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6975 2655 l gs col0 s gr % Polyline 7.500 slw [60 30 15 30] 0 sd n 2610 3015 m 2610 3915 l gs col0 s gr [] 0 sd % Polyline 30.000 slw n 4815 3960 m 4817 3958 l 4822 3955 l 4829 3949 l 4840 3941 l 4854 3930 l 4870 3918 l 4888 3905 l 4907 3892 l 4926 3879 l 4945 3867 l 4964 3856 l 4983 3845 l 5003 3836 l 5022 3829 l 5040 3825 l 5062 3825 l 5074 3830 l 5076 3840 l 5071 3851 l 5062 3863 l 5054 3875 l 5049 3887 l 5051 3898 l 5063 3908 l 5085 3915 l 5106 3917 l 5128 3916 l 5150 3914 l 5172 3910 l 5194 3906 l 5216 3900 l 5237 3894 l 5257 3888 l 5275 3882 l 5290 3877 l 5301 3873 l 5307 3871 l 5310 3870 l gs col0 s gr /Palatino-Bold ff 225.00 scf sf 8235 4005 m gs 1 -1 sc (Im\(z\)=0) col0 sh gr /Times-Bold ff 480.00 scf sf 6795 2655 m gs 1 -1 sc (C) col0 sh gr /Times-Bold ff 360.00 scf sf 2205 4050 m gs 1 -1 sc (a) col0 sh gr /Times-Bold ff 360.00 scf sf 6480 4050 m gs 1 -1 sc (b) col0 sh gr /Times-Bold ff 225.00 scf sf 2385 4140 m gs 1 -1 sc (1) col0 sh gr /Times-Bold ff 225.00 scf sf 6705 4095 m gs 1 -1 sc (1) col0 sh gr /Times-Bold ff 390.00 scf sf 810 7335 m gs 1 -1 sc (J=[a , b ]) col0 sh gr /Times-Bold ff 225.00 scf sf 1530 7470 m gs 1 -1 sc (1) col0 sh gr /Times-Bold ff 225.00 scf sf 2025 7470 m gs 1 -1 sc (1) col0 sh gr /Times-Bold ff 240.00 scf sf 6210 4095 m gs 1 -1 sc (*) col0 sh gr /Times-Bold ff 240.00 scf sf 2655 4095 m gs 1 -1 sc (+) col0 sh gr /Times-Bold ff 360.00 scf sf 4950 8370 m gs 1 -1 sc (Fig. 1.) col0 sh gr /Symbol ff 240.00 scf sf 1440 7740 m gs 1 -1 sc (e) col0 sh gr /Times-Bold ff 240.00 scf sf 810 7740 m gs 1 -1 sc (*=b+i) col0 sh gr /Symbol ff 240.00 scf sf 1440 8100 m gs 1 -1 sc (e, e ) col0 sh gr /Times-Bold ff 240.00 scf sf 1890 8145 m gs 1 -1 sc (>0.) col0 sh gr /Times-Bold ff 240.00 scf sf 675 8100 m gs 1 -1 sc ( +=a+i) col0 sh gr /Times-Bold ff 390.00 scf sf 4230 3960 m gs 1 -1 sc (I) col0 sh gr /Symbol ff 270.00 scf sf 3600 3870 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 180.00 scf sf 3735 4005 m gs 1 -1 sc ( 0) col0 sh gr /Symbol ff 600.00 scf sf 720 3375 m gs 1 -1 sc ( W) col-1 sh gr /Symbol ff 600.00 scf sf 3960 4725 m gs 1 -1 sc ( ) col-1 sh gr /Symbol ff 600.00 scf sf 4590 6975 m gs 1 -1 sc ( W) col-1 sh gr /Symbol ff 600.00 scf sf 5580 6975 m gs 1 -1 sc ( W) col-1 sh gr /Symbol ff 600.00 scf sf 3915 4455 m gs 1 -1 sc ( W) col-1 sh gr $F2psEnd rs %%EndDocument endTexFig 189 2255 a Fp(Our)27 b(main)g(result)h(is)f(:)189 2431 y Fq(Theorem)j(1.5.)189 2557 y Fc(Let)19 b Ff(\025)377 2569 y Fm(0)414 2557 y Ff(;)14 b Fp(\012)p Ff(;)555 2536 y Fg(e)548 2557 y Fp(\012)p Ff(;)g(I)7 b(;)14 b(J)27 b Fc(b)r(e)19 b(as)e(ab)r(o)n(v)n(e)h(and)g Ff(W)31 b Fc(satis\014es)17 b Fq(\(H1\))p Fc(.)34 b(Then)19 b(for)f Ff(h)g Fc(su\016cien)n(tly)h(small)f(the)h(follo)n(wing)189 2681 y(holds)31 b(:)44 b(Let)31 b Ff(g)h Fp(=)d Ff(g)s Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))30 b Fc(b)r(e)i(holomorphic)e(for)g Ff(z)j Fe(2)c Fp(\012)i Fc(and)g(satis\014es)g Fe(j)p Ff(g)s Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)28 b(\024)h Fp(1)h Fc(for)h Ff(z)h Fe(2)e Fp(\012)21 b Fe(n)3420 2660 y Fg(e)3413 2681 y Fp(\012)p Ff(:)189 2806 y Fc(Let)27 b Ff(\037)c Fe(2)g Ff(C)555 2776 y Fj(1)549 2826 y Fm(0)626 2806 y Fp(\()p Ff(J)8 b Fp(\))p Ff(;)27 b Fc(b)r(e)h(indep)r(enden)n(t) f(of)g Ff(h)g Fc(with)g Ff(\037)c Fe(\021)g Fp(1)j Fc(near)p 2148 2739 44 4 v 26 w Ff(I)7 b(:)27 b Fc(Then,)g(uniformly)g(with)g (resp)r(ect)g(to)f Ff(g)s(;)787 3032 y Fp(tr)852 2965 y Fg(\002)886 3032 y Fp(\()p Ff(\037g)s Fp(\)\()p Ff(P)1130 3044 y Fm(1)1168 3032 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Fp(\()p Ff(\037g)s Fp(\)\()p Ff(P)1658 3044 y Fm(0)1696 3032 y Fp(\))1728 2965 y Fg(\003)1786 3032 y Fp(=)2085 2953 y Fg(X)1874 3155 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)2125 3163 y Fk(1)2157 3155 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)2320 3151 y Fg(e)2319 3155 y Fm(\012)2366 3163 y Fi(\000)2346 3032 y Ff(g)s Fp(\()p Ff(z)t Fp(\))g(+)g Fe(O)r Fp(\()p Ff(h)2745 2998 y Fj(\000)p Fl(n)2843 3032 y Fp(\))p Ff(;)427 b Fp(\(1)p Ff(:)p Fp(9\))189 3370 y Fc(where)436 3349 y Fg(e)429 3370 y Fp(\012)489 3382 y Fj(\000)568 3370 y Fp(=)663 3349 y Fg(e)656 3370 y Fp(\012)18 b Fe(\\)h(f)p Ff(w)25 b Fe(2)f Fd(C)15 b Fp(;)47 b Fe(=)p Fp(\()p Ff(w)r Fp(\))24 b Fe(\024)f Fp(0)p Fe(g)p Ff(:)355 3546 y Fp(In)i(the)g(follo) n(wing)f(Theorem,)h(w)n(e)f(giv)n(e)g(a)g(lo)n(w)n(er)f(b)r(ound)j(for) e(the)h(n)n(um)n(b)r(er)g(of)f(resonances)f(of)i Ff(P)3346 3558 y Fm(1)3384 3546 y Fp(\()p Ff(h)p Fp(\))189 3671 y(near)i(an)n(y)g(p)r(oin)n(t)g(of)h(the)g(analytic)f(singular)f(supp)r (ort)i(of)42 b Fg(e)-61 b Ff(w)s(:)189 3796 y Fq(Theorem)30 b(1.6.)189 3921 y Fc(Let)e Fp(\012)h Fc(b)r(e)f(as)g(ab)r(o)n(v)n(e)f (and)h(let)h Ff(E)1223 3933 y Fm(0)1285 3921 y Fe(2)24 b Fp(\012)19 b Fe(\\)g Fp(singsupp)1832 3941 y Fl(a)1872 3921 y Fp(\()c Fg(e)-61 b Ff(w)s Fp(\))p Ff(:)29 b Fc(Then)f(for)g(ev)n (ery)f Ff(h)p Fe(\000)p Fc(indep)r(enden)n(t)i(complex)189 4046 y(neigh)n(b)r(orho)r(o)r(d)716 4025 y Fg(e)709 4046 y Fp(\012)d Fc(of)g Ff(E)949 4058 y Fm(0)986 4046 y Ff(;)g Fc(there)g(exist)g Ff(h)1489 4058 y Fm(0)1549 4046 y Fp(=)d Ff(h)p Fp(\()1724 4025 y Fg(e)1717 4046 y Fp(\012\))g Ff(>)g Fp(0)i Fc(su\016cien)n(tly)h(small)g(and)g Ff(C)j Fp(=)23 b Ff(C)6 b Fp(\()3057 4025 y Fg(e)3050 4046 y Fp(\012\))23 b Ff(>)g Fp(0)j Fc(large)189 4170 y(enough)h(suc)n(h)g (that)h(for)f Ff(h)c Fe(2)p Fp(]0)p Ff(;)14 b(h)1245 4182 y Fm(0)1282 4170 y Fp([)p Ff(;)1224 4410 y Fp(#)1293 4343 y Fg(\002)1328 4410 y Fp(Res)o(\()p Ff(P)1543 4422 y Fm(1)1581 4410 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\\)1825 4390 y Fg(e)1818 4410 y Fp(\012)1878 4343 y Fg(\003)1936 4410 y Fe(\025)2108 4354 y Fp(1)p 2033 4391 190 4 v 2033 4482 a Ff(C)6 b Fp(\()2137 4461 y Fg(e)2130 4482 y Fp(\012)q(\))2252 4410 y Fe(\001)18 b Ff(h)2341 4376 y Fj(\000)p Fl(n)2438 4410 y Ff(:)823 b Fp(\(1)p Ff(:)p Fp(10\))189 4675 y Fc(Here)27 b Fp(singsupp)700 4696 y Fl(a)740 4675 y Fp(\()15 b Fg(e)-61 b Ff(w)s Fp(\))28 b Fc(denote)f(the)h(analytic)f(singular)g (supp)r(ort)g(of)h(the)g(distribution)42 b Fg(e)-61 b Ff(w)s(:)189 4826 y Fq(Remark)46 b(1.7.)76 b Fp(Note)41 b(that)h(if)f Ff(\025)1338 4838 y Fm(0)1417 4826 y Fp(is)g(a)g(simple)g (non-critical)f(energy)g(of)h Ff(P)2761 4838 y Fm(0)2840 4826 y Fp(then)g(the)h(function)189 4951 y Ff(\025)24 b Fe(7!)g Ff(w)r Fp(\()p Ff(\025)p Fp(\))30 b(is)e(analytic)f(near)g Ff(\025)1204 4963 y Fm(0)1242 4951 y Ff(:)h Fp(See)g([12,)f(Theorem)h (4.4].)37 b(Consequen)n(tly)-7 b(,)27 b(for)h Ff(R)h Fp(large)d(enough)i(the)189 5093 y(function)512 4980 y Fg(Z)558 5168 y Fj(j)p Fl(x)p Fj(j)p Fl(>R)755 5000 y Fg(h)794 5093 y Ff(w)r Fp(\()p Ff(s)13 b Fe(\000)g Ff(W)f Fp(\()p Ff(x)p Fp(\)\))h Fe(\000)g Ff(w)r Fp(\()p Ff(s)p Fp(\))1505 5000 y Fg(i)1562 5093 y Ff(dx)26 b Fp(is)f(analytic)f(near)g Ff(\025)2302 5105 y Fm(0)2340 5093 y Ff(;)h Fp(since)g Ff(W)12 b Fp(\()p Ff(x)p Fp(\))26 b(tends)f(to)g(zero)f(when)1822 5349 y(5)p eop %%Page: 6 6 6 5 bop 189 272 a Fe(j)p Ff(x)p Fe(j)28 b Fp(tends)g(to)g(in\014nit)n (y)-7 b(.)37 b(Therefore,)26 b(for)h Ff(E)i Fe(2)23 b Fd(R)34 b Fp(and)27 b Ff(R)d(>)e 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5349 y Fp(7)p eop %%Page: 8 8 8 7 bop 189 291 a Fq(Pro)s(of.)36 b Fp(Since)701 199 y Fg(\020)751 291 y Ff(E)812 303 y Fm(0)p Fl(;)p Fj(\000)921 291 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(W)e Fp(\()p Ff(x)p Fp(\))1314 199 y Fg(\021)1364 216 y Fj(\003)1425 291 y Fp(=)23 b Ff(W)12 b Fp(\()p Ff(x)p Fp(\))p Ff(E)1775 303 y Fm(0)p Fl(;)p Fm(+)1884 291 y Fp(\()p 1916 245 43 4 v Ff(z)s(;)i(h)p Fp(\))28 b(it)g(su\016ce)f(to)h(pro)n(v)n(e)e (\(2.13\).)36 b(Without)28 b(an)n(y)189 416 y(loss)f(of)g(generalit)n (y)-7 b(,)26 b(w)n(e)i(ma)n(y)f(assume)f(that)i Ff(N)k Fp(=)23 b(1.)189 565 y(Consider)39 b(the)i(op)r(erator)e Ff(H)50 b Fp(=)1280 498 y Fg(\000)1318 565 y Fp(Id)27 b Fe(\000)g Ff(h)1561 535 y Fm(2)1598 565 y Fp(\001)1667 577 y Fb(T)1712 561 y Fi(\003)1745 498 y Fg(\001)1793 488 y Fi(\000)p Fh(n)p Fi(\000)p Fh(\017)p 1793 502 152 4 v 1855 535 a Fk(2)1999 565 y Fp(on)40 b Ff(L)2184 535 y Fm(2)2221 565 y Fp(\()p Fd(T)2309 535 y Fj(\003)2347 565 y Fp(;)14 b Fd(C)h Fp(\))p Ff(:)46 b Fp(Set)41 b Ff(A)j Fp(=)g Ff(W)12 b Fp(\()p Ff(x)p Fp(\))p Ff(E)3172 577 y Fm(0)p Fl(;)p Fm(+)3281 565 y Fp(\()p Ff(z)t(;)i(h)p Fp(\),)189 689 y Ff(B)27 b Fp(=)c Ff(A)429 659 y Fj(\003)467 689 y Ff(A)28 b Fp(and)g Ff(C)h Fp(=)23 b Ff(H)971 659 y Fj(\000)p Fm(1)1059 689 y Ff(B)t(H)1202 659 y Fj(\000)p Fm(1)1292 689 y Ff(:)189 814 y Fp(Since)37 b Ff(W)50 b Fe(2)38 b Ff(S)692 784 y Fm(0)729 747 y Fg(\000)767 814 y Fd(R)821 784 y Fm(2)p Fl(n)905 814 y Ff(;)14 b(<)37 b(x)i(>)1195 784 y Fj(\000)p Fl(n)p Fj(\000)p Fl(\017)1409 747 y Fg(\001)1484 814 y Fp(and)d Ff(E)1715 826 y Fm(0)p Fl(;)p Fm(+)1823 814 y Fp(\()p Ff(k)s(;)14 b(x)p Fp(;)g Ff(z)t(;)g(h)p Fp(\))38 b Fe(2)g Ff(S)2369 784 y Fm(0)2406 747 y Fg(\000)2444 814 y Fd(T)2500 784 y Fj(\003)2500 838 y Fl(k)2565 814 y Fe(\002)24 b Fd(R)2708 784 y Fl(n)2708 835 y(x)2759 747 y Fg(\001)2797 814 y Fp(,)39 b(a)d(standard)f(result) 189 939 y(of)f Ff(h)p Fp(-pseudo-di\013eren)n(tial)e(calculus)i(sho)n (ws)f(that)h Ff(C)40 b Fe(2)35 b Ff(S)2061 908 y Fm(0)2098 939 y Fp(\()p Fd(T)2186 908 y Fj(\003)2186 962 y Fl(k)2249 939 y Fe(\002)22 b Fd(R)2390 908 y Fl(n)2390 959 y(x)2441 939 y Fp(\))p Ff(:)35 b Fp(Therefore,)f Ff(C)41 b Fp(extends)34 b(to)g(a)189 1063 y(b)r(ounded)j(op)r(erator)d(from)i Ff(L)1142 1033 y Fm(2)1179 1063 y Fp(\()p Fd(T)1267 1033 y Fj(\003)1305 1063 y Fp(;)14 b Fd(C)h Fp(\))42 b(in)n(to)36 b Ff(L)1704 1033 y Fm(2)1741 1063 y Fp(\()p Fd(T)1829 1033 y Fj(\003)1867 1063 y Fp(;)14 b Fd(C)h Fp(\))42 b(\(see)36 b([11,)i(Chapter)e(13]\).)62 b(Com)n(bining)36 b(this)189 1188 y(with)28 b(the)g(fact)g(that)g Ff(B)j Fp(is)d(p)r(ositif,)g(w)n(e)f(get)h(:)1376 1412 y(0)23 b Fe(\024)f Ff(B)28 b Fp(=)22 b Ff(H)7 b(C)f(H)30 b Fe(\024)23 b(k)p Ff(C)6 b Fe(k)14 b Ff(H)2273 1377 y Fm(2)189 1636 y Fp(whic)n(h)28 b(implies)1482 1760 y(0)22 b Fe(\024)h Ff(B)1711 1704 y Fk(1)p 1711 1713 29 4 v 1711 1746 a(2)1776 1760 y Fe(\024)1864 1685 y Fg(p)p 1947 1685 149 4 v 75 x Fe(k)p Ff(C)6 b Fe(k)13 b Ff(H)r(:)189 1943 y Fp(Since)35 b Ff(H)41 b Fp(:)35 b Ff(L)638 1913 y Fm(2)675 1943 y Fp(\()p Fd(T)763 1913 y Fj(\003)801 1943 y Fp(;)14 b Fd(C)h Fp(\))41 b Fe(!)35 b Ff(L)1140 1913 y Fm(2)1176 1943 y Fp(\()p Fd(T)1264 1913 y Fj(\003)1302 1943 y Fp(;)14 b Fd(C)h Fp(\))41 b(is)34 b(of)h(trace)f(class,)h(the)g(lemma)g(follo)n (ws)f(from)g(the)h(ab)r(o)n(v)n(e)e(in-)189 2068 y(equalit)n(y)-7 b(.)2937 b Fa(\003)189 2242 y Fq(Prop)s(osition)30 b(2.5.)36 b Fc(F)-7 b(or)27 b Ff(z)f Fe(2)d Fd(C)49 b Fc(suc)n(h)27 b(that)h Fe(=)p Fp(\()p Ff(z)t Fp(\))23 b Fe(6)p Fp(=)g(0)p Ff(;)k Fc(the)h(op)r(erator)1140 2391 y Fg(h)1180 2483 y Ff(E)1241 2495 y Fl(j;)p Fm(+)1343 2483 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(E)1596 2495 y Fl(j;)p Fj(\000)p Fm(+)1750 2483 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1942 2449 y Fj(\000)p Fm(1)2031 2483 y Ff(E)2092 2495 y Fl(j;)p Fj(\000)2195 2483 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2387 2391 y Fg(i)2426 2408 y Fm(1)2426 2541 y Fl(j)s Fm(=0)189 2728 y Fc(is)27 b(of)h(trace)f(class)f(from)i Ff(L)1023 2698 y Fm(2)1059 2728 y Fp(\()p Fd(R)1146 2698 y Fl(n)1197 2728 y Fp(\))g Fc(to)f Ff(L)1415 2698 y Fm(2)1452 2728 y Fp(\()p Fd(R)1538 2698 y Fl(n)1589 2728 y Fp(\))h Fc(and)318 2969 y Fp(tr)383 2877 y Fg(\020h)472 2969 y Ff(E)533 2981 y Fl(j;)p Fm(+)635 2969 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(E)888 2981 y Fl(j;)p Fj(\000)p Fm(+)1042 2969 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1234 2935 y Fj(\000)p Fm(1)1323 2969 y Ff(E)1384 2981 y Fl(j;)p Fj(\000)1487 2969 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1679 2877 y Fg(i)1718 2894 y Fm(1)1718 3027 y Fl(j)s Fm(=0)1837 2877 y Fg(\021)1910 2969 y Fp(=)22 b(tr)2062 2877 y Fg(\020h)2151 2969 y Ff(E)2212 2981 y Fl(j;)p Fj(\000)p Fm(+)2366 2969 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2558 2935 y Fj(\000)p Fm(1)2647 2969 y Ff(@)2691 2981 y Fl(z)2729 2969 y Ff(E)2790 2981 y Fl(j;)p Fj(\000)p Fm(+)2944 2969 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))3136 2877 y Fg(i)3175 2894 y Fm(1)3175 3027 y Fl(j)s Fm(=0)3294 2877 y Fg(\021)3344 2969 y Ff(:)3284 3109 y Fp(\(2)p Ff(:)p Fp(15\))189 3234 y Fc(Here)27 b(the)h(op)r(erator)e(in)i(the)g (righ)n(t)f(mem)n(b)r(er)g(of)h Fp(\(2.15\))f Fc(is)g(de\014ned)h(on)f Ff(L)2498 3203 y Fm(2)2535 3234 y Fp(\()p Fd(T)2623 3203 y Fj(\003)2661 3234 y Fp(;)14 b Fd(C)2752 3203 y Fl(N)2821 3234 y Fp(\))p Ff(:)189 3408 y Fq(Pro)s(of.)36 b Fp(Let)28 b Ff(z)e Fe(2)e Fd(C)48 b Fp(s.t.)37 b Fe(=)p Fp(\()p Ff(z)t Fp(\))23 b Fe(6)p Fp(=)g(0)p Ff(;)k Fp(w)n(e)g(ha)n(v)n(e)g(the) h(follo)n(wing)e(iden)n(tit)n(y)i(:)1097 3557 y Fg(h)1136 3649 y Ff(E)1197 3661 y Fl(j;)p Fm(+)1299 3649 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(E)1552 3661 y Fl(j;)p Fj(\000)p Fm(+)1706 3649 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1898 3615 y Fj(\000)p Fm(1)1987 3649 y Ff(E)2048 3661 y Fl(j;)p Fj(\000)2151 3649 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2343 3557 y Fg(i)2382 3574 y Fm(1)2382 3707 y Fl(j)s Fm(=0)2524 3649 y Fp(=)695 b(\(2)p Ff(:)p Fp(16\))1041 3830 y Fg(h)1080 3855 y(\000)1118 3922 y Fp([)p Ff(E)1202 3934 y Fl(j;)p Fm(+)1305 3922 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\)])1520 3888 y Fm(1)1520 3942 y Fl(j)s Fm(=0)1639 3855 y Fg(\001)1677 3922 y Ff(E)1738 3934 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1898 3922 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2090 3888 y Fj(\000)p Fm(1)2178 3922 y Ff(E)2239 3934 y Fm(1)p Fl(;)p Fj(\000)2349 3922 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2541 3830 y Fg(i)2579 3922 y Fp(+)1041 4046 y Fg(h)1080 4138 y Ff(E)1141 4150 y Fm(0)p Fl(;)p Fm(+)1250 4138 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(E)1503 4150 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1662 4138 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1854 4103 y Fj(\000)p Fm(1)1943 4070 y Fg(\000)1981 4138 y Fp([)p Ff(E)2065 4150 y Fl(j;)p Fj(\000)2168 4138 y Fp(\()p Ff(z)t(;)g(h)p Fp(\)])2383 4103 y Fm(1)2383 4158 y Fl(j)s Fm(=0)2502 4070 y Fg(\001)2540 4046 y(i)2579 4138 y Fe(\000)605 4261 y Fg(h)644 4354 y Ff(E)705 4366 y Fm(0)p Fl(;)p Fm(+)813 4354 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(E)1066 4366 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1226 4354 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1418 4319 y Fj(\000)p Fm(1)1506 4286 y Fg(\000)1545 4354 y Fp([)p Ff(E)1629 4366 y Fl(j;)p Fj(\000)p Fm(+)1783 4354 y Fp(\()p Ff(z)t(;)g(h)p Fp(\)])1998 4319 y Fm(1)1998 4374 y Fl(j)s Fm(=0)2117 4286 y Fg(\001)2155 4354 y Ff(E)2216 4366 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2376 4354 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2568 4319 y Fj(\000)p Fm(1)2656 4354 y Ff(E)2717 4366 y Fm(1)p Fl(;)p Fj(\000)2827 4354 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))3019 4261 y Fg(i)3057 4354 y Ff(:)189 4536 y Fp(According)29 b(to)g(Lemma)g(2.3)g(and)h(Prop) r(osition)e(2.4,)h(all)g(the)h(term)g(of)f(the)h(righ)n(t)f(mem)n(b)r (er)h(in)f(the)h(last)189 4661 y(equalit)n(y)c(are)g(of)h(trace)f (class.)36 b(Using)27 b(the)g(cyclicit)n(y)g(of)g(the)g(trace)g(and)f (iden)n(tit)n(y)i(\(2.8\),)e(w)n(e)h(obtain)g(the)189 4785 y(prop)r(osition.)3431 4910 y Fa(\003)1822 5349 y Fp(8)p eop %%Page: 9 9 9 8 bop 355 272 a Fp(The)28 b(main)f(result)h(in)f(this)h(section)f(is) 189 440 y Fq(Prop)s(osition)34 b(2.6.)47 b Fc(Let)32 b Ff( )g Fe(2)e Ff(C)1295 410 y Fj(1)1289 461 y Fm(0)1365 440 y Fp(\()p Fd(R)q Fp(\))37 b Fc(and)32 b(let)1825 418 y Fg(e)1810 440 y Ff( )j Fc(b)r(e)d(an)f(almost)f(analytic)h (extension)g(of)g Ff( )s(:)h Fc(Then)189 565 y(the)26 b(op)r(erator)f Fp([)p Ff( )s Fp(\()p Ff(P)829 577 y Fm(1)867 565 y Fp(\()p Ff(h)p Fp(\)\))15 b Fe(\000)g Ff( )s Fp(\()p Ff(P)1248 577 y Fm(0)1286 565 y Fp(\)])27 b Fc(is)e(of)h(trace)f(class)h(as)f(an)h(op)r(erator)e(from)h Ff(L)2740 535 y Fm(2)2777 565 y Fp(\()p Fd(R)2863 535 y Fl(n)2915 565 y Fp(\))h Fc(to)g Ff(L)3130 535 y Fm(2)3167 565 y Fp(\()p Fd(R)3253 535 y Fl(n)3304 565 y Fp(\))g Fc(and)315 761 y Fp(tr[)p Ff( )s Fp(\()p Ff(P)545 773 y Fm(1)583 761 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff( )s Fp(\()p Ff(P)971 773 y Fm(0)1009 761 y Fp(\)])24 b(=)e(tr[)p Ff( )s Fp(\()1369 740 y Fg(b)1352 761 y Ff(P)1405 773 y Fm(1)1443 761 y Fp(\()p Ff(h)p Fp(\)\))d Fe(\000)f Ff( )s Fp(\()1795 740 y Fg(b)1778 761 y Ff(P)1831 773 y Fm(0)1869 761 y Fp(\()p Ff(h)p Fp(\)\)])24 b(=)1159 b(\(2)p Ff(:)p Fp(17\))1579 975 y Fe(\000)1658 919 y Fp(1)p 1654 956 51 4 v 1654 1032 a Ff(\031)1728 862 y Fg(Z)1774 1051 y Fb(C)p 1834 907 49 4 v 1834 975 a Ff(@)1897 953 y Fg(e)1883 975 y Ff( )s Fp(\()p Ff(z)t Fp(\)tr)2111 883 y Fg(\020)2161 908 y(\002)2196 975 y Ff(E)2257 987 y Fl(j;)p Fj(\000)p Fm(+)2411 975 y Fp(\()p Ff(z)t Fp(\))2518 941 y Fj(\000)p Fm(1)2607 975 y Ff(@)2651 987 y Fl(z)2689 975 y Ff(E)2750 987 y Fl(j;)p Fj(\000)p Fm(+)2904 908 y Fg(\003)2939 925 y Fm(1)2939 1008 y Fl(j)s Fm(=0)3058 883 y Fg(\021)3121 975 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))p Ff(:)189 1218 y Fc(Here)p 385 1151 V 27 w Ff(@)28 b Fp(=)576 1162 y Ff(@)p 555 1199 91 4 v 555 1275 a(@)p 604 1230 43 4 v 5 w(z)655 1218 y(:)189 1387 y Fq(Pro)s(of.)36 b Fp(By)28 b(Hel\013er-Sj\177)-42 b(ostrand)26 b(form)n(ula)h(\(see)g ([15]\),)h(w)n(e)f(ha)n(v)n(e)466 1601 y Ff( )s Fp(\()572 1580 y Fg(b)555 1601 y Ff(P)608 1613 y Fm(1)646 1601 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff( )s Fp(\()998 1580 y Fg(b)981 1601 y Ff(P)1034 1613 y Fm(0)1072 1601 y Fp(\()p Ff(h)p Fp(\)\))24 b(=)e Fe(\000)1406 1544 y Fp(1)p 1402 1581 51 4 v 1402 1657 a Ff(\031)1476 1488 y Fg(Z)1522 1676 y Fb(C)p 1582 1533 49 4 v 1582 1601 a Ff(@)1646 1579 y Fg(e)1631 1601 y Ff( )s Fp(\()p Ff(z)t Fp(\))1795 1508 y Fg(h)1834 1601 y Fp(\()p Ff(z)g Fe(\000)2026 1580 y Fg(b)2010 1601 y Ff(P)2063 1613 y Fm(1)2101 1601 y Fp(\()p Ff(h)p Fp(\)\))2245 1566 y Fj(\000)p Fm(1)2353 1601 y Fe(\000)c Fp(\()p Ff(z)k Fe(\000)2628 1580 y Fg(b)2612 1601 y Ff(P)2665 1613 y Fm(0)2702 1601 y Fp(\()p Ff(h)p Fp(\)\))2846 1566 y Fj(\000)p Fm(1)2936 1508 y Fg(i)2989 1601 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))p Ff(:)189 1819 y Fp(Com)n(bining)27 b(this)h(with)g(\(2.6\),)f(w)n(e)h(obtain)868 2032 y Ff( )s Fp(\()974 2011 y Fg(b)957 2032 y Ff(P)1010 2044 y Fm(1)1048 2032 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff( )s Fp(\()1400 2011 y Fg(b)1383 2032 y Ff(P)1436 2044 y Fm(0)1474 2032 y Fp(\()p Ff(h)p Fp(\)\))24 b(=)1744 1976 y(1)p 1739 2013 51 4 v 1739 2089 a Ff(\031)1813 1919 y Fg(Z)1860 2108 y Fb(C)p 1920 1965 49 4 v 1920 2032 a Ff(@)1983 2011 y Fg(e)1968 2032 y Ff( )s Fp(\()p Ff(z)t Fp(\))2132 1965 y Fg(\002)2167 2032 y Ff(E)2228 2044 y Fl(j)2263 2032 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\)])2478 1998 y Fm(1)2478 2053 y Fl(j)s Fm(=0)2610 2032 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))467 b(\(2)p Ff(:)p Fp(18\))785 2301 y Fe(\000)864 2245 y Fp(1)p 860 2282 51 4 v 860 2358 a Ff(\031)934 2188 y Fg(Z)980 2377 y Fb(C)p 1040 2234 49 4 v 1040 2301 a Ff(@)1103 2279 y Fg(e)1088 2301 y Ff( )t Fp(\()p Ff(z)t Fp(\))1253 2209 y Fg(h)1292 2301 y Ff(E)1353 2313 y Fl(j;)p Fm(+)1455 2301 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(E)1708 2313 y Fl(j;)p Fj(\000)p Fm(+)1862 2301 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2054 2267 y Fj(\000)p Fm(1)2142 2301 y Ff(E)2203 2313 y Fl(j;)p Fj(\000)2307 2301 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2499 2209 y Fg(i)2538 2226 y Fm(1)2538 2359 y Fl(j)s Fm(=0)2670 2301 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))p Ff(:)189 2513 y Fp(Since)31 b Ff(E)470 2525 y Fl(j)505 2513 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(;)27 b(j)33 b Fp(=)27 b(0)p Ff(;)14 b Fp(1)29 b(is)i(holomorphic)e(in)h(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)i(supp\() 2632 2491 y Fg(e)2617 2513 y Ff( )s Fp(\))p Ff(;)g Fp(the)g(\014rst)f (term)g(in)h(the)189 2637 y(righ)n(t)c(mem)n(b)r(er)g(of)h(\(2.18\))f (v)-5 b(anishes.)36 b(Consequen)n(tly)-7 b(,)354 2851 y Ff( )s Fp(\()460 2830 y Fg(b)443 2851 y Ff(P)496 2863 y Fm(1)534 2851 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff( )s Fp(\()886 2830 y Fg(b)869 2851 y Ff(P)922 2863 y Fm(0)960 2851 y Fp(\()p Ff(h)p Fp(\)\))24 b(=)f Fe(\000)1295 2795 y Fp(1)p 1291 2832 51 4 v 1291 2908 a Ff(\031)1364 2738 y Fg(Z)1410 2927 y Fb(C)p 1470 2784 49 4 v 1470 2851 a Ff(@)1534 2829 y Fg(e)1519 2851 y Ff( )s Fp(\()p Ff(z)t Fp(\))1683 2759 y Fg(h)1722 2851 y Ff(E)1783 2863 y Fl(j;)p Fm(+)1886 2851 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(E)2139 2863 y Fl(j;)p Fj(\000)p Fm(+)2292 2851 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2484 2817 y Fj(\000)p Fm(1)2573 2851 y Ff(E)2634 2863 y Fl(j;)p Fj(\000)2737 2851 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2929 2759 y Fg(i)2968 2776 y Fm(1)2968 2909 y Fl(j)s Fm(=0)3101 2851 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))p Ff(:)189 3088 y Fp(Using)24 b(Prop)r(osition)f (2.5,)h(w)n(e)g(conclude)g(that)h([)p Ff( )s Fp(\()1779 3067 y Fg(b)1763 3088 y Ff(P)1816 3100 y Fm(1)1854 3088 y Fp(\()p Ff(h)p Fp(\)\))12 b Fe(\000)g Ff( )s Fp(\()2193 3067 y Fg(b)2176 3088 y Ff(P)2229 3100 y Fm(0)2267 3088 y Fp(\()p Ff(h)p Fp(\)\)])25 b(is)f(of)h(trace)e(class)h(and)g (applying)189 3213 y(\(2.15\),)i(w)n(e)g(obtain)g(the)h(second)f (equalit)n(y)g(of)g(\(2.17\).)36 b(The)27 b(\014rst)f(equalit)n(y)g (follo)n(ws)f(from)h(the)h(fact)g(that)205 3316 y Fg(b)189 3337 y Ff(P)242 3349 y Fl(j)277 3337 y Fp(\()p Ff(h)p Fp(\))h(is)g(unitary)f(equiv)-5 b(alen)n(t)27 b(to)h Ff(P)1343 3349 y Fl(j)1378 3337 y Fp(\()p Ff(h)p Fp(\))p Ff(;)g(j)g Fp(=)23 b(0)p Ff(;)14 b Fp(1)1619 b Fa(\003)189 3549 y Fq(3.)42 b(Pro)s(of)31 b(of)h(Theorem)f(1.3)355 3717 y Fp(The)37 b(pro)r(of)f(of)g(Theorem)g(1.3)g(is)h(a)f(simple)h (consequence)e(of)i(Prop)r(osition)e(2.6)h(\(with)h Ff( )k Fp(=)d Ff(f)9 b Fp(\))189 3842 y(and)38 b(sym)n(b)r(olic)g(calculus.)68 b(Here,)41 b(w)n(e)d(only)g(giv)n(e)f(an)h(outline)h(of)f(the)h(pro)r (of.)68 b(F)-7 b(or)38 b(the)h(details,)i(w)n(e)189 3966 y(refer)32 b(to)h([9].)53 b(Fix)33 b Ff(\016)k Fp(in)c(]0)p Ff(;)1098 3933 y Fm(1)p 1098 3947 34 4 v 1098 3995 a(2)1141 3966 y Fp([.)54 b(The)33 b(in)n(tegral)f(\(2.17\))g(o)n(v)n(er)f Fe(f)p Ff(z)k Fe(2)e Fd(C)15 b Fp(;)53 b Fe(j=)p Ff(z)t Fe(j)31 b(\024)h Ff(h)2812 3936 y Fl(\016)2848 3966 y Fe(g)h Fp(is)g Fe(O)r Fp(\()p Ff(h)3160 3936 y Fj(1)3230 3966 y Fp(\),)i(since)p 189 4023 49 4 v 189 4091 a Ff(@)253 4069 y Fg(e)238 4091 y Ff(f)8 b Fp(\()p Ff(z)t Fp(\))23 b(=)g Fe(O)r Fp(\()p Fe(j=)p Ff(z)t Fe(j)754 4060 y Fj(1)824 4091 y Fp(\))28 b(and)1004 4208 y Fg(\015)1004 4258 y(\015)1004 4308 y(\015)1050 4212 y(h)1089 4304 y Ff(E)1150 4316 y Fl(j;)p Fj(\000)p Fm(+)1305 4304 y Fp(\()p Ff(z)t Fp(\))1412 4270 y Fj(\000)p Fm(1)1501 4304 y Ff(@)1545 4316 y Fl(z)1583 4304 y Ff(E)1644 4316 y Fl(j;)p Fj(\000)p Fm(+)1798 4304 y Fp(\()p Ff(z)t Fp(\))1905 4212 y Fg(i)1944 4229 y Fm(1)1944 4362 y Fl(j)s Fm(=0)2063 4208 y Fg(\015)2063 4258 y(\015)2063 4308 y(\015)2109 4362 y Fm(tr)2188 4304 y Fp(=)23 b Fe(O)2344 4237 y Fg(\000)2382 4304 y Fe(j=)p Ff(z)t Fe(j)2531 4270 y Fj(\000)p Fm(1)2620 4237 y Fg(\001)2658 4304 y Ff(:)189 4566 y Fp(On)f(the)h(other)e(hand,)898 4474 y Fg(h)937 4566 y Ff(E)998 4578 y Fl(j;)p Fj(\000)p Fm(+)1152 4566 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1344 4536 y Fj(\000)p Fm(1)1433 4566 y Ff(@)1477 4578 y Fl(z)1515 4566 y Ff(E)1576 4578 y Fl(j;)p Fj(\000)p Fm(+)1730 4566 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1922 4474 y Fg(i)1961 4491 y Fm(1)1961 4624 y Fl(j)s Fm(=0)2102 4566 y Fp(has)22 b(an)g(asymptotic)g(expansion)f(in)i(p)r(o) n(w)n(ers)189 4725 y(of)32 b Ff(h)g Fp(uniformly)g(on)g Ff(z)j Fp(in)e Fe(f)p Ff(z)g Fe(2)e Fp(supp)1434 4704 y Fg(e)1418 4725 y Ff(f)9 b Fp(;)46 b Fe(j=)p Fp(\()p Ff(z)t Fp(\))p Fe(j)30 b(\025)g Ff(h)1923 4695 y Fl(\016)1960 4725 y Fe(g)h Fp(\(see)h([9]\).)51 b(Therefore,)32 b(as)f(in)i([11,)f (Theorem)189 4850 y(13.28],)26 b(w)n(e)h(ha)n(v)n(e)495 5064 y Fe(\000)574 5008 y Fp(1)p 570 5045 51 4 v 570 5121 a Ff(\031)644 4951 y Fg(Z)690 5140 y Fb(C)p 750 4996 49 4 v 750 5064 a Ff(@)814 5042 y Fg(e)799 5064 y Ff(f)8 b Fp(\()p Ff(z)t Fp(\)tr)1020 4972 y Fg(\020)1069 4997 y(\002)1104 5064 y Ff(E)1165 5076 y Fl(j;)p Fj(\000)p Fm(+)1319 5064 y Fp(\()p Ff(z)t Fp(\))1426 5030 y Fj(\000)p Fm(1)1515 5064 y Ff(@)1559 5076 y Fl(z)1598 5064 y Ff(E)1659 5076 y Fl(j;)p Fj(\000)p Fm(+)1813 5064 y Fp(\()p Ff(z)t Fp(\))1920 4997 y Fg(\003)1954 5014 y Fm(1)1954 5097 y Fl(j)s Fm(=0)2073 4972 y Fg(\021)2137 5064 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))22 b Fe(\030)2454 4985 y Fg(X)2456 5163 y Fl(j)s Fj(\025)p Fm(0)2588 5064 y Ff(a)2632 5076 y Fl(j)2667 5064 y Ff(h)2715 5030 y Fl(j)s Fj(\000)p Fl(n)2842 5064 y Ff(;)42 b Fp(\()p Ff(h)23 b Fe(&)g Fp(0\))1822 5349 y(9)p eop %%Page: 10 10 10 9 bop 189 272 a Fp(with)672 397 y Ff(a)716 409 y Fm(0)776 397 y Fp(=)23 b(\(2)p Ff(\031)s Fp(\))1020 362 y Fj(\000)p Fl(n)1131 318 y Fg(X)1172 496 y Fl(k)1265 284 y Fg(Z)1311 472 y Fb(R)1358 456 y Fh(n)1358 489 y(x)1411 304 y Fg(\020)1475 284 y(Z)1521 472 y Fl(E)1573 456 y Fi(\003)1626 329 y Fg(\002)1660 397 y Ff(f)1710 329 y Fg(\000)1748 397 y Ff(W)12 b Fp(\()p Ff(x)p Fp(\))19 b(+)f Ff(\025)2099 409 y Fl(k)2140 397 y Fp(\()p Ff(\030)t Fp(\))2244 329 y Fg(\001)2302 397 y Fe(\000)g Ff(f)2435 329 y Fg(\000)2472 397 y Ff(\025)2520 409 y Fl(k)2562 397 y Fp(\()p Ff(\030)t Fp(\))2666 329 y Fg(\001\003)2753 397 y Ff(d\030)2836 304 y Fg(\021)2900 397 y Ff(dx:)189 620 y Fp(Note)27 b(that,)g(the)h(sum)f(in)g(the)g(last)g(equalit)n(y)f(is)h(\014nite,)g (since)54 b(lim)2139 674 y Fl(k)q Fj(!1)2322 620 y Ff(\025)2370 632 y Fl(k)2411 620 y Fp(\()p Ff(\030)t Fp(\))24 b(=)f(+)p Fe(1)j Fp(and)h Ff(W)39 b Fp(is)27 b(b)r(ounded.)189 753 y(It)h(remains)f(only)g(to)g(sho)n(w)g(\(1.5\).)37 b(Clearly)376 965 y Ff(a)420 977 y Fm(0)480 965 y Fp(=)22 b(\(2)p Ff(\031)s Fp(\))723 931 y Fj(\000)p Fl(n)835 886 y Fg(X)876 1065 y Fl(k)968 852 y Fg(Z)1015 1041 y Fb(R)1062 1024 y Fh(n)1062 1057 y(x)1115 873 y Fg(\020)1179 852 y(Z)1225 1041 y Fl(E)1277 1024 y Fi(\003)1329 852 y Fg(Z)1376 1041 y Fb(R)1423 1049 y Fh(t)1463 898 y Fg(\002)1498 965 y Fp(1)1540 980 y Fm([)p Fl(\025)1598 989 y Fh(k)1634 980 y Fm(\()p Fl(\030)r Fm(\))p Fl(;)p Fm(+)p Fj(1)p Fm([)1878 965 y Fp(\()p Ff(t)p Fp(\))d Fe(\000)f Fp(1)2116 980 y Fm([)p Fl(\025)2174 989 y Fh(k)2210 980 y Fm(\()p Fl(\030)r Fm(\)+)p Fl(W)9 b Fm(\()p Fl(x)p Fm(\))p Fl(;)p Fm(+)p Fj(1)p Fm([)2665 965 y Fp(\()p Ff(t)p Fp(\))2759 898 y Fg(\003)2794 965 y Ff(f)2844 931 y Fj(0)2867 965 y Fp(\()p Ff(t)p Fp(\))14 b Ff(dtd\030)3131 873 y Fg(\021)3196 965 y Ff(dx:)189 1213 y Fp(Remem)n(b)r(ering)27 b(the)h(de\014nition)g (of)g Ff(\032)p Fp(\()p Ff(t)p Fp(\),)g(w)n(e)f(get)1021 1425 y Ff(a)1065 1437 y Fm(0)1125 1425 y Fp(=)1213 1312 y Fg(Z)1259 1501 y Fb(R)1306 1484 y Fh(n)1306 1517 y(x)1360 1312 y Fg(Z)1406 1501 y Fb(R)1453 1509 y Fh(t)1494 1333 y Fg(\020)1543 1425 y Ff(\032)1586 1358 y Fg(\000)1624 1425 y Ff(t)1654 1358 y Fg(\001)1711 1425 y Fe(\000)18 b Ff(\032)1837 1358 y Fg(\000)1875 1425 y Ff(t)g Fe(\000)g Ff(W)12 b Fp(\()p Ff(x)p Fp(\))2207 1358 y Fg(\001)2246 1333 y(\021)2296 1425 y Ff(f)2346 1391 y Fj(0)2368 1425 y Fp(\()p Ff(t)p Fp(\))i Ff(dtdx:)189 1708 y Fp(In)n(tegrating)26 b(b)n(y)h(parts,)g(w)n(e)h(obtain)f Ff(a)1394 1720 y Fm(0)1454 1708 y Fp(=)1542 1595 y Fg(Z)1588 1784 y Fb(R)1648 1708 y Ff(f)9 b Fp(\()p Ff(s)p Fp(\))15 b Fg(e)-61 b Ff(w)s Fp(\()p Ff(s)p Fp(\))14 b Ff(ds:)1346 b Fa(\003)189 1893 y Fq(4.)42 b(Pro)s(of)31 b(of)h(the)g(lo)s(cal)f(trace)i(form)m (ula)355 2061 y Fp(Let)28 b Ff(\025)552 2073 y Fm(0)590 2061 y Ff(;)14 b Fp(\012)p Ff(;)731 2040 y Fg(e)724 2061 y Fp(\012)p Ff(;)g(I)35 b Fp(and)27 b Ff(J)37 b Fp(b)r(e)28 b(as)f(in)i(Theorem)e(1.5.)37 b(Applying)28 b(Prop)r(osition)f(2.6,)g (\(with)i Ff( )e Fp(:=)c Ff(\037g)s Fp(\),)189 2185 y(w)n(e)k(obtain) 1198 2310 y Fe(I)i Fp(=)23 b(tr[\()p Ff(\037g)s Fp(\)\()p Ff(P)12 b Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Fp(\()p Ff(\037g)s Fp(\)\()p Ff(P)2194 2322 y Fm(0)2232 2310 y Fp(\()p Ff(h)p Fp(\)\)])24 b(=)837 b(\(4)p Ff(:)p Fp(1\))708 2497 y Fe(\000)787 2440 y Fp(1)p 783 2478 51 4 v 783 2554 a Ff(\031)857 2384 y Fg(Z)903 2572 y Fb(C)963 2429 y Fg(\000)p 1001 2429 49 4 v 68 x Ff(@)12 b Fg(e)-53 b Ff(\037)1102 2429 y Fg(\001)1140 2497 y Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1461 2404 y Fg(\020)1510 2429 y(\002)1545 2497 y Ff(E)1606 2509 y Fl(j;)p Fj(\000)p Fm(+)1760 2497 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1952 2462 y Fj(\000)p Fm(1)2041 2497 y Ff(@)2085 2509 y Fl(z)2123 2497 y Ff(E)2184 2509 y Fl(j;)p Fj(\000)p Fm(+)2339 2497 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2531 2429 y Fg(\003)2565 2447 y Fm(1)2565 2530 y Fl(j)s Fm(=0)2684 2404 y Fg(\021)2747 2497 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))p Ff(;)189 2699 y Fp(where)33 b Ff(\037)g Fe(2)g Ff(C)673 2669 y Fj(1)667 2719 y Fm(0)744 2699 y Fp(\()p Ff(J)8 b Fp(\))p Ff(;)48 b Fe(\021)32 b Fp(1)h(near)p 1296 2632 44 4 v 33 w Ff(I)7 b(;)33 b Fp(and)h Ff(h)p Fe(\000)p Fp(indep)r(enden)n(t.)54 b(Here)33 b Ff(g)j Fp(=)c Ff(g)s Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))33 b(is)g(an)g(holomorphic)189 2823 y(function)28 b(in)g(\012)g(and)f(satis\014es)g Fe(j)p Ff(g)s Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)22 b(\024)h Fp(1)k(for)g Ff(z)f Fe(2)e Fp(\012)18 b Fe(n)2042 2802 y Fg(e)2035 2823 y Fp(\012)p Ff(:)355 2969 y Fp(Since)h(supp\()p 766 2902 49 4 v Ff(@)12 b Fg(e)-53 b Ff(\037)p Fp(\))23 b Fe(\032)g(f)p Ff(w)i Fe(2)f Fd(C)14 b Fp(;)39 b Fe(j)p Ff(g)s Fp(\()p Ff(w)r Fp(\))p Fe(j)24 b(\024)f Fp(1)p Fe(g)f(\032)g Fp(\012)q Fe(n)1959 2948 y Fg(e)1953 2969 y Fp(\012)o Ff(;)d Fp(the)g(in)n(tegral)e(\(4.1\))i(o)n(v)n(er)e Fe(f)p Ff(z)25 b Fe(2)f Fd(C)15 b Fp(;)34 b Fe(=)p Fp(\()p Ff(z)t Fp(\))22 b Ff(<)h Fp(0)p Fe(g)189 3094 y Fp(is)k Fe(O)r Fp(\()p Ff(h)420 3064 y Fj(\000)p Fl(n)518 3094 y Fp(\))h(uniformly)f(with)i(resp)r(ect)e(to)g Ff(g)s(:)h Fp(Consequen)n(tly)-7 b(,)371 3306 y Fe(I)30 b Fp(=)22 b Fe(\000)612 3249 y Fp(1)p 608 3287 51 4 v 608 3363 a Ff(\031)682 3193 y Fg(Z)728 3381 y Fb(C)764 3389 y Fk(+)835 3238 y Fg(\000)p 873 3238 49 4 v 68 x Ff(@)12 b Fg(e)-53 b Ff(\037)973 3238 y Fg(\001)1012 3306 y Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1333 3213 y Fg(\020)1382 3238 y(\002)1417 3306 y Ff(E)1478 3318 y Fl(j;)p Fj(\000)p Fm(+)1632 3306 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1824 3271 y Fj(\000)p Fm(1)1913 3306 y Ff(@)1957 3318 y Fl(z)1995 3306 y Ff(E)2056 3318 y Fl(j;)p Fj(\000)p Fm(+)2210 3306 y Fp(\()p Ff(z)t Fp(\))2317 3238 y Fg(\003)2352 3256 y Fm(1)2352 3339 y Fl(j)s Fm(=0)2471 3213 y Fg(\021)2534 3306 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))k(+)g Fe(O)r Fp(\()p Ff(h)2990 3271 y Fj(\000)p Fl(n)3088 3306 y Fp(\))p Ff(;)182 b Fp(\(4)p Ff(:)p Fp(2\))189 3537 y(where)27 b Fd(C)483 3549 y Fm(+)567 3537 y Fp(=)c Fe(f)p Ff(w)i Fe(2)e Fd(C)15 b Fp(;)48 b Fe(=)p Fp(\()p Ff(w)r Fp(\))24 b Fe(\025)f Fp(0)p Fe(g)p Ff(:)355 3683 y Fp(Let)31 b(~)-45 b Ff(\015)27 b Fp(:)c([0)p Ff(;)14 b Fp(1])23 b Fe(!)g Fd(C)970 3695 y Fm(+)1031 3683 y Ff(;)k(t)c Fe(7!)g Fp([)p Ff(ta)18 b Fp(+)g(\(1)g Fe(\000)g Ff(t)p Fp(\))p Ff(b)p Fp(])g(+)g Ff(i\017:)27 b Fp(In)g(the)h(follo)n(wing,)f(w)n(e)g(\014x)g Ff(\017)c(>)g Fp(0)k(small)g(enough)189 3807 y(\(indep)r(enden)n(t)i (of)e Ff(h)p Fp(\))h(suc)n(h)f(that)h Fe(f)p Ff(a)18 b Fp(+)g Ff(i\017;)c(b)j Fp(+)h Ff(i\017)p Fe(g)k(2)i Fp(\012)18 b Fe(n)p 2028 3716 60 4 v 2035 3786 a Fg(e)2028 3807 y Fp(\012)p Ff(:)28 b Fp(See)f Fq(Fig.1)p Fp(.)36 b(Green's)27 b(form)n(ula)g(giv)n(es)f(:)735 4019 y Fe(\000)814 3963 y Fp(1)p 810 4000 51 4 v 810 4076 a Ff(\031)883 3906 y Fg(Z)929 4095 y Fb(C)965 4103 y Fk(+)1036 3952 y Fg(\000)p 1074 3952 49 4 v 67 x Ff(@)12 b Fg(e)-53 b Ff(\037)1175 3952 y Fg(\001)1213 4019 y Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\))1470 3952 y Fg(\002)1504 4019 y Ff(E)1565 4031 y Fl(j;)p Fj(\000)p Fm(+)1719 4019 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1911 3985 y Fj(\000)p Fm(1)2000 4019 y Ff(@)2044 4031 y Fl(z)2082 4019 y Ff(E)2143 4031 y Fl(j;)p Fj(\000)p Fm(+)2297 4019 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2489 3952 y Fg(\003)2523 3969 y Fm(1)2523 4052 y Fl(j)s Fm(=0)2656 4019 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))23 b(=)374 b(\(4)p Ff(:)p Fp(3\))673 4301 y Fe(\000)752 4245 y Fp(1)p 748 4282 51 4 v 748 4358 a Ff(\031)822 4188 y Fg(Z)868 4377 y Fm(0)p Fj(\024=)p Fm(\()p Fl(z)r Fm(\))p Fl(<\017)1185 4234 y Fg(\000)p 1223 4233 49 4 v 67 x Ff(@)12 b Fg(e)-53 b Ff(\037)1324 4234 y Fg(\001)1362 4301 y Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\))1619 4234 y Fg(\002)1653 4301 y Ff(E)1714 4313 y Fl(j;)p Fj(\000)p Fm(+)1868 4301 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2060 4267 y Fj(\000)p Fm(1)2148 4301 y Ff(@)2192 4313 y Fl(z)2231 4301 y Ff(E)2292 4313 y Fl(j;)p Fj(\000)p Fm(+)2446 4301 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2638 4234 y Fg(\003)2672 4251 y Fm(1)2672 4334 y Fl(j)s Fm(=0)2805 4301 y Ff(L)p Fp(\()p Ff(dz)t Fp(\))880 4557 y(+)994 4501 y(1)p 955 4538 121 4 v 955 4614 a(2)p Ff(\031)s(i)1099 4444 y Fg(Z)1147 4633 y Fm(~)-35 b Fl(\015)1209 4557 y Fg(e)-54 b Ff(\037)p Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\))1510 4490 y Fg(\002)1544 4557 y Ff(E)1605 4569 y Fl(j;)p Fj(\000)p Fm(+)1759 4557 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1951 4523 y Fj(\000)p Fm(1)2040 4557 y Ff(@)2084 4569 y Fl(z)2122 4557 y Ff(E)2183 4569 y Fl(j;)p Fj(\000)p Fm(+)2338 4557 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2530 4490 y Fg(\003)2564 4507 y Fm(1)2564 4590 y Fl(j)s Fm(=0)2697 4557 y Ff(dz)t(:)189 4760 y Fp(The)26 b(in)n(tegrand)e(in)i(the)g(\014rst)g(in)n(tegral)e (in)i(the)g(righ)n(t)f(mem)n(b)r(er)g(of)h(\(4.3\))f(has)g(its)h(supp)r (ort)g(in)f(the)h(region)189 4884 y(where)h Ff(g)f Fp(=)c Ff(O)r Fp(\(1\))p Ff(;)29 b Fp(so)e(w)n(e)g(get)g(\(uniformly)h(with)g (resp)r(ect)f(to)h Ff(g)s Fp(\))560 5096 y Fe(I)h Fp(=)771 5040 y(1)p 732 5077 V 732 5153 a(2)p Ff(\031)s(i)876 4983 y Fg(Z)925 5172 y Fm(~)-36 b Fl(\015)986 5096 y Fg(e)-54 b Ff(\037)p Fp(\()p Ff(z)t Fp(\))p Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1351 5004 y Fg(\020)1401 5029 y(\002)1435 5096 y Ff(E)1496 5108 y Fl(j;)p Fj(\000)p Fm(+)1651 5096 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1843 5062 y Fj(\000)p Fm(1)1931 5096 y Ff(@)1975 5108 y Fl(z)2014 5096 y Ff(E)2075 5108 y Fl(j;)p Fj(\000)p Fm(+)2229 5096 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2421 5029 y Fg(\003)2455 5046 y Fm(1)2455 5129 y Fl(j)s Fm(=0)2574 5004 y Fg(\021)2638 5096 y Ff(dz)22 b Fp(+)c Fe(O)r Fp(\()p Ff(h)2973 5062 y Fj(\000)p Fl(n)3070 5096 y Fp(\))p Ff(:)1801 5349 y Fp(10)p eop %%Page: 11 11 11 10 bop 189 272 a Fp(Since)28 b(supp\(1)18 b Fe(\000)26 b Fg(e)-54 b Ff(\037)p Fp(\))23 b Fe(\032)g(f)p Ff(w)i Fe(2)f Fd(C)15 b Fp(;)47 b Fe(j)p Ff(g)s Fp(\()p Ff(w)r Fp(\))p Fe(j)24 b(\024)f Fp(1)p Fe(g)p Ff(;)k Fp(w)n(e)g(ma)n(y)g (replace)f Fe(I)34 b Fp(b)n(y)28 b(:)639 511 y Fe(I)h Fp(=)850 455 y(1)p 811 492 121 4 v 811 568 a(2)p Ff(\031)s(i)955 398 y Fg(Z)1004 587 y Fm(~)-36 b Fl(\015)1058 511 y Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1272 419 y Fg(\020)1321 444 y(\002)1356 511 y Ff(E)1417 523 y Fl(j;)p Fj(\000)p Fm(+)1571 511 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1763 477 y Fj(\000)p Fm(1)1852 511 y Ff(@)1896 523 y Fl(z)1934 511 y Ff(E)1995 523 y Fl(j;)p Fj(\000)p Fm(+)2150 511 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2342 444 y Fg(\003)2376 461 y Fm(1)2376 544 y Fl(j)s Fm(=0)2495 419 y Fg(\021)2558 511 y Ff(dz)22 b Fp(+)c Fe(O)r Fp(\()p Ff(h)2893 477 y Fj(\000)p Fl(n)2991 511 y Fp(\))p Ff(:)279 b Fp(\(4)p Ff(:)p Fp(4\))189 766 y(In)34 b(~)-45 b Ff(\015)5 b(;)45 b Fe(=)p Fp(\()p Ff(z)t Fp(\))29 b(=)g Ff(\017)f(>)h Fp(0)i(\014xed.)47 b(Since)32 b Ff(E)1461 736 y Fl(s)1456 787 y(j;)p Fj(\000)p Fm(+)1610 766 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))31 b(is)g(an)g(analytic)f(t)n(yp)r(e-A) i(family)f(on)g Ff(D)r Fp(\()p Ff(t)3148 778 y Fm(0)3185 766 y Fp(\))f(=)e Fe(f)p Ff(t)h Fe(2)189 890 y Fd(C)15 b Fp(;)f Fe(j)p Ff(t)p Fe(j)34 b(\024)29 b Ff(t)514 902 y Fm(0)551 890 y Fe(g)h Fp(and)h Ff(E)849 902 y Fl(j;)p Fj(\000)p Fm(+)1003 890 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))31 b(is)g(in)n(v)n(ertible,)g(there)f(exist)h Ff(t)2152 902 y Fm(1)2218 890 y Ff(>)d Fp(0)j(su\016cien)n(tly)f(small)h(suc)n(h) f(that)i(for)189 1015 y(all)h Ff(s)h Fe(2)g Ff(D)r Fp(\()p Ff(t)605 1027 y Fm(1)642 1015 y Fp(\))p Ff(;)g(E)797 985 y Fl(s)792 1037 y(j;)p Fj(\000)p Fm(+)947 1015 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))33 b(is)h(in)n(v)n(ertible)f(and)g Ff(E)1868 985 y Fl(s)1863 1037 y(j;)p Fj(\000)p Fm(+)2018 1015 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2210 985 y Fj(\000)p Fm(1)2332 1015 y Fp(is)34 b(an)f(analytic)h(t)n(yp)r(e-A)g(family)f(on) 189 1139 y Ff(D)r Fp(\()p Ff(t)322 1151 y Fm(1)360 1139 y Fp(\))p Ff(:)28 b Fp(Applying)f(this)h(with)g(the)g(fact)g(that)g Fe(U)1692 1151 y Fl(s)1755 1139 y Fp(is)g(unitary)f(when)g Ff(s)h Fp(is)f(real,)g(w)n(e)g(get)202 1378 y(tr)267 1286 y Fg(\020h)356 1378 y Ff(E)417 1390 y Fl(j;)p Fj(\000)p Fm(+)571 1378 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))763 1344 y Fj(\000)p Fm(1)852 1378 y Ff(@)896 1390 y Fl(z)934 1378 y Ff(E)995 1390 y Fl(j;)p Fj(\000)p Fm(+)1149 1378 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1341 1286 y Fg(i)1380 1303 y Fm(1)1380 1436 y Fl(j)s Fm(=0)1499 1286 y Fg(\021)1572 1378 y Fp(=)22 b(tr)1724 1286 y Fg(\020h)1813 1378 y Ff(E)1879 1344 y Fl(s)1874 1399 y(j;)p Fj(\000)p Fm(+)2028 1378 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2220 1344 y Fj(\000)p Fm(1)2309 1378 y Ff(@)2353 1390 y Fl(z)2391 1378 y Ff(E)2457 1344 y Fl(s)2452 1399 y(j;)p Fj(\000)p Fm(+)2607 1378 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2799 1286 y Fg(i)2837 1303 y Fm(1)2837 1436 y Fl(j)s Fm(=0)2956 1286 y Fg(\021)3006 1378 y Ff(;)41 b Fe(8)p Ff(s)23 b Fe(2)g Ff(D)r Fp(\()p Ff(t)3390 1390 y Fm(1)3428 1378 y Fp(\))p Ff(:)3325 1518 y Fp(\(4)p Ff(:)p Fp(5\))189 1643 y(On)c(the)h(other)f(hand,)j(since)d (the)h(righ)n(t)f(hand)h(side)f(of)h(\(4.5\))f(is)h(analytic)f(for)g Ff(s)k Fe(2)g Ff(D)r Fp(\()p Ff(t)2822 1655 y Fm(0)2860 1643 y Fp(\))s Fe(\\)s(f)p Ff(s)f Fe(2)h Fd(C)15 b Fp(;)34 b Fe(=)p Fp(\()p Ff(s)p Fp(\))23 b Fe(\025)189 1767 y Fp(0)p Fe(g)30 b Fp(and)i Ff(z)g Fe(2)h Fp(~)-45 b Ff(\015)5 b(;)31 b Fp(then)h(the)g(ab)r(o)n(v)n(e)e(equalit)n(y)g(remains)h(true) g(for)g(all)g Ff(s)e Fe(2)h Ff(D)r Fp(\()p Ff(t)2654 1779 y Fm(0)2692 1767 y Fp(\))21 b Fe(\\)g(f)p Ff(s)29 b Fe(2)h Fd(C)15 b Fp(;)33 b Fe(=)p Fp(\()p Ff(s)p Fp(\))d Fe(\025)f Fp(0)p Fe(g)189 1892 y Fp(and)e Ff(z)g Fe(2)f Fp(~)-45 b Ff(\015)5 b(:)355 2041 y Fp(F)-7 b(rom)33 b(no)n(w)h(on,)h(w)n(e)e(\014x)h Ff(s)g Fe(2)g Ff(i)p Fp(]0)p Ff(;)14 b(t)1480 2053 y Fm(0)1516 2041 y Fp([)p Ff(:)34 b Fp(The)g(iden)n(tit)n(y)g(\(4.4\))f(b)r(ecomes)h(\(uniformly) g(with)g(resp)r(ect)189 2165 y(to)27 b Ff(g)s Fp(\))639 2307 y Fe(I)i Fp(=)850 2251 y(1)p 811 2288 V 811 2364 a(2)p Ff(\031)s(i)955 2194 y Fg(Z)1004 2383 y Fm(~)-36 b Fl(\015)1058 2307 y Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1272 2215 y Fg(\020)1321 2240 y(\002)1356 2307 y Ff(E)1422 2273 y Fl(s)1417 2328 y(j;)p Fj(\000)p Fm(+)1571 2307 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1763 2273 y Fj(\000)p Fm(1)1852 2307 y Ff(@)1896 2319 y Fl(z)1934 2307 y Ff(E)2000 2273 y Fl(s)1995 2328 y(j;)p Fj(\000)p Fm(+)2150 2307 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2342 2240 y Fg(\003)2376 2257 y Fm(1)2376 2341 y Fl(j)s Fm(=0)2495 2215 y Fg(\021)2558 2307 y Ff(dz)22 b Fp(+)c Fe(O)r Fp(\()p Ff(h)2893 2273 y Fj(\000)p Fl(n)2991 2307 y Fp(\))p Ff(:)279 b Fp(\(4)p Ff(:)p Fp(6\))355 2579 y(Since)24 b(the)h(op)r(erator)1039 2512 y Fg(\000)1077 2579 y Ff(E)1143 2549 y Fl(s)1138 2600 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1299 2579 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))e Fe(\000)g Ff(E)1646 2549 y Fl(s)1641 2600 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1799 2579 y Fp(\()p Ff(z)t(;)i(h)p Fp(\))1991 2512 y Fg(\001)2029 2579 y Ff(E)2095 2549 y Fl(s)2090 2600 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2250 2579 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2442 2549 y Fj(\000)p Fm(1)2555 2579 y Fp(is)24 b(compact,)g(w)n(e)g(decomp) r(ose)189 2704 y(it)k(in)g(sum)g(of)f(t)n(w)n(o)g(op)r(erators,)f(one)h (of)g(them)i(is)e(of)h(\014nite)g(rank)e(and)i(the)g(other)f(has)g(a)g (small)g(norm.)189 2853 y Fq(Lemma)j(4.1.)35 b Fc(There)27 b(exist)g(t)n(w)n(o)g(op)r(erators)f Ff(K)1745 2865 y Fm(1)1782 2853 y Ff(;)41 b(K)1917 2865 y Fm(2)1982 2853 y Fc(from)27 b Ff(L)2235 2823 y Fm(2)2272 2853 y Fp(\()p Fd(T)2360 2823 y Fj(\003)2397 2853 y Fp(;)14 b Fd(C)2488 2823 y Fl(N)2557 2853 y Fp(\))28 b Fc(in)n(to)g(its)f(self,)h(suc)n(h)f (that)869 3075 y Fp(Id)18 b(+)g Ff(K)1117 3087 y Fm(1)1182 3075 y Fp(is)27 b(in)n(v)n(ertible)o Ff(;)42 b(K)1740 3087 y Fm(2)1804 3075 y Fp(is)28 b(of)f(\014nite)i(rank)d(\(=)d Fe(O)r Fp(\()p Ff(h)2654 3041 y Fj(\000)p Fl(n)2752 3075 y Fp(\)\))509 b(\(4)p Ff(:)p Fp(7\))189 3297 y Fc(and)40 b Ff(K)434 3309 y Fm(1)497 3297 y Fp(+)27 b Ff(K)660 3309 y Fm(2)740 3297 y Fp(=)849 3229 y Fg(\000)887 3297 y Ff(E)953 3266 y Fl(s)948 3317 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1108 3297 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))26 b Fe(\000)g Ff(E)1483 3266 y Fl(s)1478 3317 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1639 3297 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1831 3229 y Fg(\001)1868 3297 y Ff(E)1934 3266 y Fl(s)1929 3317 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2089 3297 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2281 3266 y Fj(\000)p Fm(1)2370 3297 y Ff(:)40 b Fc(Here)g Fp(Id)g Fc(designe)g(the)g(iden)n (tit)n(y)189 3421 y(op)r(erator)26 b(on)h Ff(L)696 3391 y Fm(2)733 3421 y Fp(\()p Fd(T)821 3391 y Fj(\003)859 3421 y Fp(;)14 b Fd(C)950 3391 y Fl(N)1019 3421 y Fp(\))p Ff(:)189 3595 y Fq(Pro)s(of.)36 b Fp(Without)29 b(an)n(y)d(loss)h(of)h (generalit)n(y)-7 b(,)26 b(w)n(e)h(ma)n(y)g(assume)g(that)h Ff(N)k Fp(=)22 b(1)p Ff(:)355 3744 y Fp(Consider)29 b(the)i(op)r (erator)d(on)i Ff(L)1362 3714 y Fm(2)1399 3744 y Fp(\()p Fd(T)1487 3714 y Fj(\003)1525 3744 y Fp(\))p Ff(;)g(A)1672 3756 y Fm(0)1737 3744 y Fp(=)d Fe(\000)p Ff(h)1942 3714 y Fm(2)1978 3744 y Fp(\001)2047 3756 y Fb(T)2092 3740 y Fi(\003)2125 3744 y Ff(:)k Fp(The)f(sp)r(ectrum)g(of)g Ff(A)2876 3756 y Fm(0)2944 3744 y Fp(is)g(discrete.)44 b(Let)189 3869 y Ff(\037)23 b Fe(2)h Ff(C)408 3838 y Fj(1)402 3889 y Fm(0)479 3869 y Fp(\()p Fd(R)p Fp(\))34 b(suc)n(h)27 b(that)h Ff(\037)c Fe(\021)e Fp(1)28 b(in)g([)p Fe(\000)p Ff(C)q(;)14 b(C)6 b Fp(])28 b(and)g Ff(\037)23 b Fe(\021)g Fp(0)k(in)h Fd(R)d Fe(n)18 b Fp([)p Fe(\000)p Fp(2)p Ff(C)q(;)c Fp(2)p Ff(C)6 b Fp(])p Ff(:)27 b Fp(By)h(functional)g (calculus,)189 3993 y(w)n(e)f(de\014ne)h(the)g(op)r(erator)e Ff(A)d Fp(=)g Ff(\037)p Fp(\()p Fe(\000)p Ff(h)1399 3963 y Fm(2)1436 3993 y Fp(\001)1505 4005 y Fb(T)1550 3989 y Fi(\003)1583 3993 y Fp(\))p Ff(:)28 b Fp(The)f(op)r(erator)f Ff(A)i Fp(is)f(of)h(\014nite)g(rank,)2866 3926 y Fg(\000)2927 3993 y Fp(=)23 b Fe(O)r Fp(\()p Ff(h)3163 3963 y Fj(\000)p Fl(n)3260 3993 y Fp(\))3292 3926 y Fg(\001)3331 3993 y Ff(:)189 4118 y Fp(Standard)k Ff(h)p Fe(\000)p Fp(pseudo)r (di\013eren)n(tial)g(calculus)g(on)g(the)h(torus)f(and)g(the)h (assumption)f Fq(\(H.1\))g Fp(on)h Ff(W)39 b Fp(giv)n(e)445 4244 y Fg(\015)445 4294 y(\015)445 4343 y(\015)491 4339 y Fp(\(Id)19 b Fe(\000)f Ff(A)p Fp(\))795 4247 y Fg(h)835 4339 y Ff(E)901 4305 y Fl(s)896 4360 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1056 4339 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))k Fe(\000)g Ff(E)1415 4305 y Fl(s)1410 4360 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1570 4339 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))1762 4247 y Fg(i)1801 4339 y Ff(E)1867 4305 y Fl(s)1862 4360 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2022 4339 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2214 4305 y Fj(\000)p Fm(1)2303 4244 y Fg(\015)2303 4294 y(\015)2303 4343 y(\015)2372 4339 y Fe(!)23 b Fp(0)83 b(when)g Ff(C)29 b Fe(!)23 b Fp(+)p Fe(1)p Ff(:)355 4561 y Fp(T)-7 b(o)27 b(\014nish)h(the)g(pro)r(of)f(of)h(the)g(lemma,)f(it)h (su\016ces)g(to)f(tak)n(e)g(:)828 4783 y Ff(K)899 4795 y Fm(1)958 4783 y Fp(=)c(\(Id)c Fe(\000)f Ff(A)p Fp(\))1350 4690 y Fg(h)1390 4783 y Ff(E)1456 4748 y Fl(s)1451 4803 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1611 4783 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))k Fe(\000)g Ff(E)1970 4748 y Fl(s)1965 4803 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2125 4783 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))2317 4690 y Fg(i)2356 4783 y Ff(E)2422 4748 y Fl(s)2417 4803 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2577 4783 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2769 4748 y Fj(\000)p Fm(1)3325 4783 y Fp(\(4)p Ff(:)p Fp(8\))189 5004 y(and)937 5129 y Ff(K)1008 5141 y Fm(2)1068 5129 y Fp(=)23 b Ff(A)1218 5037 y Fg(h)1257 5129 y Ff(E)1323 5095 y Fl(s)1318 5149 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1478 5129 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))k Fe(\000)g Ff(E)1837 5095 y Fl(s)1832 5149 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1992 5129 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))2184 5037 y Fg(i)2223 5129 y Ff(E)2289 5095 y Fl(s)2284 5149 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2444 5129 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2636 5095 y Fj(\000)p Fm(1)2725 5129 y Ff(:)577 b Fp(\(4)p Ff(:)p Fp(9\))1801 5349 y(11)p eop %%Page: 12 12 12 11 bop 355 272 a Fp(The)26 b(remainder)f(of)g(the)h(pro)r(of)g(of)f (the)h(theorem)g(1.5,)f(is)h(close)f(to)g(the)h(one)g(in)g([27,)f (Section)h(4],)g(for)189 397 y(that)i(w)n(e)f(omit)h(the)g(details.)355 544 y(Set)g Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))22 b(=)h Ff(K)948 556 y Fm(2)984 544 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1176 477 y Fg(\000)1214 544 y Fp(Id)19 b(+)f Ff(K)1463 556 y Fm(1)1499 544 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))1691 477 y Fg(\001)1729 494 y Fj(\000)p Fm(1)1818 544 y Ff(:)28 b Fp(A)g(simple)g(calculus)f(giv)n(es)f(:)927 750 y Ff(E)993 716 y Fl(s)988 771 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)1148 750 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))23 b(=)1451 683 y Fg(\000)1489 750 y Fp(Id)18 b(+)g Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1935 683 y Fg(\001)1972 750 y Fp(\(Id)19 b(+)f Ff(K)2253 762 y Fm(1)2290 750 y Fp(\))p Ff(E)2388 716 y Fl(s)2383 771 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2543 750 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))p Ff(:)526 b Fp(\(4)p Ff(:)p Fp(10\))189 956 y(Di\013eren)n(tiating)34 b(the)g(last)g(iden)n(tit)n(y)g(with)g(resp)r(ect)g(to)g Ff(z)j Fp(and)d(inserting)f(it)i(in)f(the)g(in)n(tegrand)f(of)h(the)189 1081 y(righ)n(t)27 b(mem)n(b)r(er)g(of)h(\(4.6\),)f(w)n(e)g(get)1207 1211 y Fg(h)1247 1304 y Ff(E)1313 1269 y Fl(s)1308 1324 y(j;)p Fj(\000)p Fm(+)1462 1304 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1654 1269 y Fj(\000)p Fm(1)1743 1304 y Ff(@)1787 1316 y Fl(z)1825 1304 y Ff(E)1891 1269 y Fl(s)1886 1324 y(j;)p Fj(\000)p Fm(+)2040 1304 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2232 1211 y Fg(i)2271 1229 y Fm(1)2271 1362 y Fl(j)s Fm(=0)2413 1304 y Fp(=)806 b(\(4)p Ff(:)p Fp(11\))472 1550 y Ff(E)538 1516 y Fl(s)533 1571 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)693 1550 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))885 1516 y Fj(\000)p Fm(1)974 1483 y Fg(\000)1012 1550 y Fp(Id)19 b(+)f Ff(K)1261 1562 y Fm(1)1297 1483 y Fg(\001)1335 1500 y Fj(\000)p Fm(1)1425 1550 y Fp(\(Id)g(+)g Ff(K)6 b Fp(\))1743 1516 y Fj(\000)p Fm(1)1832 1550 y Ff(@)1876 1562 y Fl(z)1915 1550 y Ff(K)g Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2184 1483 y Fg(\000)2221 1550 y Fp(Id)k(+)g Ff(K)2469 1562 y Fm(1)2506 1483 y Fg(\000)2544 1550 y Ff(z)t(;)c(h)p Fp(\)\))p Ff(E)2802 1516 y Fl(s)2797 1571 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2957 1550 y Fp(\()p Ff(z)t(;)g(h)p Fp(\)+)862 1725 y Ff(E)928 1691 y Fl(s)923 1745 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1083 1725 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))1275 1691 y Fj(\000)p Fm(1)1364 1658 y Fg(\000)1402 1725 y Fp(Id)k(+)g Ff(K)1650 1737 y Fm(1)1687 1725 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))1879 1658 y Fg(\001)1917 1675 y Fj(\000)p Fm(1)2006 1725 y Ff(@)2050 1737 y Fl(z)2088 1725 y Ff(K)2159 1737 y Fm(1)2196 1725 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(E)2454 1691 y Fl(s)2449 1745 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)2609 1725 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(:)189 1899 y Fp(Clearly)-7 b(,)25 b(all)g(the)g(terms)g(in)g (\(4.11\))g(are)f(of)h(trace)f(class.)35 b(Using)25 b(the)h(cyclicit)n (y)e(of)i(the)f(trace)f(w)n(e)h(obtain)g(:)1130 2106 y(tr)1195 2013 y Fg(\020)1244 2038 y(\002)1279 2106 y Ff(E)1345 2071 y Fl(s)1340 2126 y(j;)p Fj(\000)p Fm(+)1494 2106 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1686 2071 y Fj(\000)p Fm(1)1775 2106 y Ff(@)1819 2118 y Fl(z)1857 2106 y Ff(E)1923 2071 y Fl(s)1918 2126 y(j;)p Fj(\000)p Fm(+)2073 2106 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2265 2038 y Fg(\003)2299 2055 y Fm(1)2299 2139 y Fl(j)s Fm(=0)2418 2013 y Fg(\021)2490 2106 y Fp(=)729 b(\(4)p Ff(:)p Fp(12\))752 2345 y(tr)817 2253 y Fg(\020)866 2278 y(\002\000)939 2345 y Fp(Id)19 b(+)f Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1386 2278 y Fg(\001)1423 2295 y Fj(\000)p Fm(1)1512 2345 y Ff(@)1556 2357 y Fl(z)1594 2345 y Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1863 2278 y Fg(\003)1897 2253 y(\021)1965 2345 y Fp(+)k(tr)2113 2253 y Fg(\020)2162 2278 y(\002)2197 2345 y Fp(\(Id)h(+)f Ff(K)2478 2357 y Fm(1)2514 2345 y Fp(\))2546 2311 y Fj(\000)p Fm(1)2636 2345 y Ff(@)2680 2357 y Fl(z)2718 2345 y Ff(K)2789 2357 y Fm(1)2826 2278 y Fg(\003)2860 2253 y(\021)2910 2345 y Ff(:)189 2519 y Fp(Here)35 b(the)h(last)f(term)h(is)f(holomorphic)f(in)i(\012)p Ff(;)g Fp(so)f(committing)h(another)e(error)g Fe(O)r Fp(\()p Ff(h)2947 2489 y Fj(\000)p Fl(n)3044 2519 y Fp(\))p Ff(;)i Fp(w)n(e)f(obtain)189 2644 y(\(uniformly)28 b(with)g(resp)r(ect) f(to)h Ff(g)s Fp(\))766 2867 y Fe(I)h Fp(=)977 2811 y(1)p 938 2848 121 4 v 938 2924 a(2)p Ff(\031)s(i)1082 2754 y Fg(Z)1131 2943 y Fm(~)-36 b Fl(\015)1185 2867 y Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1399 2800 y Fg(\002)1433 2867 y Fp(\(Id)19 b(+)f Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\)\))1944 2833 y Fj(\000)p Fm(1)2033 2867 y Ff(@)2077 2879 y Fl(z)2115 2867 y Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2384 2800 y Fg(\003)2431 2867 y Ff(dz)22 b Fp(+)c Fe(O)r Fp(\()p Ff(h)2766 2833 y Fj(\000)p Fl(n)2864 2867 y Fp(\))p Ff(:)365 b Fp(\(4)p Ff(:)p Fp(13\))355 3129 y(It)37 b(follo)n(ws)e(from)h(Prop)r(osition)e(2.1)i(\(see)g(also) f([10]\))h(that,)i Ff(z)i Fp(is)c(a)g(resonance)e(of)i Ff(P)3091 3141 y Fm(1)3129 3129 y Fp(\()p Ff(h)p Fp(\))h(if)f(and)189 3254 y(only)d(if)g(0)f Fe(2)h Ff(\033)670 3187 y Fg(\000)708 3254 y Ff(E)774 3224 y Fl(s)769 3275 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)929 3254 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1121 3187 y Fg(\001)1192 3254 y Fp(with)34 b(the)g(same)e(m)n(ultiplicit)n (y)-7 b(.)54 b(Com)n(bining)33 b(this)g(with)h(the)g(fact)f(that)189 3378 y(\(Id)24 b(+)f Ff(K)480 3390 y Fm(1)517 3378 y Fp(\))p Ff(E)615 3348 y Fl(s)610 3399 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)805 3378 y Fp(is)35 b(holomorphic)f(in)i(\012)f(and)g (ha)n(v)n(e)f(a)h(b)r(ounded)g(in)n(v)n(erse)f(ev)n(erywhere,)h(w)n(e)g (deduce)189 3503 y(that)i(the)f(zeros)f(of)i Ff(D)r Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))37 b(:=)g(det)1390 3436 y Fg(\000)1428 3503 y Fp(Id)25 b(+)f Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))1887 3436 y Fg(\001)1960 3503 y Fp(are)35 b(the)i(resonances)d (of)j Ff(P)2836 3515 y Fm(1)2873 3503 y Fp(\()p Ff(h)p Fp(\))g(and)f(that)h(the)189 3638 y(m)n(ultiplicities)30 b(agree.)39 b(Therefore,)29 b(for)f(all)h(\012)1647 3650 y Fm(1)1709 3638 y Fe(\032\032)c Fp(\012)k(suc)n(h)g(that)g Ff(@)5 b Fp(\012)2432 3650 y Fm(1)2494 3638 y Fp(=)29 b(~)-46 b Ff(\015)24 b Fe([)f Fp(~)-45 b Ff(\015)2769 3650 y Fm(1)2806 3638 y Ff(;)30 b Fp(where)h(~)-45 b Ff(\015)3143 3650 y Fm(1)3206 3638 y Fe(\032)25 b Fp(\012)19 b Fe(n)p 3436 3547 60 4 v 3443 3617 a Fg(e)3436 3638 y Fp(\012)189 3762 y(and)27 b(a)n(v)n(oiding)f(the)i(resonances)e(of)h Ff(P)1382 3774 y Fm(1)1420 3762 y Fp(\()p Ff(h)p Fp(\))p Ff(;)h Fp(w)n(e)f(ha)n(v)n(e)998 3930 y(1)p 958 3967 121 4 v 958 4043 a(2)p Ff(\031)s(i)1102 3873 y Fg(Z)1151 4062 y Fm(~)-36 b Fl(\015)t Fj([)s Fm(~)g Fl(\015)1267 4070 y Fk(1)1317 3986 y Ff(g)s Fp(\()p Ff(z)t Fp(\)tr)1531 3919 y Fg(\002)1566 3986 y Fp(\(Id)19 b(+)f Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\)\))2077 3952 y Fj(\000)p Fm(1)2165 3986 y Ff(@)2209 3998 y Fl(z)2247 3986 y Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))2516 3919 y Fg(\003)2564 3986 y Ff(dz)26 b Fp(=)547 b(\(4)p Ff(:)p Fp(14\))824 4277 y(tr)889 4184 y Fg(h)977 4220 y Fp(1)p 938 4258 V 938 4334 a(2)p Ff(\031)s(i)1082 4164 y Fg(Z)1131 4352 y Fm(~)-36 b Fl(\015)t Fj([)s Fm(~)g Fl(\015)1247 4360 y Fk(1)1297 4277 y Ff(g)s Fp(\()p Ff(z)t Fp(\))1477 4220 y Ff(d)p 1457 4258 86 4 v 1457 4334 a(dz)1552 4209 y Fg(\000)1604 4277 y Fp(log)14 b Ff(D)r Fp(\()p Ff(z)t(;)g(h)p Fp(\))1988 4209 y Fg(\001)2039 4277 y Ff(dz)2125 4184 y Fg(i)2187 4277 y Fp(=)2477 4198 y Fg(X)2274 4380 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)2525 4388 y Fk(1)2558 4380 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)p Fm(\012)2767 4388 y Fk(1)2688 4277 y Ff(g)s Fp(\()p Ff(z)t Fp(\))p Ff(:)189 4531 y Fp(Since)28 b Ff(K)477 4543 y Fm(2)541 4531 y Fp(is)g(of)f(\014nite)h(rank)1123 4464 y Fg(\000)1184 4531 y Fp(=)23 b Fe(O)r Fp(\()p Ff(h)1420 4501 y Fj(\000)p Fl(n)1517 4531 y Fp(\))1549 4464 y Fg(\001)1588 4531 y Ff(;)k Fp(w)n(e)h(ha)n(v)n(e)1220 4738 y Ff(D)r Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))23 b(=)g Ff(O)r Fp(\(1\))p Ff(e)1804 4703 y Fj(O)r Fm(\(1\))p Fl(h)1983 4678 y Fi(\000)p Fh(n)2072 4738 y Ff(;)97 b Fe(8)p Ff(z)25 b Fe(2)f Fp(\012)p Ff(:)819 b Fp(\(4)p Ff(:)p Fp(15\))189 4944 y(In)28 b(\012)353 4956 y Fm(+)p Fl(;)435 4934 y Fh(\017)p 433 4943 29 4 v 433 4976 a Fk(2)499 4944 y Fp(:=)22 b Fe(f)p Ff(z)k Fe(2)e Fp(\012;)j Fe(=)p Fp(\()p Ff(z)t Fp(\))c Fe(\025)1196 4911 y Fl(\017)p 1193 4925 34 4 v 1193 4972 a Fm(2)1236 4944 y Fe(g)p Ff(;)k Fp(the)h(op)r(erator)e Ff(E)1872 4913 y Fl(s)1867 4964 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)2027 4944 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))27 b(is)h(uniformly)f(in)n(v)n (ertible,)g(hence)499 5082 y Fg(\000)537 5150 y Fp(Id)19 b(+)f Ff(K)6 b Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))984 5082 y Fg(\001)1021 5100 y Fj(\000)p Fm(1)1133 5150 y Fp(=)1221 5082 y Fg(\000)1259 5150 y Fp(Id)k(+)g Ff(K)1507 5162 y Fm(1)1544 5150 y Fp(\()p Ff(z)t(;)c(h)p Fp(\))1736 5082 y Fg(\001)1774 5150 y Ff(E)1840 5115 y Fl(s)1835 5170 y Fm(0)p Fl(;)p Fj(\000)p Fm(+)1995 5150 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(E)2253 5115 y Fl(s)2248 5170 y Fm(1)p Fl(;)p Fj(\000)p Fm(+)2408 5150 y Fp(\()p Ff(z)t(;)g(h)p Fp(\))2600 5115 y Fj(\000)p Fm(1)2711 5150 y Fp(=)23 b Fe(O)r Fp(\(1\))311 b(\(4)p Ff(:)p Fp(16\))1801 5349 y(12)p eop %%Page: 13 13 13 12 bop 189 272 a Fp(as)27 b(an)g(op)r(erator)f(on)h Ff(L)913 242 y Fm(2)950 272 y Fp(\()p Fd(T)1038 242 y Fj(\003)1076 272 y Ff(;)14 b Fd(C)1167 242 y Fl(n)1218 272 y Fp(\))28 b(and)1262 486 y Fe(j)p Ff(D)r Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)23 b(\025)g Ff(e)1721 452 y Fj(\000)p Fl(C)t(h)1864 427 y Fi(\000)p Fh(n)1953 486 y Ff(;)97 b(z)26 b Fe(2)e Fp(\012)2277 498 y Fm(+)p Fl(;)2359 476 y Fh(\017)p 2357 485 29 4 v 2357 518 a Fk(2)2399 486 y Ff(:)862 b Fp(\(4)p Ff(:)p Fp(17\))189 700 y(Let)23 b Ff(N)9 b Fp(\()p Ff(P)r(;)14 b Fp(\012;)g Ff(h)p Fp(\))23 b(b)r(e)g(the)h(n)n(um)n(b)r(er)e(of)h(resonances)e(in) i(\012)f(coun)n(ted)h(with)g(their)g(m)n(ultiplicit)n(y)-7 b(.)36 b(Com)n(bining)189 824 y(\(4.15\),)27 b(\(4.17\))g(and)g (Jensen's)g(inequalit)n(y)-7 b(,)27 b(w)n(e)h(get)1482 1038 y Ff(N)9 b Fp(\()p Ff(P)r(;)14 b Fp(\012;)g Ff(h)p Fp(\))23 b Fe(\024)g Ff(C)6 b(h)2083 1004 y Fj(\000)p Fl(n)2180 1038 y Ff(:)1081 b Fp(\(4)p Ff(:)p Fp(18\))355 1276 y(Let)28 b Ff(z)543 1288 y Fl(j)577 1276 y Ff(;)42 b(j)28 b Fp(=)23 b(1)p Ff(;)14 b Fe(\001)g(\001)g(\001)f Ff(;)h(M)31 b Fp(=)23 b Ff(N)9 b Fp(\()p Ff(P)r(;)14 b Fp(\012;)g Ff(h)p Fp(\))28 b(rep)r(eated)f(according)f(to)i(their)f (m)n(ultiplicit)n(y)h(and)g(put)1420 1549 y Ff(D)1489 1561 y Fl(w)1543 1549 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))22 b(=)1868 1445 y Fl(M)1849 1470 y Fg(Y)1845 1647 y Fl(j)s Fm(=1)1960 1549 y Fp(\()p Ff(z)g Fe(\000)c Ff(z)2175 1561 y Fl(j)2209 1549 y Fp(\))p Ff(:)1020 b Fp(\(4)p Ff(:)p Fp(19\))189 1827 y(The)28 b(iden)n(tit)n(y)f(\(4.18\))g(implies) 809 2041 y Fe(j)p Ff(D)901 2053 y Fl(w)955 2041 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)22 b(\024)h Ff(e)1319 2006 y Fl(C)t(h)1410 1981 y Fi(\000)p Fh(n)1527 2041 y Fp(in)28 b(\012)p Ff(;)96 b Fe(j)p Ff(D)1895 2053 y Fl(w)1949 2041 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)23 b(\025)f Ff(e)2313 2006 y Fj(\000)p Fl(C)t(h)2456 1981 y Fi(\000)p Fh(n)2573 2041 y Fp(in)28 b(\012)2730 2053 y Fm(+)p Fl(;)2812 2031 y Fh(\017)p 2810 2040 V 2810 2073 a Fk(2)2853 2041 y Ff(:)408 b Fp(\(4)p Ff(:)p Fp(20\))189 2255 y(Using)27 b([27,)g(Lemma)h(4.3],)e(w)n(e)i(ma)n(y)f(c)n(ho)r(ose) i(~)-45 b Ff(\015)1664 2267 y Fm(1)1729 2255 y Fp(suc)n(h)27 b(that)1288 2469 y Fe(j)p Ff(D)1380 2481 y Fl(w)1434 2469 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)22 b(\025)h Ff(e)1798 2434 y Fj(\000)p Fl(C)t(h)1941 2409 y Fi(\000)p Fh(n)2030 2469 y Ff(;)97 b(z)26 b Fe(2)h Fp(~)-45 b Ff(\015)2337 2481 y Fm(1)2374 2469 y Ff(:)887 b Fp(\(4)p Ff(:)p Fp(21\))189 2682 y(Here)27 b(the)h(idea)f(is)h(to)f(a)n(v)n(oid)g(small)g(\014xed)g (neigh)n(b)r(orho)r(o)r(ds)g(of)g(resonances)f(of)h Ff(P)2733 2694 y Fm(1)2771 2682 y Fp(\()p Ff(h)p Fp(\))h(in)g(\012)18 b Fe(n)p 3146 2592 60 4 v 3153 2661 a Fg(e)3146 2682 y Fp(\012)p Ff(:)189 2831 y Fp(W)-7 b(e)28 b(factorize)f Ff(D)i Fp(:)1197 2955 y Ff(D)r Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))23 b(=)f Ff(D)1639 2967 y Fl(w)1693 2955 y Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Ff(G)p Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Ff(;)96 b(z)27 b Fe(2)c Fp(\012)p Ff(;)796 b Fp(\(4)p Ff(:)p Fp(22\))189 3148 y(where)27 b Ff(G)h Fp(and)705 3091 y(1)p 693 3129 66 4 v 693 3205 a Ff(G)796 3148 y Fp(are)f(holomorphic)f(in)i(\012)p Ff(:)g Fp(Com)n(bining)f (\(4.20\),)g(\(4.21\))g(and)g(\(4.22\),)g(w)n(e)g(get)1205 3389 y Fe(j)p Ff(G)p Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)23 b(\024)g Ff(e)1658 3355 y Fl(C)t(h)1749 3330 y Fi(\000)p Fh(n)1838 3389 y Ff(;)97 b(z)26 b Fe(2)d Fp(\012)2161 3401 y Fm(+)p Fl(;)2243 3379 y Fh(\017)p 2242 3388 29 4 v 2242 3422 a Fk(2)2303 3389 y Fe([)f Fp(~)-45 b Ff(\015)2420 3401 y Fm(1)2457 3389 y Ff(:)804 b Fp(\(4)p Ff(:)p Fp(23\))189 3617 y(F)-7 b(or)25 b(\012)396 3629 y Fm(1)457 3617 y Fe(\032\032)d Fp(\012)k(a)f(simply)h(connected)g (relativ)n(ely)f(op)r(en)h Ff(h)p Fe(\000)p Fp(indep)r(enden)n(t)g(set) g(with)p 2868 3526 60 4 v 2875 3596 a Fg(e)2868 3617 y Fp(\012)g(in)h(its)f(\(relativ)n(e\))189 3742 y(in)n(terior,)g(b)n(y) i(maxim)n(um)f(principle,)h(w)n(e)f(ha)n(v)n(e)1334 3955 y Fe(j)p Ff(G)p Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)23 b(\024)g Ff(e)1787 3921 y Fl(C)t(h)1878 3896 y Fi(\000)p Fh(n)1967 3955 y Ff(;)97 b(z)26 b Fe(2)e Fp(\012)2291 3967 y Fm(1)2328 3955 y Ff(:)933 b Fp(\(4)p Ff(:)p Fp(24\))189 4169 y(Thanks)27 b(to)g(\(4.17\),)g(\(4.20\))g(and)h(\(4.22\),)e(w)n(e) i(ha)n(v)n(e)1265 4383 y Fe(j)p Ff(G)p Fp(\()p Ff(z)t(;)14 b(h)p Fp(\))p Fe(j)23 b(\025)g Ff(e)1718 4349 y Fj(\000)p Fl(C)t(h)1861 4324 y Fi(\000)p Fh(n)1950 4383 y Ff(;)97 b(z)26 b Fe(2)e Fp(\012)2274 4395 y Fm(+)p Fl(;)2356 4373 y Fh(\017)p 2354 4382 29 4 v 2354 4416 a Fk(2)2397 4383 y Ff(:)864 b Fp(\(4)p Ff(:)p Fp(25\))189 4597 y(F)-7 b(or)27 b Ff(C)397 4609 y Fm(0)458 4597 y Ff(>)22 b Fp(0)27 b(large)g(enough,)g(put)1176 4811 y Ff(F)12 b Fp(\()p Ff(z)t(;)i(h)p Fp(\))23 b(:=)g Ff(C)1626 4823 y Fm(0)1663 4811 y Ff(h)1711 4777 y Fj(\000)p Fl(n)1826 4811 y Fe(\000)18 b Fp(log)d Fe(j)p Ff(G)p Fp(\()p Ff(z)t(;)f(h)p Fp(\))p Fe(j)23 b(\025)f Fp(0)p Ff(:)775 b Fp(\(4)p Ff(:)p Fp(26\))189 5025 y(\(4.25\))27 b(imply)h(that)1296 5150 y Ff(F)12 b Fp(\()p Ff(z)t(;)i(h)p Fp(\))22 b Fe(\024)h Ff(C)6 b(h)1776 5115 y Fj(\000)p Fl(n)1873 5150 y Ff(;)97 b Fe(8)p Ff(z)26 b Fe(2)d Fp(\012)2243 5162 y Fm(+)p Fl(;)2325 5140 y Fh(\017)p 2324 5149 V 2324 5182 a Fk(2)2366 5150 y Ff(:)895 b Fp(\(4)p Ff(:)p Fp(27\))1801 5349 y(13)p eop %%Page: 14 14 14 13 bop 189 272 a Fp(Notice)31 b(that)h Ff(F)44 b Fp(is)31 b(harmonic.)48 b(Harnac)n(k's)29 b(inequalit)n(y)i(for)g(non-negativ)n (e)f(harmonic)h(functions)g(and)189 397 y(\(4.27\))c(giv)n(e)f(\(after) i(an)f(arbitrarily)f(small)h(decrease)f(of)i(\012)2043 409 y Fm(1)2108 397 y Fp(\))g(:)1339 629 y Ff(F)12 b Fp(\()p Ff(z)t(;)i(h)p Fp(\))22 b Fe(\024)h Ff(C)6 b(h)1819 595 y Fj(\000)p Fl(n)1916 629 y Ff(;)97 b Fe(8)p Ff(z)26 b Fe(2)d Fp(\012)2286 641 y Fm(1)2323 629 y Ff(:)938 b Fp(\(4)p Ff(:)p Fp(28\))189 861 y(Hence,)28 b(log)14 b Fe(j)p Ff(G)p Fe(j)23 b(\025)g(\000)p Ff(C)6 b(h)980 831 y Fj(\000)p Fl(n)1104 861 y Fp(on)28 b(\012)1280 873 y Fm(1)1317 861 y Ff(:)g Fp(Using)f(this)h(with)g(\(4.24\),)f(w)n (e)g(get)1372 1023 y Fg(\014)1372 1073 y(\014)1413 1094 y Fp(log)15 b Fe(j)p Ff(G)p Fe(j)1646 1023 y Fg(\014)1646 1073 y(\014)1697 1094 y Fe(\024)22 b Ff(C)6 b(h)1897 1060 y Fj(\000)p Fl(n)2078 1094 y Fp(on)27 b(\012)2253 1106 y Fm(1)2290 1094 y Ff(:)971 b Fp(\(4)p Ff(:)p Fp(29\))189 1326 y(Since)27 b(log)14 b Fe(j)p Ff(G)p Fp(\()p Ff(z)t(;)g(h)p Fp(\))p Fe(j)23 b Fp(=)f Fe(<)p Fp(\(log)15 b Ff(G)p Fp(\))27 b(is)f(harmonic,)g(w)n(e)g(get)h(after)f(an)g(arbitrarily)e (small)j(decrease)e(of)h(\012)3409 1338 y Fm(1)3473 1326 y Fp(:)1261 1559 y Fe(r)1330 1492 y Fg(\000)1368 1559 y Fe(<)p Fp(\(log)14 b Ff(G)p Fp(\))1678 1492 y Fg(\001)1740 1559 y Fp(=)23 b Fe(O)r Fp(\()p Ff(h)1976 1525 y Fj(\000)p Fl(n)2073 1559 y Fp(\))84 b(on)27 b(\012)2364 1571 y Fm(1)2401 1559 y Ff(:)860 b Fp(\(4)p Ff(:)p Fp(30\))189 1791 y(The)28 b(Cauc)n(h)n(y)e(Riemann)i(equations)f(for)g(log)14 b Ff(G)28 b Fp(imply)1367 1981 y Ff(d)p 1346 2018 86 4 v 1346 2094 a(dz)1455 2037 y Fp(log)14 b Ff(G)24 b Fp(=)e Fe(O)r Fp(\()p Ff(h)1900 2003 y Fj(\000)p Fl(n)1998 2037 y Fp(\))83 b(on)28 b(\012)2289 2049 y Fm(1)2326 2037 y Ff(:)935 b Fp(\(4)p Ff(:)p Fp(31\))189 2339 y(F)-7 b(or)27 b(the)h(curv)n(e)i(~)-45 b Ff(\015)746 2351 y Fm(1)806 2339 y Fe(2)23 b Fp(\012)c Fe(n)p 1023 2248 60 4 v 1030 2318 a Fg(e)1023 2339 y Fp(\012)27 b(\014xed)h(as)f(ab)r(o) n(v)n(e)f(and)h(from)h(\(4.18\),)f(\(4.19\),)g(w)n(e)g(get)1169 2476 y Fg(Z)1218 2665 y Fm(~)-35 b Fl(\015)1251 2673 y Fk(1)1301 2589 y Ff(g)s Fp(\()p Ff(z)t Fp(\))1481 2533 y Ff(d)p 1461 2570 86 4 v 1461 2646 a(dz)1569 2589 y Fp(log)15 b Ff(D)1760 2601 y Fl(w)1813 2589 y Fp(\()p Ff(z)t(;)f(h)p Fp(\))g Ff(dz)26 b Fp(=)d Fe(O)r Fp(\()p Ff(h)2363 2555 y Fj(\000)p Fl(n)2460 2589 y Fp(\))p Ff(:)769 b Fp(\(4)p Ff(:)p Fp(32\))189 2855 y(\(4.22\),)27 b(\(4.31\))g(and)g (\(4.32\))g(imply)h(that)1195 2993 y Fg(Z)1244 3181 y Fm(~)-36 b Fl(\015)1276 3189 y Fk(1)1327 3106 y Ff(g)s Fp(\()p Ff(z)t Fp(\))1507 3049 y Ff(d)p 1487 3087 V 1487 3163 a(dz)1595 3106 y Fp(log)14 b Ff(D)r Fp(\()p Ff(z)t(;)g(h)p Fp(\))g Ff(dz)26 b Fp(=)d Fe(O)r Fp(\()p Ff(h)2337 3071 y Fj(\000)p Fl(n)2435 3106 y Fp(\))p Ff(:)794 b Fp(\(4)p Ff(:)p Fp(33\))189 3371 y(Com)n(bining)27 b(\(4.13\),)g(\(4.14\))g(and) g(\(4.33\),)g(w)n(e)g(obtain)h(\(uniformly)f(with)h(resp)r(ect)g(to)f Ff(g)s Fp(\))1258 3604 y Fe(I)i Fp(=)1622 3525 y Fg(X)1420 3707 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)1671 3715 y Fk(1)1703 3707 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)p Fm(\012)1912 3715 y Fk(1)1875 3604 y Ff(g)s Fp(\()p Ff(z)t Fp(\))18 b(+)g Fe(O)r Fp(\()p Ff(h)2274 3569 y Fj(\000)p Fl(n)2372 3604 y Fp(\))p Ff(:)857 b Fp(\(4)p Ff(:)p Fp(34\))189 3927 y(Since)32 b Fe(j)p Ff(g)s Fe(j)f(\024)f Fp(1)h(on)h(\012)878 3939 y Fm(1)937 3927 y Fe(n)1007 3906 y Fg(e)1000 3927 y Fp(\012)g(and)g(the)h(fact)f(that)g(the)h(n)n(um)n(b)r(er)f(of)g(the) g(resonances)e(of)i Ff(P)3027 3939 y Fm(1)3065 3927 y Fp(\()p Ff(h)p Fp(\))g(in)h(\012)3371 3939 y Fm(1)3440 3927 y Fp(is)189 4051 y Fe(O)r Fp(\()p Ff(h)337 4021 y Fj(\000)p Fl(n)435 4051 y Fp(\),)28 b(w)n(e)f(ma)n(y)g(replace)f(the) i(sum)g(in)g(\(4.34\))f(b)n(y)1731 4205 y Fg(X)1520 4407 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)1771 4415 y Fk(1)1804 4407 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)1967 4403 y Fg(e)1966 4407 y Fm(\012)2013 4415 y Fi(\000)1992 4284 y Ff(g)s Fp(\()p Ff(z)t Fp(\))p Ff(;)189 4604 y Fp(whic)n(h)h(pro)n(v)n(es)d (Theorem)i(1.5.)2267 b Fa(\003)189 4842 y Fq(5.)42 b(Pro)s(of)31 b(of)h(the)g(lo)m(w)m(er)g(b)s(ound)355 5025 y Fp(As)25 b(indicated)h(in)f(the)h(in)n(tro)r(duction,)f(the)h(pro)r(of)f(is)g(a) g(simple)g(consequence)f(of)h(Theorem)g(1.3)f(and)189 5150 y(Theorem)f(1.5.)35 b(Let)25 b(\012)f(b)r(e)h(as)e(in)i(Theorem)f (1.5)f(and)h(let)h Ff(E)2047 5162 y Fm(0)2107 5150 y Fe(2)f Fp(\012)p Ff(:)g Fp(Supp)r(ose)g(that)h Ff(E)2853 5162 y Fm(0)2914 5150 y Fe(2)e Fp(singsupp)3307 5170 y Fl(a)3347 5150 y Fp(\()15 b Fg(e)-61 b Ff(w)s Fp(\))p Ff(:)1801 5349 y Fp(14)p eop %%Page: 15 15 15 14 bop 355 272 a Fp(Since)51 b Fg(e)-61 b Ff(w)39 b Fp(is)c(real,)i(it)g(is)e(clear)g(that)h(\()p Ff(E)1635 284 y Fm(0)1673 272 y Ff(;)14 b Fp(1\))36 b(and)g(\()p Ff(E)2083 284 y Fm(0)2120 272 y Ff(;)14 b Fe(\000)p Fp(1\))36 b(are)e(in)j(WF)2723 284 y Fl(a)2763 272 y Fp(\()15 b Fg(e)-61 b Ff(w)s Fp(\))p Ff(:)36 b Fp(It)h(follo)n(ws)d(from)189 397 y(the)d(de\014nition)f(of)h(the)f(analytic)g(w)n(a)n(v)n(e)f(fron)n (t)h(set)g(b)n(y)g(the)h(F.B.I)f(transformation)f(that)i(:)42 b(there)30 b(exist)189 521 y(sequences)d(\()p Ff(\013)651 533 y Fl(j)686 521 y Ff(;)14 b(\014)770 533 y Fl(j)805 521 y Fp(\))23 b Fe(!)g Fp(\()p Ff(E)1059 533 y Fm(0)1097 521 y Ff(;)14 b Fp(1\))p Ff(;)28 b(\025)1307 533 y Fl(j)1365 521 y Fe(!)23 b Fp(+)p Fe(1)28 b Fp(and)f Ff(\017)1842 533 y Fl(j)1900 521 y Fe(!)c Fp(0)k(suc)n(h)g(that)1241 674 y Fg(\014)1241 724 y(\014)1241 773 y(\014)1282 656 y(Z)1379 702 y(\000)1417 769 y Ff(f)1458 781 y Fl(j)1493 769 y Ff(\037)1545 781 y Fm(0)1582 702 y Fg(\001)1620 769 y Fp(\()p Ff(E)5 b Fp(\))t Fg(e)-50 b Ff(!)s Fp(\()p Ff(E)5 b Fp(\))14 b Ff(dE)2058 674 y Fg(\014)2058 724 y(\014)2058 773 y(\014)2110 769 y Fe(\025)22 b Ff(e)2236 735 y Fj(\000)p Fl(\017)2316 743 y Fh(j)2347 735 y Fl(\025)2386 743 y Fh(j)2421 769 y Ff(;)881 b Fp(\(5)p Ff(:)p Fp(1\))189 1052 y(where)29 b Ff(f)472 1064 y Fl(j)506 1052 y Fp(\()p Ff(E)5 b Fp(\))27 b(=)e Ff(e)792 1022 y Fl(i\025)854 1030 y Fh(j)885 1022 y Fm(\()p Fl(\013)954 1030 y Fh(j)986 1022 y Fj(\000)p Fl(E)s Fm(\))p Fl(\014)1154 1030 y Fh(j)1188 1052 y Ff(e)1227 1022 y Fj(\000)1289 981 y Fh(\025)1324 994 y(j)p 1289 1009 66 4 v 1307 1042 a Fk(2)1365 1022 y Fm(\()p Fl(\013)1434 1030 y Fh(j)1465 1022 y Fj(\000)p Fl(E)s Fm(\))1595 997 y Fk(2)1660 1052 y Fp(and)k Ff(\037)1875 1064 y Fm(0)1942 1052 y Fp(is)g(a)g(cut-o\013)g(function)g(de\014ned)h (in)f(a)g(small)g(real)189 1177 y(neigh)n(b)r(orho)r(o)r(d)d(of)i Ff(E)866 1189 y Fm(0)903 1177 y Ff(;)g Fe(\021)23 b Fp(1)k(near)g Ff(E)1357 1189 y Fm(0)1394 1177 y Ff(:)h Fp(See)g([24].)355 1328 y(Let)40 b Ff(a)f Fp(b)r(e)h(a)f(small)h(p)r(ositiv)n(e)f(constan) n(t.)72 b(Set)40 b(\012)1972 1340 y Fm(0)2052 1328 y Fp(:=])p Ff(E)2224 1340 y Fm(0)2288 1328 y Fe(\000)26 b Fp(2)p Ff(a;)14 b(E)2563 1340 y Fm(0)2626 1328 y Fp(+)26 b(2)p Ff(a)p Fp([+)p Ff(i)p Fp(])f Fe(\000)h Fp(2)p Ff(a)3145 1298 y Fm(2)3182 1328 y Ff(;)14 b(a)3263 1298 y Fm(2)3300 1328 y Fp(])39 b(and)196 1432 y Fg(e)189 1453 y Fp(\012)23 b(:=])p Ff(E)444 1465 y Fm(0)500 1453 y Fe(\000)18 b Ff(a;)c(E)725 1465 y Fm(0)780 1453 y Fp(+)k Ff(a)p Fp([+)p Ff(i)p Fp(])g Fe(\000)g Ff(a)1192 1423 y Fm(2)1229 1453 y Ff(;)c(a)1310 1423 y Fm(2)1347 1453 y Fp(])p Ff(:)28 b Fp(By)f(construction,)g(there)h(exist)f Ff(C)2518 1465 y Fm(0)2579 1453 y Ff(>)22 b Fp(0)28 b(suc)n(h)f(that)997 1610 y Fg(\014)997 1660 y(\014)1025 1681 y Ff(f)1066 1693 y Fl(j)1101 1681 y Fp(\()p Ff(E)5 b Fp(\))1231 1610 y Fg(\014)1231 1660 y(\014)1282 1681 y Fe(\024)23 b Ff(e)1409 1639 y Fj(\000)1493 1617 y Fk(1)p 1470 1626 74 4 v 1470 1659 a Fh(C)1511 1671 y Fk(0)1554 1639 y Fl(\025)1593 1647 y Fh(j)1711 1681 y Fp(uniformly)29 b(on)83 b Ff(E)28 b Fe(2)23 b Fp(\012)2489 1693 y Fm(0)2545 1681 y Fe(n)2612 1660 y Fg(e)2605 1681 y Fp(\012)p Ff(:)637 b Fp(\(5)p Ff(:)p Fp(2\))189 1908 y(Let)31 b Ff(\037)e Fe(2)g Ff(C)571 1878 y Fj(1)565 1929 y Fm(0)642 1841 y Fg(\000)680 1908 y Fp(])p Ff(E)764 1920 y Fm(0)822 1908 y Fe(\000)20 b Fp(2)p Ff(a;)14 b(E)1091 1920 y Fm(0)1149 1908 y Fp(+)20 b(2)p Ff(a)p Fp([;)14 b([0)p Ff(;)g Fp(1])1547 1841 y Fg(\001)1584 1908 y Ff(;)45 b Fe(\021)28 b Fp(1)i(on)h(])p Ff(E)2020 1920 y Fm(0)2079 1908 y Fe(\000)20 b Ff(a;)14 b(E)2306 1920 y Fm(0)2364 1908 y Fp(+)20 b Ff(a)p Fp([)p Ff(:)31 b Fp(Then)g(the)h(inequalit)n(y)e(\(5.1\))189 2033 y(remains)d(true)g(when)h(w)n(e)f(replace)g Ff(\037)1347 2045 y Fm(0)1412 2033 y Fp(b)n(y)g Ff(\037:)189 2185 y Fp(Theorem)g(1.3)g(giv)n(es)699 2425 y(tr)764 2358 y Fg(\002)799 2425 y Ff(f)840 2437 y Fl(j)874 2425 y Fp(\()p Ff(P)959 2437 y Fm(1)997 2425 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff(f)1284 2437 y Fl(j)1319 2425 y Fp(\()p Ff(P)1404 2437 y Fm(0)1442 2425 y Fp(\))1474 2358 y Fg(\003)1532 2425 y Fp(=)k Ff(h)1667 2391 y Fj(\000)p Fl(n)1778 2312 y Fg(Z)1875 2358 y(\000)1913 2425 y Ff(f)1954 2437 y Fl(j)1989 2425 y Ff(\037)2041 2358 y Fg(\001)2079 2425 y Fp(\()p Ff(E)5 b Fp(\))t Fg(e)-50 b Ff(!)s Fp(\()p Ff(E)5 b Fp(\))14 b Ff(dE)24 b Fp(+)18 b Fe(O)2685 2437 y Fl(j)2720 2425 y Fp(\()p Ff(h)2800 2391 y Fm(1)p Fj(\000)p Fl(n)2930 2425 y Fp(\))p Ff(:)340 b Fp(\(5)p Ff(:)p Fp(3\))189 2666 y(On)27 b(the)h(other)f(hand,)h(applying)f(Theorem)g(1.5)f(with)j Ff(g)c Fp(=)e Ff(f)2125 2678 y Fl(j)2187 2666 y Fp(and)28 b(\012)f(replaced)g(b)n(y)g(\012)2939 2678 y Fm(0)2976 2666 y Ff(;)h Fp(w)n(e)f(ha)n(v)n(e)662 2894 y(tr)727 2826 y Fg(\002)762 2894 y Ff(f)803 2906 y Fl(j)837 2894 y Fp(\()p Ff(P)922 2906 y Fm(1)960 2894 y Fp(\()p Ff(h)p Fp(\)\))19 b Fe(\000)f Ff(f)1247 2906 y Fl(j)1282 2894 y Fp(\()p Ff(P)1367 2906 y Fm(0)1405 2894 y Fp(\))1437 2826 y Fg(\003)1495 2894 y Fp(=)1793 2815 y Fg(X)1582 3017 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)1833 3025 y Fk(1)1866 3017 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)2029 3013 y Fg(e)2028 3017 y Fm(\012)2075 3025 y Fi(\000)2138 2894 y Ff(f)2179 2906 y Fl(j)2213 2894 y Fp(\()p Ff(z)t Fp(\))h(+)f Fe(O)r Fp(\(1\))p Ff(h)2644 2859 y Fj(\000)p Fl(n)2741 2894 y Ff(e)2780 2852 y Fj(\000)2864 2830 y Fk(1)p 2841 2839 V 2841 2872 a Fh(C)2882 2884 y Fk(0)2925 2852 y Fl(\025)2964 2860 y Fh(j)3000 2894 y Ff(;)302 b Fp(\(5)p Ff(:)p Fp(4\))189 3233 y(where)436 3212 y Fg(e)429 3233 y Fp(\012)489 3245 y Fj(\000)568 3233 y Fp(=)23 b Fe(f)p Ff(w)i Fe(2)867 3212 y Fg(e)860 3233 y Fp(\012;)42 b Fe(=)p Fp(\()p Ff(w)r Fp(\))24 b Fe(\024)f Fp(0)p Fe(g)p Ff(:)k Fp(Com)n(bining)g(\(5.3\))g (and)g(\(5.4\))h(w)n(e)f(get)647 3399 y Fg(X)437 3602 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)688 3610 y Fk(1)720 3602 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)883 3598 y Fg(e)882 3602 y Fm(\012)929 3610 y Fi(\000)867 3478 y Ff(f)908 3490 y Fl(j)943 3478 y Fp(\()p Ff(z)t Fp(\))c(=)g Ff(h)1209 3444 y Fj(\000)p Fl(n)1278 3365 y Fg(Z)1375 3411 y(\000)1413 3478 y Ff(f)1454 3490 y Fl(j)1488 3478 y Ff(\037)1540 3411 y Fg(\001)1578 3478 y Fp(\()p Ff(E)5 b Fp(\))t Fg(e)-50 b Ff(!)t Fp(\()p Ff(E)5 b Fp(\))14 b Ff(dE)24 b Fp(+)18 b Fe(O)r Fp(\()p Ff(h)2267 3444 y Fj(\000)p Fl(n)2365 3478 y Fp(\))p Ff(e)2436 3437 y Fj(\000)2520 3414 y Fk(1)p 2497 3423 V 2497 3457 a Fh(C)2538 3469 y Fk(0)2581 3437 y Fl(\025)2620 3445 y Fh(j)2674 3478 y Fp(+)g Fe(O)2823 3490 y Fl(j)2858 3478 y Fp(\()p Ff(h)2938 3444 y Fm(1)p Fj(\000)p Fl(n)3068 3478 y Fp(\))p Ff(;)189 3799 y Fp(whic)n(h)28 b(together)e(with)i(\(5.1\))g(imply)659 3931 y Fg(\014)659 3981 y(\014)659 4031 y(\014)911 3948 y(X)700 4150 y Fl(z)r Fj(2)p Fm(Res\()p Fl(P)951 4158 y Fk(1)984 4150 y Fm(\()p Fl(h)p Fm(\)\))p Fj(\\)1147 4146 y Fg(e)1146 4150 y Fm(\012)1193 4158 y Fi(\000)1255 4027 y Ff(f)1296 4039 y Fl(j)1331 4027 y Fp(\()p Ff(z)t Fp(\))1438 3931 y Fg(\014)1438 3981 y(\014)1438 4031 y(\014)1489 4027 y Fe(\025)22 b Ff(h)1624 3992 y Fj(\000)p Fl(n)1721 3934 y Fg(h)1760 4027 y Ff(e)1799 3992 y Fj(\000)p Fl(\017)1879 4000 y Fh(j)1910 3992 y Fl(\025)1949 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y Fl(j)735 4901 y Fp(\(with)g Ff(j)41 b Fp(\014xed\))35 b(is)h(indep)r(enden)n(t)g(of)g Ff(h:)f Fp(The)h(inequalit)n(y)f(\(5.5\))g(giv)n(es)f(the)i(required) 189 5025 y(lo)n(w)n(er)27 b(b)r(ound)i(on)g(the)g(n)n(um)n(b)r(er)f(of) h(resonances)e(in)1842 5004 y Fg(e)1835 5025 y Fp(\012)1895 5037 y Fj(\000)1980 5025 y Fp(and)h(since)h(w)n(e)f(can)h(c)n(ho)r(ose) 2895 5004 y Fg(e)2888 5025 y Fp(\012)g(as)f(small)g(as)h(w)n(e)189 5150 y(lik)n(e,)e(w)n(e)g(get)h(the)g(theorem)f(1.6.)2210 b Fa(\003)1801 5349 y Fp(15)p eop %%Page: 16 16 16 15 bop 189 272 a Fo(A)l(cknow)t(le)l(dgements)34 b Fp(:)48 b(The)33 b(authors)f(are)g(grateful)g(to)h(J.)g(Sj\177)-42 b(ostrand)32 b(for)h(v)-5 b(aluable)32 b(commen)n(ts)h(con-)189 397 y(cerning)f(this)h(w)n(ork.)51 b(W)-7 b(e)33 b(w)n(ould)g(lik)n(e)f (to)h(express)e(our)h(gratitude)h(to)f(V.)i(P)n(etk)n(o)n(v,)e(A.)h(V) -7 b(asy)32 b(and)h(M.)189 521 y(Zw)n(orski)19 b(for)i(in)n(vitations)g (to)g(\\Programme)d(in)j(Scattering)g(Theory")e(at)i(Erwin)g(Sc)n (hr\177)-42 b(odinger)19 b(Institute)189 646 y(\(Vienna\))28 b(where)f(a)g(part)h(of)f(this)h(w)n(ork)e(w)n(as)h(done.)189 826 y Fq(References)k(:)189 979 y Fp([1])25 b(J.)h(Aguilar)e(and)i (J-M.)f(Com)n(b)r(es,)g(A)h(class)f(of)g(analytic)g(p)r(erturbations)g (for)g(one-b)r(o)r(dy)g(Sc)n(hr\177)-42 b(odinger)189 1104 y(Hamiltonians.)36 b Fo(Comm.)k(Math.)f(Phys.)h Fq(22)p Fp(\(1971\),)26 b(269{279.)189 1257 y([2])f(E.)g(Bardos,)g (J-C.)g(Guillot)g(and)h(J.)f(Ralston,)g(La)g(relation)g(de)h(P)n (oisson)d(p)r(our)i(l')n(\023)-39 b(equation)24 b(des)i(ondes)189 1382 y(dans)h(un)h(ouv)n(ert)f(non-b)r(orn)n(\023)-39 b(e.)34 b Fo(Comm.)40 b(PDE.)30 b Fq(7)p Fo(,)p Fp(\(1982\),)d (905{958.)189 1535 y([3])e(J.-F,)g(Bon)n(y)-7 b(,)25 b(Ma)5 b(joration)23 b(du)j(nom)n(bre)e(de)h(r)n(\023)-39 b(esonances)23 b(dans)i(un)g(domaine)g(de)g(taille)g Ff(h:)h Fo(to)h(app)l(e)l(ar.)189 1689 y Fp([4])32 b(J.-F.)f(Bon)n(y)g (and)h(J.)g(Sj\177)-42 b(ostrand,)32 b(T)-7 b(race)31 b(form)n(ula)g(for)g(resonances)f(in)i(small)g(domains,)g Fo(J.)i(F)-6 b(unct.)189 1814 y(A)n(nal.)39 b Fq(184)p Fo(,)p Fp(no.)d(2)27 b(\(2001\),)f(402{418.)189 1967 y([5])34 b(V.)h(Bruneau)f(and)g(V.)h(P)n(etk)n(o)n(v,)f(Represen)n (tation)f(of)i(the)f(sp)r(ectral)g(shift)h(function)g(and)g(sp)r (ectral)189 2092 y(asymptotics)27 b(for)g(trapping)g(p)r(erturbations)g Fo(to)i(app)l(e)l(ar)j(in)d(Commun.)39 b(P.D.E.)189 2245 y Fp([6])e(V.)h(Bruneau)f(and)h(V.)g(P)n(etk)n(o)n(v,)g(Meromorphic)e (con)n(tin)n(uation)h(of)g(the)h(sp)r(ectral)f(shift)i(function.)189 2370 y Fo(Pr)l(eprint)30 b(Mar)l(ch)h(2001.)189 2523 y Fp([7])25 b(V.)g(S.)h(Buslaev,)e(Semiclassical)g(appro)n(ximation)f (for)i(equations)f(with)i(p)r(erio)r(dic)f(co)r(e\016cien)n(ts.)35 b Fo(R)n(us-)189 2648 y(sian.)k(Math.)h(Surveys,)30 b Fq(42)p Fp(\(1987\),)c(97{125.)189 2801 y([8])39 b(H-L.)g(Cycon,)i Fo(R)l(esonanc)l(es)f(de\014ne)l(d)h(by)f(mo)l(di\014e)l(d)i (dilations.)72 b Fp(Helv.)f(Ph)n(ys.)f(Acta)39 b(58\(1985\),)189 2926 y(969{981.)189 3079 y([9])32 b(M.)h(Dimassi,)h(D)n(\023)-39 b(ev)n(elopp)r(emen)n(ts)31 b(asymptotiques)h(des)g(p)r(erturbations)g (len)n(tes)g(de)h(l'op)n(\023)-39 b(erateur)30 b(de)189 3204 y(Sc)n(hr\177)-42 b(odinger)26 b(p)n(\023)-39 b(erio)r(dique.)35 b(Comm.)i(P)-7 b(.D.E.,)27 b Fq(18)p Fp(\(5&)g(6\),)g(\(1993\),)g (771{803.)189 3357 y([10])36 b(M.)g(Dimassi,)j(Resonances)c(for)h(a)g (slo)n(wly)g(v)-5 b(arying)35 b(p)r(erturbation)h(of)g(a)h(p)r(erio)r (dic)f(Sc)n(hr\177)-42 b(odinger)189 3482 y(op)r(erator.)35 b Fo(to)30 b(app)l(e)l(ar)h(in)f(Can.)39 b(J.)30 b(Math.)189 3635 y Fp([11])f(M.)g(Dimassi)h(and)f(J.)g(Sj\177)-42 b(ostrand,)30 b(Sp)r(ectral)f(asymptotics)g(in)h(the)g(semi-classical)d (limit.)44 b Fo(L)l(ondon)189 3760 y(Math.)c(So)l(c.)e(L)l(e)l(ctur)l (e)29 b(Note)h(Series,)g Fq(268)p Fo(.)p Fp(Cam)n(bridge)c(Univ)n (ersit)n(y)h(Press,)f(Cam)n(bridge,)g(\(1999\).)189 3913 y([12])36 b(C.)h(G)n(\023)-39 b(erard,)37 b(Resonance)e(theory)h(for)g (p)r(erio)r(dic)h(Sc)n(hr\177)-42 b(odinger)35 b(op)r(erators.)62 b Fo(Bul)t(l.)j(So)l(c.)f(Math.)189 4038 y(F)-6 b(r)l(anc)l(e,)30 b Fq(118)p Fp(\(1990\),)c(27{54.)189 4191 y([13])33 b(C.)h(G)n(\023)-39 b(erard,)33 b(A.)h(Martinez)g(and)f(J.)h(Sj\177)-42 b(ostrand,)35 b(A)f(Mathematical)g(Approac)n(h)e(to)i(the)g(e\013ectiv)n(e)189 4316 y(Hamiltonian)26 b(in)h(p)r(erturb)r(ed)g(p)r(erio)r(dic)f (Problems.)35 b Fo(Commun.)k(Math.)g(Phys.,)31 b Fq(142)p Fp(\(1991\),)24 b(217{244.)189 4469 y([14])d(J.)g(C.)h(Guillot,)h(J.)e (Ralston)g(and)g(E.)g(T)-7 b(rub)r(o)n(witz,)23 b(Semi-classical)d (metho)r(ds)i(in)f(solid)g(state)h(ph)n(ysics.)189 4594 y Fo(Comm.)39 b(Math.)h(Phys.,)31 b Fq(116)p Fp(\(1988\),)26 b(401{415.)189 4747 y([15])19 b(B.)g(Hel\013er)h(and)g(J.)f(Sj\177)-42 b(ostrand,)1334 4726 y(\023)1327 4747 y(Equation)19 b(de)g(Sc)n(hr\177) -42 b(odinger)18 b(a)n(v)n(ec)g(c)n(hamp)i(magn)n(\023)-39 b(etique)18 b(et)f(\023)-39 b(equation)189 4872 y(de)28 b(Harp)r(er.)36 b Fo(Springer)30 b(L)l(e)l(ctur)l(e)f(Notes)g(in)h (Physics)i Fq(354)p Fp(\(1989\),)25 b(118{197.)189 5025 y([16])i(B.)i(Hel\013er)f(and)g(J.)g(Sj\177)-42 b(ostrand,)28 b(On)g(diamagnetism)f(and)h(Haas-v)-5 b(an)27 b(Alphen)i(e\013ect.)39 b Fo(A)n(nn.)g(Inst.)189 5150 y(H.)30 b(Poinc)l(ar)n(\023)-40 b(e)31 b(Phys.)40 b(Th)n(\023)-40 b(eor.)40 b Fq(52)p Fp(\(1990\),)26 b(303{375.)1801 5349 y(16)p eop %%Page: 17 17 17 16 bop 189 272 a Fp([17])28 b(P)-7 b(.)28 b(Kuc)n(hmen)n(t,)h(B.)g (V)-7 b(ain)n(b)r(erg,)28 b(On)h(absence)f(of)g(em)n(b)r(edded)h(eigen) n(v)-5 b(alues)28 b(for)g(Sc)n(hr\177)-42 b(odinger)27 b(op)r(er-)189 397 y(ators)f(with)i(p)r(erturb)r(ed)g(p)r(erio)r(dic)g (p)r(oten)n(tials.)36 b Fo(Comm.)k(P.D.E.,)31 b Fq(25)p Fp(\(2000\),)26 b(1809{1826.)189 546 y([18])h(P)-7 b(.)27 b(Lax,)g(R.)h(Phillips,)f(Scattering)g(theory)-7 b(.)36 b Fo(A)l(c)l(ademic)31 b(Pr)l(ess,)g(New)e(Y)-6 b(ork,)p Fp(\(1967\).)189 695 y([19])27 b(R-B.)g(Melrose,)g(Scattering)f(theory) h(and)h(the)g(trace)f(of)g(w)n(a)n(v)n(e)f(group.)36 b Fo(J.)30 b(F)-6 b(unct.)37 b(A)n(n.)h Fq(45)p Fp(\(1982\),)189 820 y(429{440.)189 969 y([20])32 b(L.)h(Nedelec,)h(Lo)r(calisation)e (of)g(resonances)f(for)i(matrix)f(Sc)n(hr\177)-42 b(odinger)31 b(op)r(erators.)51 b Fo(Duke)34 b(Math.)189 1094 y(J.)c Fq(106)p Fo(,)f Fp(no.)37 b(2)27 b(\(2001\),)f(209{236.)189 1243 y([21])k(V.)h(P)n(etk)n(o)n(v)d(and)i(M.)h(Zw)n(orski,)e (Semi-classical)g(estimates)h(on)h(the)f(scattering)g(determinan)n(t.) 45 b Fo(to)189 1368 y(app)l(e)l(ar)31 b(in)f(A)n(nn.)38 b(Inst.)f(H.)30 b(Poinc)l(ar)n(\023)-40 b(e.)189 1517 y Fp([22])28 b(M.)h(Reed)g(and)g(B.Simon,)g(Metho)r(ds)h(of)e(Mo)r (dern)h(Mathematical)g(Ph)n(ysics,)f(IV,)h(analysis)f(op)r(era-)189 1642 y(tors.)36 b Fo(A)l(c)l(ademic)31 b(Pr)l(ess,)f(New)g(Y)-6 b(ork-L)l(ondon,)p Fp(\(1978\).)189 1791 y([23])18 b(M-A.)h(Sh)n(ubin,) i(The)e(sp)r(ectral)f(theory)g(and)g(the)h(index)g(of)g(elliptic)g(op)r (erators)e(with)i(almost)f(p)r(erio)r(dic)189 1916 y(co)r(e\016cien)n (ts.)36 b Fo(R)n(ussian)30 b(Math.)39 b(Surveys)30 b Fq(34)p Fp(\(2\),)d(\(1979\))f(109{157.)189 2065 y([24])h(J.)g(Sj\177) -42 b(ostrand,)27 b(Singularit)n(\023)-39 b(es)26 b(analytiques)h (microlo)r(cales.)35 b Fo(SMF.)30 b(Ast)n(\023)-40 b(erisque)30 b Fq(95)p Fp(\(1982\).)189 2215 y([25])j(J.)h(Sj\177)-42 b(ostrand,)35 b(Microlo)r(cal)e(analysis)f(for)i(the)g(p)r(erio)r(dic)g (magnetic)g(Sc)n(hr\177)-42 b(odinger)32 b(equation)h(and)189 2339 y(related)27 b(questions.)36 b Fo(Springer)31 b(V)-6 b(erl.)38 b(L)l(e)l(ct.)g(Not.)h(in)29 b(Math.,)j Fq(1495)d Fp(\(1991\),)d(237-332.)189 2489 y([26])32 b(J.)h(Sj\177)-42 b(ostrand,)35 b(A)e(trace)g(form)n(ula)f(and)h(review)f(of)i(somme)e (estimates)h(for)g(resonances.)52 b Fo(In)34 b(Mi-)189 2613 y(cr)l(olo)l(c)l(al)d(analysis)g(and)f(sp)l(e)l(ctr)l(al)g(the)l (ory,)h(Kluwer,)f(NA)-6 b(TO)29 b(ASI)g(Series)i(C)f Fq(490)p Fp(\(1997\),)25 b(377{437.)189 2763 y([27])41 b(J.)h(Sj\177)-42 b(ostrand,)46 b(Resonances)41 b(for)g(the)i(b)r (ottles)f(and)g(trace)g(form)n(ulae.)79 b Fo(Math.)h(Nachrichten,)189 2887 y Fq(221)p Fp(\(2001\),)26 b(95{149.)189 3037 y([28])20 b(J.)h(Sj\177)-42 b(ostrand,)-8 b(A)21 b(trace)f(form)n(ula)g(for)h (resonances)e(and)i(application)g(to)g(semi-classical)e(Sc)n(hr\177)-42 b(odinger)189 3161 y(op)r(erators.)35 b Fo(S)n(\023)-40 b(eminair)l(e)28 b(\023)-40 b(equations)30 b(aux)g(d)n(\023)-40 b(eriv)n(\023)g(ees)31 b(p)l(artiel)t(les,)h(exp)l(os)n(\023)-40 b(e)p Fp(no)28 b(11)e(\(1996-97\).)189 3311 y([29])32 b(J.)g(Sj\177)-42 b(ostrand)32 b(and)g(M.)h(Zw)n(orski,)f(Lo)n(w)n(er)e (b)r(ounds)j(on)f(the)h(n)n(um)n(b)r(er)f(of)g(scattering)g(p)r(oles,)h (I)r(I.)g Fo(J.)189 3435 y(F)-6 b(unct.)37 b(A)n(n.,)30 b Fq(123)p Fp(\(2\),)d(\(1994\))g(336{367.)189 3585 y([30])19 b(M.)h(Zw)n(orski,)g(P)n(oisson)e(form)n(ula)h(for)g(resonances)f(in)i (ev)n(en)g(dimensions.)34 b Fo(Asian)23 b(J.)f(Math.)38 b Fq(2)p Fp(\(1998\),)189 3709 y(615{624.)189 3883 y Fq(Classi\014cation)31 b(AMS)c Fp(:)h(35P99,)d(47A60,)h(47A40.)189 4008 y Fq(KEYW)m(ORDS)h Fp(:)h(p)r(erio)r(dic,)f(resonance,)f(trace.) 1801 5349 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0112210438758--