(o)1733 3394 y Fo(\()p Fn(x;)14 b(\030)t(;)g(z)t Fo(\))23 b Fn(dz)k Fo(=)2262 3338 y Fn(i)p 2230 3375 V 2230 3451 a Fo(2)p Fn(\031)2346 3281 y Fc(I)2392 3469 y Fk(\015)t Fl(\()p Fk(x)p Fl(\))2524 3394 y Fo(\()p Fn(V)19 b Fo(\()p Fn(x)p Fo(\))h Fm(\000)e Fn(z)t Fo(\))2912 3359 y Fi(\000)p Fl(1)3024 3394 y Fn(dz)789 3630 y Fo(^)-46 b Fn(\031)832 3642 y Fk(j)867 3630 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))84 b(=)1328 3574 y Fn(i)p 1297 3611 V 1297 3687 a Fo(2)p Fn(\031)1412 3517 y Fc(I)1458 3706 y Fl(\000\()p Fk(x;\030)r Fl(\))1659 3630 y Fn(q)1696 3642 y Fk(j)1731 3630 y Fo(\()p Fn(x;)14 b(\030)t(;)g(z)t Fo(\))23 b Fn(dz)1063 b Fo(\(3.12\))515 3874 y(Then)26 b(^)-47 b Fn(\031)25 b Fo(satis\014es)c(\(3.1\))g(and)g(\(3.2\))h(and,) g(b)n(y)f(making)g(the)h(c)n(hange)e(of)i(v)-5 b(ariable)20 b Fn(z)26 b Fm(!)d Fn(z)10 b Fm(\000)c Fn(\030)3342 3843 y Fl(2)515 3973 y Fo(in)41 b(the)g(in)n(tegral)e(\(3.12\))h(and)g(ha)n (ving)g(in)h(mind)g(\(3.11\),)i(one)d(can)g(easily)g(c)n(hec)n(k)g (that)519 4115 y(^)-46 b Fn(\031)562 4127 y Fk(j)597 4115 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))29 b(=)948 4059 y Fn(i)p 916 4096 V 916 4172 a Fo(2)p Fn(\031)1032 4002 y Fc(I)1078 4191 y Fk(\015)t Fl(\()p Fk(x)p Fl(\))1224 4115 y Fn(q)1261 4127 y Fk(j)1296 4115 y Fo(\()p Fn(x;)14 b(\030)t(;)g(z)24 b Fo(+)c Fn(\030)1677 4081 y Fl(2)1714 4115 y Fo(\))28 b Fn(dz)k Fo(=)2003 4036 y Fc(X)1980 4218 y Fi(j)p Fk(\013)p Fi(j\024)p Fk(j)2159 4115 y Fn(a)2203 4127 y Fk(\013;j)2301 4115 y Fo(\()p Fn(x)p Fo(\))p Fn(\030)2452 4081 y Fk(\013)2531 4115 y Fo(are)d(p)r(olynomials)h(matrix)515 4320 y(v)-5 b(alued)41 b(functions)h(of)f(degree)f Fn(j)5 b Fo(,)45 b(with)d(smo)r(oth)f(co)r(e\016cien)n(ts)g(deca)n(ying)f(as)h Fm(h)p Fn(x)p Fm(i)3174 4290 y Fi(\000)p Fk(\016)3305 4320 y Fo(as)515 4420 y Fm(j)p Fn(x)p Fm(j)27 b(!)f(1)p Fo(,)31 b(if)f Fn(j)h Fm(\025)26 b Fo(1.)43 b(Moreo)n(v)n(er,)28 b(if)i(\()p Fn(H)7 b Fo(1)p Fn(A)p Fo(\))30 b(and)f(\()p Fn(H)7 b Fo(3)p Fn(A)p Fo(\))2373 4432 y Fk(\016)2439 4420 y Fo(hold)30 b(for)f(some)g Fn(\016)h Fm(\025)c Fo(0,)k(then)515 4519 y(the)24 b Fn(a)698 4531 y Fk(\013;j)819 4519 y Fo(are)f(analytic)g(on)g(\000)1429 4531 y Fk(a)1493 4519 y Fo(and)g(\(3.6\))h(holds)f(uniformly)h(on)f(\000)2596 4531 y Fk(a)2632 4515 y Fb(0)2658 4519 y Fo(,)i(for)e(an)n(y)g(0)g Fn(<)f(a)3178 4489 y Fi(0)3224 4519 y Fn(<)h(a)p Fo(.)p 523 4631 34 63 v 515 4718 a Fd(Pr)l(o)l(of)36 b(of)g(The)l(or)l(em)f (2.1)g Fo(Let)e(us)h(consider)e(the)h Fn(h)p Fo(-di\013eren)n(tial)g (op)r(erator)e(giv)n(en)i(b)n(y)g(the)515 4818 y(W)-7 b(eyl)28 b(quan)n(tization)e(of)i(the)g(sym)n(b)r(ols)j(^)-46 b Fn(\031)1806 4830 y Fk(j)1869 4818 y Fo(i.e)1603 4985 y(^)1593 5006 y(\005)1655 5018 y Fk(j)1713 5006 y Fo(=)23 b(Op)1911 4970 y Fk(w)1911 5027 y(h)1965 5006 y Fo(\()t(^)-46 b Fn(\031)2044 5018 y Fk(j)2080 5006 y Fo(\()p Fn(x;)14 b(\030)t Fo(\)\))867 b(\(3.13\))1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fo(No)n(w)33 b(w)n(e)g(c)n(ho)r(ose)f(\010)h Fm(2)g Fn(C)1353 493 y Fi(1)1347 544 y Fl(0)1424 523 y Fo(\()p Fj(R)p Fo(\))g(and)h(w)n(e)f(consider)f(the)i(family)g(of)f (b)r(ounded)h(op)r(erator)515 623 y(\010\()p Fn(P)12 b Fo(\))714 602 y(^)704 623 y(\005)766 635 y Fk(j)801 623 y Fo(.)52 b(Then)32 b(w)n(e)g(can)g(resum)g(in)h(a)f(standard)f(w)n (a)n(y)h(the)g(formal)g(series)f(of)i(op)r(erator)515 742 y(\010\()p Fn(P)12 b Fo(\)\005)766 754 y Fl(0)814 742 y Fo(+)890 663 y Fc(X)892 841 y Fk(j)s Fi(\025)p Fl(1)1023 742 y Fo(\010\()p Fn(P)g Fo(\))1222 721 y(^)1212 742 y(\005)1274 754 y Fk(j)1310 742 y Fn(h)1358 708 y Fk(j)1417 742 y Fo(and)23 b(\005)1636 754 y Fl(0)1674 742 y Fo(\010\()p Fn(P)12 b Fo(\))f(+)1950 663 y Fc(X)1952 841 y Fk(j)s Fi(\025)p Fl(1)2093 721 y Fo(^)2083 742 y(\005)2145 754 y Fk(j)2180 742 y Fo(\010\()p Fn(P)h Fo(\))p Fn(h)2417 708 y Fk(j)2476 742 y Fo(and)24 b(w)n(e)f(denote)h(b) n(y)f(\010\()p Fn(P)12 b Fo(\))3326 721 y(^)3316 742 y(\005)515 948 y(and)687 927 y(^)676 948 y(\005\010\()p Fn(P)g Fo(\))28 b(the)g(resulting)f(op)r(erators.)35 b(Let)28 b(us)g(de\014ne)718 1109 y(^)707 1130 y(\005)769 1142 y Fl(\010)844 1130 y Fo(:=)23 b(\010\()p Fn(P)12 b Fo(\))1154 1109 y(^)1144 1130 y(\005)19 b(+)f(\(1)g Fm(\000)g Fo(\010\()p Fn(P)12 b Fo(\)\))1714 1109 y(^)1704 1130 y(\005)q(\010\()p Fn(P)g Fo(\))19 b(+)f(\(1)g Fm(\000)g Fo(\010\()p Fn(P)12 b Fo(\)\)\005)2516 1142 y Fl(0)2554 1130 y Fo(\(1)19 b Fm(\000)f Fo(\010\()p Fn(P)12 b Fo(\)\))p Fn(:)193 b Fo(\(3.14\))515 1313 y(Then)742 1292 y(^)732 1313 y(\005)794 1325 y Fl(\010)873 1313 y Fo(satisfy)973 1496 y Fm(k)1025 1475 y Fo(^)1015 1496 y(\005)1077 1508 y Fl(\010)1147 1496 y Fm(\000)18 b Fo(\005)1292 1508 y Fl(0)1329 1496 y Fm(k)1371 1508 y Fk(\016)1426 1496 y Fo(+)g Fm(k)1561 1475 y Fo(^)1551 1496 y(\005)1613 1461 y Fl(2)1613 1516 y(\010)1683 1496 y Fm(\000)1776 1475 y Fo(^)1766 1496 y(\005)1828 1508 y Fl(\010)1879 1496 y Fm(k)1921 1508 y Fk(\016)1976 1496 y Fo(+)g Fm(k)p Fo([)2134 1475 y(^)2124 1496 y(\005)2186 1508 y Fl(\010)2237 1496 y Fn(;)c(P)e Fo(])p Fm(k)2404 1508 y Fk(\016)2463 1496 y Fo(=)23 b Fm(O)r Fo(\()p Fn(h)p Fo(\))p Fn(:)515 1678 y Fo(In)33 b(particular,)f(the)h(sp)r(ectrum)g(of)1662 1657 y(^)1652 1678 y(\005)1714 1690 y Fl(\010)1798 1678 y Fo(is)g(close)f(to)g Fm(f)p Fo(0)p Fn(;)14 b Fo(1)p Fm(g)31 b Fo(up)i(to)g Fm(O)r Fo(\()p Fn(h)p Fo(\))g(and,)h(b)n(y)e (using)515 1778 y(the)24 b(standard)g(calculus)g(for)g(op)r(erator-v)-5 b(alued)22 b(pseudo)r(di\013eren)n(tial)i(op)r(erators,)f(\(see)h(also) 515 1878 y(the)30 b(direct)g(pro)r(of)f(of)h(Theorem)f(2.2\))g(it)i(is) f(easy)f(to)g(c)n(hec)n(k)g(that,)i(for)f Fn(\037)c Fm(2)i Fn(C)2983 1847 y Fi(1)2977 1898 y Fl(0)3053 1878 y Fo(\()p Fj(R)p Fo(\))i(suc)n(h)515 1977 y(that)e Fn(\037)p Fo(\010)23 b(=)f Fn(\037)p Fo(,)28 b(then,)g(for)f(an)n(y)g Fn(N)32 b Fm(2)23 b Fj(N)p Fo(,)28 b(w)n(e)f(ha)n(v)n(e)1054 2160 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)1310 2139 y(^)1300 2160 y(\005)1362 2172 y Fl(\010)1414 2160 y Fn(;)i(P)e Fo(])p Fm(k)1581 2172 y Fk(\016)1635 2160 y Fo(+)18 b Fm(k)p Fo([)1793 2139 y(^)1783 2160 y(\005)1845 2172 y Fl(\010)1896 2160 y Fn(;)c(P)e Fo(])p Fn(\037)p Fo(\()p Fn(P)g Fo(\))p Fm(k)2244 2172 y Fk(\016)2303 2160 y Fo(=)23 b Fm(O)r Fo(\()p Fn(h)2539 2126 y Fk(N)6 b Fl(+1)2687 2160 y Fo(\))448 b(\(3.15\))1054 2284 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()1319 2263 y(^)1309 2284 y(\005)1371 2250 y Fl(2)1371 2305 y(\010)1441 2284 y Fm(\000)1535 2263 y Fo(^)1524 2284 y(\005)1586 2296 y Fl(\010)1638 2284 y Fo(\))p Fm(k)1712 2296 y Fk(\016)1767 2284 y Fo(+)18 b Fm(k)p Fo(\()1934 2263 y(^)1924 2284 y(\005)1986 2250 y Fl(2)1986 2305 y(\010)2056 2284 y Fm(\000)2149 2263 y Fo(^)2139 2284 y(\005)2201 2296 y Fl(\010)2253 2284 y Fo(\))p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\))p Fm(k)2508 2296 y Fk(\016)2567 2284 y Fo(=)23 b Fm(O)r Fo(\()p Fn(h)2803 2250 y Fk(N)6 b Fl(+1)2951 2284 y Fo(\))p Fn(:)161 b Fo(\(3.16\))515 2467 y(T)-7 b(aking)1315 2605 y(\005)23 b(:=)1553 2548 y Fn(i)p 1521 2585 92 4 v 1521 2661 a Fo(2)p Fn(\031)1637 2492 y Fc(Z)1683 2680 y Fi(j)p Fk(z)r Fi(\000)p Fl(1)p Fi(j)p Fl(=1)p Fk(=)p Fl(2)2010 2512 y Fc(\020)2070 2584 y Fo(^)2060 2605 y(\005)2122 2617 y Fl(\010)2192 2605 y Fm(\000)18 b Fn(z)2318 2512 y Fc(\021)2367 2530 y Fi(\000)p Fl(1)2493 2605 y Fn(dz)515 2815 y Fo(and)27 b(recalling)f(\(see)i([Ne])g(and)f ([NeSo]\))h(\))g(that)515 3040 y(\005)7 b Fm(\000)667 3019 y Fo(^)656 3040 y(\005)718 3052 y Fl(\010)793 3040 y Fo(=)922 2984 y Fn(i)p 891 3021 V 891 3097 a Fo(2)p Fn(\031)992 3040 y Fo(\()1034 3019 y(^)1024 3040 y(\005)1086 3006 y Fl(2)1086 3061 y(\010)1146 3040 y Fm(\000)1228 3019 y Fo(^)1218 3040 y(\005)1280 3052 y Fl(\010)1332 3040 y Fo(\))1378 2927 y Fc(I)1424 3116 y Fi(j)p Fk(z)r Fi(\000)p Fl(1)p Fi(j)p Fl(=1)p Fk(=)p Fl(2)1738 3040 y Fo(\()1780 3019 y(^)1770 3040 y(\005)1832 3052 y Fl(\010)1891 3040 y Fm(\000)g Fn(z)t Fo(\))2038 3006 y Fi(\000)p Fl(1)2127 3040 y Fo(\(2)2211 3019 y(^)2201 3040 y(\005)2263 3052 y Fl(\010)2322 3040 y Fm(\000)g Fo(1\)\(1)g Fm(\000)2631 3019 y Fo(^)2621 3040 y(\005)2683 3052 y Fl(\010)2741 3040 y Fm(\000)g Fn(z)t Fo(\))2888 3006 y Fi(\000)p Fl(1)2977 3040 y Fo(\(1)g Fm(\000)g Fn(z)t Fo(\))3205 3006 y Fi(\000)p Fl(1)3294 3040 y Fn(dz)3167 3201 y Fo(\(3.17\))515 3301 y(it)28 b(is)f(easy)g(to)g(pro)n(v)n(e)f(that,)i(there)g(exists)f Fn(C)i(>)23 b Fo(0)k(indep)r(enden)n(t)h(of)g Fn(N)9 b Fo(suc)n(h)27 b(that)1190 3483 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\(\005)19 b Fm(\000)1619 3462 y Fo(^)1609 3483 y(\005)1671 3495 y Fl(\010)1723 3483 y Fo(\))p Fm(k)1797 3495 y Fk(\016)1856 3483 y Fm(\024)k Fn(C)6 b Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()2274 3462 y(^)2264 3483 y(\005)2326 3449 y Fl(2)2326 3504 y(\010)2396 3483 y Fm(\000)2490 3462 y Fo(^)2480 3483 y(\005)2542 3495 y Fl(\010)2593 3483 y Fo(\))p Fm(k)2667 3495 y Fk(\016)3167 3483 y Fo(\(3.18\))515 3666 y(Using)27 b(\(3.16\))g(and)g(\(3.18\),)g (one)h(easily)e(obtain)i(\(2.5\))f(and)g(\(2.6\).)p 3337 3678 34 63 v 515 3765 a(No)n(w,)g(let)g(us)g(consider)f(the)i(analytic) f(case.)36 b(In)27 b(suc)n(h)g(a)g(case,)f(in)i(order)d(to)j(get)f(exp) r(onen-)515 3865 y(tially)k(small)g(error)f(terms,)i(the)g(idea)f(is)g (to)g(tak)n(e)g(instead)g(of)h(a)f(classical)f(resummation)515 3977 y(of)k(\010\()p Fn(P)12 b Fo(\)\005)867 3989 y Fl(0)928 3977 y Fo(+)1016 3898 y Fc(X)1018 4076 y Fk(j)s Fi(\025)p Fl(1)1150 3977 y Fo(\010\()p Fn(P)g Fo(\))1349 3956 y(^)1339 3977 y(\005)1401 3989 y Fk(j)1436 3977 y Fn(h)1484 3942 y Fk(j)1519 3977 y Fo(,)36 b(an)f(analytic)f(t)n(yp)r(e)g(resummation.) 57 b(This)35 b(means)f(that)515 4256 y(w)n(e)29 b(w)n(an)n(t)g(to)g (consider)g(\010\()p Fn(P)12 b Fo(\)\005)1524 4268 y Fl(0)1582 4256 y Fo(+)1696 4152 y Fk(N)1666 4177 y Fc(X)1668 4354 y Fk(j)s Fl(=1)1800 4256 y Fo(\010\()p Fn(P)g Fo(\))1999 4235 y(^)1989 4256 y(\005)2051 4268 y Fk(j)2086 4256 y Fn(h)2134 4221 y Fk(j)2199 4256 y Fo(when)30 b Fn(N)38 b Fo(can)29 b(b)r(e)h Fn(h)g Fo(dep)r(enden)n(t)g(i.e.)515 4452 y Fn(N)37 b Fo(=)712 4385 y Fc(\002)786 4419 y Fl(1)p 757 4433 92 4 v 757 4481 a Fk(B)s(h)859 4385 y Fc(\003)924 4452 y Fo(with)32 b Fn(B)j Fo(large)30 b(enough)g(\(see)h([BoKr)n(,)g (Sj)q(,)g(Ne]\).)47 b(Due)32 b(to)f(the)g(fact)g(that)g(w)n(e)515 4577 y(ha)n(v)n(e)g(in)n(tro)r(duced)h(cuto\013)h(functions)g(in)g(the) g(de\014nition)g(of)g(\010\()p Fn(P)12 b Fo(\)\005)2707 4589 y Fl(0)2766 4577 y Fo(+)2853 4498 y Fc(X)2855 4676 y Fk(j)s Fi(\025)p Fl(1)2987 4577 y Fo(\010\()p Fn(P)g Fo(\))3186 4556 y(^)3176 4577 y(\005)3238 4589 y Fk(j)3273 4577 y Fn(h)3321 4542 y Fk(j)3356 4577 y Fo(,)515 4762 y(w)n(e)30 b(cannot)g(used)g(the)h(standard)f(calculus)g(for)g (analytic)g(pseudo)r(di\013eren)n(tial)g(op)r(erators)515 4861 y(and)i(w)n(e)h(need)g(to)f(con)n(trol)g(directly)g(the)h(constan) n(ts)f(in)n(v)n(olv)n(ed.)51 b(Let)33 b(us)f(start)g(with)i(the)515 4961 y(follo)n(wing)26 b(lemma:)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fj(Lemma)29 b(3.2)41 b Fd(L)l(et)1504 623 y Fn(q)1541 635 y Fk(j)1576 623 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))24 b(=)1899 544 y Fc(X)1876 726 y Fi(j)p Fk(\013)p Fi(j\024)p Fk(j)2055 623 y Fn(q)2092 635 y Fk(\013;j)2190 623 y Fo(\()p Fn(x)p Fo(\))p Fn(\030)2341 588 y Fk(\013)3167 623 y Fo(\(3.19\))515 859 y Fd(b)l(e)29 b(p)l(olynomial)j(functions)d(of)h(de)l(gr)l(e)l(e)g Fn(j)k Fd(with)c(r)l(esp)l(e)l(ct)f(to)h Fn(\030)t Fd(,)g(with)g(c)l(o) l(e\016cients)g Fn(q)3091 871 y Fk(\013;j)3189 859 y Fo(\()p Fn(x)p Fo(\))24 b Fm(2)515 959 y(L)p Fo(\()p Fm(H)q Fo(\))30 b Fd(analytic)h(on)f Fo(\000)1222 971 y Fk(a)1292 959 y Fd(such)g(that)1298 1125 y Fm(k)p Fn(@)1389 1091 y Fk(\015)1384 1146 y(x)1431 1125 y Fn(q)1468 1137 y Fk(\013;j)1566 1125 y Fo(\()p Fn(x)p Fo(\))p Fm(k)1719 1140 y Fi(L)p Fl(\()p Fi(H)p Fl(\))1901 1125 y Fm(\024)23 b Fn(C)2054 1091 y Fk(j)s Fl(+)p Fk(\015)t Fl(+1)2263 1125 y Fn(j)5 b Fo(!)p Fn(\015)g Fo(!)p Fm(h)p Fn(x)p Fm(i)2507 1091 y Fi(\000)p Fk(\016)515 1292 y Fd(for)43 b(some)f Fn(C)52 b(>)45 b Fo(0)c Fd(and)i Fn(\016)48 b Fm(\025)d Fo(0)c Fd(uniformly)i(on)f Fo(\000)2214 1304 y Fk(a)2296 1292 y Fd(and)h(let)f(us)f(set)h Fn(Q)2927 1304 y Fk(j)2962 1292 y Fo(\()p Fn(x;)14 b(hD)3195 1304 y Fk(x)3237 1292 y Fo(\))45 b(=)515 1391 y(Op)625 1355 y Fk(w)625 1412 y(h)679 1391 y Fo(\()p Fn(q)748 1403 y Fk(j)784 1391 y Fo(\()p Fn(x;)14 b(\030)t Fo(\)\))41 b Fd(the)g(Weyl)g(quantization)g(of)g Fn(q)2046 1403 y Fk(j)2081 1391 y Fd(.)71 b(Then,)44 b(for)d(any)g Fo(\010)i Fm(2)f Fn(C)3010 1361 y Fi(1)3004 1412 y Fl(0)3081 1391 y Fo(\()p Fj(R)p Fo(\))p Fd(,)i(we)515 1491 y(have,)1308 1591 y Fm(k)p Fo(\010\()p Fn(P)12 b Fo(\))p Fn(Q)1605 1603 y Fk(j)1640 1591 y Fm(k)1682 1603 y Fk(\016)1736 1591 y Fo(+)18 b Fm(k)p Fn(Q)1927 1603 y Fk(j)1961 1591 y Fo(\010\()p Fn(P)12 b Fo(\))p Fm(k)2192 1603 y Fk(\016)2252 1591 y Fm(\024)22 b Fn(C)2404 1556 y Fk(j)s Fl(+1)2524 1591 y Fn(j)5 b Fo(!)581 b(\(3.20\))515 1730 y Fd(for)30 b(some)h(c)l(onstant)d Fn(C)i(>)22 b Fo(0)30 b Fd(indep)l(endent)g(of)h Fn(h)e Fd(and)h Fn(j)5 b Fd(,)31 b(as)f(far)g(as)g Fn(j)e Fm(\024)2823 1663 y Fc(\002)2858 1730 y Fo(1)p Fn(=h)2988 1663 y Fc(\003)3023 1730 y Fd(.)515 1900 y(Pr)l(o)l(of)p Fo(:)38 b(T)-7 b(aking)27 b Fn(a)c Fm(2)g Fj(R)p Fo(,)k Fn(a)c(>>)f Fo(1)28 b(suc)n(h)f(that)1986 1879 y(^)1968 1900 y Fn(P)35 b Fo(=:)22 b Fn(P)31 b Fo(+)18 b Fn(a)27 b Fo(is)h(in)n(v)n(ertible,)f(one)g(can)g(write)538 2087 y Fm(h)p Fn(x)p Fm(i)649 2052 y Fk(\016)686 2087 y Fo(\010\()p Fn(P)12 b Fo(\))p Fn(Q)941 2099 y Fk(j)999 2087 y Fo(=)1087 1994 y Fc(\020)1136 2087 y Fm(h)p Fn(x)p Fm(i)1247 2052 y Fk(\016)1285 2087 y Fo(\010\()p Fn(P)g Fo(\))1493 2066 y(^)1474 2087 y Fn(P)1539 2052 y Fk(j)s(=)p Fl(2)1641 2087 y Fm(h)p Fn(x)p Fm(i)1752 2052 y Fi(\000)p Fk(\016)1841 1994 y Fc(\021)i(\020)1954 2087 y Fm(h)p Fn(x)p Fm(i)2065 2052 y Fk(\016)2121 2066 y Fo(^)2103 2087 y Fn(P)2168 2052 y Fi(\000)p Fk(j)s(=)p Fl(2)2340 2066 y Fo(^)2322 2087 y Fn(P)2387 2043 y Fk(j)s(=)p Fl(2)2375 2109 y(0)2489 2087 y Fm(h)p Fn(x)p Fm(i)2600 2052 y Fi(\000)p Fk(\016)2689 1994 y Fc(\021)f(\020)2802 2087 y Fm(h)p Fn(x)p Fm(i)2913 2052 y Fk(\016)2950 2087 y Fn(P)3015 2043 y Fi(\000)p Fk(j)s(=)p Fl(2)3003 2109 y(0)3169 2087 y Fn(Q)3235 2099 y Fk(j)3270 1994 y Fc(\021)3333 2087 y Fn(:)515 2292 y Fo(If)598 2271 y(~)589 2292 y(\010)23 b Fm(2)g Fn(C)815 2262 y Fi(1)809 2313 y Fl(0)886 2292 y Fo(\()p Fj(C)p Fo(\))d(is)e(an)h(almost)f(analytic)h(extension)f(of)h(\010)g(suc)n(h)f (that)h Fm(j)p 2682 2224 49 4 v Fn(@)2740 2271 y Fo(~)2731 2292 y(\010\()p Fn(z)t Fo(\))p Fm(j)k(\024)g Fn(C)3091 2304 y Fl(1)3128 2292 y Fm(j)p Fo(Im)q Fn(z)t Fm(j)3317 2262 y Fk(m)515 2392 y Fo(for)k(a)g(\014xed)h Fn(m)23 b Fm(\025)f Fo(1,)27 b(w)n(e)h(ha)n(v)n(e)e(\(see)i([DiSj)q(]\))1060 2601 y(\010\()p Fn(P)12 b Fo(\))1268 2580 y(^)1249 2601 y Fn(P)1314 2566 y Fk(j)s(=)p Fl(2)1439 2601 y Fo(=)1541 2544 y(1)p 1537 2581 51 4 v 1537 2657 a Fn(\031)1611 2488 y Fc(Z)p 1708 2533 49 4 v 1708 2601 a Fn(@)1766 2580 y Fo(~)1756 2601 y(\010\()p Fn(z)t Fo(\)\()p Fn(z)22 b Fo(+)c Fn(a)p Fo(\))2175 2566 y Fk(j)s(=)p Fl(2)2278 2601 y Fo(\()p Fn(P)30 b Fm(\000)18 b Fn(z)t Fo(\))2551 2566 y Fi(\000)p Fl(1)2640 2601 y Fn(dz)26 b(d)5 b Fo(\026)-47 b Fn(z)515 2805 y Fo(and)27 b(then)515 3014 y Fm(kh)p Fn(x)p Fm(i)668 2980 y Fk(\016)705 3014 y Fo(\010\()p Fn(P)12 b Fo(\))913 2993 y(^)894 3014 y Fn(P)959 2980 y Fk(j)s(=)p Fl(2)1061 3014 y Fm(h)p Fn(x)p Fm(i)1172 2980 y Fi(\000)p Fk(\016)1261 3014 y Fm(k)23 b(\024)1424 2958 y Fn(C)1489 2918 y Fk(j)1483 2980 y Fl(2)1524 2958 y Fn(C)1583 2970 y Fl(1)p 1424 2995 197 4 v 1497 3071 a Fn(\031)1644 2901 y Fc(Z)1690 3090 y Fl(supp)1834 3075 y(~)1827 3090 y(\010)1892 3014 y Fm(j)p Fo(Im)p Fn(z)t Fm(j)2080 2980 y Fk(m)2143 3014 y Fm(kh)p Fn(x)p Fm(i)2296 2980 y Fk(\016)2333 3014 y Fo(\()p Fn(P)12 b Fm(\000)p Fn(z)t Fo(\))2570 2980 y Fi(\000)p Fl(1)2658 3014 y Fm(h)p Fn(x)p Fm(i)2769 2980 y Fi(\000)p Fk(\016)2858 3014 y Fm(k)p Fn(dz)26 b(d)5 b Fo(\026)-47 b Fn(z)27 b Fm(\024)22 b Fn(C)3263 3026 y Fl(3)3301 3014 y Fn(C)3366 2975 y Fk(j)3360 3037 y Fl(2)3167 3172 y Fo(\(3.21\))515 3272 y(for)27 b(some)g Fn(C)909 3284 y Fl(1)946 3272 y Fn(;)14 b(C)1042 3284 y Fl(2)1080 3272 y Fn(;)g(C)1176 3284 y Fl(3)1236 3272 y Fn(>)23 b Fo(0)k(indep)r(enden)n(t)i(of)e Fn(j)5 b Fo(.)37 b(Moreo)n(v)n(er,)25 b(let)j(us)g(pro)n(v)n(e:)1391 3450 y Fm(kh)p Fn(x)p Fm(i)1544 3415 y Fk(\016)1599 3429 y Fo(^)1581 3450 y Fn(P)1646 3415 y Fi(\000)p Fk(j)s(=)p Fl(2)1818 3429 y Fo(^)1799 3450 y Fn(P)1864 3406 y Fk(j)s(=)p Fl(2)1852 3472 y(0)1966 3450 y Fm(h)p Fn(x)p Fm(i)2077 3415 y Fi(\000)p Fk(\016)2167 3450 y Fm(k)22 b(\024)h Fn(C)2384 3410 y Fk(j)s Fl(+1)2378 3472 y(4)3167 3450 y Fo(\(3.22\))515 3616 y(for)34 b(some)g Fn(C)923 3628 y Fl(4)995 3616 y Fn(>)g Fo(0)g(indep)r(enden)n(t)h(of)g Fn(j)k Fo(as)34 b(far)g(as)g Fn(j)39 b Fm(\024)2346 3549 y Fc(\002)2381 3616 y Fo(1)p Fn(=h)2512 3549 y Fc(\003)2546 3616 y Fo(.)58 b(T)-7 b(o)34 b(pro)n(v)n(e)f(\(3.22\))g(it)i(is)515 3730 y(su\016cien)n(t)24 b(to)g(sho)n(w)g(b)n(y)g(induction)h(on)f Fn(j)5 b Fo(,)25 b(that,)g(if)g Fn(j)j Fm(\024)2261 3662 y Fc(\002)2296 3730 y Fo(1)p Fn(=h)2426 3662 y Fc(\003)2461 3730 y Fo(,)d(then)g(there)f(exists)3149 3709 y(~)3130 3730 y Fn(C)3189 3742 y Fl(4)3250 3730 y Fn(>)e Fo(0)515 3829 y(indep)r(enden)n(t)28 b(of)g Fn(j)k Fo(suc)n(h)c(that)1272 3929 y Fc(X)1250 4111 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(j)1428 4008 y Fm(kh)p Fn(x)p Fm(i)1581 3974 y Fk(\016)1618 4008 y Fo(\()p Fn(hD)1767 4020 y Fk(x)1809 4008 y Fo(\))1841 3974 y Fk(\013)1907 3987 y Fo(^)1888 4008 y Fn(P)1953 3974 y Fi(\000)p Fk(j)s(=)p Fl(2)2107 4008 y Fm(h)p Fn(x)p Fm(i)2218 3974 y Fi(\000)p Fk(\016)2307 4008 y Fm(k)23 b(\024)2478 3987 y Fo(~)2460 4008 y Fn(C)2525 3968 y Fk(j)s Fl(+1)2519 4030 y(4)3167 4008 y Fo(\(3.23\))515 4271 y(Supp)r(ose)31 b(that)g(\(3.23\))f(holds)g(for)g Fn(j)k Fm(\024)27 b Fn(k)d Fm(\000)c Fo(1)30 b(and)h(let)g(us)g(pro)n (v)n(e)e(it)i(for)f Fn(j)j Fo(=)28 b Fn(k)s Fo(.)47 b(Setting)537 4366 y(~)515 4387 y Fn(K)28 b Fo(=)724 4308 y Fc(X)702 4490 y Fi(j)p Fk(\014)s Fi(j)p Fl(=2)880 4387 y Fm(kh)p Fn(x)p Fm(i)1033 4353 y Fk(\016)1088 4366 y Fo(^)1070 4387 y Fn(P)1135 4353 y Fi(\000)p Fl(1)1224 4387 y Fo(\()p Fn(hD)1373 4399 y Fk(x)1414 4387 y Fo(\))1446 4353 y Fk(\014)1492 4387 y Fm(h)p Fn(x)p Fm(i)1603 4353 y Fi(\000)p Fk(\016)1692 4387 y Fm(k)f Fo(and)g(writing)540 4597 y Fc(X)515 4779 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)685 4676 y Fm(h)p Fn(x)p Fm(i)796 4641 y Fk(\016)833 4676 y Fo(\()p Fn(hD)982 4688 y Fk(x)1024 4676 y Fo(\))1056 4641 y Fk(\013)1122 4655 y Fo(^)1104 4676 y Fn(P)1169 4641 y Fi(\000)1230 4619 y Ff(k)p 1230 4628 33 4 v 1232 4661 a Fe(2)1277 4676 y Fm(h)p Fn(x)p Fm(i)1388 4641 y Fi(\000)p Fk(\016)1560 4676 y Fo(=)1730 4597 y Fc(X)1708 4779 y Fi(j)p Fk(\014)s Fi(j)p Fl(=2)1872 4676 y Fm(h)p Fn(x)p Fm(i)1983 4641 y Fk(\016)2039 4655 y Fo(^)2020 4676 y Fn(P)2085 4641 y Fi(\000)p Fl(1)2174 4676 y Fo(\()p Fn(hD)2323 4688 y Fk(x)2365 4676 y Fo(\))2397 4641 y Fk(\014)2442 4676 y Fm(h)p Fn(x)p Fm(i)2553 4641 y Fi(\000)p Fk(\016)2724 4597 y Fc(X)2656 4779 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)q Fi(\000)p Fl(2)2911 4676 y Fm(h)p Fn(x)p Fm(i)3022 4641 y Fk(\016)3060 4676 y Fo(\()p Fn(hD)3209 4688 y Fk(x)3250 4676 y Fo(\))3282 4641 y Fk(\013)3349 4655 y Fo(^)3330 4676 y Fn(P)3395 4641 y Fi(\000)3457 4619 y Ff(k)p 3457 4628 V 3459 4661 a Fe(2)3499 4641 y Fl(+1)3587 4676 y Fm(h)p Fn(x)p Fm(i)3698 4641 y Fi(\000)p Fk(\016)1708 4922 y Fo(+)1811 4843 y Fc(X)1787 5025 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)1956 4922 y Fm(h)p Fn(x)p Fm(i)2067 4888 y Fk(\016)2105 4922 y Fo([)2146 4901 y(^)2128 4922 y Fn(P)2193 4888 y Fi(\000)p Fl(1)2282 4922 y Fn(;)14 b Fo(\()p Fn(hD)2468 4934 y Fk(x)2509 4922 y Fo(\))2541 4888 y Fk(\013)2589 4922 y Fo(])2631 4901 y(^)2612 4922 y Fn(P)2677 4888 y Fi(\000)2739 4865 y Ff(k)p 2739 4874 V 2741 4908 a Fe(2)2781 4888 y Fl(+1)2869 4922 y Fm(h)p Fn(x)p Fm(i)2980 4888 y Fi(\000)p Fk(\016)1926 5255 y Fo(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fo(b)n(y)27 b(inductiv)n(e)h(assumptions,)f(one)g (has)706 639 y Fc(X)681 821 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)865 718 y Fm(kh)p Fn(x)p Fm(i)1018 683 y Fk(\016)1055 718 y Fo(\()p Fn(hD)1204 730 y Fk(x)1246 718 y Fo(\))1278 683 y Fk(\013)1344 697 y Fo(^)1325 718 y Fn(P)1390 683 y Fi(\000)p Fk(k)q(=)p Fl(2)1550 718 y Fm(h)p Fn(x)p Fm(i)1661 683 y Fi(\000)p Fk(\016)1750 718 y Fm(k)847 991 y(\024)22 b Fn(K)1030 970 y Fo(~)1011 991 y Fn(C)1076 955 y Fk(k)q Fi(\000)p Fl(1)1070 1013 y(4)1220 991 y Fo(+)1329 912 y Fc(X)1303 1094 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)1568 912 y Fc(X)1488 1090 y Fk(\014)s Fi(\024)p Fk(\013;\014)s Fi(6)p Fl(=0)1937 934 y Fn(\013)p Fo(!)p 1791 972 369 4 v 1791 1048 a(\()p Fn(\013)e Fm(\000)e Fn(\014)t Fo(\)!)p Fn(\014)t Fo(!)2170 991 y Fn(h)2218 956 y Fi(j)p Fk(\014)s Fi(j)2302 991 y Fm(kh)p Fn(x)p Fm(i)2455 956 y Fk(\016)2510 970 y Fo(^)2492 991 y Fn(P)2557 956 y Fi(\000)p Fl(1)2646 991 y Fo(\()p Fn(@)2727 956 y Fk(\014)2722 1011 y(x)2771 991 y Fn(V)h Fo(\)\()p Fn(hD)3019 1003 y Fk(x)3061 991 y Fo(\))3093 956 y Fk(\013)p Fi(\000)p Fk(\014)3252 970 y Fo(^)3233 991 y Fn(P)3298 956 y Fi(\000)3360 934 y Ff(k)p 3360 943 33 4 v 3362 976 a Fe(2)3406 991 y Fm(h)p Fn(x)p Fm(i)3517 956 y Fi(\000)p Fk(\016)3607 991 y Fm(k)p Fn(:)515 1288 y Fo(Observ)n(e)29 b(no)n(w)h(that,)i(since)f Fn(V)50 b Fo(is)30 b(analytic)g(on)h(\000)2101 1300 y Fk(a)2141 1288 y Fo(,)h(w)n(e)e(ha)n(v)n(e)g Fm(kh)p Fn(x)p Fm(i)2669 1258 y Fk(\016)2706 1288 y Fn(@)2755 1258 y Fk(\014)2750 1309 y(x)2799 1288 y Fn(V)19 b Fm(k)28 b(\024)g Fn(K)3106 1248 y Fk(\014)3100 1310 y Fl(1)3150 1288 y Fn(\014)t Fo(!,)33 b(for)515 1432 y(some)c Fn(K)796 1444 y Fl(1)860 1432 y Fn(>)d Fo(0.)43 b(Moreo)n(v)n(er)27 b(since)j Fn(\013)d Fm(\024)g(j)p Fn(\013)p Fm(j)g(\024)f Fn(k)k Fm(\024)2186 1365 y Fc(\002)2220 1432 y Fo(1)p Fn(=h)2351 1365 y Fc(\003)2386 1432 y Fo(,)g(w)n(e)g(ha)n(v)n(e)2876 1376 y Fn(\013)p Fo(!)p 2767 1413 294 4 v 2767 1489 a(\()p Fn(\013)19 b Fm(\000)f Fn(\014)t Fo(\)!)3098 1432 y Fm(\024)26 b Fn(\013)3242 1398 y Fk(\014)3314 1432 y Fm(\024)515 1545 y Fc(\002)549 1612 y Fo(1)p Fn(=h)680 1545 y Fc(\003)715 1562 y Fi(j)p Fk(\014)s Fi(j)822 1612 y Fm(\024)d Fn(h)958 1578 y Fi(\000j)p Fk(\014)s Fi(j)1094 1612 y Fo(.)37 b(Hence,)27 b(setting)h Fn(c)1732 1582 y Fl(2)1792 1612 y Fo(=)23 b Fm(kh)p Fn(x)p Fm(i)2033 1582 y Fk(\016)2088 1591 y Fo(^)2070 1612 y Fn(P)2135 1582 y Fi(\000)p Fl(1)2224 1612 y Fm(h)p Fn(x)p Fm(i)2335 1582 y Fi(\000)p Fk(\016)2424 1612 y Fm(k)p Fo(,)k(one)h(has)540 1741 y Fc(X)515 1923 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)699 1820 y Fm(kh)p Fn(x)p Fm(i)852 1785 y Fk(\016)889 1820 y Fo(\()p Fn(hD)1038 1832 y Fk(x)1079 1820 y Fo(\))1111 1785 y Fk(\013)1178 1799 y Fo(^)1159 1820 y Fn(P)1224 1785 y Fi(\000)p Fk(k)q(=)p Fl(2)1384 1820 y Fm(h)p Fn(x)p Fm(i)1495 1785 y Fi(\000)p Fk(\016)1584 1820 y Fm(k)83 b(\024)f Fn(K)1951 1799 y Fo(~)1933 1820 y Fn(C)1998 1784 y Fk(k)q Fi(\000)p Fl(1)1992 1842 y(4)1709 2074 y Fo(+)g Fn(c)1892 2040 y Fl(2)1968 1995 y Fc(X)1943 2177 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)2194 1995 y Fc(X)2127 2177 y Fl(0)p Fk(<)p Fi(j)p Fk(\014)s Fi(j\024)p Fk(k)2395 2074 y Fo(\()p Fn(cK)2534 2086 y Fl(1)2571 2074 y Fo(\))2603 2032 y Fi(j)p Fk(\014)s Fi(j)2701 2074 y Fm(kh)p Fn(x)p Fm(i)2854 2040 y Fk(\016)2891 2074 y Fo(\()p Fn(hD)3040 2086 y Fk(x)3082 2074 y Fo(\))3114 2040 y Fk(\013)p Fi(\000)p Fk(\014)3272 2053 y Fo(^)3254 2074 y Fn(P)3329 2010 y Fb(\000)p Fe(\()p Ff(k)q Fb(\000j)p Ff(\014)r Fb(j)p Fe(\))p 3329 2027 241 4 v 3435 2060 a(2)3583 2074 y Fm(h)p Fn(x)p Fm(i)3694 2040 y Fi(\000)p Fk(\016)3783 2074 y Fm(k)515 2349 y Fo(and)27 b(then,)h(b)n(y)g (inductiv)n(e)f(assumption)795 2521 y Fc(X)770 2703 y Fi(j)p Fk(\013)p Fi(j)p Fl(=)p Fk(k)954 2600 y Fm(kh)p Fn(x)p Fm(i)1107 2566 y Fk(\016)1144 2600 y Fo(\()p Fn(hD)1293 2612 y Fk(x)1335 2600 y Fo(\))1367 2566 y Fk(\013)1433 2579 y Fo(^)1415 2600 y Fn(P)1480 2566 y Fi(\000)p Fk(k)q(=)p Fl(2)1639 2600 y Fm(h)p Fn(x)p Fm(i)1750 2566 y Fi(\000)p Fk(\016)1840 2600 y Fm(k)22 b(\024)h Fn(K)2087 2579 y Fo(~)2069 2600 y Fn(C)2134 2564 y Fk(k)q Fi(\000)p Fl(1)2128 2622 y(4)2278 2600 y Fo(+)2380 2579 y(~)2361 2600 y Fn(C)2426 2564 y Fk(k)q Fl(+1)2420 2622 y(4)2551 2600 y Fn(c)2587 2566 y Fl(2)2705 2521 y Fc(X)2638 2703 y Fl(0)p Fk(<)p Fi(j)p Fk(\014)s Fi(j\024)p Fk(k)2906 2483 y Fc(\022)2977 2544 y Fn(K)3048 2556 y Fl(1)3084 2544 y Fn(c)p 2977 2581 144 4 v 3019 2646 a Fo(~)3000 2667 y Fn(C)3059 2679 y Fl(4)3130 2483 y Fc(\023)3191 2500 y Fi(j)p Fk(\014)s Fi(j)770 2879 y Fm(\024)877 2858 y Fo(~)858 2879 y Fn(C)923 2844 y Fk(k)q Fl(+1)917 2901 y(4)1062 2762 y Fc(\022)1146 2823 y Fn(K)p 1133 2860 103 4 v 1152 2926 a Fo(~)1133 2947 y Fn(C)1198 2918 y Fl(2)1192 2969 y(4)1264 2879 y Fo(+)18 b Fn(c)1383 2845 y Fl(2)1434 2762 y Fc(\022)1495 2879 y Fo(\(1)g Fm(\000)1680 2823 y Fn(K)1751 2835 y Fl(1)1788 2823 y Fn(c)p 1680 2860 144 4 v 1723 2926 a Fo(~)1704 2947 y Fn(C)1763 2959 y Fl(4)1834 2879 y Fo(\))1866 2845 y Fi(\000)p Fk(n)1982 2879 y Fm(\000)g Fo(1)2107 2762 y Fc(\023)o(\023)2252 2879 y Fm(\024)2358 2858 y Fo(~)2339 2879 y Fn(C)2404 2844 y Fk(k)q Fl(+1)2398 2901 y(4)515 3180 y Fo(if)608 3159 y(~)589 3180 y Fn(C)648 3192 y Fl(4)712 3180 y Fo(is)25 b(c)n(hosen)g(in)i(suc)n(h)e(a)h(w)n(a) n(y)e(that)1773 3124 y Fn(K)p 1760 3161 103 4 v 1779 3227 a Fo(~)1760 3248 y Fn(C)1825 3219 y Fl(2)1819 3270 y(4)1888 3180 y Fo(+)14 b Fn(c)2017 3063 y Fc(\022)2078 3180 y Fo(\(1)k Fm(\000)2263 3124 y Fn(K)2334 3136 y Fl(1)2371 3124 y Fn(c)p 2263 3161 144 4 v 2306 3227 a Fo(~)2287 3248 y Fn(C)2346 3260 y Fl(4)2417 3180 y Fo(\))2449 3146 y Fi(\000)p Fk(n)2565 3180 y Fm(\000)g Fo(1)2690 3063 y Fc(\023)2774 3180 y Fn(<)k Fo(1.)36 b(This)26 b(pro)n(v)n(es)515 3340 y(\(3.23\))h(and)g(then)h(\(3.22\).)36 b(Finally)-7 b(,)28 b(since)f(one)h(can)f(write)1161 3607 y Fn(Q)1227 3619 y Fk(j)1261 3607 y Fo(\()p Fn(x;)14 b(hD)1494 3619 y Fk(x)1536 3607 y Fo(\))24 b(=)1724 3499 y Fk(j)1680 3528 y Fc(X)1679 3707 y Fk(k)q Fl(=0)1814 3607 y Fn(h)1862 3573 y Fk(k)1982 3528 y Fc(X)1917 3710 y Fi(j)p Fk(\014)s Fi(j\024)p Fk(j)s Fi(\000)p Fk(k)2168 3607 y Fo(\()p Fn(hD)2317 3619 y Fk(x)2359 3607 y Fo(\))2391 3573 y Fk(\014)2436 3607 y Fn(q)2473 3619 y Fk(\014)s(;j;k)2621 3607 y Fo(\()p Fn(x)p Fo(\))515 3882 y(for)j(some)g(smo)r(oth)g (functions)h Fn(q)1538 3894 y Fk(\014)s(;j;k)1714 3882 y Fo(analytic)f(on)g(\000)2197 3894 y Fk(a)2265 3882 y Fo(and)g(satisfying)1175 4075 y Fm(k)p Fn(@)1266 4041 y Fk(\015)1261 4095 y(x)1308 4075 y Fn(q)1345 4087 y Fk(\014)s(;j;k)1493 4075 y Fo(\()p Fn(x)p Fo(\))p Fm(k)1646 4090 y Fi(L)p Fl(\()p Fi(H)p Fl(\))1828 4075 y Fm(\024)c Fn(C)1981 4041 y Fk(j)s Fl(+)p Fk(k)q Fl(+)p Fi(j)p Fk(\015)t Fi(j)p Fl(+1)2317 4075 y Fn(j)5 b Fo(!)p Fn(k)s Fo(!)p Fn(\015)g Fo(!)p Fm(h)p Fn(x)p Fm(i)2630 4041 y Fi(\000)p Fk(\016)515 4257 y Fo(for)32 b(some)h Fn(C)38 b(>)32 b Fo(0)g(and)h Fn(\016)i Fm(\025)d Fo(0)h(uniformly)g(on)f(\000)2096 4269 y Fk(a)2170 4257 y Fo(is)g(easy)h(to)f(c)n(hec)n(k)h(that,)h (there)f(exist)515 4357 y Fn(c)551 4369 y Fl(1)588 4357 y Fn(;)14 b(c)661 4369 y Fl(2)698 4357 y Fn(;)g(C)794 4369 y Fl(5)855 4357 y Fn(>)22 b Fo(0)27 b(suc)n(h)h(that)713 4619 y Fm(kh)p Fn(x)p Fm(i)866 4585 y Fk(\016)921 4598 y Fo(^)903 4619 y Fn(P)968 4576 y Fi(\000)p Fk(j)s(=)p Fl(2)956 4641 y(0)1122 4619 y Fn(Q)1188 4631 y Fk(j)1222 4619 y Fm(k)23 b(\024)1419 4511 y Fk(j)1375 4540 y Fc(X)1374 4719 y Fk(k)q Fl(=0)1509 4619 y Fn(h)1557 4585 y Fk(k)1677 4540 y Fc(X)1612 4722 y Fi(j)p Fk(\014)s Fi(j\024)p Fk(j)s Fi(\000)p Fk(k)1877 4619 y Fm(kh)p Fn(x)p Fm(i)2030 4585 y Fk(\016)2085 4598 y Fo(^)2067 4619 y Fn(P)2132 4576 y Fi(\000)p Fk(j)s(=)p Fl(2)2120 4641 y(0)2286 4619 y Fo(\()p Fn(hD)2435 4631 y Fk(x)2476 4619 y Fo(\))2508 4585 y Fk(\014)2553 4619 y Fm(h)p Fn(x)p Fm(i)2664 4585 y Fi(\000)p Fk(\016)2754 4619 y Fm(k)f(kh)p Fn(x)p Fm(i)2971 4585 y Fk(\016)3008 4619 y Fn(q)3045 4631 y Fk(\014)s(;j;k)3193 4619 y Fo(\()p Fn(x)p Fo(\))p Fm(k)713 4933 y(\024)g Fn(c)836 4893 y Fk(j)s Fl(+1)836 4955 y(1)955 4933 y Fn(j)5 b Fo(!\()1095 4825 y Fk(j)1049 4854 y Fc(X)1049 5032 y Fk(k)q Fl(=0)1184 4933 y Fn(h)1232 4898 y Fk(k)1273 4933 y Fn(c)1309 4898 y Fk(k)1309 4953 y Fl(2)1350 4933 y Fn(k)s Fo(!\))23 b Fm(\024)g Fn(C)1627 4893 y Fk(j)s Fl(+1)1621 4955 y(5)1746 4933 y Fn(j)5 b Fo(!)1359 b(\(3.24\))1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fo(if)28 b Fn(j)g Fm(\024)22 b Fo([1)p Fn(=h)p Fo(].)36 b(Then)28 b(b)n(y)g(\(3.22\),)e(\(3.23\))h(and)h (\(3.24\))f(one)g(has)g(that)h(that)1508 666 y Fm(kh)p Fn(x)p Fm(i)1661 632 y Fk(\016)1698 666 y Fo(\010\()p Fn(P)12 b Fo(\))p Fn(Q)1953 678 y Fk(j)1988 666 y Fm(k)22 b(\024)h Fn(C)2205 632 y Fk(j)s Fl(+1)2324 666 y Fn(j)5 b Fo(!)515 809 y(for)22 b(some)g(constan)n(t)g Fn(C)29 b(>)23 b Fo(0)f(indep)r(enden)n(t)h(of)g Fn(j)5 b Fo(.)35 b(In)22 b(the)h(same)f(w)n(a)n(y)g(one)g(can)g(easily)g(pro)n(v)n(e)515 909 y(that)28 b(also)1508 1008 y Fm(k)p Fo(\010\()p Fn(P)12 b Fo(\))p Fn(Q)1805 1020 y Fk(j)1839 1008 y Fm(h)p Fn(x)p Fm(i)1950 974 y Fk(\016)1988 1008 y Fm(k)22 b(\024)h Fn(C)2205 974 y Fk(j)s Fl(+1)2324 1008 y Fn(j)5 b Fo(!)515 1134 y(and)27 b(then,)h(b)n(y)g(in)n(terp)r(olation,)e(one)i(obtains)f (\(3.20\).)p 3337 1146 34 63 v 515 1273 a Fd(Pr)l(o)l(of)i(of)h(The)l (or)l(em)f(2.1:)38 b Fo(Let)27 b(us)f(de\014ne)1850 1252 y(^)1840 1273 y(\005)1902 1285 y Fk(j)1963 1273 y Fo(as)g(in)g (\(3.13\))g(and,)g(for)g Fn(N)32 b Fo(=)2894 1206 y Fc(\002)2975 1217 y Fo(1)p 2938 1254 115 4 v 2938 1330 a Fn(B)t(h)3063 1206 y Fc(\003)3124 1273 y Fo(with)27 b Fn(B)515 1396 y Fo(su\016cien)n(tly)g(large,)g(let)g(us)h(set)957 1594 y(^)946 1615 y(\005)1008 1581 y Fk(B)1008 1636 y Fl(\010)1089 1615 y Fo(:=)23 b(\005)1262 1627 y Fl(0)1318 1615 y Fo(+)1431 1512 y Fk(N)1401 1537 y Fc(X)1403 1713 y Fk(j)s Fl(=1)1534 1615 y Fo(\010\()p Fn(H)7 b Fo(\))1744 1594 y(^)1734 1615 y(\005)1796 1627 y Fk(j)1832 1615 y Fn(h)1880 1581 y Fk(j)1933 1615 y Fo(+)2047 1512 y Fk(N)2016 1537 y Fc(X)2019 1713 y Fk(j)s Fl(=1)2136 1615 y Fo(\(1)18 b Fm(\000)g Fo(\010\()p Fn(H)7 b Fo(\)\))2553 1594 y(^)2543 1615 y(\005)2605 1627 y Fk(j)2641 1615 y Fo(\010\()p Fn(H)g Fo(\))p Fn(h)2889 1581 y Fk(j)2924 1615 y Fn(:)220 b Fo(\(3.25\))515 1842 y(Observ)n(e)31 b(that,)j(b)n(y)e(Lemma)h(3.2,)g (there)f(exists)g(a)g(constan)n(t)g Fn(C)38 b(>)31 b Fo(0)h(indep)r(enden)n(t)i(of)e Fn(j)515 1942 y Fo(suc)n(h)27 b(that,)h(w)n(e)f(ha)n(v)n(e)g(that,)h(for)f(1)22 b Fm(\024)h Fn(j)28 b Fm(\024)22 b Fo([1)p Fn(=B)t(h)p Fo(],)27 b Fn(B)g Fm(\025)c Fo(1)1300 2085 y Fm(k)p Fo(\010\()p Fn(P)12 b Fo(\))1541 2064 y(^)1531 2085 y(\005)1593 2097 y Fk(j)1628 2085 y Fm(k)1670 2097 y Fk(\016)1725 2085 y Fo(+)18 b Fm(k)1860 2064 y Fo(^)1850 2085 y(\005)1912 2097 y Fk(j)1946 2085 y Fo(\010\()p Fn(P)12 b Fo(\))p Fm(k)2177 2097 y Fk(\016)2237 2085 y Fm(\024)22 b Fn(C)2389 2051 y Fk(j)s Fl(+1)2509 2085 y Fn(j)5 b Fo(!)p Fn(:)573 b Fo(\(3.26\))515 2239 y(Denoting)27 b(here)g(\005)1115 2251 y Fl(0)1176 2239 y Fo(=)1274 2218 y(^)1264 2239 y(\005)1326 2251 y Fl(0)1363 2239 y Fo(,)h(w)n(e)f(ha)n(v)n(e)720 2459 y Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)934 2438 y(^)924 2459 y(\005)986 2425 y Fk(B)986 2479 y Fl(\010)1044 2459 y Fn(;)i(P)e Fo(])23 b(=)f Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)1514 2355 y Fk(N)1483 2380 y Fc(X)1486 2557 y Fk(j)s Fl(=0)1628 2438 y Fo(^)1618 2459 y(\005)1680 2471 y Fk(j)1715 2459 y Fn(h)1763 2425 y Fk(j)1798 2459 y Fn(;)i(P)e Fo(])23 b(=)2060 2355 y Fl(2)p Fk(N)6 b Fl(+2)2089 2380 y Fc(X)2033 2559 y Fk(k)q Fl(=)p Fk(N)g Fl(+1)2278 2459 y Fn(h)2326 2425 y Fk(k)2366 2459 y Fn(\037)p Fo(\()p Fn(P)12 b Fo(\))p Fn(Q)2613 2471 y Fk(k)2654 2459 y Fo(\()p Fn(x;)i(hD)2887 2471 y Fk(x)2929 2459 y Fo(\))206 b(\(3.27\))515 2691 y(where)23 b Fn(Q)817 2703 y Fk(k)882 2691 y Fo(is)h(a)g(di\013eren)n(tial)g(op)r(erator)e (of)i(order)f Fn(k)k Fo(satisfying)d(for)f(\()p Fn(x;)14 b(\030)t Fo(\))25 b Fm(2)e Fo(\000)2979 2703 y Fk(a)3031 2691 y Fm(\002)12 b Fj(C)3177 2660 y Fk(n)3245 2691 y Fo(and)515 2790 y(for)27 b(an)n(y)g(m)n(ulti-index)g Fn(\013;)14 b(\014)1051 2933 y Fm(j)p Fn(@)1123 2899 y Fk(\013)1118 2954 y(x)1170 2933 y Fn(@)1219 2893 y Fk(\014)1214 2958 y(\030)1264 2933 y Fn(Q)1330 2945 y Fk(k)1370 2933 y Fo(\()p Fn(x;)g(\030)t Fo(\))p Fm(j)24 b(\024)f Fn(A)1755 2899 y Fk(k)q Fl(+)p Fi(j)p Fk(\013)p Fi(j)p Fl(+)p Fi(j)p Fk(\014)s Fi(j)p Fl(+1)2145 2933 y Fn(k)s Fo(!)p Fn(\013)p Fo(!)p Fn(\014)t Fo(!)p Fm(h)p Fn(\030)t Fm(i)2468 2899 y Fk(k)q Fi(\000j)p Fk(\014)s Fi(j)2642 2933 y Fm(h)p Fn(x)p Fm(i)2753 2899 y Fi(\000)p Fk(\016)3167 2933 y Fo(\(3.28\))515 3076 y(for)30 b(some)g(constan)n(t) h Fn(A)d(>)g Fo(0)j(indep)r(enden)n(t)h(of)e Fn(h;)14 b(\013;)g(\014)t(;)g(k)s Fo(.)47 b(Hence,)32 b(b)n(y)f(Lemma)g(3.2,)f (one)515 3210 y(obtain)d(that,)h(if)g Fn(N)k Fo(=)1239 3143 y Fc(\002)1320 3154 y Fo(1)p 1283 3191 V 1283 3267 a Fn(B)t(h)1408 3143 y Fc(\003)1470 3210 y Fo(and)c Fn(B)k Fo(is)27 b(su\016cien)n(tly)h(large,)965 3471 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)1221 3450 y(^)1211 3471 y(\005)1273 3437 y Fk(B)1273 3491 y Fl(\010)1331 3471 y Fn(;)i(P)e Fo(])p Fm(k)1498 3483 y Fk(\016)1552 3471 y Fo(+)18 b Fm(k)p Fo([)1710 3450 y(^)1700 3471 y(\005)1762 3437 y Fk(B)1762 3491 y Fl(\010)1819 3471 y Fn(;)c(P)e Fo(])p Fn(\037)p Fo(\()p Fn(P)g Fo(\))p Fm(k)2167 3483 y Fk(\016)2226 3471 y Fm(\024)23 b Fn(C)6 b Fo(\()2436 3367 y Fl(2)p Fk(N)g Fl(+4)2464 3392 y Fc(X)2411 3570 y Fk(s)p Fl(=)p Fk(N)g Fl(+1)2650 3471 y Fn(h)2698 3437 y Fk(s)2733 3471 y Fn(C)2798 3437 y Fk(s)2834 3471 y Fn(s)p Fo(!\))239 b(\(3.29\))515 3699 y(for)29 b(some)h(constan)n(t)f Fn(C)k(>)27 b Fo(0)i(indep)r(enden)n(t)i(of)f Fn(h)g Fo(and)f Fn(s)p Fo(.)44 b(If)31 b(\010)p Fn(;)14 b Fo(\010)2607 3711 y Fl(1)2671 3699 y Fm(2)27 b Fn(C)2818 3669 y Fi(1)2812 3720 y Fl(0)2889 3699 y Fo(\()p Fj(R)p Fo(\),)j(are)f(suc)n(h)515 3799 y(that)f Fn(\037)p Fo(\010)23 b(=)f Fn(\037)p Fo(,)28 b(\010\010)1140 3811 y Fl(1)1200 3799 y Fo(=)23 b(\010)k(one)h(has)692 4019 y Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\(\()947 3998 y(^)937 4019 y(\005)999 3984 y Fk(B)999 4039 y Fl(\010)1057 4019 y Fo(\))1089 3984 y Fl(2)1145 4019 y Fm(\000)1238 3998 y Fo(^)1228 4019 y(\005)1290 3984 y Fk(B)1290 4039 y Fl(\010)1347 4019 y Fo(\))24 b(=)e Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()1718 3915 y Fl(2)p Fk(N)1703 3940 y Fc(X)1705 4116 y Fk(s)p Fl(=0)1838 4019 y Fn(h)1886 3984 y Fk(s)1979 3915 y(s)1935 3940 y Fc(X)1935 4119 y Fk(k)q Fl(=0)2080 3998 y Fo(^)2069 4019 y(\005)2131 4031 y Fk(k)2183 3998 y Fo(^)2172 4019 y(\005)2234 4031 y Fk(s)p Fi(\000)p Fk(k)2377 4019 y Fm(\000)2491 3915 y Fk(N)2460 3940 y Fc(X)2463 4117 y Fk(j)s Fl(=0)2594 4019 y Fn(h)2642 3984 y Fk(j)2687 3998 y Fo(^)2677 4019 y(\005)2739 4031 y Fk(j)2774 4019 y Fo(\)\010)2866 4031 y Fl(1)2904 4019 y Fo(\()p Fn(P)g Fo(\))692 4321 y(+)p Fn(\037)p Fo(\()p Fn(P)g Fo(\))1008 4217 y Fk(N)977 4242 y Fc(X)952 4421 y Fk(k)q(;s)p Fl(=0)1137 4321 y Fn(h)1185 4287 y Fk(k)q Fl(+)p Fk(s)1308 4321 y Fo([)1341 4300 y(^)1331 4321 y(\005)1393 4333 y Fk(k)1434 4321 y Fn(;)i Fo(\010\()p Fn(P)e Fo(\)])1693 4300 y(^)1683 4321 y(\005)1745 4333 y Fk(s)1781 4321 y Fo(\010)1841 4333 y Fl(1)1878 4321 y Fo(\()p Fn(P)g Fo(\))19 b Fm(\000)f Fn(\037)p Fo(\()p Fn(P)12 b Fo(\))2361 4217 y Fk(N)2330 4242 y Fc(X)2304 4421 y Fk(k)q(;s)p Fl(=0)2490 4321 y Fn(h)2538 4287 y Fk(k)q Fl(+)p Fk(s)2671 4300 y Fo(^)2661 4321 y(\005)2723 4333 y Fk(k)2764 4321 y Fo(\010\()p Fn(P)g Fo(\)[)2986 4300 y(^)2976 4321 y(\005)3038 4333 y Fk(s)3074 4321 y Fn(;)i Fo(\010)3171 4333 y Fl(1)3208 4321 y Fo(\()p Fn(P)e Fo(\)])692 4626 y Fm(\000)p Fn(\037)p Fo(\()p Fn(P)g Fo(\))1008 4522 y Fk(N)977 4547 y Fc(X)952 4726 y Fk(s;k)q Fl(=0)1137 4626 y Fn(h)1185 4591 y Fk(s)p Fl(+)p Fk(k)1308 4626 y Fo([)1341 4605 y(^)1331 4626 y(\005)1393 4638 y Fk(k)1434 4626 y Fn(;)i Fo(\010\()p Fn(P)e Fo(\)])1693 4605 y(^)1683 4626 y(\005)1745 4638 y Fk(s)1781 4626 y Fo(\010\()p Fn(P)g Fo(\))19 b Fm(\000)f Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()2316 4522 y Fk(N)2285 4547 y Fc(X)2288 4724 y Fk(j)s Fl(=0)2419 4626 y Fn(h)2467 4591 y Fk(j)2502 4626 y Fo([)2535 4605 y(^)2525 4626 y(\005)2587 4638 y Fk(j)2623 4626 y Fn(;)i Fo(\010\()p Fn(P)e Fo(\)])2882 4605 y(^)2872 4626 y(\005)2934 4638 y Fl(0)2971 4626 y Fo(\(1)19 b Fm(\000)f Fo(\010\()p Fn(P)12 b Fo(\)\))692 4930 y(+)p Fn(\037)p Fo(\()p Fn(P)g Fo(\))982 4826 y Fk(N)951 4851 y Fc(X)954 5028 y Fk(j)s Fl(=0)1085 4930 y Fn(h)1133 4896 y Fk(j)1168 4930 y Fo([)1201 4909 y(^)1191 4930 y(\005)1253 4942 y Fk(j)1288 4930 y Fn(;)i Fo(\010)1385 4942 y Fl(1)1422 4930 y Fo(\()p Fn(P)e Fo(\)])1593 b(\(3.30\))1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fo(Observ)n(e)26 b(that)1151 634 y Fl(2)p Fk(N)1137 659 y Fc(X)1139 834 y Fk(s)p Fl(=0)1270 737 y Fn(h)1318 703 y Fk(s)1412 634 y(s)1368 659 y Fc(X)1368 837 y Fk(k)q Fl(=0)1513 716 y Fo(^)1502 737 y(\005)1564 749 y Fk(k)1616 716 y Fo(^)1605 737 y(\005)1667 749 y Fk(s)p Fi(\000)p Fk(k)1810 737 y Fm(\000)1924 634 y Fk(N)1893 659 y Fc(X)1896 835 y Fk(j)s Fl(=0)2027 737 y Fn(h)2075 703 y Fk(j)2120 716 y Fo(^)2110 737 y(\005)2172 749 y Fk(j)2230 737 y Fo(=)2387 634 y Fl(4)p Fk(N)2373 659 y Fc(X)2318 837 y Fk(k)q Fl(=)p Fk(N)6 b Fl(+1)2562 737 y Fn(h)2610 703 y Fk(k)2670 716 y Fo(~)2651 737 y Fn(Q)2717 749 y Fk(k)515 994 y Fo(where)766 973 y(~)747 994 y Fn(Q)813 1006 y Fk(k)873 994 y Fo(is)20 b(a)g(di\013eren)n(tial)f(op)r(erator)f (of)i(order)f Fn(k)j Fo(satisfying)e(\(3.28\).)33 b(Hence,)22 b(b)n(y)d(Lemma)515 1094 y(3.2,)753 1324 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()1022 1220 y Fl(2)p Fk(N)1008 1245 y Fc(X)1010 1421 y Fk(s)p Fl(=0)1142 1324 y Fn(h)1190 1290 y Fk(s)1284 1220 y(s)1239 1245 y Fc(X)1239 1424 y Fk(k)q Fl(=0)1384 1303 y Fo(^)1374 1324 y(\005)1436 1336 y Fk(k)1487 1303 y Fo(^)1477 1324 y(\005)1539 1336 y Fk(s)p Fi(\000)p Fk(k)1681 1324 y Fm(\000)1795 1220 y Fk(N)1765 1245 y Fc(X)1767 1422 y Fk(j)s Fl(=0)1898 1324 y Fn(h)1946 1290 y Fk(j)1991 1303 y Fo(^)1981 1324 y(\005)2043 1336 y Fk(j)2078 1324 y Fo(\)\010)2170 1336 y Fl(1)2208 1324 y Fo(\()p Fn(P)g Fo(\))p Fm(k)2379 1336 y Fk(\016)2438 1324 y Fm(\024)23 b Fn(C)6 b Fo(\()2690 1220 y Fl(4)p Fk(N)2676 1245 y Fc(X)2623 1423 y Fk(s)p Fl(=)p Fk(N)g Fl(+1)2862 1324 y Fn(h)2910 1290 y Fk(s)2946 1324 y Fn(C)3011 1290 y Fk(s)3047 1324 y Fn(s)p Fo(!\))515 1614 y(for)38 b(some)g(constan)n(t)h Fn(C)48 b(>)41 b Fo(0)e(indep)r(enden)n(t)h(of)e Fn(h)h Fo(and)g Fn(s)p Fo(,)j(if)d Fn(N)51 b Fo(=)2775 1547 y Fc(\002)2857 1558 y Fo(1)p 2820 1595 115 4 v 2820 1671 a Fn(B)t(h)2945 1547 y Fc(\003)2979 1614 y Fo(,)42 b(and)d Fn(B)k Fo(is)515 1737 y(su\016cien)n(tly)27 b(large.)36 b(Moreo)n(v)n(er,)25 b(writing)515 1967 y Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)750 1863 y Fk(N)719 1888 y Fc(X)719 2067 y Fk(k)q Fl(=0)854 1967 y Fn(h)902 1933 y Fk(k)953 1946 y Fo(^)942 1967 y(\005)1004 1979 y Fk(k)1046 1967 y Fn(;)i Fo(\010\()p Fn(P)e Fo(\)])23 b(=)g Fm(\000)1484 1911 y Fo(1)p 1481 1948 51 4 v 1481 2024 a Fn(\031)1554 1854 y Fc(Z)p 1651 1900 49 4 v 1651 1967 a Fn(@)1709 1946 y Fo(~)1700 1967 y(\010\()p Fn(z)t Fo(\)\()p Fn(P)15 b Fm(\000)s Fn(z)t Fo(\))2110 1933 y Fi(\000)p Fl(1)2198 1967 y Fn(\037)p Fo(\()p Fn(P)d Fo(\)[)2433 1863 y Fk(N)2402 1888 y Fc(X)2402 2067 y Fk(k)q Fl(=0)2537 1967 y Fn(h)2585 1933 y Fk(k)2636 1946 y Fo(^)2626 1967 y(\005)2688 1979 y Fk(k)2729 1967 y Fn(;)i(P)e Fo(]\()p Fn(P)j Fm(\000)s Fn(z)t Fo(\))3097 1933 y Fi(\000)p Fl(1)3185 1967 y Fn(dz)27 b(d)5 b Fo(\026)-47 b Fn(z)515 2200 y Fo(and)27 b(taking)g(in)n(to)h(accoun)n(t)e(\(3.29\)) h(one)g(easily)g(obtains)563 2435 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)840 2332 y Fk(N)809 2356 y Fc(X)809 2535 y Fk(k)q Fl(=0)943 2435 y Fn(h)991 2401 y Fk(k)1043 2414 y Fo(^)1032 2435 y(\005)1094 2447 y Fk(k)1135 2435 y Fn(;)i Fo(\010\()p Fn(P)e Fo(\)])1429 2332 y Fk(N)1398 2356 y Fc(X)1401 2532 y Fk(s)p Fl(=0)1532 2435 y Fn(h)1580 2401 y Fk(s)1626 2414 y Fo(^)1616 2435 y(\005)1678 2447 y Fk(s)1713 2435 y Fo(\010)1773 2447 y Fl(1)1810 2435 y Fo(\()p Fn(P)g Fo(\))p Fm(k)1981 2447 y Fk(\016)2101 2435 y Fm(\024)82 b Fn(C)6 b Fo(\()2373 2332 y Fl(2)p Fk(N)g Fl(+4)2401 2356 y Fc(X)2345 2535 y Fk(k)q Fl(=)p Fk(N)g Fl(+1)2590 2435 y Fn(h)2638 2401 y Fk(k)2679 2435 y Fn(C)2744 2401 y Fk(k)2785 2435 y Fn(k)s Fo(!\)\()2949 2332 y Fk(N)2918 2356 y Fc(X)2920 2532 y Fk(s)p Fl(=0)3052 2435 y Fn(h)3100 2401 y Fk(s)3136 2435 y Fn(C)3201 2401 y Fk(s)3195 2456 y Fl(1)3237 2435 y Fn(s)p Fo(!\))2101 2735 y Fm(\024)82 b Fn(C)2307 2747 y Fl(2)2345 2735 y Fo(\()2405 2632 y Fl(2)p Fk(N)6 b Fl(+4)2433 2657 y Fc(X)2377 2835 y Fk(k)q Fl(=)p Fk(N)g Fl(+1)2622 2735 y Fn(h)2670 2701 y Fk(k)2710 2735 y Fn(C)2775 2701 y Fk(k)2816 2735 y Fn(k)s Fo(!\))p Fn(:)227 b Fo(\(3.31\))515 2971 y(for)29 b(some)h(new)g(constan)n(ts)f Fn(C)q(;)14 b(C)1554 2983 y Fl(1)1593 2971 y Fn(;)g(C)1689 2983 y Fl(2)1756 2971 y Fo(and)30 b Fn(C)k(>)27 b Fo(0)j(indep)r(enden)n(t)h(of)f Fn(h)g Fo(and)g Fn(s)p Fo(,)h(for)e Fn(N)36 b Fo(=)515 3038 y Fc(\002)596 3049 y Fo(1)p 559 3086 115 4 v 559 3162 a Fn(B)t(h)684 3038 y Fc(\003)719 3105 y Fo(,)29 b(and)f Fn(B)h Fm(\025)24 b Fn(C)1173 3117 y Fl(1)1239 3105 y Fo(su\016cien)n(tly)k(large.)38 b(The)29 b(other)f(terms)g(in)h (\(3.30\))f(con)g(b)r(e)g(treated)515 3228 y(in)g(the)g(same)f(w)n(a)n (y)f(and)h(one)h(gets)1169 3458 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\(\()1466 3437 y(^)1456 3458 y(\005)1518 3424 y Fk(B)1518 3479 y Fl(\010)1576 3458 y Fo(\))1608 3424 y Fl(2)1664 3458 y Fm(\000)1757 3437 y Fo(^)1747 3458 y(\005)1809 3424 y Fk(B)1809 3479 y Fl(\010)1866 3458 y Fo(\))p Fm(k)1940 3470 y Fk(\016)1999 3458 y Fm(\024)23 b Fn(C)6 b Fo(\()2212 3354 y Fl(4)p Fk(N)g Fl(+4)2240 3379 y Fc(X)2184 3558 y Fk(k)q Fl(=)p Fk(N)g Fl(+1)2429 3458 y Fn(h)2477 3424 y Fk(k)2518 3458 y Fn(C)2583 3424 y Fk(k)2624 3458 y Fn(k)s Fo(!\))442 b(\(3.32\))515 3745 y(for)28 b(a)f(new)i(constan)n(t)e Fn(C)k(>)24 b Fo(0)k(indep)r(enden)n (t)h(of)f Fn(h)g Fo(and)g Fn(s)h Fo(and)f Fn(N)33 b Fo(=)2691 3678 y Fc(\002)2772 3689 y Fo(1)p 2735 3726 V 2735 3802 a Fn(B)t(h)2860 3678 y Fc(\003)2895 3745 y Fo(,with)c Fn(B)j Fo(su\016-)515 3868 y(cien)n(tly)26 b(large.)35 b(By)26 b(the)h(same)e(kind)i(of)f(argumen)n(ts)f(one)h(can)g(easily)g (pro)n(v)n(e)e(an)i(estimates)515 3968 y(analogous)31 b(to)i(\(3.32\))g(for)g(\(\()1464 3947 y(^)1454 3968 y(\005)1516 3938 y Fk(B)1516 3991 y Fl(\010)1574 3968 y Fo(\))1606 3938 y Fl(2)1666 3968 y Fm(\000)1763 3947 y Fo(^)1753 3968 y(\005)1815 3938 y Fk(B)1815 3991 y Fl(\010)1872 3968 y Fo(\))p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\).)56 b(Hence,)35 b(setting)2730 3947 y(^)2720 3968 y(\005)2782 3980 y Fl(\010)2866 3968 y Fo(=)2974 3947 y(^)2964 3968 y(\005)3026 3938 y Fk(B)3026 3991 y Fl(\010)3117 3968 y Fo(with)f Fn(B)515 4068 y Fo(large)26 b(enough)h(and)g(using)h(Stirling)f(form)n(ula,)g(one)g(gets)1101 4221 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)[)1357 4201 y(^)1347 4221 y(\005)1409 4233 y Fl(\010)1460 4221 y Fn(;)i(P)e Fo(])p Fm(k)18 b Fo(+)g Fm(k)p Fo([)1803 4201 y(^)1793 4221 y(\005)1855 4233 y Fl(\010)1907 4221 y Fn(;)c(P)e Fo(])p Fn(\037)p Fo(\()p Fn(P)g Fo(\))p Fm(k)22 b Fo(=)h Fm(O)r Fo(\()p Fn(e)2504 4187 y Fi(\000)p Fk(c=h)2663 4221 y Fo(\))1101 4346 y Fm(k)p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\)\()1366 4325 y(^)1356 4346 y(\005)1418 4312 y Fl(2)1418 4367 y(\010)1488 4346 y Fm(\000)1581 4325 y Fo(^)1571 4346 y(\005)1633 4358 y Fl(\010)1685 4346 y Fo(\))p Fm(k)18 b Fo(+)g Fm(k)p Fo(\()1944 4325 y(^)1934 4346 y(\005)1996 4312 y Fl(2)1996 4367 y(\010)2066 4346 y Fm(\000)2160 4325 y Fo(^)2149 4346 y(\005)2211 4358 y Fl(\010)2263 4346 y Fo(\))p Fn(\037)p Fo(\()p Fn(P)12 b Fo(\))p Fm(k)23 b Fo(=)g Fm(O)r Fo(\()p Fn(e)2768 4312 y Fi(\000)p Fk(c=h)2927 4346 y Fo(\))208 b(\(3.33\))515 4500 y(for)27 b(some)g(constan)n(t)g Fn(c)c(>)f Fo(0.)37 b(T)-7 b(aking)1315 4692 y(\005)23 b(:=)1553 4635 y Fn(i)p 1521 4673 92 4 v 1521 4749 a Fo(2)p Fn(\031)1637 4579 y Fc(Z)1683 4767 y Fi(j)p Fk(z)r Fi(\000)p Fl(1)p Fi(j)p Fl(=1)p Fk(=)p Fl(2)2010 4599 y Fc(\020)2070 4671 y Fo(^)2060 4692 y(\005)2122 4704 y Fl(\010)2192 4692 y Fm(\000)18 b Fn(z)2318 4599 y Fc(\021)2367 4617 y Fi(\000)p Fl(1)2493 4692 y Fn(dz)515 4907 y Fo(b)n(y)27 b(\(3.17\))g(and)g(\(3.33\))g(one)g (obtains)g(\(2.7\))h(and)f(\(2.8\).)p 3337 4919 34 63 v 1905 5255 a(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fp(4)134 b(An)30 b(application)h(to)f(scattering)i (theory)e(for)g(matrix-)716 672 y(v)-7 b(alued)45 b(Sc)l(hr\177)-67 b(odinger)45 b(op)t(erators)515 854 y Fo(In)24 b(this)g(section)f(w)n (e)h(giv)n(e)e(an)i(application)f(of)h(the)g(results)f(obtained)h(in)g (the)g(previous)e(one)515 954 y(to)27 b(the)h(scattering)f(theory)g (for)g(t)n(w)n(o)g(comp)r(onen)n(ts)g(Sc)n(hr\177)-42 b(odinger)26 b(systems.)36 b(W)-7 b(e)28 b(consider)515 1054 y(the)g(t)n(w)n(o-comp)r(onen)n(t)e(Sc)n(hr\177)-42 b(odinger)26 b(Hamiltonian)1490 1219 y Fn(P)12 b Fo(\()p Fn(h)p Fo(\))23 b(=)g Fm(\000)p Fn(h)1891 1185 y Fl(2)1927 1219 y Fo(\001)1996 1231 y Fk(x)2038 1219 y Fj(1)2086 1231 y Fl(2)2142 1219 y Fo(+)18 b Fn(V)h Fo(\()p Fn(x)p Fo(\))805 b(\(4.1\))515 1384 y(acting)25 b(on)h Fm(H)e Fo(:=)f Fn(L)1138 1354 y Fl(2)1175 1384 y Fo(\()p Fj(R)1279 1354 y Fk(n)1324 1384 y Fo(\))16 b Fm(\010)f Fn(L)1509 1354 y Fl(2)1546 1384 y Fo(\()p Fj(R)1650 1354 y Fk(n)1695 1384 y Fo(\).)37 b(Here)26 b Fj(1)2030 1396 y Fl(2)2093 1384 y Fo(is)g(the)h(2)15 b Fm(\002)g Fo(2)26 b(iden)n(tit)n(y)h (matrix)e(and,)i(for)515 1484 y Fn(x)c Fm(2)h Fj(R)736 1454 y Fk(n)780 1484 y Fo(,)k Fn(V)19 b Fo(\()p Fn(x)p Fo(\))29 b(is)e(a)g(2)18 b Fm(\002)g Fo(2)28 b(symmetric)f(matrix.)36 b(W)-7 b(e)28 b(assume)f(that)515 1584 y(\()p Fn(A)p Fo(1\))83 b(the)28 b(map)g Fj(R)1166 1553 y Fk(n)1233 1584 y Fm(3)c Fn(x)f Fm(!)g Fn(V)c Fo(\()p Fn(x)p Fo(\))29 b(admits)f(an)f(analytic)g(extension)g(in)h(some)f(strip)1246 1749 y(\000)1298 1761 y Fk(a)1361 1749 y Fo(=)c Fm(f)p Fn(x)g Fm(2)g Fj(C)1708 1715 y Fk(n)1777 1749 y Fo(;)37 b Fm(j)p Fo(Im)p Fn(x)p Fm(j)23 b Fn(<)g(a)p Fm(g)p Fn(;)179 b(a)23 b(>)g Fo(0)p Fn(:)560 b Fo(\(4.2\))515 1914 y(Concerning)26 b(the)i(sp)r(ectrum)g(of)f Fn(V)19 b Fo(\()p Fn(x)p Fo(\),)29 b(w)n(e)e(assume)g(that)515 2014 y(\(A2\))83 b(the)28 b(t)n(w)n(o)f(real)g(eigen)n(v)-5 b(alues)26 b Fn(\025)1705 2026 y Fk(i)1733 2014 y Fo(\()p Fn(x)p Fo(\))j(of)e(the)h(matrix)f Fn(V)19 b Fo(\()p Fn(x)p Fo(\))29 b(satisfy:)1408 2179 y(Inf)1373 2233 y Fk(x)p Fi(2)p Fg(R)1512 2216 y Ff(n)1566 2179 y Fo(\()p Fn(\025)1646 2191 y Fl(2)1684 2179 y Fo(\()p Fn(x)p Fo(\))19 b Fm(\000)f Fn(\025)1945 2191 y Fl(1)1983 2179 y Fo(\()p Fn(x)p Fo(\)\))24 b(:=)f(2)p Fn(d)g(>)f Fo(0)p Fn(:)687 b Fo(\(4.3\))515 2382 y(P)n(ossibly)22 b(b)n(y)i(shrinking)f(\000)1359 2394 y Fk(a)1399 2382 y Fo(,)i(one)e(can)h(supp)r(ose)f(that)50 b(Inf)2228 2435 y Fk(x)p Fi(2)p Fl(\000)2352 2443 y Ff(a)2402 2382 y Fo(\()p Fn(\025)2482 2394 y Fl(2)2520 2382 y Fo(\()p Fn(x)p Fo(\))19 b Fm(\000)f Fn(\025)2781 2394 y Fl(1)2819 2382 y Fo(\()p Fn(x)p Fo(\)\))24 b Fm(\025)f Fn(d:)h Fo(More-)515 2523 y(o)n(v)n(er)i(w)n(e)h(assume)g(that)h Fn(V)18 b Fo(\()p Fn(x)p Fo(\))29 b(is)1572 2493 y Fi(00)1642 2523 y Fo(short)e(range)2055 2493 y Fi(00)2123 2523 y Fo(i.e.)515 2622 y(\()p Fn(A)p Fo(3\))683 2634 y Fk(\016)803 2622 y Fo(there)g(exists)g(a)g(real)g(symmetric)g(2)18 b Fm(\002)g Fo(2)27 b(matrix)h Fn(V)2410 2634 y Fi(1)2508 2622 y Fo(suc)n(h)f(that)1528 2788 y Fm(j)p Fn(V)19 b Fo(\()p Fn(x)p Fo(\))h Fm(\000)e Fn(V)1880 2800 y Fi(1)1950 2788 y Fm(j)23 b Fo(=)g Fm(O)r Fo(\()p Fm(h)p Fn(x)p Fm(i)2295 2753 y Fk(\016)2333 2788 y Fo(\))843 b(\(4.4\))515 2953 y(with)33 b Fn(\016)i Fm(\025)c Fo(0,)j(uniformly)f(for)f Fn(x)h Fm(2)f Fo(\000)1711 2965 y Fk(a)1751 2953 y Fo(.)53 b(Without)33 b(loss)f(of)h(generalit)n(y)-7 b(,)33 b(w)n(e)f(can)h (assume)515 3053 y(that)28 b Fn(V)743 3065 y Fi(1)841 3053 y Fo(is)g(diagonal)e(i.e.)1486 3263 y Fn(V)1534 3275 y Fi(1)1628 3263 y Fo(:=)1739 3146 y Fc(\022)1842 3213 y Fn(\025)1890 3225 y Fl(1)p Fk(;)p Fi(1)2161 3213 y Fo(0)1907 3312 y(0)147 b Fn(\025)2144 3324 y Fl(2)p Fk(;)p Fi(1)2309 3146 y Fc(\023)2384 3263 y Fn(:)801 b Fo(\(4.5\))515 3478 y(Let)27 b Fn(\015)5 b Fo(\()p Fn(x)p Fo(\))24 b(=)f Fm(f)p Fn(z)j Fm(2)d Fj(C)g Fo(;)37 b Fm(j)p Fn(z)22 b Fm(\000)c Fn(\025)1486 3490 y Fl(1)1523 3478 y Fo(\()p Fn(x)p Fo(\))p Fm(j)25 b Fo(=)d Fn(d=)p Fo(2)p Fm(g)27 b Fo(and)g(let)h(us)g(de\014ne)1330 3686 y(\005)1392 3698 y Fl(0)1429 3686 y Fo(\()p Fn(x)p Fo(\))c(:=)1717 3630 y Fn(i)p 1685 3667 92 4 v 1685 3743 a Fo(2)p Fn(\031)1801 3573 y Fc(I)1847 3762 y Fk(\015)t Fl(\()p Fk(x)p Fl(\))1979 3686 y Fo(\()p Fn(V)19 b Fo(\()p Fn(x)p Fo(\))h Fm(\000)e Fn(z)t Fo(\))2367 3652 y Fi(\000)p Fl(1)2478 3686 y Fn(dz)515 3971 y Fo(Due)28 b(to)f(\(4.4\))h(and)f(\(4.5\),)g(w)n(e)h(ha)n(v)n(e,) e(setting)1991 3950 y(~)1981 3971 y(\005)2043 3983 y Fl(0)2103 3971 y Fo(:=)2214 3854 y Fc(\022)2317 3920 y Fo(1)82 b(0)2317 4020 y(0)g(0)2524 3854 y Fc(\023)2585 3971 y Fo(,)1477 4200 y(Sup)1467 4270 y Fk(x)p Fi(2)p Fl(\000)1591 4278 y Ff(a)1626 4200 y Fm(h)p Fn(x)p Fm(i)1737 4166 y Fk(\016)1775 4200 y Fm(j)p Fo(\005)1860 4212 y Fl(0)1897 4200 y Fo(\()p Fn(x)p Fo(\))20 b Fm(\000)2121 4179 y Fo(~)2111 4200 y(\005)2173 4212 y Fl(0)2210 4200 y Fm(j)j Fn(<)g Fm(1)781 b Fo(\(4.6\))515 4423 y(Let)21 b(us)h(\014x)f(\010)i Fm(2)g Fn(C)1095 4393 y Fi(1)1089 4443 y Fl(0)1166 4423 y Fo(\()p Fj(R)p Fo(\).)35 b(Applying)22 b(the)f(results)g(of)g(the)h(previous)e(section)h(to)h Fn(P)12 b Fo(\()p Fn(h)p Fo(\))21 b(and,)515 4522 y(in)32 b(particular,)g(Theorem)f(2.2,)h(one)f(can)h(construct)f(a)g(pro)5 b(jector)31 b(\005\()p Fn(h)p Fo(\))h(\(that)h(dep)r(ends)515 4622 y(on)f(the)h(c)n(hoice)f(of)g(\010\))h(on)f Fm(H)i Fo(satisfying)e(\(2.7\))g(and)h(\(2.8\).)51 b(Since)33 b Fm(k)p Fo(\005)21 b Fm(\000)h Fo(\005)2969 4634 y Fl(0)3006 4622 y Fm(k)31 b Fo(=)g Fm(O)r Fo(\()p Fn(h)p Fo(\),)515 4722 y(one)25 b(can)f(de\014ne)i(\(see)f([Ka)o(,)h(NeSo]\))f(the)h (transformation)d(matrix)i(corresp)r(onding)f(to)h(the)515 4821 y(pair)i(\005)p Fn(;)14 b Fo(\005)847 4833 y Fl(0)884 4821 y Fo(:)1018 5006 y Fn(U)31 b Fo(:=)1217 4939 y Fc(\000)1255 5006 y Fo(1)18 b Fm(\000)g Fo(\(\005)h Fm(\000)f Fo(\005)1656 5018 y Fl(0)1694 5006 y Fo(\))1726 4972 y Fl(2)1763 4939 y Fc(\001)1801 4955 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)1971 5006 y Fo(\()q(\005\005)2128 5018 y Fl(0)2184 5006 y Fo(+)g(\(1)g Fm(\000)g Fo(\005\)\(1)h Fm(\000)f Fo(\005)2774 5018 y Fl(0)2812 5006 y Fo(\)\))332 b(\(4.7\))1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fo(ha)n(ving)26 b(the)i(prop)r(erties)1707 694 y Fn(U)1773 660 y Fi(\003)1811 694 y Fn(U)k Fo(=)22 b Fn(U)9 b(U)2119 660 y Fi(\003)2180 694 y Fo(=)23 b Fj(1)2316 706 y Fl(2)3208 694 y Fo(\(4.8\))1707 819 y Fn(U)9 b Fo(\005)1835 831 y Fl(0)1895 819 y Fo(=)23 b(\005)p Fn(U:)1083 b Fo(\(4.9\))515 990 y(F)-7 b(urthermore,)24 b(since)h Fj(R)1290 960 y Fk(n)1361 990 y Fo(is)g(con)n(tractible,)f (and,)i(b)n(y)f(assumptions,)g Fn(V)19 b Fo(\()p Fn(x)p Fo(\))26 b(is)f(real)g(on)g(the)515 1089 y(real)h(axis,)h(one)g(can)h (\014nd)g(a)f(2)18 b Fm(\002)g Fo(2)27 b(matrix)g Fn(Q)p Fo(\()p Fn(x)p Fo(\),)i(unitary)e(for)g(real)f Fn(x)p Fo(,)j(satisfying)1539 1261 y(\005)1601 1273 y Fl(0)1639 1261 y Fo(\()p Fn(x)p Fo(\))24 b(=)e Fn(Q)p Fo(\()p Fn(x)p Fo(\))2048 1240 y(~)2038 1261 y(\005)2100 1273 y Fl(0)2139 1261 y Fn(Q)2205 1226 y Fi(\003)2242 1261 y Fo(\()p Fn(x)p Fo(\))515 1432 y(Hence,)28 b(de\014ning)f(the)h(unitary)f(op)r(erator) 1784 1603 y(\012)c(:=)g Fn(U)9 b(Q)515 1774 y Fo(and)27 b(setting)1755 1853 y(~)1736 1874 y Fn(P)35 b Fo(:=)23 b(\012)1995 1839 y Fi(\003)2033 1874 y Fn(P)12 b Fo(\012)515 2016 y(one)24 b(has)g(that,)i(for)f(an)n(y)f Fn(\037)f Fm(2)g Fn(C)1507 1986 y Fi(1)1501 2037 y Fl(0)1578 2016 y Fo(\()p Fj(R)p Fo(\))i(suc)n(h)f(that)h(\010)p Fn(\037)e Fo(=)g Fn(\037)p Fo(,)i(then)2628 1995 y(~)2610 2016 y Fn(P)12 b(\037)p Fo(\()2777 1995 y(~)2759 2016 y Fn(P)g Fo(\))25 b(has)f(an)h(almost)515 2116 y(blo)r(c)n(k)i(diagonal)f(form.) 36 b(More)27 b(precisely)g(w)n(e)g(ha)n(v)n(e)f(\(see)i([NeSo]\):)515 2273 y Fj(Prop)s(osition)i(4.1)40 b Fd(Assume)28 b(that)h Fo(\()p Fn(A)p Fo(1\))p Fn(;)14 b Fo(\()p Fn(A)p Fo(2\))29 b Fd(and)g Fo(\()p Fn(A)p Fo(3\))2402 2285 y Fk(\016)2468 2273 y Fd(hold)h(true)e(for)h(some)g Fn(\016)d Fm(\025)d Fo(0)p Fd(.)515 2372 y(Then,)34 b(for)f(any)f Fo(\010)27 b Fm(2)h Fn(C)1291 2342 y Fi(1)1285 2393 y Fl(0)1362 2372 y Fo(\()p Fj(R)p Fo(\))p Fd(,)33 b(ther)l(e)f(exist)g(a)h(b)l (ounde)l(d)f(op)l(er)l(ator)2703 2351 y Fo(~)2683 2372 y Fn(D)2752 2384 y Fk(j)2819 2372 y Fd(on)g Fn(L)2997 2342 y Fl(2)3034 2372 y Fo(\()p Fj(R)3138 2342 y Fk(n)3183 2372 y Fo(\))g Fd(and)515 2472 y(a)e(b)l(ounde)l(d)g(op)l(er)l(ator)h Fn(R)f Fd(on)g Fm(H)h Fd(such)e(that)1760 2622 y Fo(~)1741 2643 y Fn(P)35 b Fo(=)23 b Fn(D)d Fo(+)e Fn(R)1015 b Fo(\(4.10\))515 2814 y Fd(with)1003 2943 y Fn(D)25 b Fo(:=)1207 2826 y Fc(\022)1310 2892 y Fn(D)1379 2904 y Fl(1)1531 2892 y Fo(0)1342 2992 y(0)115 b Fn(D)1568 3004 y Fl(2)1646 2826 y Fc(\023)1721 2943 y Fn(;)99 b(D)1912 2955 y Fk(j)1970 2943 y Fo(:=)23 b Fm(\000)p Fn(h)2194 2909 y Fl(2)2230 2943 y Fo(\001)2299 2955 y Fk(x)2360 2943 y Fo(+)18 b Fn(\025)2491 2955 y Fk(j)2526 2943 y Fo(\()p Fn(x)p Fo(\))i(+)e Fn(h)2807 2922 y Fo(~)2788 2943 y Fn(D)2857 2955 y Fk(j)3167 2943 y Fo(\(4.11\))515 3135 y Fd(and,)30 b(for)h(any)f Fn(\037)23 b Fm(2)h Fn(C)1212 3105 y Fi(1)1206 3156 y Fl(0)1282 3135 y Fo(\()p Fj(R)p Fo(\))30 b Fd(satisfying)h Fn(\037)p Fo(\010)23 b(=)g(\010)p Fd(,)30 b(one)g(has)1311 3306 y Fm(k)p Fn(R)q(\037)p Fo(\()1519 3285 y(~)1501 3306 y Fn(P)11 b Fo(\))p Fm(k)1639 3318 y Fk(\016)1694 3306 y Fo(+)18 b Fm(k)p Fn(\037)p Fo(\()1921 3285 y(~)1903 3306 y Fn(P)12 b Fo(\))p Fn(R)q Fm(k)2106 3318 y Fk(\016)2165 3306 y Fo(=)22 b Fm(O)r Fo(\()p Fn(e)2391 3272 y Fi(\000)p Fk(c=h)2550 3306 y Fo(\))585 b(\(4.12\))515 3477 y Fd(uniformly)30 b(on)g Fn(h)p Fd(,)g(for)h Fn(h)23 b(>)f Fo(0)30 b Fd(smal)t(l)g(enough.)515 3634 y Fo(Let)e(us)g(apply)g(the)h(result)f(of)g(Prop)r(osition)e(4.1)i (to)g(scattering)f(theory)-7 b(.)38 b(Let)28 b(us)g(consider)515 3734 y(the)g(w)n(a)n(v)n(e)e(op)r(erator)g(asso)r(ciated)g(to)h(the)h (pairs)f(\()p Fn(P)r(;)14 b(P)2216 3746 y Fl(0)2254 3734 y Fo(\),)28 b(where)995 3955 y Fn(P)1048 3967 y Fl(0)1109 3955 y Fo(=)1196 3838 y Fc(\022)1299 3904 y Fn(P)1352 3916 y Fl(0)p Fk(;)p Fl(1)1576 3904 y Fo(0)1350 4004 y(0)133 b Fn(P)1578 4016 y Fl(0)p Fk(;)p Fl(2)1710 3838 y Fc(\023)1794 3955 y Fo(:=)23 b Fm(\000)p Fn(h)2018 3921 y Fl(2)2055 3955 y Fo(\001)2124 3967 y Fk(x)2184 3955 y Fo(+)2267 3838 y Fc(\022)2370 3904 y Fn(\025)2418 3916 y Fl(1)p Fk(;)p Fi(1)2689 3904 y Fo(0)2435 4004 y(0)147 b Fn(\025)2672 4016 y Fl(1)p Fk(;)p Fi(1)2838 3838 y Fc(\023)515 4169 y Fo(i.e.)1282 4340 y Fn(W)1360 4352 y Fi(\006)1416 4340 y Fo(\()p Fn(P)r(;)14 b(P)1593 4352 y Fl(0)1631 4340 y Fo(\))84 b(=)e(s)18 b Fm(\000)65 b Fo(lim)2028 4390 y Fk(t)p Fi(!\0061)2252 4340 y Fn(e)2291 4306 y Fk(itP)2394 4340 y Fn(e)2433 4306 y Fi(\000)p Fk(itP)2575 4314 y Fe(0)515 4636 y Fo(and,)44 b(similarly)-7 b(,)44 b(the)d(w)n(a)n(v)n(e)f(op)r(erators)f Fn(W)1921 4648 y Fi(\006)1977 4636 y Fo(\()2028 4615 y(~)2009 4636 y Fn(P)12 b(;)i(P)2164 4648 y Fl(0)2202 4636 y Fo(\),)44 b Fn(W)2379 4648 y Fi(\006)2436 4636 y Fo(\()p Fn(D)r(;)14 b(P)2629 4648 y Fl(0)2667 4636 y Fo(\))41 b(asso)r(ciated)f(to)h(the) 515 4735 y(pairs)33 b(\()776 4714 y(~)757 4735 y Fn(P)13 b(;)h(P)913 4747 y Fl(0)950 4735 y Fo(\))35 b(and)g(\()p Fn(D)r(;)14 b(P)1379 4747 y Fl(0)1416 4735 y Fo(\))35 b(resp)r(ectiv)n(ely)-7 b(.)58 b(If)35 b Fn(\016)j(>)c Fo(1,)i(it)f(is)g(easy)f(to)g(c)n(hec)n(k)g(that)h(this)515 4835 y(w)n(a)n(v)n(e)26 b(op)r(erators)f(exist)j(and)f(are)g(complete)g (so)g(w)n(e)g(can)h(de\014ne)g(the)g(scattering)e(matrix)646 5006 y Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)879 5018 y Fl(0)917 5006 y Fo(\))24 b(=)e Fn(W)1138 5018 y Fl(+)1194 5006 y Fo(\()p Fn(P)r(;)14 b(P)1371 5018 y Fl(0)1409 5006 y Fo(\))1441 4972 y Fi(\003)1479 5006 y Fn(W)1557 5018 y Fi(\000)1614 5006 y Fo(\()p Fn(P)r(;)g(P)1791 5018 y Fl(0)1829 5006 y Fo(\))p Fn(;)97 b(S)5 b Fo(\()2088 4985 y(~)2069 5006 y Fn(P)12 b(;)i(P)2224 5018 y Fl(0)2262 5006 y Fo(\))23 b(=)g Fn(W)2483 5018 y Fl(+)2538 5006 y Fo(\()2589 4985 y(~)2570 5006 y Fn(P)12 b(;)i(P)2725 5018 y Fl(0)2763 5006 y Fo(\))2795 4972 y Fi(\003)2833 5006 y Fn(W)2911 5018 y Fi(\000)2968 5006 y Fo(\()3019 4985 y(~)3000 5006 y Fn(P)e(;)i(P)3155 5018 y Fl(0)3192 5006 y Fo(\))p Fn(;)1905 5255 y Fo(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fo(and)698 735 y Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)947 747 y Fl(0)985 735 y Fo(\))23 b(=)1128 618 y Fc(\022)1231 684 y Fn(S)5 b Fo(\()p Fn(D)1388 696 y Fl(1)1425 684 y Fn(;)14 b(P)1515 696 y Fl(0)p Fk(;)p Fl(1)1605 684 y Fo(\))266 b(0)1413 784 y(0)f Fn(S)5 b Fo(\()p Fn(D)1877 796 y Fl(2)1914 784 y Fn(;)14 b(P)2004 796 y Fl(0)p Fk(;)p Fl(2)2094 784 y Fo(\))2168 618 y Fc(\023)2252 735 y Fo(=)23 b Fn(W)2418 747 y Fl(+)2473 735 y Fo(\()p Fn(D)r(;)14 b(P)2666 747 y Fl(0)2704 735 y Fo(\))2736 700 y Fi(\003)2775 735 y Fn(W)2853 747 y Fi(\000)2909 735 y Fo(\()p Fn(D)r(;)g(P)3102 747 y Fl(0)3140 735 y Fo(\))p Fn(:)515 962 y Fo(Since)1715 1061 y Fm(k)p Fo(\012)p Fm(k)1859 1073 y Fk(\016)1917 1061 y Fo(=)23 b Fm(O)r Fo(\(1\))988 b(\(4.13\))515 1211 y(one)27 b(can)g(easily)g (pro)n(v)n(e)f(\(see)h(Lemma)h(3.6)e(of)i([NeSo]\))g(that)1456 1393 y Fn(W)1534 1405 y Fi(\006)1590 1393 y Fo(\()p Fn(P)r(;)14 b(P)1767 1405 y Fl(0)1806 1393 y Fo(\))23 b(=)g(\012)2009 1359 y Fi(\003)2047 1393 y Fn(W)2125 1405 y Fi(\006)2181 1393 y Fo(\()2232 1372 y(~)2213 1393 y Fn(P)12 b(;)i(P)2368 1405 y Fl(0)2406 1393 y Fo(\))515 1576 y(and)27 b(then,)h(since)g(\012) f(is)h(an)f(unitary)g(op)r(erator,)1572 1759 y Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1805 1771 y Fl(0)1843 1759 y Fo(\))23 b(=)g Fn(S)5 b Fo(\()2093 1738 y(~)2074 1759 y Fn(P)12 b(;)i(P)2229 1771 y Fl(0)2266 1759 y Fo(\))p Fn(:)515 1941 y Fo(Using)35 b(the)g(same)g(argumen)n(t)f(of)h(the)g (pro)r(of)g(of)g(Theorem)f(3.10)g(in)h([NeSo],)i(one)e(easily)515 2041 y(obtains)515 2207 y Fj(Theorem)30 b(4.2)41 b Fd(Assume)c(that)g Fo(\()p Fn(A)p Fo(1\))p Fn(;)14 b Fo(\()p Fn(A)p Fo(2\))39 b Fd(and)f Fo(\()p Fn(A)p Fo(3\))2328 2219 y Fk(\016)2402 2207 y Fd(hold)i(true)d(for)h(some)g Fn(\016)j(>)c Fo(1)p Fd(.)515 2307 y(Then,)27 b(for)f(any)f Fo(\010)e Fm(2)g Fn(C)1261 2277 y Fi(1)1255 2327 y Fl(0)1332 2307 y Fo(\()p Fj(R)p Fo(\))p Fd(,)k(denoting)e(by)h Fn(D)g Fd(the)g(diagonal)h(op)l (er)l(ator)f(in)f(\(4.11\),)j(ther)l(e)515 2415 y(exists)d Fn(c)774 2427 y Fl(\010)849 2415 y Fn(>)d Fo(0)k Fd(\(indep)l(endent)g (of)h Fn(h)p Fd(\))f(such)g(that)g(for)h(any)f Fn(\036)2380 2427 y Fk(k)2445 2415 y Fm(2)d(H)q Fd(,)2657 2393 y Fo(^)2646 2415 y Fn(\036)2695 2427 y Fk(k)2759 2415 y Fm(2)h(C)2887 2385 y Fi(1)2882 2436 y Fl(0)2957 2415 y Fo(\()p Fj(R)3061 2385 y Fk(n)3106 2415 y Fm(nf)p Fo(0)p Fm(g)p Fo(\))10 b Fm(\010)515 2515 y(C)564 2485 y Fi(1)559 2535 y Fl(0)634 2515 y Fo(\()p Fj(R)738 2485 y Fk(n)783 2515 y Fm(nf)p Fo(0)p Fm(g)p Fo(\))p Fn(;)k(k)24 b Fo(=)e(1)p Fn(;)14 b Fo(2)p Fd(,)1005 2698 y Fm(jh)p Fo(\()q Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1326 2710 y Fl(0)1364 2698 y Fo(\))k Fm(\000)g Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)1746 2710 y Fl(0)1784 2698 y Fo(\)\))g Fn(\036)1911 2710 y Fl(1)1949 2698 y Fn(;)g(\036)2035 2710 y Fl(2)2073 2698 y Fm(ij)23 b(\024)g Fn(C)2298 2710 y Fl(\010)2350 2698 y Fo(\()p Fn(\036)2431 2710 y Fl(1)2469 2698 y Fn(;)14 b(\036)2555 2710 y Fl(2)2592 2698 y Fo(\))p Fn(e)2663 2663 y Fi(\000)p Fk(c)2745 2671 y Fe(\010)2789 2663 y Fk(=h)2866 2698 y Fn(:)278 b Fo(\(4.14\))515 2880 y Fd(for)25 b(some)g(c)l(onstant)f Fn(C)1234 2892 y Fl(\010)1286 2880 y Fo(\()p Fn(\036)1367 2892 y Fl(1)1405 2880 y Fn(;)14 b(\036)1491 2892 y Fl(2)1528 2880 y Fo(\))24 b Fn(>)e Fo(0)i Fd(dep)l(ending)i(on)f Fo(\010)p Fn(;)14 b(\036)2378 2892 y Fl(1)2415 2880 y Fn(;)g(\036)2501 2892 y Fl(2)2563 2880 y Fd(but)24 b(not)g(on)h Fn(h)p Fd(,)h(for)f Fn(h)e(>)f Fo(0)515 2980 y Fd(smal)t(l)30 b(enough.)515 3146 y Fo(Due)25 b(to)f(the)h(p)r(ossible)g(presence)e (of)i(resonances,)e(it)i(seems)g(hard)f(to)g(pro)n(v)n(e)f(an)h (analogous)515 3245 y(result)f(for)h(the)g(scattering)f(matrix)h Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1885 3257 y Fl(0)1923 3245 y Fn(;)g(\025)p Fo(\))25 b(without)f(an)n(y)f(additional)h (assumptions.)515 3345 y(On)h(the)i(con)n(trary)-7 b(,)24 b(if)i(w)n(e)g(assume)f(that)h(w)n(e)g(sta)n(y)f(close)g(to)h(a)f (non-trapping)g(energy)g(lev)n(el)515 3445 y Fn(\025)563 3457 y Fl(0)600 3445 y Fo(,)34 b(w)n(e)f(are)e(able)h(to)h(pro)n(v)n(e) e(that)i(the)f(scattering)g(matrix)g Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)2673 3457 y Fl(0)2711 3445 y Fn(;)g(\025)p Fo(\))33 b(is)g(diagonal)e(up)515 3544 y(to)i(exp)r(onen)n(tially)h (small)f(terms,)i(obtaining)f(in)g(particular)e(\(with)j(complete)f (di\013eren)n(t)515 3644 y(tec)n(hniques\))27 b(the)h(results)f(of)h ([MNS)q(].)37 b(Let)27 b(us)h(giv)n(e)e(the)i(follo)n(wing)f (de\014nition:)515 3810 y Fj(De\014nition)k(4.3)40 b Fd(We)21 b(say)h(that)g Fn(\025)1614 3822 y Fl(0)1674 3810 y Fn(>)h(\025)1810 3822 y Fl(2)p Fk(;)p Fi(1)1955 3810 y Fd(is)e(non-tr)l(apping)h(if,)i(denoting)e(by)g Fn(p)3088 3822 y Fk(j)3123 3810 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))24 b(=)515 3910 y Fn(\030)555 3880 y Fl(2)607 3910 y Fo(+)15 b Fn(\025)735 3922 y Fk(j)770 3910 y Fo(\()p Fn(x)p Fo(\))p Fd(,)30 b Fn(j)e Fo(=)23 b(1)p Fn(;)14 b Fo(2)27 b Fd(and)h(by)h Fn(H)1569 3922 y Fk(p)1603 3930 y Ff(j)1666 3910 y Fd(the)g(Hamiltonian)g(\015ows)f(gener)l(ate)l (d)h(by)f Fn(p)3004 3922 y Fk(j)3039 3910 y Fd(,)h(then)f(for)515 4009 y(any)i Fo(\()p Fn(x;)14 b(\030)t Fo(\))24 b Fm(2)f Fj(R)1036 3979 y Fl(2)p Fk(n)1114 4009 y Fd(,)30 b(such)g(that)g Fn(p)1570 4021 y Fk(j)1605 4009 y Fo(\()p Fn(x;)14 b(\030)t Fo(\))24 b(=)f Fn(\025)1953 4021 y Fl(0)2020 4009 y Fd(one)30 b(has:)1299 4192 y Fm(j)p Fo(exp\()p Fn(tH)1580 4204 y Fk(p)1614 4212 y Ff(j)1650 4192 y Fo(\)\()p Fn(x;)14 b(\030)t Fo(\))p Fm(j)24 b(!)f(1)p Fn(;)99 b Fo(if)30 b Fm(j)p Fn(t)p Fm(j)23 b(!)g(1)573 b Fo(\(4.15\))515 4375 y Fd(for)30 b Fn(j)e Fo(=)23 b(1)p Fn(;)14 b Fo(2)p Fd(.)515 4586 y Fo(In)20 b(the)h(follo)n(wing)f(w)n(e)g(denote)h(b)n(y) f Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1808 4598 y Fl(0)1846 4586 y Fn(;)g(\025)p Fo(\))24 b(=)e Fn(S)5 b Fo(\()2181 4565 y(~)2162 4586 y Fn(P)12 b(;)i(P)2317 4598 y Fl(0)2354 4586 y Fn(;)g(\025)p Fo(\))24 b(=)2583 4469 y Fc(\022)2685 4535 y Fn(S)2736 4547 y Fl(1)p Fk(;)p Fl(1)2826 4535 y Fo(\()p Fn(\025)p Fo(\))84 b Fn(S)3073 4547 y Fl(1)p Fk(;)p Fl(2)3163 4535 y Fo(\()p Fn(\025)p Fo(\))2685 4635 y Fn(S)2736 4647 y Fl(2)p Fk(;)p Fl(1)2826 4635 y Fo(\()p Fn(\025)p Fo(\))g Fn(S)3073 4647 y Fl(2)p Fk(;)p Fl(2)3163 4635 y Fo(\()p Fn(\025)p Fo(\))3318 4469 y Fc(\023)515 4793 y Fo(and)30 b Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)928 4805 y Fl(0)965 4793 y Fn(;)g(\025)p Fo(\))28 b(=)1202 4676 y Fc(\022)1305 4743 y Fn(S)5 b Fo(\()p Fn(D)1462 4755 y Fl(1)1499 4743 y Fn(;)14 b(P)1589 4755 y Fl(0)1626 4743 y Fn(;)g(\025)p Fo(\))282 b(0)1503 4842 y(0)g Fn(S)5 b Fo(\()p Fn(D)1984 4854 y Fl(2)2021 4842 y Fn(;)14 b(P)2111 4854 y Fl(0)2148 4842 y Fn(;)g(\025)p Fo(\))2307 4676 y Fc(\023)2399 4793 y Fo(the)30 b(scattering)f(matrix)h (asso-)515 4943 y(ciated)d(to)h(the)g(scattering)e(op)r(erator)g Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1955 4955 y Fl(0)1993 4943 y Fo(\))28 b(and)f Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)2463 4955 y Fl(0)2501 4943 y Fo(\))28 b(resp)r(ectiv)n(ely)-7 b(.)1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fj(Theorem)30 b(4.4)41 b Fd(Assume)25 b(that)h Fo(\()p Fn(A)p Fo(1\))p Fn(;)14 b Fo(\()p Fn(A)p Fo(2\))27 b Fd(and)g Fo(\()p Fn(A)p Fo(3\))2282 535 y Fk(\016)2345 523 y Fd(hold)h(true)d(for)i(some)g Fn(\016)f Fm(\025)d Fo(1)i Fd(and)515 623 y(let)35 b(us)g(c)l(onsider)i(a)f (non-tr)l(apping)f(ener)l(gy)h(level)h(if)f Fn(\025)2254 635 y Fl(0)2325 623 y Fn(>)d(\025)2471 635 y Fl(2)p Fk(;)p Fi(1)2595 623 y Fd(.)56 b(Then,)38 b(ther)l(e)d(exist)h(a)515 722 y(neighb)l(orho)l(o)l(d)c Fn(I)37 b Fd(of)30 b Fn(\025)1226 734 y Fl(0)1294 722 y Fd(and)g(c)l(onstants)f Fn(C)q(;)14 b(c)23 b(>)g Fo(0)29 b Fd(\(indep)l(endent)i(of)f Fn(h)p Fd(\))g(such)g(that)789 888 y Fm(k)p Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)1064 900 y Fl(0)1102 888 y Fn(;)g(\025)p Fo(\))19 b Fm(\000)f Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)1570 900 y Fl(0)1607 888 y Fn(;)g(\025)p Fo(\))p Fm(k)1766 903 y Fi(L)p Fl(\()p Fk(L)1884 887 y Fe(2)1916 903 y Fl(\()p Fk(S)1986 887 y Ff(n)p Fb(\000)p Fe(1)2100 903 y Fl(\))p Fi(\010)p Fk(L)2224 887 y Fe(2)2256 903 y Fl(\()p Fk(S)2326 887 y Ff(n)p Fb(\000)p Fe(1)2440 903 y Fl(\)\))2519 888 y Fm(\024)23 b Fn(C)6 b(e)2711 853 y Fi(\000)p Fk(c=h)2869 888 y Fn(:)275 b Fo(\(4.16\))515 1053 y Fd(for)30 b Fn(\025)24 b Fm(2)f Fn(I)37 b Fd(and)30 b(for)h Fn(h)23 b Fm(2)g Fo(\(0)p Fn(;)14 b(h)1472 1065 y Fl(0)1509 1053 y Fo(])30 b Fd(smal)t(l)g(enough.)39 b(In)30 b(p)l(articular,)h(for)g(smal)t(l)f Fn(h)1245 1218 y Fm(k)p Fn(S)1338 1230 y Fk(i;j)1416 1218 y Fo(\()p Fn(\025)p Fo(\))p Fm(k)1570 1234 y Fi(L)p Fl(\()p Fk(L)1688 1217 y Fe(2)1720 1234 y Fl(\()p Fk(S)1790 1217 y Ff(n)p Fb(\000)p Fe(1)1904 1234 y Fl(\)\))1983 1218 y Fm(\024)23 b Fn(C)6 b(e)2175 1184 y Fi(\000)p Fk(c=h)2334 1218 y Fn(;)98 b(\025)24 b Fm(2)f Fn(I)526 b Fo(\(4.17\))515 1384 y Fd(Pr)l(o)l(of:)49 b Fo(Let)33 b(us)g(consider)f(a)h(function)h(\010)e Fm(2)g Fn(C)2026 1354 y Fi(1)2020 1404 y Fl(0)2097 1384 y Fo(\()p Fj(R)p Fo(\))h(suc)n(h)g(that)g(\010\()p Fn(\025)p Fo(\))h(=)d(1)i(for)g Fn(\025)g Fo(in)g(a)515 1483 y(neigh)n(b)r(orho)r(o)r(d)27 b Fn(J)36 b Fo(of)28 b Fn(\025)1262 1495 y Fl(0)1300 1483 y Fo(.)39 b(Asso)r(ciated)27 b(to)h(\010,)h(let)f(us)g(consider) 2562 1462 y(~)2543 1483 y Fn(P)41 b Fo(\(dep)r(ending)28 b(on)g(\010\))h(as)515 1583 y(in)f(Prop)r(osition)e(4.1)g(and)i(let)g (us)f(write)556 1748 y Fn(S)5 b Fo(\()p Fn(P)r(;)14 b(P)789 1760 y Fl(0)827 1748 y Fo(\))24 b(=)e Fn(S)5 b Fo(\()1077 1727 y(~)1058 1748 y Fn(P)12 b(;)i(P)1213 1760 y Fl(0)1251 1748 y Fo(\))23 b(=)f Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)1642 1760 y Fl(0)1680 1748 y Fo(\))9 b Fm(\000)g Fn(W)1885 1714 y Fi(\003)1873 1769 y Fl(+)1929 1748 y Fo(\()1980 1727 y(~)1961 1748 y Fn(P)j(;)i(P)2116 1760 y Fl(0)2153 1748 y Fo(\)\()p Fn(S)5 b Fo(\()p Fn(D)r(;)2432 1727 y Fo(~)2413 1748 y Fn(P)13 b Fo(\))c Fm(\000)g Fj(1)2642 1760 y Fl(2)2679 1748 y Fo(\))p Fn(W)2789 1760 y Fi(\000)2845 1748 y Fo(\()2896 1727 y(~)2877 1748 y Fn(P)k(;)h(P)3033 1760 y Fl(0)3070 1748 y Fo(\))p Fn(:)42 b Fo(\(4.18\))515 1914 y(Denoting)27 b(b)n(y)h Fn(F)1042 1926 y Fl(0)1079 1914 y Fo(\()p Fn(\025)p Fo(\))h(the)f(sp)r(ectral)f(represen)n(tation) f(of)h Fn(P)2365 1926 y Fl(0)2431 1914 y Fo(and)g(b)n(y)1315 2079 y Fn(F)1368 2091 y Fi(\006)1424 2079 y Fo(\()p Fn(\025)p Fo(\))d(:=)f Fn(F)1724 2091 y Fl(0)1762 2079 y Fo(\()p Fn(\025)p Fo(\))p Fn(W)1964 2045 y Fi(\003)1952 2099 y(\006)2009 2079 y Fo(\()2060 2058 y(~)2041 2079 y Fn(P)12 b(;)i(P)2196 2091 y Fl(0)2234 2079 y Fo(\))p Fn(;)97 b(\025)23 b Fm(2)h Fn(I)515 2244 y Fo(w)n(e)j(can)g(write)556 2410 y Fn(S)5 b Fo(\()663 2389 y(~)644 2410 y Fn(P)12 b(;)i(P)799 2422 y Fl(0)837 2410 y Fn(;)g(\025)p Fo(\))23 b(=)g Fn(S)5 b Fo(\()p Fn(D)r(;)14 b(P)1314 2422 y Fl(0)1351 2410 y Fn(;)g(\025)p Fo(\))19 b Fm(\000)e Fn(F)1622 2422 y Fl(+)1678 2410 y Fo(\()p Fn(\025)p Fo(\))p Fn(F)1843 2422 y Fl(0)1882 2410 y Fo(\()p Fn(\025)p Fo(\))1994 2375 y Fi(\003)2033 2410 y Fo(\()p Fn(S)5 b Fo(\()p Fn(D)r(;)2279 2389 y Fo(~)2261 2410 y Fn(P)12 b(;)i(\025)p Fo(\))k Fm(\000)g Fj(1)2592 2422 y Fl(2)2629 2410 y Fo(\))p Fn(F)2714 2422 y Fl(0)2752 2410 y Fo(\()p Fn(\025)p Fo(\))p Fn(F)2917 2422 y Fi(\000)2974 2410 y Fo(\()p Fn(\025)p Fo(\))3086 2375 y Fi(\003)3167 2410 y Fo(\(4.19\))515 2575 y(Observ)n(e)29 b(that)h(\(see)h([Na])f(and)h([MNS]\))g Fn(F)1880 2587 y Fl(+)1936 2575 y Fo(\()p Fn(\025)p Fo(\))g(is)g(a)f(sp)r(ectral)g (represen)n(tation)e(of)3215 2554 y(~)3196 2575 y Fn(P)42 b Fo(on)515 2675 y(Ran)p Fn(E)725 2687 y Fk(J)771 2675 y Fo(\()822 2654 y(~)803 2675 y Fn(P)12 b Fo(\))25 b(\(here)f Fn(E)1195 2687 y Fk(J)1242 2675 y Fo(\()1293 2654 y(~)1274 2675 y Fn(P)12 b Fo(\))25 b(denotes)f(the)h(sp)r(ectral)f(pro)5 b(jection)23 b(of)h(an)g(op)r(erator)3087 2654 y(~)3068 2675 y Fn(P)36 b Fo(on)25 b Fn(J)8 b Fo(\).)515 2774 y(Moreo)n(v)n(er,)25 b(denoting)i(b)n(y)1432 2940 y Fm(H)1503 2905 y Fk(\013)1573 2940 y Fo(:=)c Fn(L)1741 2905 y Fl(2)p Fk(;\013)1841 2940 y Fo(\()p Fj(R)1945 2905 y Fk(n)1990 2940 y Fo(\))c Fm(\010)f Fn(L)2181 2905 y Fl(2)p Fk(;\013)2280 2940 y Fo(\()p Fj(R)2384 2905 y Fk(n)2429 2940 y Fo(\))515 3105 y(with)h Fn(L)752 3075 y Fl(2)p Fk(;\013)851 3105 y Fo(\()p Fj(R)955 3075 y Fk(n)1000 3105 y Fo(\))24 b(:=)e Fm(h)p Fn(x)p Fm(i)1277 3075 y Fi(\000)p Fk(\013)1378 3105 y Fn(L)1435 3075 y Fl(2)1471 3105 y Fo(\()p Fj(R)1575 3075 y Fk(n)1620 3105 y Fo(\))d(and)f(using)g(a)g(stationary)f (represen)n(tation)g(of)h Fn(W)3174 3117 y Fl(+)3230 3105 y Fo(\()3281 3084 y(~)3262 3105 y Fn(P)12 b(;)i(P)3417 3117 y Fl(0)3454 3105 y Fo(\))515 3205 y(and)19 b(semiclassical)f (resolv)n(en)n(t)g(estimates)h(for)1977 3184 y(~)1958 3205 y Fn(P)32 b Fo(as)19 b(in)h([JeNa)o(],)i(one)d(can)g(sho)n(w)g (that)g Fn(F)3210 3217 y Fl(+)3266 3205 y Fo(\()p Fn(\025)p Fo(\))515 3304 y(is)k(b)r(ounded)h(from)f Fm(H)1192 3274 y Fk(\013)1262 3304 y Fo(to)g Fn(L)1416 3274 y Fl(2)1453 3304 y Fo(\()p Fn(S)1541 3274 y Fk(n)p Fi(\000)p Fl(1)1671 3304 y Fo(\))10 b Fm(\010)g Fn(L)1845 3274 y Fl(2)1882 3304 y Fo(\()p Fn(S)1970 3274 y Fk(n)p Fi(\000)p Fl(1)2100 3304 y Fo(\))24 b(for)e Fn(\013)i(>)f Fo(1)p Fn(=)p Fo(2)e(and)j(its)f (op)r(erator)f(norm)515 3404 y(is)h(b)r(ounded)g(b)n(y)g Fn(ch)1123 3374 y Fi(\000)p Fl(2)1235 3404 y Fo(uniformly)g(for)g Fn(\025)g Fm(2)g Fn(J)8 b Fo(.)36 b(Hence)23 b(w)n(e)g(can)g(apply)f (the)i(represen)n(tation)515 3503 y(form)n(ula)i(for)h(the)h (scattering)f(matrix)774 3669 y Fn(S)5 b Fo(\()p Fn(D)r(;)988 3648 y Fo(~)970 3669 y Fn(P)11 b(;)j(\025)p Fo(\))24 b(=)f Fj(1)1311 3681 y Fl(2)1366 3669 y Fo(+)18 b(2)p Fn(\031)s(iF)1623 3681 y Fl(+)1678 3669 y Fo(\()p Fn(\025)p Fo(\)\()p Fn(R)i Fo(+)f Fn(R)q Fo(\()p Fn(D)h Fm(\000)e Fn(\025)h Fm(\000)f Fn(i)p Fo(0\))2510 3634 y Fi(\000)p Fl(1)2598 3669 y Fn(R)q Fo(\))p Fn(F)2747 3681 y Fl(+)2803 3669 y Fo(\()p Fn(\025)p Fo(\))2915 3634 y Fi(\003)3167 3669 y Fo(\(4.20\))515 3834 y(with)39 b Fn(R)k Fo(=)f Fn(R)992 3804 y Fi(\003)1072 3834 y Fo(=)1198 3813 y(~)1179 3834 y Fn(P)c Fm(\000)26 b Fn(D)r(:)39 b Fo(Observ)n(e,)h(that,)j(if)c Fn(\037)k Fm(2)f Fn(C)2412 3804 y Fi(1)2406 3855 y Fl(0)2483 3834 y Fo(\()p Fj(R)p Fo(\))d(is)g(c)n(hosen)f(in)i(suc)n(h)e(a)515 3934 y(w)n(a)n(y)32 b(that)i Fn(\037)p Fo(\()p Fn(\025)p Fo(\))h(=)e(1)g(for)g Fn(\025)h Fm(2)g Fn(I)40 b Fm(\032)33 b Fn(J)42 b Fo(and)34 b Fn(\037)p Fo(\010)f(=)g Fn(\037)p Fo(,)j(then)e Fm(k)p Fn(R)q(\037)p Fo(\()2738 3913 y(~)2720 3934 y Fn(P)11 b Fo(\))p Fm(k)2858 3946 y Fk(\016)2928 3934 y Fo(=)33 b Fm(O)r Fo(\()p Fn(e)3165 3904 y Fi(\000)p Fk(c=h)3324 3934 y Fo(\),)515 4033 y(for)21 b(some)f(constan)n(t)h Fn(c)i(>)g Fo(0)d(indep)r(enden)n(t)j(of)e Fn(h)p Fo(.)35 b(As)21 b(a)g(consequence,)h(b)n(y)f(\(4.12\))f(for)h Fn(\025)j Fm(2)f Fn(I)7 b Fo(,)515 4133 y Fm(kh)p Fn(x)p Fm(i)668 4103 y Fk(\016)r(=)p Fl(2)772 4133 y Fn(R)q(F)889 4145 y Fl(1)926 4133 y Fo(\()p Fn(\025)p Fo(\))1038 4103 y Fi(\003)1077 4133 y Fm(k)1119 4149 y Fi(L)p Fl(\()p Fk(L)1237 4132 y Fe(2)1269 4149 y Fl(\()p Fk(S)1339 4132 y Ff(n)p Fb(\000)p Fe(1)1453 4149 y Fl(\))p Fi(\010)p Fk(L)1577 4132 y Fe(2)1608 4149 y Fl(\()p Fk(S)1678 4132 y Ff(n)p Fb(\000)p Fe(1)1792 4149 y Fl(\))p Fk(;)p Fi(H)p Fl(\))1925 4133 y Fo(+)p Fm(k)p Fn(F)2085 4145 y Fl(1)2121 4133 y Fo(\()p Fn(\025)p Fo(\))p Fn(R)q Fm(h)p Fn(x)p Fm(i)2408 4103 y Fk(\016)r(=)p Fl(2)2514 4133 y Fm(k)2556 4149 y Fi(L)p Fl(\()p Fi(H)p Fk(;L)2751 4132 y Fe(2)2782 4149 y Fl(\()p Fk(S)2852 4132 y Ff(n)p Fb(\000)p Fe(1)2966 4149 y Fl(\))p Fi(\010)p Fk(L)3090 4132 y Fe(2)3121 4149 y Fl(\()p Fk(S)3191 4132 y Ff(n)p Fb(\000)p Fe(1)3305 4149 y Fl(\)\))3384 4133 y Fo(=)515 4245 y Fm(O)r Fo(\()p Fn(e)654 4215 y Fi(\000)p Fk(c=h)813 4245 y Fo(\))28 b(and)f(using)g(the)h(resolv)n(en)n(t)e(estimates)821 4411 y Fm(kh)p Fn(x)p Fm(i)974 4376 y Fi(\000)p Fk(\016)r(=)p Fl(2)1130 4411 y Fo(\()p Fn(D)21 b Fm(\000)d Fn(\025)h Fm(\000)f Fn(i)p Fo(0\))1588 4376 y Fi(\000)p Fl(1)1676 4411 y Fm(h)p Fn(x)p Fm(i)1787 4376 y Fi(\000)p Fk(\016)r(=)p Fl(2)1944 4411 y Fm(k)1986 4426 y Fi(L)p Fl(\()p Fi(H)p Fl(\))2167 4411 y Fo(=)23 b Fm(O)r Fo(\()p Fn(h)2403 4376 y Fi(\000)p Fl(1)2492 4411 y Fo(\))p Fn(;)98 b(\025)23 b Fm(2)h Fn(I)7 b(:)306 b Fo(\(4.21\))515 4576 y(and)27 b(\(4.20\))g(w)n(e)g(can)g(conclude)h(that)1038 4741 y Fm(k)p Fn(S)5 b Fo(\()p Fn(D)r(;)1294 4720 y Fo(~)1276 4741 y Fn(P)11 b(;)j(\025)p Fo(\))19 b Fm(\000)f Fj(1)1607 4753 y Fl(2)1644 4741 y Fm(k)1686 4757 y Fi(L)p Fl(\()p Fk(L)1804 4740 y Fe(2)1836 4757 y Fl(\()p Fk(S)1906 4740 y Ff(n)p Fb(\000)p Fe(1)2019 4757 y Fl(\))p Fi(\010)p Fk(L)2143 4740 y Fe(2)2175 4757 y Fl(\()p Fk(S)2245 4740 y Ff(n)p Fb(\000)p Fe(1)2359 4757 y Fl(\)\))2438 4741 y Fo(=)23 b Fm(O)r Fo(\()p Fn(e)2665 4707 y Fi(\000)p Fk(c=h)2824 4741 y Fo(\))311 b(\(4.22\))515 4907 y(uniformly)27 b(with)h(resp)r(ect)g(to)f Fn(h)h Fo(and)f(then,)h(b)n(y)g(\(4.19\),)e (Theorem)h(4.4)g(follo)n(ws.)p 3337 4919 34 63 v 1905 5255 a(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fp(References)515 705 y Fo([Ba])240 b Fa(Balazard-K)n(onlein)p Fo(,)p Fd(Calcul)22 b(fonctionel)j(p)l(our)e (des)g(op)n(\023)-40 b(er)l(ateurs)24 b(h-)f(admis-)902 805 y(sibles)28 b(\022)-42 b(a)28 b(symb)l(ole)g(op)n(\023)-40 b(er)l(ateur)28 b(et)f(applic)l(ations)p Fo(,)i(Th)n(\022)-39 b(ese)23 b(de)j(3)2821 774 y Fl(0)2883 805 y Fo(cycle,)f(Univ)n(er-)902 904 y(sit)n(\023)-39 b(e)26 b(de)i(Nan)n(tes,)f(1985.)515 1070 y([BeMa])127 b Fa(M.)26 b(Bencha)n(ou,)i(A.Mar)-6 b(tinez)p Fo(,)24 b Fd(Estimations)j(exp)l(onentiel)t(les)g(en)f(th)n (\023)-40 b(eorie)902 1170 y(de)37 b(la)g(di\013usion)g(p)l(our)g(des)g (op)n(\023)-40 b(er)l(ateurs)37 b(de)g(Schr\177)-42 b(odinger)38 b(matriciels)e Fo(Ann.)902 1269 y(Inst.)27 b(H.)h(P)n(oincar)1484 1252 y(\023)1484 1269 y(71)d(\(6\),)j(p.561-594)c(\(1999\).)515 1435 y([BrNo])145 b Fa(R.)45 b(Br)n(ummelhuis,J.)f(Nourriga)-6 b(t)p Fo(,)44 b Fd(Sc)l(attering)e(amplitude)h(for)f(Dir)l(ac)902 1535 y(op)l(er)l(ators)p Fo(,)32 b(Comm)n(un.)f(P)n(artial)e(Di\013er.) i(Equations)f(24,)h(No.1-2,)f(p.377-394)902 1635 y(\(1999\))515 1801 y([BoKr])142 b Fa(L.)23 b(Boutet)h(de)g(Monvel,)h(P.)e(Kr)2077 1794 y(\023)2077 1801 y(ee)p Fo(,)p Fd(Pseudo-di\013er)l(ential)j(op)l (er)l(ators)e(and)902 1900 y(Gevr)l(ey)30 b(classes)p Fo(,)f(Ann.)f(Inst.)g(F)-7 b(ourier,17)26 b(\(1\))h(p.295-323)e (\(1967\).)515 2066 y([DiSj])184 b Fa(M.Dimassi,)59 b(J.Sj)1568 2059 y(\177)1565 2066 y(ostrand)p Fo(,)110 b(Sp)r(ectral)49 b(Asymptotics)h(in)g(the)g(Semi-)902 2166 y(Classical)24 b(Limit,)j(London)e(Math.)h(So)r(c.)f(Lecture)h(Notes)f(Series)g(268)f (\(1999\).)515 2332 y([JeNa])157 b Fa(A.)24 b(Jensen,)j(S.)e(Nakamura)p Fo(,)44 b Fd(Mapping)27 b(Pr)l(op)l(erties)f(for)f(Wave)g(and)h(Sc)l (at-)902 2432 y(tering)35 b(Op)l(er)l(ators)g(for)h(Two-Bo)l(dy)h (Schr\177)-42 b(odinger)37 b(Op)l(er)l(ators)p Fo(,)e(Lett.)f(Math.)902 2531 y(Ph)n(ys.)26 b(24,)h(295-305)d(\(1992\))515 2697 y([HeSj])171 b Fa(B.Helffer,J.Sj)1524 2690 y(\177)1521 2697 y(ostrand)20 b Fo(,)j Fd(A)n(nalyse)h(semiclassique)h(p)l(our)f (l'e)l(quation)h(de)902 2797 y(Harp)l(er)30 b(II)p Fo(,)e(Mem.)g(So)r (c.)f(Math.)h(F)-7 b(r.,)27 b(Nouv.)h(Ser.)f(40,)g(139)f(p.)i (\(1990\).)515 2963 y([Ka])234 b Fa(T.Ka)-6 b(to)p Fo(,)36 b Fd(Perturb)l(ation)g(the)l(ory)g(for)h(line)l(ar)g(op)l(er)l(ators)p Fo(,)g(2nd)d(ed.)g(Classics)902 3063 y(in)27 b(Mathematics.)h(Berlin:) 36 b(Springer-V)-7 b(erlag)25 b(\(1980\))515 3229 y([MaSo])135 b Fa(A.)144 b(Mar)-6 b(tinez,V.Sordoni,)p Fd(On)126 b(the)k(time-dep)l (endent)g(Born-)902 3328 y(Opp)l(enheimer)77 b(Appr)l(oximation)g(with) g(Smo)l(oth)f(Potential)p Fo(,)91 b(Preprin)n(t)902 3428 y(Univ)n(ersit\022)-42 b(a)26 b(degli)i(studi)g(di)f(Bologna)f(\(4\))i (\(2001\).)515 3594 y([MNS])157 b Fa(A.)25 b(Mar)-6 b(tinez,)27 b(S.)e(Nakamura,)g(V.Sordoni)p Fo(,)45 b Fd(Phase)26 b(Sp)l(ac)l(e)f(T)-6 b(unneling)902 3694 y(and)33 b(Multistate)h(Sc)l (attering)p Fo(,)e(Preprin)n(t)e(\(2001\))g(Dipartimen)n(to)i(di)f (Matem-)902 3793 y(atica,)c(Univ)n(ersit\022)-42 b(a)26 b(degli)i(studi)g(di)f(Bologna,)515 3959 y([Na])237 b Fa(S.)23 b(Nakamura)p Fo(,)f Fd(T)-6 b(unneling)23 b(e\013e)l(cts)h(in) g(momentum)f(sp)l(ac)l(e)h(and)h(sc)l(attering.)902 4059 y Fo(M.)36 b(Ik)-5 b(a)n(w)n(a)34 b(\(ed.\),)39 b(Sp)r(ectral)c(and)h (scattering)f(theory)-7 b(.)35 b(Pro)r(ceedings)f(of)i(the)902 4159 y(T)-7 b(aniguc)n(hi)26 b(in)n(ternational)f(w)n(orkshop,)h(Lect.) h(Notes)f(Pure)g(Appl.)i(Math.)f(M.)902 4258 y(Dekk)n(er,)f(161,)h (131-151)d(\(1994\))515 4424 y([Ne])242 b Fa(G.Nenciu)38 b Fd(Line)l(ar)j(A)l(diab)l(atic)g(The)l(ory.)h(Exp)l(onential)f (Estimates)p Fo(,)i(Com-)902 4524 y(m)n(un.)27 b(Math.)h(Ph)n(ys.)f (152,)f(No.3,)h(p.479-496)e(\(1993\).)515 4690 y([NeSo])154 b Fa(V.Sordoni,)44 b(G.Nenciu)p Fo(,)80 b Fd(Semiclassic)l(al)43 b(limit)e(for)g(multistate)f(Klein-)902 4790 y(Gor)l(don)g(systems:)59 b(almist)40 b(invariant)g(subsp)l(ac)l(es)g(and)h(sc)l(attering)e(the)l (ory)p Fo(,)902 4889 y(Preprin)n(t)27 b(\(2001\))f(Dipartimen)n(to)i (di)h(Matematica,)e(Univ)n(ersit\022)-42 b(a)27 b(degli)h(Studi)902 4989 y(di)f(Bologna.)1905 5255 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fo([Sj])270 b Fa(J.)44 b(Sj)1109 516 y(\177)1106 523 y(ostrand)c Fd(Pr)l(oje)l(cteurs)h(adiab)l(atique)j (du)d(p)l(oint)h(de)g(vue)f(pseudo)l(dif-)902 623 y(fer)n(\023)-40 b(entiel)p Fo(,)28 b(C.R.Acad.Sci.)g(P)n(aris,)e(t.)i(317,)e(S)n(\023) -39 b(erie)26 b(I,)i(p.)g(217-220)c(\(1993\).)515 789 y([T)-7 b(a])246 b Fa(M.E.T)-7 b(a)h(ylor)p Fo(,)27 b(Pseudo)r (di\013eren)n(tial)h(op)r(erators)f(Princeton)h(Mathematical)902 888 y(Series,)39 b(34.)e(Princeton,)i(New)f(Jersey:)56 b(Princeton)37 b(Univ)n(ersit)n(y)f(Press.)h(XI)902 988 y(\(1981\).)1905 5255 y(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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