Content-Type: multipart/mixed; boundary="-------------0305130249990" This is a multi-part message in MIME format. ---------------0305130249990 Content-Type: text/plain; name="03-220.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-220.keywords" spectral shift function, resonances, Stark hamiltonians ---------------0305130249990 Content-Type: application/postscript; name="ssfgen21.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ssfgen21.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ssfgen21.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o ssfgen21.ps ssfgen21 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.13:0911 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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Fl(Q)1902 1329 y Fk(#)1902 1400 y Fg(j)1965 1374 y Fn(\(1)i Fj(\000)f Fl(')p Fn(\))p Fl(u;)37 b(u)25 b Fj(2)g(D)2601 1329 y Fk(#)2598 1400 y Fg(j)2663 1374 y Fl(;)0 1533 y Fn(with)k(domain)1083 1653 y Fj(D)1156 1608 y Fk(#)1153 1678 y Fg(j)1243 1653 y Fn(=)c Fj(f)p Fl(u)h Fj(2)f(H)1626 1608 y Fk(#)1625 1678 y Fg(j)1714 1653 y Fn(:)46 b Fl('u)25 b Fj(2)g(D)2077 1667 y Fg(j)2114 1653 y Fl(;)35 b Fn(\(1)21 b Fj(\000)f Fl(')p Fn(\))p Fl(u)26 b Fj(2)f Fl(H)2707 1615 y Fk(2)2747 1653 y Fj(g)p Fl(;)0 1815 y Fn(where)30 b Fl(')c Fj(2)e Fl(C)505 1782 y Fi(1)498 1838 y Fk(0)580 1815 y Fn(\()p Fl(B)5 b Fn(\(0)p Fl(;)15 b(R)q Fn(\);)g([0)p Fl(;)g Fn(1]\))34 b(is)29 b(equal)h(to)h(1)g(near)p 1923 1737 339 4 v 30 w Fl(B)5 b Fn(\(0)p Fl(;)15 b(R)2186 1829 y Fk(0)2226 1815 y Fn(\))q Fl(:)150 1937 y Fn(Denote)39 b(b)m(y)e Fl(N)10 b Fn(\()p Fl(L)782 1892 y Fk(#)782 1962 y Fg(j)845 1937 y Fl(;)15 b Fn([)p Fj(\000)p Fl(\025;)g(\025)p Fn(]\))39 b(the)e(n)m(um)m(b)s(er)f(of)h(eigen)m(v)-5 b(alues)37 b(of)h Fl(L)2487 1892 y Fk(#)2487 1962 y Fg(j)2587 1937 y Fn(in)e(the)h(in)m(terv)-5 b(al)36 b([)p Fj(\000)p Fl(\025;)15 b(\025)p Fn(].)62 b(Then)36 b(w)m(e)0 2047 y(assume)30 b(that)1002 2250 y Fl(N)10 b Fn(\()p Fl(L)1182 2206 y Fk(#)1182 2276 y Fg(j)1246 2250 y Fl(;)15 b Fn([)p Fj(\000)p Fl(\025;)g(\025)p Fn(]\))26 b(=)f Fj(O)s Fn(\()1820 2156 y Ff(\020)1899 2188 y Fl(\025)p 1880 2229 92 4 v 1880 2312 a(h)1932 2286 y Fk(2)1982 2156 y Ff(\021)2031 2172 y Fg(n)2074 2139 y Ft(#)2074 2195 y Fq(j)2128 2172 y Fg(=)p Fk(2)2203 2250 y Fn(\))p Fl(;)36 b(n)2354 2206 y Fk(#)2354 2276 y Fg(j)2442 2250 y Fj(\025)25 b Fl(n;)35 b(\025)25 b Fj(\025)g Fn(1)p Fl(:)816 b Fn(\(2.9\))150 2454 y(Giv)m(en)33 b Fl(f)38 b Fj(2)29 b Fl(C)661 2421 y Fi(1)654 2478 y Fk(0)735 2454 y Fn(\()p Fh(R)s Fn(\),)40 b(indep)s(enden)m(t)30 b(on)j Fl(h)p Fn(,)g(and)g Fl(\037)c Fj(2)f Fl(C)2113 2421 y Fi(1)2106 2478 y Fk(0)2187 2454 y Fn(\()p Fh(R)2282 2421 y Fg(n)2335 2454 y Fn(\))33 b(equal)g(to)g(1)g(on)p 2966 2376 339 4 v 33 w Fl(B)5 b Fn(\(0)p Fl(;)15 b(R)3229 2468 y Fk(0)3269 2454 y Fn(\),)34 b(w)m(e)f(can)g(de\014ne)0 2562 y(tr)71 2577 y Fk(bb)154 2562 y Fn([)p Fl(f)10 b Fn(\()p Fl(L)331 2576 y Fg(j)367 2562 y Fn(\)])427 2529 y Fk(2)427 2587 y Fg(j)t Fk(=1)554 2562 y Fn(,)31 b(as)f(in)f([34)q(],)i([35)q(],)g(b)m(y)g(the)f(equalit) m(y)470 2762 y(tr)541 2777 y Fk(bb)624 2668 y Ff(\020)673 2762 y Fl(f)10 b Fn(\()p Fl(L)825 2776 y Fk(2)864 2762 y Fn(\))21 b Fj(\000)f Fl(f)10 b Fn(\()p Fl(L)1163 2776 y Fk(1)1202 2762 y Fn(\))1237 2668 y Ff(\021)1312 2762 y Fn(=)25 b([tr\()p Fl(\037f)10 b Fn(\()p Fl(L)1748 2776 y Fg(j)1784 2762 y Fn(\))p Fl(\037)21 b Fn(+)f Fl(\037f)10 b Fn(\()p Fl(L)2197 2776 y Fg(j)2232 2762 y Fn(\)\(1)22 b Fj(\000)e Fl(\037)p Fn(\))g(+)g(\(1)h Fj(\000)f Fl(\037)p Fn(\))p Fl(f)10 b Fn(\()p Fl(L)3099 2776 y Fg(j)3135 2762 y Fn(\))p Fl(\037)p Fn(\)])3287 2724 y Fk(2)3287 2784 y Fg(j)t Fk(=1)1375 2970 y Fn(+tr)o([\(1)21 b Fj(\000)f Fl(\037)p Fn(\))p Fl(f)10 b Fn(\()p Fl(L)1977 2984 y Fg(j)2014 2970 y Fn(\)\(1)21 b Fj(\000)f Fl(\037)p Fn(\)])2358 2933 y Fk(2)2358 2993 y Fg(j)t Fk(=1)2500 2970 y Fl(:)0 3123 y Fn(F)-8 b(ollo)m(wing)29 b([34)q(],)i([35)q(],)f(w)m(e)h (de\014ne)e(the)h(resonances)g Fl(w)e Fj(2)p 1981 3051 66 4 v 25 w Fh(C)2046 3137 y Fi(\000)2135 3123 y Fn(b)m(y)i(the)g (complex)f(scaling)g(metho)s(d)h(as)g(the)g(eigen-)0 3231 y(v)-5 b(alues)34 b(of)h(the)g(complex)g(scaling)f(op)s(erators)h Fl(L)1677 3246 y Fg(j;\022)1764 3231 y Fl(;)j(j)h Fn(=)32 b(1)p Fl(;)15 b Fn(2.)56 b(W)-8 b(e)36 b(denote)f(b)m(y)g(Res)23 b Fl(L)3034 3245 y Fg(j)3071 3231 y Fn(\()p Fl(h)p Fn(\))p Fl(;)39 b(j)g Fn(=)32 b(1)p Fl(;)15 b Fn(2)p Fl(;)36 b Fn(the)f(set)0 3358 y(of)h(resonances)h(and)e(set)i Fl(n)951 3325 y Fk(#)1049 3358 y Fn(=)d(max)p Fj(f)p Fl(n)1423 3313 y Fk(#)1423 3382 y(1)1510 3358 y Fl(;)15 b(n)1605 3313 y Fk(#)1605 3382 y(2)1668 3358 y Fj(g)p Fl(:)37 b Fn(In)f(the)g(pap)s(er)f([)p Fl(a)2391 3372 y Fg(j)2428 3358 y Fn(])2453 3325 y Fk(2)2453 3382 y Fg(j)t Fk(=1)2616 3358 y Fn(means)h Fl(a)2948 3372 y Fk(2)3011 3358 y Fj(\000)24 b Fl(a)3154 3372 y Fk(1)3230 3358 y Fn(and)35 b Fl(C)43 b Fn(denotes)37 b(a)0 3466 y(p)s(ositiv)m(e)29 b(constan)m(t)j(whic)m(h)d(ma)m(y)i(c)m(hange)h (from)e(line)e(to)j(line.)150 3682 y(The)f(sp)s(ectral)g(shift)f (function)g Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))26 b Fj(2)f(D)1684 3649 y Fi(0)1707 3682 y Fn(\()p Fh(R)s Fn(\))37 b(related)30 b(to)h Fl(L)2350 3696 y Fk(1)2390 3682 y Fl(;)40 b(L)2517 3696 y Fk(2)2587 3682 y Fn(is)29 b(de\014ned)g(b)m(y)930 3867 y Fj(h)p Fl(\030)1009 3829 y Fi(0)1033 3867 y Fn(\()p Fl(\025;)15 b(h)p Fn(\))p Fl(;)g(f)10 b Fn(\()p Fl(\025)p Fn(\))p Fj(i)28 b Fn(=)c(tr)1695 3882 y Fk(bb)1778 3773 y Ff(\020)1828 3867 y Fl(f)10 b Fn(\()p Fl(L)1980 3881 y Fk(2)2019 3867 y Fn(\))20 b Fj(\000)g Fl(f)10 b Fn(\()p Fl(L)2317 3881 y Fk(1)2356 3867 y Fn(\))2391 3773 y Ff(\021)2441 3867 y Fl(;)56 b(f)34 b Fj(2)25 b Fl(C)2759 3829 y Fi(1)2752 3889 y Fk(0)2833 3867 y Fn(\()p Fh(R)s Fn(\))0 4052 y(and)30 b(our)g(result)f(concerning)h(the)g(deriv)-5 b(ativ)m(e)30 b(of)h Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))32 b(is)d(the)h(follo) m(wing.)0 4220 y Fx(Theorem)k(1.)42 b Fd(Assume)32 b(that)i Fl(L)1154 4234 y Fg(j)1190 4220 y Fn(\()p Fl(h)p Fn(\))p Fl(;)j(j)31 b Fn(=)25 b(1)p Fl(;)15 b Fn(2)p Fl(;)34 b Fd(satisfy)f(assumptions)i Fn(\(2)p Fl(:)p Fn(1\))22 b Fj(\000)e Fn(\(2)p Fl(:)p Fn(9\))p Fl(:)35 b Fd(L)-5 b(et)1176 4392 y Fn(\012)25 b Fj(\032\032)g Fl(e)1476 4354 y Fg(i)p Fk(])p Fi(\000)p Fk(2)p Fg(\022)1643 4363 y Ft(0)1678 4354 y Fg(;)p Fk(2)p Fg(\022)1766 4363 y Ft(0)1800 4354 y Fk([)1824 4392 y Fn(]0)p Fl(;)15 b Fn(+)p Fj(1)p Fn([)p Fl(;)36 b Fn(0)26 b Fl(<)f(\022)2392 4406 y Fk(0)2456 4392 y Fl(<)g(\031)s(=)p Fn(2)p Fl(;)0 4549 y Fd(b)-5 b(e)25 b(an)g(op)-5 b(en)27 b(simply)f(c)-5 b(onne)g(cte)g(d)27 b(set)e(and)h(let)f Fl(W)38 b Fj(\032\032)24 b Fn(\012)h Fd(b)-5 b(e)25 b(an)g(op)-5 b(en)27 b(simply)f(c)-5 b(onne)g(cte)g(d)26 b(and)g(r)-5 b(elatively)27 b(c)-5 b(omp)g(act)0 4657 y(set)37 b(which)g(is)g(symmetric)g(with)h(r)-5 b(esp)g(e)g(ct)38 b(to)f Fh(R)s Fd(.)59 b(Assume)37 b(that)g Fl(J)42 b Fn(=)32 b(\012)23 b Fj(\\)f Fh(R)2675 4624 y Fk(+)2740 4657 y Fl(;)62 b(I)39 b Fn(=)32 b Fl(W)k Fj(\\)22 b Fh(R)3274 4624 y Fk(+)3376 4657 y Fd(ar)-5 b(e)37 b(intervals.)0 4765 y(Then)c(for)g Fl(\025)25 b Fj(2)g Fl(I)40 b Fd(we)33 b(have)g(the)g(r)-5 b(epr)g(esentation)493 4964 y Fl(\030)537 4926 y Fi(0)560 4964 y Fn(\()p Fl(\025;)15 b(h)p Fn(\))27 b(=)913 4902 y(1)p 908 4943 56 4 v 908 5026 a Fl(\031)988 4964 y Fn(Im)15 b Fl(r)s Fn(\()p Fl(\025;)g(h)p Fn(\))21 b(+)1483 4870 y Ff(h)1660 4883 y(X)1532 5055 y Fq(w)q Fr(2)p Ft(Res)15 b Fq(L)1771 5071 y(j)1804 5055 y Fr(\\)p Ft(\012)p Fq(;)1617 5125 y Ft(Im)c Fq(w)q Fr(6)p Ft(=0)1993 4902 y Fj(\000)k Fn(Im)f Fl(w)p 1943 4943 377 4 v 1943 5026 a(\031)s Fj(j)p Fl(\025)20 b Fj(\000)g Fl(w)r Fj(j)2279 5000 y Fk(2)2350 4964 y Fn(+)2580 4883 y Ff(X)2441 5067 y Fg(w)r Fi(2)p Fk(Res)15 b Fg(L)2715 5077 y Fq(j)2747 5067 y Fi(\\)p Fg(J)2854 4964 y Fl(\016)s Fn(\()p Fl(\025)22 b Fj(\000)e Fl(w)r Fn(\))3200 4870 y Ff(i)3240 4885 y Fk(2)3240 5027 y Fg(j)t Fk(=1)3367 4964 y Fl(;)276 b Fn(\(2.10\))p eop %%Page: 5 5 5 4 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g(RESONANCES) 1035 b(5)0 267 y Fd(wher)-5 b(e)35 b Fl(r)s Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))28 b(=)f Fl(g)679 281 y Fk(+)738 267 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))23 b Fj(\000)p 1061 189 313 4 v 21 w Fl(g)1104 281 y Fk(+)1163 267 y Fn(\()p 1198 218 47 4 v Fl(z)5 b(;)15 b(h)p Fn(\))q Fl(;)57 b(g)1498 281 y Fk(+)1557 267 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))36 b Fd(is)d(a)h(function)g(holomorphic)j(in)d Fn(\012)f Fd(and)h Fl(g)3288 281 y Fk(+)3348 267 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))35 b Fd(satis\014es)0 375 y(the)e(estimate)1315 535 y Fj(j)p Fl(g)1383 549 y Fk(+)1443 535 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))p Fj(j)27 b(\024)e Fl(C)7 b Fn(\()p Fl(W)13 b Fn(\))p Fl(h)2092 497 y Fi(\000)p Fg(n)2190 474 y Ft(#)2248 535 y Fl(;)56 b(z)29 b Fj(2)c Fl(W)1096 b Fn(\(2.11\))0 699 y Fd(with)33 b Fl(C)7 b Fn(\()p Fl(W)13 b Fn(\))25 b Fl(>)g Fn(0)33 b Fd(indep)-5 b(endent)34 b(on)g Fl(h)25 b Fj(2)p Fn(]0)p Fl(;)15 b(h)1572 713 y Fk(0)1613 699 y Fn(])p Fl(:)170 922 y Fn(3.)46 b Fm(Represent)-6 b(a)g(tion)32 b(of)h(the)h(deriv)-8 b(a)i(tive)33 b(of)h(the)f (spectral)f(shift)i(function)f(f)n(or)g(non)1451 1030 y(semi-bounded)g(opera)-6 b(tors)150 1192 y Fn(Let)24 b Fl(L)368 1206 y Fg(j)404 1192 y Fl(;)e(j)31 b Fn(=)25 b(1)p Fl(;)15 b Fn(2)25 b(b)s(e)e(t)m(w)m(o)i(op)s(erators)e (satisfying)f(the)i(assumptions)e(\(2.1\))j(-)f(\(2.9\).)40 b(Giv)m(en)24 b(0)h Fl(<)g(\022)j Fj(\024)d Fl(\022)3652 1206 y Fk(0)3716 1192 y Fl(<)3822 1156 y Fg(\031)p 3822 1171 43 4 v 3826 1223 a Fk(2)3875 1192 y Fl(;)0 1300 y Fn(w)m(e)31 b(c)m(ho)s(ose)g(0)26 b Fl(<)f(\024)h(<)e Fn(1)31 b(so)g(that)g(2)p Fl(\022)d(<)d(\024\031)s Fn(.)41 b(Consider)28 b(the)j(functions)126 1485 y Fl(\033)178 1499 y Fi(\006)237 1485 y Fn(\()p Fl(z)t Fn(\))26 b(=)f(\()p Fl(z)556 1447 y Fk(2)616 1485 y Fn(+)20 b(1)h Fj(\000)e Fn(2)p Fl(z)h Fn(cos)q(\()p Fl(\024\031)s Fn(\)\))1304 1447 y Fg(m)1387 1485 y Fn(tr)1458 1500 y Fk(bb)1541 1390 y Ff(h)1580 1485 y Fn(\()p Fl(L)1677 1499 y Fg(j)1734 1485 y Fj(\000)g Fl(e)1867 1447 y Fg(i\024\031)1979 1485 y Fn(\))2014 1447 y Fi(\000)p Fg(m)2136 1485 y Fn(\()p Fl(L)2233 1499 y Fg(j)2290 1485 y Fj(\000)g Fl(e)2423 1447 y Fi(\000)p Fg(i\024\031)2590 1485 y Fn(\))2625 1447 y Fi(\000)p Fg(m)2746 1485 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(L)3001 1499 y Fg(j)3038 1485 y Fn(\))3073 1447 y Fi(\000)p Fk(1)3167 1390 y Ff(i)3206 1406 y Fk(2)3206 1548 y(1)3246 1485 y Fl(;)56 b Fj(\006)15 b Fn(Im)f Fl(z)30 b(>)25 b Fn(0)p Fl(;)0 1676 y Fn(where)j Fl(m)e(>)f(n=)p Fn(2)k(is)f(an)h(in)m(teger.)40 b(W)-8 b(e)31 b(will)26 b(sho)m(w)j(b)s(elo)m(w)f(that)h Fl(\033)2240 1690 y Fi(\006)2299 1676 y Fn(\()p Fl(z)t Fn(\))h(is)e(w)m(ell)g(de\014ned)g (and)g(it)h(is)f(clear)h(that)g(w)m(e)0 1784 y(ha)m(v)m(e)1450 1905 y Fl(\033)1502 1919 y Fk(+)1561 1905 y Fn(\()p Fl(z)t Fn(\))d(=)p 1799 1827 228 4 v 25 w Fl(\033)1851 1919 y Fi(\000)1910 1905 y Fn(\()5 b(\026)-50 b Fl(z)5 b Fn(\))p Fl(;)36 b Fn(Im)15 b Fl(z)29 b(>)c Fn(0)p Fl(:)150 2052 y Fn(Let)d(\012)i Fj(\032\032)h Fl(e)603 2019 y Fg(i)p Fk(])p Fi(\000)p Fk(2)p Fg(\022)r(;)p Fk(2)p Fg(\022)r Fk([)886 2052 y Fn(]0)p Fl(;)15 b Fn(+)p Fj(1)p Fn([)22 b(b)s(e)e(a)h(simply)e(connected)j(op)s(en)e(relativ)m(ely)h(compact)h (set)g(suc)m(h)e(that)i(\012)r Fj(\\)r Fh(R)3739 2019 y Fk(+)3829 2052 y Fn(=)0 2162 y Fl(J)35 b Fn(is)24 b(an)h(in)m(terv)-5 b(al.)39 b(The)24 b(sp)s(ectrum)h(of)g Fl(L)1389 2177 y Fg(j;\022)1502 2162 y Fn(outside)f(of)i Fl(e)1951 2129 y Fi(\000)p Fk(2)p Fg(i\022)2104 2162 y Fn([0)p Fl(;)15 b Fn(+)p Fj(1)p Fn([)27 b(consists)e(of)g(the)h(negativ)m(e)h(eigen)m (v)-5 b(alues)25 b(of)0 2281 y Fl(L)62 2295 y Fg(j)128 2281 y Fn(and)j(the)i(eigen)m(v)-5 b(alues)29 b(in)f Fl(e)1075 2248 y Fi(\000)p Fg(i)p Fk([0)p Fg(;)p Fk(2)p Fg(\022)r Fk([)1323 2281 y Fn(]0)p Fl(;)15 b 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Fl(L)1582 3880 y Fg(:;\022)1680 3865 y Fj(\000)20 b Fl(L)1833 3880 y Fg(:;\022)1931 3865 y Fn(has)40 b(rank)20 b Fj(O)s Fn(\()p Fl(h)2465 3827 y Fi(\000)p Fg(n)2563 3804 y Ft(#)2622 3865 y Fn(\))0 4024 y(and)30 b(for)g(all)f Fl(N)5 b(;)36 b(M)f Fj(2)25 b Fh(N)43 b Fn(w)m(e)31 b(ha)m(v)m(e)1289 4185 y Fl(K)1366 4200 y Fg(:;\022)1470 4185 y Fn(=)24 b Fj(O)s Fn(\(1\))j(:)e Fj(D)s Fn(\()p Fl(L)2002 4147 y Fg(N)2002 4207 y(:)2069 4185 y Fn(\))h Fj(\000)-15 b(!)25 b(D)s Fn(\()p Fl(L)2472 4147 y Fg(M)2472 4207 y(:)2551 4185 y Fn(\))p Fl(:)0 4344 y Fn(Secondly)-8 b(,)31 b Fl(K)477 4359 y Fg(:;\022)587 4344 y Fn(is)f(compactly)i(supp) s(orted,)e(that)i(is)e(if)g Fl(\037)d Fj(2)g Fl(C)2189 4311 y Fi(1)2182 4367 y Fk(0)2263 4344 y Fn(\()p Fh(R)2358 4311 y Fg(n)2411 4344 y Fn(\))k(is)g(equal)f(to)j(1)e(on)g Fl(B)5 b Fn(\(0)p Fl(;)15 b(R)q Fn(\))32 b(for)f Fl(R)d Fj(\025)f Fl(R)3861 4358 y Fk(0)0 4452 y Fn(large)j(enough,)h(w)m(e)g (ha)m(v)m(e)g Fl(K)981 4467 y Fg(:;\022)1085 4452 y Fn(=)25 b 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Fi(\000)p Fk(1)1885 4961 y Fn(+)g(\()p Fl(L)2073 4976 y Fg(:;\022)2171 4961 y Fj(\000)g Fl(z)t Fn(\))2343 4923 y Fi(\000)p Fk(1)2438 4961 y Fl(K)2515 4976 y Fg(:;\022)2594 4961 y Fn(\()2640 4938 y(^)2629 4961 y Fl(L)2691 4976 y Fg(:;\022)2789 4961 y Fj(\000)g Fl(z)t Fn(\))2961 4923 y Fi(\000)p Fk(1)3056 4961 y Fl(:)632 b Fn(\(3.2\))150 5143 y(The)30 b(con)m(tributions)f(related)h(to)h(the)g(resolv)m(en)m (t)g(\()1890 5120 y(^)1879 5143 y Fl(L)1941 5158 y Fg(:;\022)2039 5143 y Fj(\000)20 b Fl(z)t Fn(\))2211 5110 y Fi(\000)p Fk(1)2337 5143 y Fn(are)30 b(examined)g(in)f(the)h(follo)m(wing.)p eop %%Page: 6 6 6 5 bop 0 66 a Fk(6)1488 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)0 266 y Fx(Prop)s(osition)29 b(1.)37 b Fd(Ther)-5 b(e)27 b(exists)h(a)f(function)g Fl(a)1648 280 y Fk(+)1707 266 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))28 b Fd(holomorphic)j(in)26 b Fn(\012)h Fd(such)g(that)h(for)f Fl(z)j Fj(2)25 b Fn(\012)8 b Fj(\\)g(f)p Fn(Im)14 b Fl(z)29 b(>)c Fn(0)p Fj(g)0 374 y Fd(we)33 b(have)857 550 y Fl(\033)909 564 y Fk(+)968 550 y Fn(\()p Fl(z)t Fn(\))26 b(=)f(tr)1277 456 y Ff(h)1317 550 y Fn(\()p Fl(L)1414 565 y Fg(j;\022)1521 550 y Fj(\000)20 b Fl(z)t Fn(\))1693 513 y Fi(\000)p Fk(1)1788 550 y Fl(K)1865 565 y Fg(j;\022)1952 550 y Fn(\()1998 527 y(^)1987 550 y Fl(L)2049 565 y Fg(j;\022)2157 550 y Fj(\000)g Fl(z)t Fn(\))2329 513 y Fi(\000)p Fk(1)2424 456 y Ff(i)2463 472 y Fk(2)2463 614 y Fg(j)t Fk(=1)2610 550 y Fn(+)g Fl(a)2749 564 y Fk(+)2808 550 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))p Fl(:)672 b Fn(\(3.3\))0 744 y Fd(Mor)-5 b(e)g(over,)1346 914 y Fj(j)p Fl(a)1419 928 y Fk(+)1478 914 y Fn(\()p Fl(z)t(;)15 b(h)p Fn(\))p Fj(j)27 b(\024)e Fl(C)7 b Fn(\(\012\))p Fl(h)2094 876 y Fi(\000)p Fg(n)2192 853 y Ft(#)2250 914 y Fl(;)56 b(z)30 b Fj(2)25 b Fn(\012)1158 b(\(3.4\))0 1074 y Fd(with)33 b(a)g(c)-5 b(onstant)35 b Fl(C)7 b Fn(\(\012\))32 b Fd(indep)-5 b(endent)34 b(on)f Fl(h)26 b Fj(2)p Fn(]0)p Fl(;)15 b(h)1814 1088 y Fk(0)1855 1074 y Fn(])p Fl(:)150 1241 y Fd(Pr)-5 b(o)g(of.)39 b Fn(The)22 b(pro)s(of)g(is)g(a)h(mo)s(di\014cation)e(of)h(that)h(of)g(Prop)s (osition)e(2)i(in)e([8)q(])h(and)g(for)h(the)f(sak)m(e)i(of)f (completeness)0 1349 y(w)m(e)31 b(will)d(exp)s(ose)i(the)h(main)e (steps.)40 b(According)31 b(to)g(\(3.2\),)h(for)e Fl(z)g Fj(2)25 b Fn(\012)19 b Fj(\\)h(f)p Fn(Im)15 b Fl(z)30 b Fj(\025)25 b Fl(\016)s Fj(g)32 b Fn(w)m(e)e(ha)m(v)m(e)555 1549 y Fl(\033)607 1563 y Fk(+)666 1549 y Fn(\()p Fl(z)t Fn(\))d(=)905 1455 y Ff(\020)954 1549 y Fn(\()p Fl(z)e Fj(\000)20 b Fl(z)1189 1563 y Fk(0)1229 1549 y Fn(\)\()p Fl(z)25 b Fj(\000)g Fn(\026)-50 b Fl(z)1499 1563 y Fk(0)1538 1549 y Fn(\))1573 1455 y Ff(\021)1623 1471 y Fg(m)1705 1549 y Fn(tr)1776 1564 y Fk(bb)1859 1455 y Ff(h)1898 1549 y Fn(\()1944 1526 y(^)1933 1549 y Fl(L)1995 1564 y Fg(j;\022)2102 1549 y Fj(\000)20 b Fl(z)t Fn(\))2274 1511 y Fi(\000)p Fk(1)2369 1549 y Fn(\()p Fl(L)2466 1564 y Fg(j;\022)2574 1549 y Fj(\000)g Fl(z)2707 1563 y Fk(0)2746 1549 y Fn(\))2781 1511 y Fi(\000)p Fg(m)2903 1549 y Fn(\()p 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b Fl(k)c Fj(2)c Fh(N)45 b Fd(lar)-5 b(ge)33 b(enough)g(the)g(op)-5 b(er)g(ator)1363 4996 y Fn(\()p Fl(i)21 b Fj(\000)f Fl(P)1599 5010 y Fk(2)1638 4996 y Fn(\()p Fl(h)p Fn(\)\))1795 4958 y Fi(\000)p Fg(k)1915 4996 y Fj(\000)g Fn(\()p Fl(i)g Fj(\000)g Fl(P)2241 5010 y Fk(1)2281 4996 y Fn(\()p Fl(h)p Fn(\)\))2438 4958 y Fi(\000)p Fg(k)0 5143 y Fd(is)33 b(tr)-5 b(ac)g(e)33 b(class)h(one)f(and)g(its)g(tr)-5 b(ac)g(e)34 b(norm)g(is)e Fj(O)s Fn(\()p Fl(h)1738 5110 y Fi(\000)p Fg(n)1841 5143 y Fn(\))p Fd(.)p eop %%Page: 9 9 9 8 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g(RESONANCES) 1035 b(9)150 266 y Fd(ii\))32 b(Mor)-5 b(e)g(over,)34 b(for)f Fn(Im)15 b Fl(z)30 b Fj(6)p Fn(=)25 b(0)33 b Fd(we)f(have)437 474 y Fj(k)p Fn(\()p Fl(i)21 b Fj(\000)f Fl(P)718 488 y Fk(2)758 474 y Fn(\()p Fl(h)p Fn(\)\))915 436 y Fi(\000)p Fg(k)1014 474 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)1265 488 y Fk(2)1304 474 y Fn(\()p Fl(h)p Fn(\)\))1461 436 y Fi(\000)p Fk(1)1577 474 y Fj(\000)g Fn(\()p Fl(i)h 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Fk(2)2191 1310 y Fn(\()p Fl(h)p Fn(\)\))2348 1277 y Fi(\000)p Fg(l)2430 1310 y Fl(V)52 b Fn(and)31 b Fl(V)20 b Fn(\()p Fl(i)i Fj(\000)f Fl(P)3024 1324 y Fk(1)3064 1310 y Fn(\()p Fl(h)p Fn(\)\))3221 1277 y Fi(\000)p Fg(l)3335 1310 y Fn(are)32 b(trace)h(class)0 1421 y(ones)f(for)f Fl(l)i Fn(large)e(enough.)44 b(On)30 b(the)i(other)f(hand,)g(b)m(y)g(dualit)m(y)-8 b(,)31 b(w)m(e)h(m)m(ust)f(sho)m(w)h(only)e(that)i Fl(V)20 b Fn(\()p Fl(i)i Fj(\000)e Fl(P)3531 1435 y Fg(j)3568 1421 y Fn(\()p Fl(h)p Fn(\)\))3725 1388 y Fi(\000)p Fg(l)3839 1421 y Fn(is)0 1529 y(trace)32 b(class.)150 1636 y(Since)42 b(the)h(op)s(erators)f Fl(P)1041 1650 y Fg(j)1078 1636 y Fn(\()p Fl(h)p Fn(\))p Fl(;)15 b(j)53 b Fn(=)46 b(1)p Fl(;)15 b Fn(2)p Fl(;)44 b Fn(are)f(not)g(elliptic,)g(w)m(e)h(cannot)f (use)f(the)h Fl(h)p Fn(-pseudo)s(di\013eren)m(tial)0 1750 y(calculus)37 b(for)h(the)h(analysis)e(of)h(\()p Fl(i)27 b Fj(\000)e Fl(P)1372 1764 y Fg(j)1409 1750 y Fn(\()p Fl(h)p Fn(\)\))1566 1717 y Fi(\000)p Fg(l)1648 1750 y Fn(.)65 b(T)-8 b(o)39 b(o)m(v)m(ercome)i(this)c(di\016cult)m(y) -8 b(,)40 b(w)m(e)e(will)e(decomp)s(ose)j Fl(V)20 b Fn(\()p Fl(i)26 b Fj(\000)0 1864 y Fl(P)58 1878 y Fg(j)95 1864 y Fn(\()p Fl(h)p Fn(\)\))252 1831 y Fi(\000)p Fg(l)365 1864 y Fn(as)k(a)h(sum)e(of)i(three)f(terms:)460 2025 y Fl(V)21 b Fn(\()p Fl(i)g Fj(\000)e Fl(P)769 2039 y Fg(j)806 2025 y Fn(\()p Fl(h)p Fn(\)\))963 1988 y Fi(\000)p Fg(l)1071 2025 y Fn(=)25 b Fl(g)s(V)c Fn(\()p Fl(i)g Fj(\000)f Fl(P)1523 2039 y Fg(j)1559 2025 y Fn(\()p Fl(h)p Fn(\)\))1716 1988 y Fi(\000)p Fg(l)1819 2025 y Fn(+)g(\(1)h Fj(\000)f Fl(g)s Fn(\))p Fl(V)h Fn(\(1)g Fj(\000)f Fl(\015)2501 1988 y Fg(w)2558 2025 y Fn(\()p Fl(x;)15 b(hD)2812 2039 y Fg(x)2857 2025 y Fn(\)\)\()p Fl(i)21 b Fj(\000)f Fl(P)3163 2039 y Fg(j)3200 2025 y Fn(\()p Fl(h)p Fn(\)\))3357 1988 y Fi(\000)p Fg(l)670 2190 y Fn(+\(1)g Fj(\000)g Fl(g)s Fn(\))p Fl(V)i(\015)1140 2153 y Fg(w)1196 2190 y Fn(\()p Fl(x;)15 b(hD)1450 2204 y Fg(x)1495 2190 y Fn(\)\()p Fl(i)22 b Fj(\000)e 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y(Fix)26 b Fl(\022)h Fj(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)706 4585 y Fk(0)746 4571 y Fn(\))27 b(with)e Fl(\022)1054 4585 y Fk(0)1118 4571 y Fj(\024)g Fl(\016)1254 4585 y Fk(0)1294 4571 y Fl(;)39 b Fn(Im)15 b Fl(\022)27 b(<)e Fn(0)i(and)f(\014x)f(the)i(constan)m(ts)g Fl(R)2637 4585 y Fk(0)2702 4571 y Fl(>)e Fn(0)h(and)g Fl(\017)f(>)g Fn(0)i(in)e(the)h(de\014nition)0 4679 y(of)31 b Fl(v)s Fn(\()p Fl(x)238 4693 y Fk(1)278 4679 y Fn(\).)41 b(Consider)28 b(an)i(op)s(en)g(simply)e(connected)j(relativ)m(ely)f (compact)i(domain)579 4902 y(\012)645 4917 y Fg(\022)r(;\013)774 4902 y Fj(\032)25 b(f)p Fl(z)30 b Fj(2)25 b Fh(C)49 b Fn(:)65 b(Re)16 b Fl(z)29 b Fj(\024)c Fl(R)1613 4916 y Fk(0)1673 4902 y Fj(\000)20 b Fn(3)p Fl(\017;)36 b Fn(Im)15 b Fl(z)29 b Fj(\025)c Fl(\013)p Fn(\(1)d Fj(\000)e Fl(e)2491 4864 y Fi(\000)p Fg(\017)2578 4902 y Fn(\))15 b(Im)h Fl(\022)s Fj(g)p Fl(;)35 b Fn(0)26 b Fl(<)f(\013)g(<)g Fn(1)p Fl(:)150 5035 y Fn(W)-8 b(e)45 b(assume)e(that)h(\012)923 5050 y Fg(\022)r(;\013)1070 5035 y Fn(is)f(indep)s(enden)m(t)e(of)j Fl(h)p Fn(.)80 b(The)43 b(domain)g(\012)2580 5050 y Fg(\022)r(;\013) 2727 5035 y Fn(dep)s(ends)e(on)j Fl(\022)h Fn(and)e Fl(\013)h Fn(but)f(for)0 5143 y(simplicit)m(y)28 b(of)i(notation)h(w)m(e)g(will)c (write)j(b)s(elo)m(w)g(\012)f(instead)h(of)h(\012)2257 5158 y Fg(\022)r(;\013)2360 5143 y Fl(:)p eop %%Page: 12 12 12 11 bop 0 66 a Fk(12)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)0 266 y Fx(Lemma)32 b(3.)41 b Fd(Ther)-5 b(e)33 b(exist)f Fl(\022)1007 280 y Fk(0)1071 266 y Fl(>)25 b Fn(0)p Fl(;)53 b(h)1342 280 y Fk(0)1407 266 y Fl(>)25 b Fn(0)33 b Fd(smal)5 b(l)33 b(enough)f(such)g(that)h(for)g Fl(\022)27 b Fj(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)3063 280 y Fk(0)3103 266 y Fn(\))32 b Fd(with)h Fn(Im)15 b Fl(\022)27 b Fj(\024)e Fn(0)p Fl(;)35 b(h)25 b Fj(2)0 374 y Fn(]0)p Fl(;)15 b(h)162 388 y Fk(0)203 374 y Fn(])33 b Fd(we)f(have)527 574 y Fj(k)p Fn(\()p Fl(z)26 b Fj(\000)20 b Fl(P)824 589 y Fk(1)p Fg(;\022)918 574 y Fn(\))953 536 y Fi(\000)p Fk(1)1048 574 y Fj(k)25 b(\024)1864 512 y Fl(C)1929 526 y Fk(0)p 1224 553 1385 4 v 1224 636 a Fn(min)o Fj(f)p Fl(\017=)p Fn(2)p Fl(;)37 b Fn(\()p Fj(\000)p Fl(c)1755 650 y Fk(1)1810 636 y Fn(Im)14 b Fl(\022)23 b Fn(+)d(Im)14 b Fl(z)25 b Fj(\000)20 b Fl(C)2436 650 y Fk(1)2475 636 y Fl(h)p Fn(\))p Fj(g)2618 574 y Fl(;)36 b(c)2718 588 y Fk(1)2783 574 y Fn(=)25 b(1)c Fj(\000)f Fl(e)3078 536 y Fi(\000)p Fg(\017)3191 574 y Fl(>)25 b Fn(0)p Fl(;)356 b Fn(\(5.8\))0 776 y Fd(uniformly)40 b(with)g(r)-5 b(esp)g(e)g(ct)41 b(to)f Fl(z)i Fj(2)37 b Fn(\012)p Fd(,)j(pr)-5 b(ovide)g(d)42 b Fn(Im)14 b Fl(z)42 b(>)37 b(c)2080 790 y Fk(1)2135 776 y Fn(Im)14 b Fl(\022)28 b Fn(+)c Fl(C)2489 790 y Fk(1)2529 776 y Fl(h:)40 b Fd(The)f(c)-5 b(onstants)41 b Fl(C)3309 790 y Fk(0)3386 776 y Fj(\025)c Fn(1)p Fl(;)k(C)3670 790 y Fk(1)3747 776 y Fl(>)c Fn(0)0 884 y Fd(dep)-5 b(end)34 b(only)f(on)h Fn(\012)p Fl(:)150 1051 y Fx(Remark.)51 b Fn(The)34 b(estimate)g(\(5.8\))i(is)d(similar)f(to)j(that)f(in)f (Lemma)h(3)h(in)e([2],)j(where)d(Im)15 b Fl(z)38 b Fn(is)33 b(related)h(to)h Fl(\016)0 1158 y Fn(and)30 b Fl(C)7 b(h)p Fn(.)150 1374 y Fd(Pr)-5 b(o)g(of.)49 b Fn(Let)33 b Fl( )664 1341 y Fk(2)661 1397 y(0)704 1374 y Fn(\()p Fl(x)791 1388 y Fk(1)831 1374 y Fn(\))22 b(+)f Fl( )1042 1341 y Fk(2)1039 1397 y(1)1082 1374 y Fn(\()p Fl(x)1169 1388 y Fk(1)1209 1374 y Fn(\))29 b(=)g(1,)34 b(where)e Fl( )1801 1388 y Fk(0)1870 1374 y Fj(2)d Fl(C)2032 1341 y Fi(1)2106 1374 y Fn(\()p Fh(R)s Fn(\))39 b(is)31 b(equal)i(to)g(1)g (for)g Fl(x)2996 1388 y Fk(1)3064 1374 y Fj(\025)c Fl(R)3233 1388 y Fk(0)3294 1374 y Fj(\000)22 b Fl(\017;)37 b( )3545 1388 y Fk(0)3613 1374 y Fn(=)29 b(0)k(for)0 1482 y Fl(x)52 1496 y Fk(1)117 1482 y Fj(\024)25 b Fl(R)282 1496 y Fk(0)340 1482 y Fj(\000)19 b Fn(2)p Fl(\017)p Fn(.)41 b(On)29 b(the)h(supp)s(ort)e(of)i Fl( )1383 1496 y Fk(0)1422 1482 y Fn(,)g(w)m(e)h(ha)m(v)m(e)g Fl(x)1872 1496 y Fk(1)1930 1482 y Fj(\000)19 b Fn(Re)c Fl(z)30 b(>)25 b(\017)k Fn(for)h(all)f Fl(z)g Fj(2)c Fn(\012.)40 b(Com)m(bining)28 b(this)g(with)h(the)0 1590 y(fact)i(that)g(Re)16 b Fl(a)547 1605 y Fg(\022)611 1590 y Fj(\025)25 b Fn(0)31 b(for)f Fl(\022)965 1604 y Fk(0)1034 1590 y Fn(small,)f(w)m(e)i(get)304 1744 y(Re)427 1675 y Ff(\000)465 1744 y Fn(\()p Fl(P)558 1759 y Fk(1)p Fg(;\022)673 1744 y Fj(\000)20 b Fl(z)t Fn(\))p Fl( )904 1758 y Fk(0)944 1744 y Fl(u;)15 b( )1095 1758 y Fk(0)1135 1744 y Fl(u)1187 1675 y Ff(\001)1250 1744 y Fj(\025)25 b Fn(Re)1469 1675 y Ff(\000)1507 1744 y Fn(\()p Fl(h)p Fj(r)15 b Fn(Re)h Fl(a)1856 1759 y Fg(\022)1895 1744 y Fl(h)p Fj(r)k Fn(+)g Fl(x)2186 1758 y Fk(1)2246 1744 y Fj(\000)g Fn(Re)15 b Fl(z)t Fn(\))p Fl( )2599 1758 y Fk(0)2640 1744 y Fl(u;)g( )2791 1758 y Fk(0)2831 1744 y Fl(u)2883 1675 y Ff(\001)2941 1744 y Fj(\000)20 b(O)s Fn(\()p Fl(h)3194 1706 y Fk(2)3234 1744 y Fl(\022)s Fn(\))p Fj(k)p Fl( )3419 1758 y Fk(0)3459 1744 y Fl(u)p Fj(k)3556 1706 y Fk(2)1393 1918 y Fj(\025)25 b Fl(\017)p Fj(k)p Fl( )1630 1932 y Fk(0)1670 1918 y Fl(u)p Fj(k)1767 1880 y Fk(2)1827 1918 y Fj(\000)20 b(O)s Fn(\()p Fl(h)2080 1880 y Fk(2)2120 1918 y Fl(\022)s Fn(\))p Fj(k)p Fl( )2305 1932 y Fk(0)2345 1918 y Fl(u)p Fj(k)2442 1880 y Fk(2)2482 1918 y Fl(:)0 2053 y Fn(Th)m(us)29 b(for)h Fl(z)g Fj(2)25 b Fn(\012)30 b(and)g Fl(u)25 b Fj(2)g(D)33 b Fn(and)c(for)i Fl(\022)h Fn(small)d(w)m(e)i(deduce)1373 2206 y Fj(k)p Fn(\()p Fl(P)1511 2221 y Fk(1)p Fg(;\022)1627 2206 y Fj(\000)20 b Fl(z)t Fn(\))p Fl( )1858 2220 y Fk(0)1898 2206 y Fl(u)p Fj(k)26 b(\025)f Fl(\017=)p Fn(2)p Fj(k)p Fl( )2348 2220 y Fk(0)2389 2206 y Fl(u)p Fj(k)p Fl(;)1202 b Fn(\(5.9\))0 2360 y(uniformly)27 b(on)k Fl(z)e Fj(2)c Fn(\012.)150 2468 y(On)30 b(the)g(supp)s(ort)f(of)h Fl( )957 2482 y Fk(1)1027 2468 y Fn(w)m(e)h(ha)m(v)m(e)1260 2627 y Fl(v)s Fn(\()p Fl(x)1394 2641 y Fk(1)1434 2627 y Fn(\))26 b(=)f(1)20 b Fj(\000)g Fl(e)1789 2589 y Fk(\()p Fg(x)1856 2598 y Ft(1)1891 2589 y Fi(\000)p Fg(R)1999 2598 y Ft(0)2034 2589 y Fk(\))2091 2627 y Fj(\025)25 b Fn(1)20 b Fj(\000)g Fl(e)2385 2589 y Fi(\000)p Fg(\017)2499 2627 y Fl(>)k Fn(0)0 2781 y(and)1061 2901 y(Im)15 b Fl(a)1233 2916 y Fg(\022)r(;)p Fk(1)p Fg(;)p Fk(1)1407 2901 y Fn(=)1503 2807 y Ff(\020)1552 2901 y Fn(2\(Im)h Fl(\022)s Fn(\))k(+)g Fj(O)s Fn(\(Re)15 b Fl(\022)j Fn(Im)c Fl(\022)s Fn(\))2446 2807 y Ff(\021)2495 2901 y Fl(e)2537 2864 y Fk(\()p Fg(x)2604 2873 y Ft(1)2640 2864 y Fi(\000)p Fg(R)2748 2873 y Ft(0)2782 2864 y Fk(\))2814 2901 y Fl(;)1072 3133 y Fn(Im)g Fl(g)1238 3148 y Fg(\022)1278 3133 y Fn(\()p Fl(x)p Fn(\))26 b(=)1522 3039 y Ff(\020)1571 3133 y Fj(\000)1652 3072 y Fn(\(Im)15 b Fl(\022)s Fn(\))p 1652 3112 240 4 v 1749 3196 a(2)1922 3133 y Fj(\000)20 b Fn(\(Re)c Fl(\022)h Fn(Im)e Fl(\022)s Fn(\))2436 3039 y Ff(\021)2485 3133 y Fl(e)2527 3096 y Fk(\()p Fg(x)2594 3105 y Ft(1)2629 3096 y Fi(\000)p Fg(R)2737 3105 y Ft(0)2771 3096 y Fk(\))2803 3133 y Fl(:)0 3294 y Fn(Cho)s(osing)29 b Fl(\022)440 3308 y Fk(0)509 3294 y Fn(small,)h(w)m(e)g(obtain)579 3470 y(Im)687 3376 y Ff(\020)737 3470 y Fj(\000)p Fl(h)860 3433 y Fk(2)900 3470 y Fj(r)p Fl(a)1024 3485 y Fg(\022)1062 3470 y Fj(r)p Fl( )1197 3484 y Fk(1)1237 3470 y Fl(u;)15 b( )1388 3484 y Fk(1)1428 3470 y Fl(u)1480 3376 y Ff(\021)1555 3470 y Fj(\024)25 b Fn(0)p Fl(;)56 b Fj(j)15 b Fn(Im)g Fl(h)1993 3433 y Fk(2)2033 3470 y Fn(\()p Fl(g)2111 3485 y Fg(\022)2151 3470 y Fl( )2210 3484 y Fk(1)2249 3470 y Fl(u;)g( )2400 3484 y Fk(1)2440 3470 y Fl(u)p Fn(\))p Fj(j)26 b(\024)f Fl(ch)2765 3433 y Fk(2)2805 3470 y Fj(j)15 b Fn(Im)h Fl(\022)s Fj(jk)p Fl( )3145 3484 y Fk(1)3184 3470 y Fl(u)p Fj(k)3281 3433 y Fk(2)0 3646 y Fn(On)30 b(the)g(other)h(hand,)e(for)h Fl(x)989 3660 y Fk(1)1054 3646 y Fj(2)25 b Fn(supp)18 b Fl( )1406 3660 y Fk(1)1476 3646 y Fn(w)m(e)31 b(get)708 3823 y(Im)831 3753 y Ff(\000)870 3823 y Fn(\()p Fl(x)957 3837 y Fk(1)1016 3823 y Fn(+)20 b Fl(\022)s(v)s Fn(\()p Fl(x)1287 3837 y Fk(1)1327 3823 y Fn(\))g Fj(\000)g Fl(z)t Fn(\))p Fl( )1613 3837 y Fk(1)1653 3823 y Fl(u;)15 b( )1804 3837 y Fk(1)1844 3823 y Fl(u)1896 3753 y Ff(\001)1960 3823 y Fj(\024)2056 3728 y Ff(\020)2105 3823 y Fn(\()p Fl(c)2179 3837 y Fk(1)2220 3823 y Fn(\(Im)g Fl(\022)s Fn(\))k Fj(\000)h Fn(Im)15 b Fl(z)t Fn(\))p Fl( )2834 3837 y Fk(1)2874 3823 y Fl(u;)g( )3025 3837 y Fk(1)3065 3823 y Fl(u)3117 3728 y Ff(\021)3167 3823 y Fl(:)150 3999 y Fn(Consequen)m(tly)-8 b(,)30 b(for)g Fl(h)922 4013 y Fk(0)962 3999 y Fn(,)h(and)f Fl(\022)1238 4013 y Fk(0)1307 3999 y Fn(su\016cien)m(tly)f(small,)g(w)m(e)i(deduce)f(the) h(estimate)974 4152 y Fj(k)p Fn(\()p Fl(P)1112 4167 y Fk(1)p Fg(;\022)1227 4152 y Fj(\000)20 b Fl(z)t Fn(\))p Fl( )1458 4166 y Fk(1)1498 4152 y Fl(u)p Fj(k)26 b(\025)f Fn(\()p Fj(\000)p Fl(c)1862 4166 y Fk(1)1917 4152 y Fn(Im)14 b Fl(\022)23 b Fn(+)d(Im)15 b Fl(z)24 b Fj(\000)c Fl(c)2517 4166 y Fk(2)2557 4152 y Fl(h)p Fn(\))p Fj(k)p Fl( )2748 4166 y Fk(1)2789 4152 y Fl(u)p Fj(k)p Fl(;)757 b Fn(\(5.10\))0 4306 y(uniformly)27 b(on)k Fl(z)e Fj(2)c Fn(\012.)150 4419 y(Let)331 4392 y Ff(e)313 4419 y Fl( )372 4433 y Fg(j)434 4419 y Fj(2)g Fl(C)592 4386 y Fi(1)585 4442 y Fk(0)666 4419 y Fn(\()p Fh(R)s Fn(\))37 b(b)s(e)29 b(equal)h(to)h(1)g(on)f(the)h(supp)s(ort)e(of)h Fj(r)p Fl( )2241 4433 y Fg(j)2278 4419 y Fn(.)40 b(Since)457 4604 y([)p Fl(h)534 4567 y Fk(2)574 4604 y Fj(r)650 4510 y Ff(\020)700 4604 y Fl(a)748 4619 y Fg(\022)787 4604 y Fj(r)863 4510 y Ff(\021)912 4604 y Fl(;)15 b( )1011 4618 y Fg(j)1048 4604 y Fn(])26 b(=)1213 4577 y Ff(e)1195 4604 y Fl( )1254 4618 y Fg(j)1291 4604 y Fn([)p Fl(h)1368 4567 y Fk(2)1408 4604 y Fj(r)p Fl(a)1532 4619 y Fg(\022)1570 4604 y Fj(r)p Fl(;)15 b( )1745 4618 y Fg(j)1782 4604 y Fn(])26 b(=)f Fl(h)1999 4577 y Ff(e)1981 4604 y Fl( )2040 4618 y Fg(j)2077 4510 y Ff(\020)2127 4604 y Fl(h)p Fj(r)2255 4510 y Ff(\020)2304 4604 y Fl(a)2352 4619 y Fg(\022)2391 4604 y Fn(\()p Fj(r)p Fl( )2561 4618 y Fg(j)2598 4604 y Fn(\))2633 4510 y Ff(\021)2703 4604 y Fn(+)20 b(2\()p Fj(r)p Fl( )3009 4618 y Fg(j)3047 4604 y Fn(\))p Fl(a)3130 4619 y Fg(\022)3169 4604 y Fn(\()p Fl(h)p Fj(r)p Fn(\))3367 4510 y Ff(\021)3417 4604 y Fl(;)0 4786 y Fn(as)31 b(in)e(the)h(pro)s (of)g(of)g(\(5.6\),)j(w)m(e)e(obtain)921 4967 y Fj(k)p Fn([)p Fl(h)1043 4930 y Fk(2)1084 4967 y Fj(r)1160 4873 y Ff(\020)1209 4967 y Fl(a)1257 4982 y Fg(\022)1296 4967 y Fj(r)1372 4873 y Ff(\021)1422 4967 y Fl(;)k( )1541 4981 y Fg(j)1578 4967 y Fn(])p Fl(u)p Fj(k)26 b(\024)f Fl(C)1887 4981 y Fk(2)1926 4967 y Fl(h)1978 4873 y Ff(\020)2028 4967 y Fj(k)p Fn(\()p Fl(P)2166 4982 y Fk(1)p Fg(;\022)2262 4967 y Fn(\()p Fl(h)p Fn(\))c Fj(\000)f Fl(z)t Fn(\))p Fl(u)p Fj(k)h Fn(+)f Fj(k)p Fl(u)p Fj(k)2928 4873 y Ff(\021)3668 4967 y Fn(\(5.11\))0 5143 y(for)30 b Fl(z)g Fj(2)25 b Fn(\012)k(and)h Fl(u)c Fj(2)e(D)33 b Fn(with)c(a)i(constan)m(t)h Fl(C)1550 5157 y Fk(2)1615 5143 y Fl(>)24 b Fn(0)31 b(dep)s(ending)d (on)i(\012)p Fl(:)p eop %%Page: 13 13 13 12 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g (RESONANCES)999 b(13)150 266 y Fn(Com)m(bining)28 b(\(5.9\),)33 b(\(5.10\),)f(\(5.11\))h(with)c(the)i(estimate)1087 510 y Fj(k)p Fn(\()p Fl(P)1225 525 y Fk(1)p Fg(;\022)1320 510 y Fn(\()p Fl(h)p Fn(\))21 b Fj(\000)f Fl(z)t Fn(\))p Fl(u)p Fj(k)1732 472 y Fk(2)1798 510 y Fn(=)1938 404 y Fk(1)1895 429 y Ff(X)1894 611 y Fg(j)t Fk(=0)2032 510 y Fj(k)p Fl( )2136 524 y Fg(j)2173 510 y Fn(\()p Fl(P)2266 525 y Fk(1)p Fg(;\022)2360 510 y Fn(\()p Fl(h)p Fn(\))i Fj(\000)e Fl(z)t Fn(\))p Fl(u)p Fj(k)2773 472 y Fk(2)856 871 y Fj(\025)962 810 y Fn(1)p 962 850 46 4 v 962 934 a(2)1076 766 y Fk(1)1033 791 y Ff(X)1032 973 y Fg(j)t Fk(=0)1170 871 y Fj(k)p Fn(\()p Fl(P)1308 886 y Fk(1)p Fg(;\022)1403 871 y Fn(\()p Fl(h)p Fn(\))h Fj(\000)f Fl(z)t Fn(\))p Fl( )1777 885 y Fg(j)1815 871 y Fl(u)p Fj(k)1912 834 y Fk(2)1972 871 y Fj(\000)2106 766 y Fk(1)2064 791 y Ff(X)2063 973 y Fg(j)t Fk(=0)2200 871 y Fj(k)p Fn([)p Fl(h)2322 834 y Fk(2)2363 871 y Fj(r)2439 777 y Ff(\020)2488 871 y Fl(a)2536 886 y Fg(\022)2575 871 y Fj(r)2651 777 y Ff(\021)2701 871 y Fl(;)35 b( )2820 885 y Fg(j)2857 871 y Fn(])p Fl(u)p Fj(k)2979 834 y Fk(2)3019 871 y Fl(;)0 1103 y Fn(for)30 b Fl(h)h Fn(small)e(enough)h(and)g Fl(z)f Fj(2)c Fn(\012)30 b(w)m(e)h(deduce)580 1267 y Fl(C)645 1281 y Fk(0)684 1267 y Fj(k)p Fn(\()p Fl(P)822 1282 y Fk(1)p Fg(;\022)917 1267 y Fn(\()p Fl(h)p Fn(\))21 b Fj(\000)f Fl(z)t Fn(\))p Fl(u)p Fj(k)27 b(\025)e Fn(min)n Fj(f)p Fl(\017=)p Fn(2)p Fl(;)37 b Fn(\()p Fj(\000)p Fl(c)1982 1281 y Fk(1)2037 1267 y Fn(Im)15 b Fl(\022)22 b Fn(+)e(Im)15 b Fl(z)24 b Fj(\000)c Fl(C)2663 1281 y Fk(1)2702 1267 y Fl(h)p Fn(\))p Fj(gk)p Fl(u)p Fj(k)p Fl(;)58 b(u)26 b Fj(2)f(D)s Fl(:)347 b Fn(\(5.12\))150 1436 y(By)21 b(the)g(same)g(argumen)m(t,)j(w)m(e)d(pro)m(v)m(e)h(an)e (estimate)i(similar)c(to)k(\(5.12\))h(for)e(the)g(adjoin)m(t)f(op)s (erator)h Fl(P)3551 1403 y Fi(\003)3538 1462 y Fk(1)p Fg(;\022)3632 1436 y Fn(\()p Fl(h)p Fn(\))q Fj(\000)7 b Fn(\026)-51 b Fl(z)6 b Fn(.)0 1551 y(Since)37 b(\()p Fl(P)338 1566 y Fk(1)p Fg(;\022)433 1551 y Fn(\()p Fl(h)p Fn(\))26 b Fj(\000)g Fl(z)t Fn(\))38 b(is)g(closed,)i(the)f(op)s (erator)f(\()p Fl(P)1832 1566 y Fk(1)p Fg(;\022)1927 1551 y Fn(\()p Fl(h)p Fn(\))26 b Fj(\000)g Fl(z)t Fn(\))38 b(has)g(a)h(zero)g(index,)g(and)f(w)m(e)h(conclude)e(that)0 1659 y(\()p Fl(P)93 1674 y Fk(1)p Fg(;\022)188 1659 y Fn(\()p Fl(h)p Fn(\))19 b Fj(\000)f Fl(z)t Fn(\))30 b(is)f(in)m(v)m (ertible)f(for)h(ev)m(ery)h Fl(z)g Fj(2)25 b Fn(\012.)40 b(Finally)-8 b(,)28 b(\(5.8\))j(follo)m(ws)e(from)g(\(5.12\))j(and)d (this)f(completes)i(the)0 1767 y(pro)s(of.)3599 b Fc(2)150 1983 y Fn(No)m(w,)29 b(it)e(is)g(easy)h(to)h(see)f(that)g(the)g(op)s (erator)g Fl(P)1777 1998 y Fk(2)p Fg(;\022)1871 1983 y Fn(\()p Fl(h)p Fn(\))15 b Fj(\000)g Fl(z)t(;)46 b(z)30 b Fj(2)24 b Fn(\012)p Fl(;)k Fn(is)f(a)h(F)-8 b(redholm)27 b(op)s(erator)h(and)f(w)m(e)h(ha)m(v)m(e)0 2091 y(the)j(follo)m(wing.)0 2263 y Fx(Lemma)36 b(4.)44 b Fd(L)-5 b(et)35 b Fn(Im)15 b Fl(z)824 2277 y Fk(0)894 2263 y Fl(>)30 b(c)1034 2277 y Fk(1)1089 2263 y Fn(Im)15 b Fl(\022)24 b Fn(+)e Fl(C)1438 2277 y Fk(1)1477 2263 y Fl(h;)38 b(z)1634 2277 y Fk(0)1704 2263 y Fj(2)30 b Fn(\012)p Fl(:)35 b Fd(Then)h(the)g(op)-5 b(er)g(ator)39 b Fl(P)2734 2278 y Fk(2)p Fg(;\022)2828 2263 y Fn(\()p Fl(h)p Fn(\))23 b Fj(\000)f Fl(z)3108 2277 y Fk(0)3183 2263 y Fd(is)35 b(a)h(F)-7 b(r)i(e)g(dholm)39 b(one)0 2371 y(with)33 b(index)g(0.)150 2544 y(Pr)-5 b(o)g(of.)42 b Fn(F)-8 b(or)31 b Fl(z)e Fj(2)c Fn(\012)30 b(w)m(e)h(ha)m(v)m(e)0 2747 y Fl(P)58 2762 y Fk(2)p Fg(;\022)152 2747 y Fn(\()p Fl(h)p Fn(\))q Fj(\000)q Fl(z)g Fn(=)516 2652 y Ff(\020)565 2747 y Fl(I)8 b Fn(+)685 2652 y Ff(\020)735 2747 y Fl(V)21 b Fj(\016)q Fn(\010)921 2762 y Fg(\022)960 2652 y Ff(\021)1009 2747 y Fn(\(1)q Fj(\000)q Fl( )1221 2761 y Fg(z)1262 2747 y Fn(\()p Fl(x)p Fn(\)\))1419 2652 y Ff(h)1459 2747 y Fn(\()p Fl(P)1552 2762 y Fk(1)p Fg(;\022)1647 2747 y Fn(\()p Fl(h)p Fn(\))q Fj(\000)q Fl(z)t Fn(\))q(+)1996 2652 y Ff(\020)2047 2747 y Fl(V)g Fj(\016)q Fn(\010)2233 2762 y 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Fl(h)p Fn(\))h Fj(\000)f Fl(z)t Fn(\))26 b Fl(>)f Fn(0)p Fl(:)p eop %%Page: 14 14 14 13 bop 0 66 a Fk(14)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)0 266 y Fn(T)-8 b(o)28 b(examine)f(the)h(dep)s(endence)f(on)h Fl(\022)h Fn(of)f(the)g(resonances,)h(w)m(e)f(will)d(sho)m(w)i(that)i (the)f(op)s(erator)g Fl(K)3360 281 y Fg(\022)3399 266 y Fn(\()p Fl(z)3476 280 y Fk(0)3516 266 y Fn(\))g(dep)s(ends)0 374 y(analytically)i(on)i Fl(\022)d Fj(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)1020 388 y Fk(0)1060 374 y Fn(\))p Fl(:)32 b Fn(T)-8 b(o)32 b(do)g(this,)f(it)g(is)g(su\016cien)m(t)g(to)h(sho)m (w)g(that)g(the)g(resolv)m(en)m(t)g(\()p Fl(P)3358 389 y Fk(1)p Fg(;\022)3453 374 y Fn(\()p Fl(h)p Fn(\))22 b Fj(\000)f Fl(z)3731 388 y Fk(0)3770 374 y Fn(\))3805 341 y Fi(\000)p Fk(1)0 482 y Fn(is)29 b(analytic)h(with)f(resp)s(ect)i (to)g Fl(\022)s(:)f Fn(Fix)g Fl(\022)1374 496 y Fk(1)1438 482 y Fj(2)25 b Fl(D)s Fn(\(0)p Fl(;)15 b(\022)1765 496 y Fk(0)1805 482 y Fn(\))31 b(and)e(write)551 631 y Fl(P)609 646 y Fk(1)p Fg(;\022)723 631 y Fj(\000)20 b Fl(z)856 645 y Fk(0)921 631 y Fn(=)25 b Fl(P)1075 646 y Fk(1)p Fg(;\022)1163 655 y Ft(1)1222 631 y Fj(\000)20 b Fl(z)1355 645 y Fk(0)1415 631 y Fj(\000)g Fl(h)1558 593 y Fk(2)1598 631 y Fj(r)p Fn(\()p Fl(a)1757 646 y Fg(\022)1816 631 y Fj(\000)g Fl(a)1955 646 y Fg(\022)1988 655 y Ft(1)2027 631 y Fn(\))p Fj(r)g Fn(+)g(\()p Fl(\022)j Fj(\000)c Fl(\022)2483 645 y Fk(1)2523 631 y Fn(\))p Fl(v)s Fn(\()p Fl(x)2692 645 y Fk(1)2732 631 y Fn(\))i(+)e Fl(h)2930 593 y Fk(2)2970 631 y Fn(\()p Fl(g)3048 646 y Fg(\022)3108 631 y Fj(\000)h Fl(g)3242 646 y Fg(\022)3275 655 y Ft(1)3314 631 y Fn(\))765 809 y(=)25 b Fl(P)919 824 y Fk(1)p Fg(;\022)1007 833 y Ft(1)1066 809 y Fj(\000)20 b Fl(z)1199 823 y Fk(0)1259 809 y Fn(+)g Fl(B)1419 824 y Fg(\022)r(;\022)1507 833 y Ft(1)1570 809 y Fn(=)1666 715 y Ff(\020)1716 809 y Fl(I)27 b Fn(+)20 b Fl(B)1943 824 y Fg(\022)r(;\022)2031 833 y Ft(1)2069 809 y Fn(\()p Fl(P)2162 824 y Fk(1)p Fg(;\022)2250 833 y Ft(1)2309 809 y Fj(\000)g Fl(z)2442 823 y Fk(0)2482 809 y Fn(\))2517 772 y Fi(\000)p Fk(1)2612 715 y Ff(\021)2661 809 y Fn(\()p Fl(P)2754 824 y Fk(1)p Fg(;\022)2842 833 y Ft(1)2902 809 y Fj(\000)g Fl(z)3035 823 y Fk(0)3074 809 y Fn(\))p Fl(:)0 959 y Fn(Here)37 b(the)f(op)s(erator)g Fl(B)824 974 y Fg(\022)r(;\022)912 983 y Ft(1)986 959 y Fn(dep)s(ends)e(analytically)h(on)g Fl(\022)s Fn(.)57 b(On)35 b(the)h(other)h(hand,)f(it)g(is)f(easy)h(to)h (see)g(that)f(for)0 1067 y Fj(j)p Fl(\022)23 b Fj(\000)d Fl(\022)225 1081 y Fk(1)264 1067 y Fj(j)30 b Fn(small)f(enough)h(w)m(e) h(ma)m(y)g(arrange)999 1216 y Fj(k)p Fl(B)1113 1231 y Fg(\022)r(;\022)1201 1240 y Ft(1)1240 1216 y Fn(\()p Fl(P)1333 1231 y Fk(1)p Fg(;\022)1421 1240 y Ft(1)1480 1216 y Fj(\000)20 b Fl(z)1613 1230 y Fk(0)1653 1216 y Fn(\))1688 1179 y Fi(\000)p Fk(1)1783 1216 y Fj(k)25 b(\024)g Fl(C)2014 1230 y Fk(1)2054 1216 y Fn(\()p Fl(\022)2132 1230 y Fk(1)2171 1216 y Fl(;)15 b(z)2253 1230 y Fk(0)2293 1216 y Fn(\))p Fj(j)p Fl(\022)23 b Fj(\000)d Fl(\022)2553 1230 y Fk(1)2592 1216 y Fj(j)26 b(\024)f Fn(1)p Fl(=)p Fn(2)p Fl(:)0 1362 y Fn(F)-8 b(or)31 b(example,)f(to)i(estimate)f(the)f (terms)h(in)m(v)m(olving)d Fl(a)1866 1377 y Fg(\022)1906 1362 y Fl(;)35 b(a)2014 1377 y Fg(\022)2047 1386 y Ft(1)2086 1362 y Fn(,)30 b(w)m(e)h(apply)e(the)i(equalit)m(y)1313 1537 y Fl(h)1365 1499 y Fk(2)1405 1443 y Ff(\020)1454 1537 y Fj(r)p Fn(\()p Fl(a)1613 1552 y Fg(\022)1673 1537 y Fj(\000)20 b Fl(a)1812 1552 y Fg(\022)1845 1561 y Ft(1)1883 1537 y Fn(\))p Fj(r)1994 1443 y Ff(\021)2044 1537 y Fn(\()p Fl(P)2137 1552 y Fk(1)p Fg(;\022)2225 1561 y Ft(1)2285 1537 y Fj(\000)g Fl(z)2418 1551 y Fk(0)2457 1537 y Fn(\))2492 1499 y Fi(\000)p Fk(1)635 1747 y Fn(=)25 b Fl(h)783 1709 y Fk(2)823 1652 y Ff(\020)873 1747 y Fj(r)p Fn(\()p Fl(a)1032 1762 y Fg(\022)1091 1747 y Fj(\000)20 b Fl(a)1230 1762 y Fg(\022)1263 1771 y Ft(1)1302 1747 y Fn(\))p Fj(r)1413 1652 y Ff(\021)o(h)1502 1747 y Fn(\()p Fl(P)1595 1761 y Fk(1)1655 1747 y Fj(\000)g Fl(i)p Fn(\))1812 1709 y Fi(\000)p Fk(1)1927 1747 y Fn(+)g(\()p Fl(P)2111 1761 y Fk(1)2171 1747 y Fj(\000)g Fl(i)p Fn(\))2328 1709 y Fi(\000)p Fk(1)2423 1747 y Fn(\()p Fl(i)h Fj(\000)f Fl(z)t Fn(\)\()p Fl(P)2775 1762 y Fk(1)p Fg(;\022)2863 1771 y Ft(1)2923 1747 y Fj(\000)g Fl(z)3056 1761 y Fk(0)3096 1747 y Fn(\))3131 1709 y Fi(\000)p Fk(1)3225 1652 y Ff(i)0 1901 y Fn(and)30 b(w)m(e)h(use)f(the)g(b)s(ound)f(\(5.6\).)42 b(Th)m(us)818 2091 y(\()p Fl(P)911 2106 y Fk(1)p Fg(;\022)1026 2091 y Fj(\000)20 b Fl(z)1159 2105 y Fk(0)1199 2091 y Fn(\))1234 2054 y Fi(\000)p Fk(1)1354 2091 y Fn(=)25 b(\()p Fl(P)1543 2106 y Fk(1)p Fg(;\022)1631 2115 y Ft(1)1690 2091 y Fj(\000)20 b Fl(z)1823 2105 y Fk(0)1863 2091 y Fn(\))1898 2054 y Fi(\000)p Fk(1)1992 1997 y Ff(\020)2042 2091 y Fl(I)27 b Fn(+)20 b Fl(B)2269 2106 y Fg(\022)r(;\022)2357 2115 y Ft(1)2395 2091 y Fn(\()p Fl(P)2488 2106 y Fk(1)p Fg(;\022)2576 2115 y Ft(1)2636 2091 y Fj(\000)g Fl(z)2769 2105 y Fk(0)2808 2091 y Fn(\))2843 2054 y Fi(\000)p Fk(1)2938 1997 y Ff(\021)2988 2013 y Fi(\000)p Fk(1)0 2272 y Fn(and)j(w)m(e)h (obtain)f(the)h(analyticit)m(y)f(for)h(small)e Fj(j)p Fl(\022)9 b Fj(\000)e Fl(\022)1730 2286 y Fk(1)1768 2272 y Fj(j)p Fl(:)24 b Fn(Also)g(let)f(us)g(p)s(oin)m(t)g(out)g(that)i(if)d (the)i(resolv)m(en)m(t)g(\()p Fl(P)3545 2287 y Fk(2)p Fg(;\022)3647 2272 y Fj(\000)7 b Fl(z)t Fn(\))3806 2239 y Fi(\000)p Fk(1)0 2380 y Fn(exists,)30 b(w)m(e)h(ha)m(v)m(e)1125 2527 y(\()p Fl(P)1218 2542 y Fk(2)p Fg(;\022)1333 2527 y Fj(\000)20 b Fl(z)t Fn(\))1505 2490 y Fi(\000)p Fk(1)1625 2527 y Fn(=)1721 2433 y Ff(\020)1770 2527 y Fl(A)1838 2542 y Fk(1)p Fg(;\022)1932 2527 y Fn(\()p Fl(z)t Fn(\))2048 2433 y Ff(\021)2099 2449 y Fi(\000)p Fk(1)2193 2527 y Fn(\()p Fl(I)28 b Fn(+)20 b Fl(K)2464 2542 y Fg(\022)2503 2527 y Fn(\()p Fl(z)t Fn(\)\))2654 2490 y Fi(\000)p Fk(1)2750 2527 y Fl(:)0 2689 y Fn(F)-8 b(or)29 b(\014xed)e Fl(z)32 b Fn(the)c(in)m(v)m(ertibilit)m(y)d(of)j(\()p Fl(I)23 b Fn(+)15 b Fl(K)1458 2704 y Fg(\022)1497 2689 y Fn(\()p Fl(z)t 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b(the)e(left-hand)g(side)f(of)i(\(5.13\))i (admits)d(a)p eop %%Page: 15 15 15 14 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g (RESONANCES)999 b(15)0 266 y Fn(meromorphic)32 b(con)m(tin)m(uation)h (with)f(resp)s(ect)h(to)h Fl(z)j Fn(in)32 b(\012)22 b Fj(\\)g(f)p Fl(z)34 b Fj(2)c Fh(C)53 b Fn(:)g(Im)14 b Fl(z)35 b(>)29 b(c)2803 280 y Fk(1)2858 266 y Fn(Im)15 b Fl(\022)24 b Fn(+)e Fl(C)3207 280 y Fk(1)3247 266 y Fl(h)p Fj(g)p Fn(,)34 b(hence)g(this)e(is)0 374 y(true)i(for)f(the)i (righ)m(t-hand)d(side.)51 b(Let)34 b(us)f(consider)g(no)m(w)h(the)g (parameter)h Fl(\022)e Fj(2)e Fh(C)58 b Fn(satisfying)32 b(Im)15 b Fl(\022)3457 388 y Fk(1)3527 374 y Fj(\024)31 b Fn(Im)15 b Fl(\022)33 b Fj(\024)0 482 y Fn(Im)15 b Fl(\022)167 496 y Fk(2)231 482 y Fl(<)25 b Fn(0)p Fl(:)31 b Fn(In)m(tro)s(duce)f(the)h(set)544 674 y Fj(R)621 689 y Fg(\022)654 698 y Ft(2)693 674 y Fn(\()p Fl(P)786 688 y Fk(2)826 674 y Fn(\))26 b(=)1035 593 y Ff([)983 777 y Fg(f)s(;g)r Fi(2A)1179 674 y Fj(f)p Fl(z)k 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b(then)e(the)g(left-hand)f(side)g(of)h(\(5.13\))i(has)e(a)h(p)s(ole)e (for)g(some)i Fl(f)5 b(;)15 b(g)36 b Fj(2)d(A)h Fn(and)h(w)m(e)g (obtain)g(the)0 1472 y(inclusion)1238 1623 y Fj(R)1315 1638 y Fg(\022)1348 1647 y Ft(2)1387 1623 y Fn(\()p Fl(P)1480 1637 y Fk(2)1520 1623 y Fn(\))25 b Fj(\032)1914 1542 y Ff(\\)1676 1727 y Fk(Im)11 b Fg(\022)1805 1736 y Ft(1)1840 1727 y Fi(\024)p Fk(Im)g Fg(\022)r Fi(\024)p Fk(Im)f Fg(\022)2209 1736 y Ft(2)2259 1623 y Fl(\033)2311 1637 y Fk(pp)2414 1623 y Fn(\()p Fl(P)2507 1638 y Fk(2)p Fg(;\022)2601 1623 y Fn(\))p Fl(:)0 1892 y Fn(On)45 b(the)g(other)h(hand,)j(if)44 b Fl(z)1003 1906 y Fk(0)1088 1892 y Fn(is)h(a)h(p)s(ole)f(of)g(\()p Fl(P)1707 1907 y Fk(2)p Fg(;\022)1832 1892 y Fj(\000)30 b Fl(z)t Fn(\))2014 1859 y Fi(\000)p Fk(1)2109 1892 y Fn(,)50 b(then)45 b(w)m(e)h(can)g(\014nd)e Fl(')51 b Fj(6)p Fn(=)f(0)p Fl(;)c( )54 b Fj(6)p Fn(=)d(0)46 b(so)f(that)0 2004 y(\()p Fl( )s(;)15 b Fn(\()p Fl(P)230 2019 y Fk(2)p Fg(;\022)351 2004 y Fj(\000)26 b Fl(z)t 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Fi(1)1225 2011 y Fn(\()p Fh(R)s Fn(\))31 b(b)s(e)25 b(a)g(function)f(with)2147 1987 y(~)2128 2011 y Fl( )2187 2025 y Fk(1)2252 2011 y Fn(=)h(1)h(near)f(supp)2844 1988 y(~)2823 2011 y Fl(G)h Fn(and)3110 1987 y(~)3091 2011 y Fl( )3150 2025 y Fk(1)3215 2011 y Fn(=)f(0)h(near)f(supp)8 b Fl( )3835 2025 y Fk(1)3875 2011 y Fn(.)0 2119 y(Then)355 2270 y Fl( )414 2284 y Fk(1)454 2176 y Ff(\020)504 2270 y Fn(\()p Fl(z)24 b Fj(\000)c Fl(P)754 2284 y Fk(2)794 2270 y Fn(\))829 2233 y Fi(\000)p Fk(1)944 2270 y Fj(\000)g Fn(\()p Fl(z)25 b Fj(\000)1248 2247 y Fn(~)1228 2270 y Fl(P)1286 2284 y Fk(2)1326 2270 y Fn(\))1361 2233 y Fi(\000)p Fk(1)1455 2176 y Ff(\021)1505 2270 y Fn(\()p Fl(@)1588 2284 y Fg(x)1628 2293 y Ft(1)1667 2270 y Fl(V)20 b Fn(\))p Fl( )1834 2284 y Fk(1)1899 2270 y Fn(=)25 b Fj(\000)p Fl( )2125 2284 y Fk(1)2165 2270 y Fn(\()p Fl(z)g Fj(\000)19 b Fl(P)2415 2284 y Fk(2)2455 2270 y Fn(\))2490 2233 y Fi(\000)p Fk(1)2604 2246 y Fn(~)2585 2270 y Fl( )2644 2284 y Fk(1)2704 2247 y Fn(~)2684 2270 y Fl(G)p Fn(\()p Fl(z)25 b Fj(\000)2968 2247 y Fn(~)2948 2270 y Fl(P)3006 2284 y Fk(2)3046 2270 y Fn(\))3081 2233 y Fi(\000)p Fk(1)3175 2270 y Fl( )3234 2284 y Fk(1)3274 2270 y Fn(\()p Fl(@)3357 2284 y Fg(x)3397 2293 y Ft(1)3436 2270 y Fl(V)20 b Fn(\))3713 2378 y(\(7.6\))810 2547 y(=)25 b Fj(\000)p Fl( )1036 2561 y Fk(1)1076 2547 y Fn(\()p Fl(z)f Fj(\000)c Fl(P)1326 2561 y Fk(2)1366 2547 y Fn(\))1401 2509 y Fi(\000)p Fk(1)1514 2523 y Fn(~)1496 2547 y Fl( )1555 2561 y Fk(1)1594 2547 y Fn(\()p Fl(z)25 b Fj(\000)1808 2524 y Fn(~)1787 2547 y Fl(P)1845 2561 y Fk(2)1885 2547 y Fn(\))1920 2509 y Fi(\000)p Fk(1)2015 2547 y Fn([)2061 2524 y(~)2040 2547 y Fl(G;)2172 2524 y Fn(~)2151 2547 y Fl(P)2209 2561 y Fk(2)2250 2547 y Fn(]\()p Fl(z)g Fj(\000)2488 2524 y Fn(~)2468 2547 y Fl(P)2526 2561 y Fk(2)2565 2547 y Fn(\))2600 2509 y Fi(\000)p Fk(1)2695 2547 y Fl( )2754 2561 y Fk(1)2794 2547 y Fn(\()p Fl(@)2877 2561 y Fg(x)2917 2570 y Ft(1)2956 2547 y Fl(V)20 b Fn(\))p Fl(:)0 2681 y Fn(Let)31 b Fl(\037)220 2695 y Fk(1)259 2681 y Fl(;)15 b(:::;)g(\037)471 2695 y Fg(N)566 2681 y Fj(2)25 b Fl(C)724 2648 y Fi(1)717 2704 y Fk(0)798 2681 y Fn(\()p Fh(R)893 2648 y Fg(n)946 2681 y Fn(;)15 b([0)p Fl(;)g Fn(1]\))34 b(with)29 b Fl( )1501 2695 y Fk(1)1566 2681 y Fj(\036)d Fl(\037)1720 2695 y Fk(1)1785 2681 y Fj(\036)f Fl(:::)i Fj(\036)e Fl(\037)2136 2695 y Fg(N)2234 2681 y Fn(and)30 b Fl(\037)2468 2695 y Fg(i)2496 2681 y Fn([)2542 2658 y(~)2521 2681 y Fl(G)q(;)2653 2658 y Fn(~)2633 2681 y Fl(P)2691 2695 y Fk(2)2731 2681 y Fn(])c(=)f(0.)42 b(By)31 b(using)e(the)i(equalities)0 2796 y Fl(\037)57 2810 y Fk(1)96 2796 y Fl(:::\037)228 2810 y Fg(N)296 2796 y Fl( )355 2810 y Fk(1)420 2796 y Fn(=)25 b Fl( )575 2810 y Fk(1)615 2796 y Fl(;)35 b(\037)732 2811 y Fg(k)795 2796 y Fn([)841 2773 y(~)820 2796 y Fl(G)q(;)953 2773 y Fn(~)932 2796 y Fl(P)990 2810 y Fk(2)1030 2796 y Fn(])26 b(=)e(0,)31 b Fl(\037)1334 2811 y Fg(k)r Fi(\000)p Fk(1)1467 2796 y Fn([)p Fl(\037)1549 2811 y Fg(k)1592 2796 y Fl(;)36 b(P)1711 2810 y Fk(2)1750 2796 y Fn(])26 b(=)f(0)31 b(and)e(the)i(fact)g(that)997 2946 y([)p Fl(\037)1079 2961 y Fg(k)1122 2946 y Fl(;)15 b Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)1413 2960 y Fk(2)1453 2946 y Fn(\))1488 2909 y Fi(\000)p Fk(1)1582 2946 y Fn(])26 b(=)f(\()p Fl(z)g Fj(\000)20 b Fl(P)1980 2960 y Fk(2)2019 2946 y Fn(\))2054 2909 y Fi(\000)p Fk(1)2149 2946 y Fn([)p Fl(\037)2231 2961 y Fg(k)2274 2946 y Fl(;)35 b(P)2392 2960 y Fk(2)2432 2946 y Fn(]\()p Fl(z)25 b Fj(\000)20 b Fl(P)2708 2960 y Fk(2)2748 2946 y Fn(\))2783 2909 y Fi(\000)p Fk(1)2878 2946 y Fl(;)0 3093 y Fn(w)m(e)31 b(get)1560 3203 y([)1606 3180 y(~)1585 3203 y Fl(G)q(;)1718 3180 y Fn(~)1697 3203 y Fl(P)1755 3217 y Fk(2)1795 3203 y Fn(]\()p Fl(z)25 b Fj(\000)2034 3180 y Fn(~)2013 3203 y Fl(P)2071 3217 y Fk(2)2111 3203 y Fn(\))2146 3165 y Fi(\000)p Fk(1)2241 3203 y Fl( )2300 3217 y Fk(1)544 3338 y Fn(=)f([)685 3315 y(~)664 3338 y Fl(G)q(;)797 3315 y Fn(~)776 3338 y Fl(P)834 3352 y Fk(2)874 3338 y Fn(]\()p Fl(z)h Fj(\000)1113 3315 y Fn(~)1092 3338 y Fl(P)1150 3352 y Fk(2)1190 3338 y Fn(\))1225 3301 y Fi(\000)p Fk(1)1320 3338 y Fn([)p Fl(\037)1402 3352 y Fk(1)1441 3338 y Fl(;)1502 3315 y Fn(~)1481 3338 y Fl(P)1539 3352 y Fk(2)1580 3338 y Fn(]\()p Fl(z)g Fj(\000)1818 3315 y Fn(~)1798 3338 y Fl(P)1856 3352 y Fk(2)1895 3338 y Fn(\))1930 3301 y Fi(\000)p Fk(1)2025 3338 y Fl(:::)p Fn([)p Fl(\037)2182 3352 y Fg(N)2250 3338 y Fl(;)2311 3315 y Fn(~)2290 3338 y Fl(P)2348 3352 y Fk(2)2388 3338 y Fn(]\()p Fl(z)h Fj(\000)2627 3315 y Fn(~)2606 3338 y Fl(P)2664 3352 y Fk(2)2704 3338 y Fn(\))2739 3301 y Fi(\000)p Fk(1)2834 3338 y Fl( )2893 3352 y Fk(1)2958 3338 y Fn(=:)f Fl(L)3141 3352 y Fg(N)3208 3338 y Fn(\()p Fl(h)p Fn(\))p Fl(:)150 3466 y Fn(Here)941 3615 y Fl(L)1003 3629 y Fg(N)1070 3615 y Fn(\()p Fl(h)p Fn(\))i(=)e Fj(O)1387 3629 y Fg(N)1454 3615 y Fn(\(1\))1569 3520 y Ff(\020)1722 3553 y Fl(h)1774 3520 y Fg(N)p 1630 3593 303 4 v 1630 3677 a Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)1865 3651 y Fg(N)1943 3520 y Ff(\021)2018 3615 y Fn(:)25 b Fl(H)2151 3577 y Fg(s)p Fi(\000)p Fg(N)2306 3615 y Fn(\()p Fh(R)2401 3577 y Fg(n)2454 3615 y Fn(\))g Fj(!)h Fl(H)2714 3577 y Fg(s)2750 3615 y Fn(\()p Fh(R)2845 3577 y Fg(n)2898 3615 y Fn(\))p Fl(;)0 3806 y Fn(and)k(w)m(e)i(equip)e Fl(H)641 3773 y Fg(N)708 3806 y Fn(\()p Fh(R)803 3773 y Fg(n)856 3806 y Fn(\))h(with)f(the)h Fl(h)p Fn(-dep)s(enden)m(t)g (norm)f Fj(kh)p Fl(hD)s Fj(i)2293 3773 y Fg(N)2362 3806 y Fj(k)2407 3826 y Fg(L)2455 3807 y Ft(2)2494 3806 y Fn(.)43 b(Cho)s(osing)29 b Fl(N)36 b(>)27 b(n)j Fn(our)h(assumptions)0 3914 y(on)f Fl(V)51 b Fn(and)30 b(Theorem)g(9.4)h(in)e([10)q(])i(yield) 1193 4014 y Ff(\015)1193 4064 y(\015)1193 4114 y(\015)1239 4017 y(\020)1289 4111 y Fj(\000)p Fl(h)1412 4074 y Fk(2)1451 4111 y Fn(\001)20 b(+)g(1\))1718 4017 y Ff(\021)1769 4033 y Fi(\000)p Fg(N)q(=)p Fk(2)1955 4111 y Fl(@)2003 4125 y Fg(x)2043 4134 y Ft(1)2082 4111 y Fl(V)2155 4014 y Ff(\015)2155 4064 y(\015)2155 4114 y(\015)2201 4175 y Fk(tr)2286 4111 y Fn(=)25 b Fj(O)s Fn(\()p Fl(h)2544 4074 y Fi(\000)p Fg(n)2646 4111 y Fn(\))p Fl(:)0 4285 y Fn(Then)295 4459 y Fj(k)p Fn([)386 4436 y(~)365 4459 y Fl(G)q(;)498 4436 y Fn(~)477 4459 y Fl(P)535 4473 y Fk(2)575 4459 y Fn(]\()p Fl(z)g Fj(\000)813 4436 y Fn(~)793 4459 y Fl(P)851 4473 y Fk(2)891 4459 y Fn(\))926 4422 y Fi(\000)p Fk(1)1021 4459 y Fl( )1080 4473 y Fk(1)1119 4459 y Fn(\()p Fl(@)1202 4473 y Fg(x)1242 4482 y Ft(1)1281 4459 y Fl(V)20 b Fn(\))p Fj(k)1434 4473 y Fk(tr)1520 4459 y Fn(=)25 b Fj(k)p Fl(L)1723 4473 y Fg(N)1790 4459 y Fn(\()p Fl(h)p Fn(\))1912 4365 y Ff(\020)1983 4459 y Fj(\000)20 b Fl(h)2126 4422 y Fk(2)2166 4459 y Fn(\001)g(+)g(1)2398 4365 y Ff(\021)2448 4381 y Fg(N)q(=)p Fk(2)2580 4365 y Ff(\020)2649 4459 y Fj(\000)g Fl(h)2792 4422 y Fk(2)2832 4459 y Fn(\001)g(+)g(1)3064 4365 y Ff(\021)3114 4381 y Fi(\000)p Fg(N)q(=)p Fk(2)3301 4459 y Fl(@)3349 4473 y Fg(x)3389 4482 y Ft(1)3428 4459 y Fl(V)g Fj(k)3546 4473 y Fk(tr)3713 4567 y Fn(\(7.7\))783 4791 y Fj(\024)25 b Fl(C)7 b Fj(k)996 4697 y Ff(\020)1066 4791 y Fj(\000)20 b Fl(h)1209 4754 y Fk(2)1249 4791 y Fn(\001)g(+)g(1)1481 4697 y Ff(\021)1531 4713 y Fi(\000)p Fg(N)q(=)p Fk(2)1717 4791 y Fl(@)1765 4805 y Fg(x)1805 4814 y Ft(1)1844 4791 y Fl(V)g Fj(k)1962 4805 y Fk(tr)2022 4697 y Ff(\020)2173 4730 y Fl(h)2225 4697 y Fg(N)p 2081 4770 V 2081 4854 a Fj(j)15 b Fn(Im)h Fl(z)t Fj(j)2317 4827 y Fg(N)2394 4697 y Ff(\021)2469 4791 y Fj(\024)25 b Fl(C)2630 4805 y Fk(1)2670 4697 y Ff(\020)2772 4730 y Fl(h)2824 4697 y Fg(N)7 b Fi(\000)p Fg(n)p 2729 4770 V 2729 4854 a Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)2964 4827 y Fg(N)3042 4697 y Ff(\021)3091 4791 y Fl(:)150 4989 y Fn(Since)399 4965 y(\026)387 4989 y Fl(@)435 5003 y Fg(z)495 4965 y Fn(~)475 4989 y Fl(f)10 b Fn(\()p Fl(z)t Fn(\))26 b(=)f Fj(O)s Fn(\()p Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)1113 4956 y Fi(1)1189 4989 y Fn(\),)30 b(w)m(e)h(ha)m(v)m(e)1201 5119 y(\026)1189 5143 y Fl(@)1237 5157 y Fg(z)1277 5143 y Fn(\()p Fl( )1371 5157 y Fg(Y)1452 5119 y Fn(~)1432 5143 y Fl(f)10 b Fn(\)\()p Fl(z)t Fn(\))26 b(=)1780 5119 y(~)1760 5143 y Fl(f)9 b Fn(\()p Fl(z)t Fn(\))1942 5119 y(\026)1930 5143 y Fl(@)1978 5157 y Fg(z)2020 5143 y Fn(\()p Fl( )2114 5157 y Fg(Y)2175 5143 y Fn(\)\()p Fl(z)t Fn(\))21 b(+)f Fj(O)s Fn(\()p Fl(h)2600 5106 y Fi(1)2676 5143 y Fn(\))p eop %%Page: 21 21 21 20 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g (RESONANCES)999 b(21)0 266 y Fn(and,)30 b(consequen)m(tly)-8 b(,)1074 465 y Fl(I)32 b Fn(=)25 b Fj(\000)1328 403 y Fn(1)p 1323 443 56 4 v 1323 527 a Fl(\031)1403 350 y Ff(Z)1449 538 y Fg(Y)15 b Fi(\024j)d Fk(Im)f Fg(z)s Fi(j\024)p Fk(2)p Fg(Y)1894 465 y Fn(\()1941 441 y(\026)1929 465 y Fl(@)1977 479 y Fg(z)2017 465 y Fl( )2076 479 y Fg(Y)2137 465 y Fn(\)\()p Fl(z)t Fn(\))2309 441 y(~)2288 465 y Fl(f)g Fn(\()p Fl(z)t Fn(\))2467 441 y(\025)2460 465 y Fl(\022)2503 480 y Fg(h)2548 465 y Fn(\()p Fl(\034)31 b Fj(\000)20 b Fl(z)t Fn(\))887 b(\(7.8\))632 736 y Fj(\002)p Fn(tr)774 641 y Ff(h)813 736 y Fl( )872 750 y Fk(1)912 641 y Ff(\020)961 736 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)1212 750 y Fk(2)1252 736 y Fn(\()p Fl(h)p Fn(\)\))1409 698 y Fi(\000)p Fk(1)1525 736 y Fj(\000)g Fn(\()p Fl(z)25 b Fj(\000)1829 713 y Fn(~)1809 736 y Fl(P)1867 750 y Fk(2)1906 736 y Fn(\()p Fl(h)p Fn(\)\))2063 698 y Fi(\000)p Fk(1)2159 641 y Ff(\021)2209 736 y Fn(\()p Fl(@)2292 750 y Fg(x)2332 759 y Ft(1)2371 736 y Fl(V)20 b Fn(\))p Fl( )2538 750 y Fk(1)2578 641 y Ff(i)2617 736 y Fl(L)p Fn(\()p Fl(dz)t Fn(\))i(+)e Fj(O)s Fn(\()p Fl(h)3117 698 y Fi(1)3192 736 y Fn(\))15 b Fl(:)150 905 y Fn(Next)31 b(w)m(e)g(tak)m(e)h(a)f(real-v)-5 b(alued)29 b(function)g Fl(K)1674 919 y Fk(0)1739 905 y Fj(2)c Fl(C)1897 872 y Fi(1)1890 928 y Fk(0)1971 905 y Fn(\()p Fh(R)s Fn(\))37 b(suc)m(h)30 b(that)1072 1074 y Fl(K)1149 1088 y Fk(0)1214 1074 y Fn(=)25 b(1)31 b(near)f(supp)19 b Fl( )1855 1088 y Fk(1)1894 1074 y Fl(;)61 b(K)2057 1088 y Fk(0)2122 1074 y Fn(=)25 b(0)31 b(near)f(supp)2722 1050 y(~)2704 1074 y Fl( )2763 1088 y Fk(1)2802 1074 y Fl(:)0 1237 y Fn(Put)i Fl(K)j Fn(=)28 b Fl(\013K)526 1251 y Fk(0)566 1237 y Fn(,)33 b Fl(\013)c(>)g Fn(0.)46 b(By)33 b(using)e(that)i Fl(K)i Fn(=)28 b Fl(\013)33 b Fn(near)f(supp)20 b Fl( )2285 1251 y Fk(1)2357 1237 y Fn(and)31 b(the)i(fact)g(that)g Fl(K)i Fn(=)28 b(0)33 b(near)f(supp)3795 1213 y(~)3776 1237 y Fl( )3835 1251 y Fk(1)3875 1237 y Fn(,)0 1345 y(w)m(e)f(get)g(in)e(the)i(op)s(erator)g(norm)646 1521 y Fl( )705 1535 y Fk(1)744 1521 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)995 1535 y Fk(2)1035 1521 y Fn(\()p Fl(h)p Fn(\)\))1192 1483 y Fi(\000)p Fk(1)1306 1497 y Fn(~)1288 1521 y Fl( )1347 1535 y Fk(1)1412 1521 y Fn(=)25 b Fl(e)1550 1483 y Fi(\000)p Fg(\013)12 b Fk(log)1778 1456 y Ft(1)p 1775 1468 37 4 v 1775 1509 a Fq(h)1825 1521 y Fl(e)1867 1483 y Fg(K)k Fk(log)2059 1456 y Ft(1)p 2056 1468 V 2056 1509 a Fq(h)2107 1521 y Fl( )2166 1535 y Fk(1)2205 1521 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)2456 1535 y Fk(2)2496 1521 y Fn(\()p Fl(h)p Fn(\)\))2653 1483 y Fi(\000)p Fk(1)2748 1521 y Fn(\()p Fl(e)2825 1483 y Fi(\000)p Fg(K)d Fk(log)3073 1456 y Ft(1)p 3070 1468 V 3070 1509 a Fq(h)3120 1521 y Fn(\))3174 1497 y(~)3155 1521 y Fl( )3214 1535 y Fk(1)3713 1521 y Fn(\(7.9\))1051 1730 y(=)25 b Fl(e)1189 1693 y Fi(\000)p Fg(\013)13 b Fk(log)1417 1666 y Ft(1)p 1415 1678 V 1415 1719 a Fq(h)1465 1730 y Fl( )1524 1744 y Fk(1)1564 1730 y Fn(\()p Fl(z)24 b Fj(\000)c Fl(e)1798 1693 y Fg(K)d Fk(log)1990 1666 y Ft(1)p 1987 1678 V 1987 1719 a Fq(h)2038 1730 y Fl(P)2096 1744 y Fk(2)2136 1730 y Fn(\()p Fl(h)p Fn(\))p Fl(e)2300 1693 y Fi(\000)p Fg(K)g Fk(log)2548 1666 y Ft(1)p 2545 1678 V 2545 1719 a Fq(h)2595 1730 y Fn(\))2630 1693 y Fi(\000)p Fk(1)2743 1706 y Fn(~)2725 1730 y Fl( )2784 1744 y Fk(1)2823 1730 y Fl(:)0 1872 y Fn(On)30 b(the)g(other)h(hand,)e(a)i(simple)d(calculus)h(sho)m(ws)h (that)1369 2048 y Fl(e)1411 2010 y Fg(K)16 b Fk(log)1603 1983 y Ft(1)p 1600 1995 V 1600 2036 a Fq(h)1650 2048 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)1901 2062 y Fk(2)1941 2048 y Fn(\()p Fl(h)p Fn(\)\))p Fl(e)2140 2010 y Fi(\000)p Fg(K)d Fk(log)2388 1983 y Ft(1)p 2385 1995 V 2385 2036 a Fq(h)2461 2048 y Fn(=)623 2173 y Ff(\020)672 2267 y Fl(I)28 b Fn(+)831 2198 y Ff(\002)865 2267 y Fn(2)p Fl(\013h)15 b Fn(log)1183 2205 y(1)p 1179 2246 53 4 v 1179 2329 a Fl(h)1242 2267 y Fn(\()p Fj(r)p Fl(K)7 b Fn(\))20 b Fj(\001)g(r)g Fn(+)g Fj(O)s Fn(\()p Fl(\013h)15 b Fn(log)2107 2205 y(1)p 2103 2246 V 2103 2329 a Fl(h)2165 2267 y Fn(\))2200 2198 y Ff(\003)2235 2267 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)2486 2281 y Fk(2)2526 2267 y Fn(\()p Fl(h)p Fn(\)\))2683 2229 y Fi(\000)p Fk(1)2779 2173 y Ff(\021)2828 2267 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)3079 2281 y Fk(2)3119 2267 y Fn(\()p Fl(h)p Fn(\)\))0 2440 y(in)29 b(the)i(op)s(erator)f (norm)g(for)g Fl(h)c Fj(\024)f Fl(h)p Fn(\()p Fl(\013)p Fn(\),)32 b Fl(h)p Fn(\()p Fl(\013)p Fn(\))27 b Fl(>)e Fn(0)30 b(b)s(eing)f(a)i(con)m(tin)m(uous)f(function.)150 2547 y(Let)h(us)f(c)m(ho)s(ose)h Fl(\013)g Fn(as)1488 2700 y Fl(\013)25 b Fn(=)g(min)1819 2606 y Ff(\020)1927 2639 y Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)p 1878 2679 333 4 v 1899 2750 a Fn(~)1878 2773 y Fl(C)7 b(h)15 b Fn(log)2163 2737 y Fk(1)p 2160 2752 41 4 v 2160 2804 a Fg(h)2221 2700 y Fl(;)35 b(\014)2337 2606 y Ff(\021)2387 2700 y Fl(;)0 2930 y Fn(where)30 b Fl(\014)g(>)25 b Fn(0)31 b(is)e(some)i(arbitrary)e(large)i(and)f(\014xed)f(constan)m(t)j(and) 2405 2907 y(~)2385 2930 y Fl(C)37 b Fn(is)29 b(su\016cien)m(tly)g (large.)150 3040 y(Since)22 b Fj(k)p Fn(\()p Fj(r)p Fl(K)7 b Fn(\))p Fj(r)p Fn(\()p Fl(z)j Fj(\000)c Fl(P)953 3054 y Fk(2)993 3040 y Fn(\()p Fl(h)p Fn(\)\))1150 3007 y Fi(\000)p Fk(1)1246 3040 y Fj(k)26 b Fn(=)e Fj(O)s Fn(\()p Fj(j)15 b Fn(Im)h Fl(z)t Fj(j)1758 3007 y Fi(\000)p Fk(1)1853 3040 y Fn(\))23 b(and)g Fl(\013h)15 b Fn(log)2352 3004 y Fk(1)p 2349 3019 V 2349 3071 a Fg(h)2400 3040 y Fj(j)g Fn(Im)g Fl(z)t Fj(j)2635 3007 y Fi(\000)p Fk(1)2755 3040 y Fj(\024)2871 3004 y Fk(1)p 2861 3019 55 4 v 2877 3067 a(~)2861 3084 y Fg(C)2926 3040 y Fn(,)25 b(then)e(for)3328 3017 y(~)3307 3040 y Fl(C)30 b Fn(large)23 b(enough)0 3187 y(the)30 b(righ)m(t)f(hand)f(side)g(of)i(\(7.9\))h(is)e Fj(O)s Fn(\()p Fl(e)1344 3151 y Fi(\000)p Fg(\013)12 b Fk(log)1572 3124 y Ft(1)p 1569 3136 37 4 v 1569 3177 a Fq(h)1619 3187 y Fj(j)j Fn(Im)g Fl(z)t Fj(j)1854 3154 y Fi(\000)p Fk(1)1949 3187 y Fn(\))30 b(in)e Fj(L)p Fn(\()p Fl(L)2279 3154 y Fk(2)2319 3187 y Fn(\()p Fh(R)2414 3154 y Fg(n)2467 3187 y Fn(\)\).)41 b(Com)m(bining)27 b(this)h(with)g (\(7.6\),)k(\(7.7\),)0 3295 y(w)m(e)f(get)g(for)f Fj(j)15 b Fn(Im)h Fl(z)t Fj(j)25 b(\025)g(\000)p Fl(M)10 b(h)15 b Fn(log)i Fl(h)31 b Fn(with)e(a)i(new)f(constan)m(t)h Fl(C)2134 3309 y Fk(2)2199 3295 y Fl(>)25 b Fn(0)1115 3386 y Ff(\015)1115 3436 y(\015)1115 3485 y(\015)1161 3483 y Fl( )1220 3497 y Fk(1)1260 3389 y Ff(\020)1309 3483 y Fn(\()p Fl(z)g Fj(\000)20 b Fl(P)1560 3497 y Fk(2)1600 3483 y Fn(\))1635 3446 y Fi(\000)p Fk(1)1750 3483 y Fj(\000)g Fn(\()p Fl(z)25 b Fj(\000)2054 3460 y Fn(~)2033 3483 y Fl(P)2091 3497 y Fk(2)2131 3483 y Fn(\))2166 3446 y Fi(\000)p Fk(1)2261 3389 y Ff(\021)2310 3483 y Fn(\()p Fl(@)2393 3497 y Fg(x)2433 3506 y Ft(1)2473 3483 y Fl(V)20 b Fn(\))p Fl( )2640 3497 y Fk(1)2680 3386 y Ff(\015)2680 3436 y(\015)2680 3485 y(\015)2726 3547 y Fk(tr)687 3745 y Fn(=)25 b Fj(O)s Fn(\()p Fl(h)945 3707 y Fi(\000)p Fg(n)1047 3745 y Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)1282 3707 y Fi(\000)p Fk(2)1377 3745 y Fl(e)1419 3707 y Fi(\000)p Fg(\013)d Fk(log)1647 3680 y Ft(1)p 1644 3692 V 1644 3733 a Fq(h)1694 3745 y Fn(\))26 b(=)f Fj(O)1926 3651 y Ff(\020)1976 3745 y Fl(h)2028 3707 y Fi(\000)p Fg(n)2130 3745 y Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)2365 3707 y Fi(\000)p Fk(2)2475 3745 y Fn(max\()p Fl(h)2731 3707 y Fg(\014)2779 3745 y Fl(;)g(e)2861 3694 y Fi(\000)2926 3660 y Fr(j)c Ft(Im)g Fq(z)r Fr(j)p 2927 3679 163 4 v 2950 3720 a Fq(C)2994 3735 y Ft(2)3029 3720 y Fq(h)3103 3745 y Fn(\))3138 3651 y Ff(\021)3188 3745 y Fl(:)0 3909 y Fn(On)30 b(the)g(other)h(hand,)e (the)i(P)m(aley-Wiener)f(theorem)h(yields)1378 4108 y Fj(j)1410 4084 y Fn(\025)1403 4108 y Fl(\022)1446 4123 y Fg(h)1491 4108 y Fn(\()p Fl(\034)f Fj(\000)20 b Fl(z)t Fn(\))p Fj(j)26 b Fn(=)f Fj(O)s Fn(\()2038 4047 y(1)p 2035 4087 53 4 v 2035 4171 a Fl(h)2098 4108 y(e)2192 4030 y Ft(1)p 2150 4042 115 4 v 2150 4083 a Fq(C)2194 4098 y Ft(0)2229 4083 y Fq(h)2275 4057 y Fi(j)11 b Fk(Im)g Fg(z)s Fi(j)2462 4108 y Fn(\))p Fl(;)0 4310 y Fn(where)30 b(w)m(e)h(ha)m(v)m(e)g(used)f(that)h(supp)18 b Fl(\022)28 b Fj(\032)p Fn(])20 b Fj(\000)1531 4274 y Fk(1)p 1506 4289 85 4 v 1506 4342 a Fg(C)1556 4351 y Ft(0)1601 4310 y Fl(;)1676 4274 y Fk(1)p 1651 4289 V 1651 4342 a Fg(C)1701 4351 y Ft(0)1746 4310 y Fn([.)41 b(W)-8 b(e)31 b(c)m(ho)s(ose)h Fl(C)2347 4324 y Fk(0)2412 4310 y Fl(>)24 b(C)2572 4324 y Fk(2)2642 4310 y Fn(and)30 b(the)h(r.h.s)e(of)i(\(7.8\))h(b)s(ecomes) 0 4542 y Fl(I)g Fn(=)25 b Fj(O)s Fn(\(1\))373 4427 y Ff(Z)431 4583 y Fq(Y)13 b Fr(\024j)d Ft(Im)h Fq(z)r Fr(j\024)p Ft(2)p Fq(Y)440 4643 y Fr(j)f Ft(Re)h Fq(z)r Fr(j\024)p Ft(const)p Fq(:)847 4542 y Fl(Y)920 4504 y Fi(\000)p Fk(2)1014 4542 y Fl(h)1066 4504 y Fi(\000)p Fg(n)p Fi(\000)p Fk(1)1274 4542 y Fn(max)1443 4447 y Ff(\020)1493 4542 y Fl(h)1545 4504 y Fg(\014)1607 4542 y Fn(exp\()1791 4480 y Fj(j)k Fn(Im)h Fl(z)t Fj(j)p 1791 4521 236 4 v 1831 4604 a Fl(C)1896 4618 y Fk(0)1935 4604 y Fl(h)2037 4542 y Fn(\))p Fl(;)51 b Fn(exp\()p Fj(\000)2406 4480 y Fn(1)p 2403 4521 53 4 v 2403 4604 a Fl(h)2466 4542 y Fj(j)15 b Fn(Im)g Fl(z)t Fj(j)p Fn(\()p Fl(C)2808 4503 y Fi(\000)p Fk(1)2801 4566 y(2)2920 4542 y Fj(\000)j Fl(C)3081 4503 y Fi(\000)p Fk(1)3074 4566 y(0)3175 4542 y Fn(\)\))3245 4447 y Ff(\021)3295 4542 y Fl(L)p Fn(\()p Fl(dz)t Fn(\))h(+)e Fj(O)s Fn(\()p Fl(h)3789 4504 y Fi(1)3865 4542 y Fn(\))492 4852 y(=)25 b Fj(O)s Fn(\(1\))p Fl(Y)852 4814 y Fi(\000)p Fk(1)946 4852 y Fl(h)998 4814 y Fi(\000)p Fg(n)p Fi(\000)p Fk(1)1206 4852 y Fn(max)1375 4758 y Ff(\020)1425 4852 y Fl(h)1477 4801 y Fg(\014)s Fi(\000)1585 4774 y Ft(2)p Fq(M)p 1585 4786 94 4 v 1593 4827 a(C)1637 4842 y Ft(0)1693 4852 y Fl(;)35 b Fn(exp)1892 4758 y Ff(\020)1942 4852 y Fj(\000)p Fl(M)10 b Fn(\(log)2292 4790 y(1)p 2289 4831 53 4 v 2289 4914 a Fl(h)2351 4852 y Fn(\)\()p Fl(C)2493 4813 y Fi(\000)p Fk(1)2486 4876 y(2)2608 4852 y Fj(\000)20 b Fl(C)2771 4813 y Fi(\000)p Fk(1)2764 4876 y(0)2864 4852 y Fn(\))2899 4758 y Ff(\021\021)3019 4852 y Fn(+)g Fj(O)s Fn(\()p Fl(h)3272 4814 y Fi(1)3347 4852 y Fn(\))p Fl(:)0 5035 y Fn(First)g(c)m(ho)s(osing)h Fl(M)30 b Fn(su\016cien)m(tly)20 b(large)h(and)f(then)g(the)h(p)s(o)m (w)m(er)g(in)e Fl(h)2264 5002 y Fg(\014)2332 5035 y Fn(large)i(enough,) i(w)m(e)e(see)g(that)h(this)d(expression)0 5143 y(is)29 b Fj(O)163 5157 y Fg(N)231 5143 y Fn(\()p Fl(h)318 5110 y Fg(N)386 5143 y Fn(\))i(for)f(an)m(y)h Fl(N)k Fj(2)25 b Fh(N)43 b Fn(and)29 b Fl(h)d Fj(\024)f Fl(h)p Fn(\()p Fl(N)10 b Fn(\))p Fl(:)p eop %%Page: 22 22 22 21 bop 0 66 a Fk(22)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)150 266 y Fn(T)-8 b(urning)31 b(to)k(the)e(study)g(of)g(\014rst)g(term)h (in)e(the)i(r.h.s.)49 b(of)34 b(\(7.4\),)i(c)m(ho)s(ose)f(a)e(real-v)-5 b(alued)33 b(smo)s(oth)g(function)0 374 y Fl(F)13 b Fn(\()p Fl(x)158 388 y Fk(1)198 374 y Fn(\))24 b(satisfying)f Fl(F)13 b Fn(\()p Fl(x)814 388 y Fk(1)854 374 y Fn(\))25 b(=)g Fl(x)1062 388 y Fk(1)1126 374 y Fn(for)f Fl(x)1311 388 y Fk(1)1375 374 y Fl(>)h(R)1540 388 y Fk(1)1588 374 y Fj(\000)8 b Fn(1)23 b(and)h Fl(F)13 b Fn(\()p Fl(x)2064 388 y Fk(1)2104 374 y Fn(\))25 b Fl(>)g(R)2329 388 y Fk(1)2376 374 y Fj(\000)8 b Fn(2)24 b(for)g(all)f Fl(x)2829 388 y Fk(1)2894 374 y Fj(2)i Fh(R)r Fn(.)45 b(Using)23 b(the)i(inequalit)m(y)0 489 y Fl(R)69 503 y Fk(1)134 489 y Fl(>)g Fj(k)p Fl(V)20 b Fj(k)393 503 y Fi(1)489 489 y Fn(+)g Fl(E)647 503 y Fk(1)706 489 y Fn(+)g Fj(j)p Fl(\016)862 503 y Fk(1)902 489 y Fj(j)h Fn(+)f(3,)31 b(as)f(w)m(ell)g(as)g(the)h(fact)g(that)g(supp)2287 465 y(~)2268 489 y Fl(f)j Fj(\032)25 b(f)p Fl(z)30 b Fj(2)25 b Fh(C)18 b Fn(;)41 b(Re)16 b Fl(z)29 b Fj(2)c Fn([)p Fl(E)3144 503 y Fk(0)3184 489 y Fl(;)15 b(E)3291 503 y Fk(1)3331 489 y Fn(])p Fj(g)p Fn(,)31 b(w)m(e)g(get)1117 651 y Fj(j)p Fl(\030)t Fj(j)1211 614 y Fk(2)1272 651 y Fn(+)19 b Fl(F)13 b Fn(\()p Fl(x)1520 665 y Fk(1)1560 651 y Fn(\))21 b(+)f Fl(V)g Fn(\()p Fl(x)p Fn(\))h Fj(\000)f Fn(Re)15 b Fl(z)30 b(>)25 b Fn(1)20 b(+)g Fj(j)p Fl(\016)2525 665 y Fk(1)2565 651 y Fj(j)26 b(\025)f Fn(1)p Fl(;)0 818 y Fn(uniformly)i(on)k(\()p Fl(x;)36 b(\030)t Fn(\))30 b(and)g Fl(z)f Fj(2)c Fn(supp)1359 794 y(~)1340 818 y Fl(f)9 b Fn(.)41 b(In)m(tro)s(duce)30 b(the)g(op)s(erator)1417 960 y(~)1396 983 y Fl(P)39 b Fn(=)25 b Fj(\000)p Fl(h)1712 945 y Fk(2)1751 983 y Fn(\001)20 b(+)g Fl(F)13 b Fn(\()p Fl(x)2096 997 y Fk(1)2136 983 y Fn(\))20 b(+)g Fl(V)h Fn(\()p Fl(x)p Fn(\))p Fl(:)0 1148 y Fn(Clearly)-8 b(,)352 1125 y(~)332 1148 y Fl(P)40 b Fn(is)27 b(semi-b)s(ounded)e(on)j Fl(C)1291 1115 y Fi(1)1284 1171 y Fk(0)1365 1148 y Fn(\()p Fh(R)1460 1115 y Fg(n)1513 1148 y Fn(\))g(and)g(w)m(e)g(denote)g(also)g (b)m(y)2496 1125 y(~)2476 1148 y Fl(P)40 b Fn(the)28 b(selfadjoin)m(t)f(extension)h(of)3684 1125 y(~)3664 1148 y Fl(P)13 b(:)28 b Fn(By)0 1264 y(construction,)36 b Fl(z)g Fj(!)d Fn(\()p Fl(z)27 b Fj(\000)972 1241 y Fn(~)952 1264 y Fl(P)13 b Fn(\))1058 1231 y Fi(\000)p Fk(1)1187 1264 y Fn(exists)35 b(and)f(is)f(analytic)i(in)e(a)i(neigh)m (b)s(orho)s(o)s(d)d(of)j(supp)3173 1240 y(~)3154 1264 y Fl(f)9 b Fn(.)54 b(Com)m(bining)32 b(this)0 1372 y(with)d (Hel\013er-Sj\177)-45 b(ostrand)30 b(form)m(ula,)f(w)m(e)i(get)1198 1553 y(tr)1269 1459 y Ff(h)1308 1553 y Fn(\()p Fl(@)1391 1567 y Fg(x)1431 1576 y Ft(1)1471 1553 y Fl(V)20 b Fn(\))p Fl( )1638 1567 y Fk(0)1685 1529 y Fn(\025)1678 1553 y Fl(\022)1721 1568 y Fg(h)1765 1553 y Fn(\()p Fl(\034)31 b Fj(\000)20 b Fl(P)2020 1567 y Fk(2)2060 1553 y Fn(\()p Fl(h)p Fn(\)\))p Fl(f)10 b Fn(\()p Fl(P)2365 1567 y Fk(2)2405 1553 y Fn(\()p Fl(h)p Fn(\)\))p Fl( )2621 1567 y Fk(0)2663 1459 y Ff(i)3668 1553 y Fn(\(7.10\))73 1804 y(=)24 b Fj(\000)254 1742 y Fn(1)p 249 1783 56 4 v 249 1866 a Fl(\031)314 1804 y Fn(tr)385 1710 y Ff(\020)435 1689 y(Z)544 1780 y Fn(\026)533 1804 y Fl(@)581 1818 y Fg(z)621 1804 y Fn(\()p Fl( )715 1818 y Fg(Y)779 1804 y Fn(~)-48 b Fl(g)t Fn(\)\()p Fl(z)t Fn(\))981 1780 y(\025)974 1804 y Fl(\022)1017 1819 y Fg(h)1063 1804 y Fn(\()p Fl(\034)30 b Fj(\000)20 b Fl(z)t Fn(\)\()p Fl(@)1423 1818 y Fg(x)1463 1827 y Ft(1)1503 1804 y Fl(V)g Fn(\))p Fl( )1670 1818 y Fk(0)1710 1804 y Fn(\()p Fl(i)h Fj(\000)f Fl(P)1946 1818 y Fk(2)1986 1804 y Fn(\()p Fl(h)p Fn(\)\))2143 1766 y Fi(\000)p Fg(m)2266 1710 y Ff(\020)2315 1804 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)2566 1818 y Fk(2)2606 1804 y Fn(\()p Fl(h)p Fn(\)\))2763 1766 y Fi(\000)p Fk(1)2879 1804 y Fj(\000)g Fn(\()p Fl(z)k Fj(\000)3183 1781 y Fn(~)3162 1804 y Fl(P)13 b Fn(\))3268 1766 y Fi(\000)p Fk(1)3363 1710 y Ff(\021)3413 1804 y Fl( )3472 1818 y Fk(0)3511 1804 y Fl(L)p Fn(\()p Fl(dz)t Fn(\))3736 1710 y Ff(\021)3802 1804 y Fl(;)0 1998 y Fn(where)33 b(~)-48 b Fl(g)t Fn(\()p Fl(z)t Fn(\))26 b(=)567 1974 y(~)548 1998 y Fl(f)9 b Fn(\()p Fl(z)t Fn(\)\()p Fl(i)22 b Fj(\000)e Fl(z)t Fn(\))978 1965 y Fg(m)1045 1998 y Fn(.)41 b(Here)31 b Fl(m)f Fn(is)f(\014xed)h (so)g(that)h(\()p Fl(@)2138 2012 y Fg(x)2178 2021 y Ft(1)2217 1998 y Fl(V)21 b Fn(\)\()p Fl(i)g Fj(\000)f Fl(P)2562 2012 y Fk(2)2602 1998 y Fn(\()p Fl(h)p Fn(\)\))2759 1965 y Fi(\000)p Fg(m)2912 1998 y Fn(is)29 b(a)i(trace)g(class)g(op)s (erator.)150 2113 y(Since)e Fl(P)445 2127 y Fk(2)503 2113 y Fj(\000)612 2090 y Fn(~)592 2113 y Fl(P)38 b Fn(=)25 b Fl(x)836 2127 y Fk(1)894 2113 y Fj(\000)18 b Fl(F)13 b Fn(\()p Fl(x)1141 2127 y Fk(1)1181 2113 y Fn(\))26 b(=:)1384 2090 y(~)1363 2113 y Fl(F)38 b Fn(=)25 b(0)30 b(near)g(supp)17 b Fl( )2097 2127 y Fk(0)2136 2113 y Fn(,)30 b(w)m(e)g(can)g(rep)s(eat)g(the)g(argumen)m(t)g(of)f(the)h(pro) s(of)f(of)0 2221 y(\(7.6\))j(and)e(get)774 2275 y Ff(\020)823 2369 y Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)1074 2383 y Fk(2)1114 2369 y Fn(\))1149 2332 y Fi(\000)p Fk(1)1264 2369 y Fj(\000)g Fn(\()p Fl(z)k Fj(\000)1568 2346 y Fn(~)1547 2369 y Fl(P)13 b Fn(\))1653 2332 y Fi(\000)p Fk(1)1748 2275 y Ff(\021)1798 2369 y Fl( )1857 2383 y Fk(0)1922 2369 y Fn(=)24 b(\()p Fl(z)h Fj(\000)20 b Fl(P)2268 2383 y Fk(2)2308 2369 y Fn(\))2343 2332 y Fi(\000)p Fk(1)2456 2345 y Fn(~)2438 2369 y Fl( )2497 2383 y Fk(2)2557 2346 y Fn(~)2536 2369 y Fl(F)13 b Fn(\()p Fl(z)25 b Fj(\000)2821 2346 y Fn(~)2800 2369 y Fl(P)2858 2383 y Fk(2)2898 2369 y Fn(\))2933 2332 y Fi(\000)p Fk(1)3028 2369 y Fl( )3087 2383 y Fk(0)1071 2566 y Fn(=)g(\()p Fl(z)g Fj(\000)19 b Fl(P)1417 2580 y Fk(2)1457 2566 y Fn(\))1492 2528 y Fi(\000)p Fk(1)1606 2542 y Fn(~)1587 2566 y Fl( )1646 2580 y Fk(2)1686 2566 y Fn(\()p Fl(z)24 b Fj(\000)1899 2543 y Fn(~)1878 2566 y Fl(P)13 b Fn(\))1984 2528 y Fi(\000)p Fk(1)2079 2566 y Fn([)2125 2543 y(~)2104 2566 y Fl(F)g(;)2236 2543 y Fn(~)2215 2566 y Fl(P)h Fn(]\()p Fl(z)25 b Fj(\000)2525 2543 y Fn(~)2505 2566 y Fl(P)13 b Fn(\))2611 2528 y Fi(\000)p Fk(1)2705 2566 y Fl( )2764 2580 y Fk(0)2804 2566 y Fl(;)0 2713 y Fn(where)281 2689 y(~)263 2713 y Fl( )322 2727 y Fk(2)387 2713 y Fj(2)25 b Fl(C)545 2680 y Fi(1)619 2713 y Fn(\()p Fh(R)s Fn(\))37 b(with)1011 2689 y(~)993 2713 y Fl( )1052 2727 y Fk(2)1116 2713 y Fn(=)25 b(1)31 b(near)f(supp)1739 2690 y(~)1718 2713 y Fl(F)44 b Fn(and)2015 2689 y(~)1996 2713 y Fl( )2055 2727 y Fk(2)2120 2713 y Fn(=)25 b(0)31 b(near)f(supp)19 b Fl( )2761 2727 y Fk(0)2801 2713 y Fn(.)150 2828 y(T)-8 b(o)33 b(estimate)g(the)g(last)f (term,)h(notice)g(that)g(\()p Fl(z)26 b Fj(\000)1914 2805 y Fn(~)1894 2828 y Fl(P)13 b Fn(\))2000 2795 y Fi(\000)p Fk(1)2127 2828 y Fn(and)32 b(\()p Fl(h)p Fj(r)p Fn(\)\()p Fl(z)27 b Fj(\000)2720 2805 y Fn(~)2700 2828 y Fl(P)13 b Fn(\))2806 2795 y Fi(\000)p Fk(1)2929 2828 y Fj(2)29 b Fn(Op)3140 2792 y Fg(w)3140 2851 y(h)3196 2828 y Fn(\()p Fl(S)3292 2795 y Fk(0)3332 2828 y Fn(\(1\)\))34 b(uniformly)0 2944 y(for)c Fl(z)g Fj(2)25 b Fn(supp)524 2920 y(~)504 2944 y Fl(f)40 b Fn(\(see)31 b(the)g(pro)s(of)e(of)i(Lemma)f(2)h(for)f (the)h(de\014nition)d(of)i Fl(S)2533 2911 y Fk(0)2573 2944 y Fn(\(1\)\).)42 b(Consequen)m(tly)-8 b(,)1139 3110 y(\()p Fl(z)25 b Fj(\000)1352 3087 y Fn(~)1332 3110 y Fl(P)13 b Fn(\))1438 3072 y Fi(\000)p Fk(1)1533 3110 y Fn([)1578 3087 y(~)1558 3110 y Fl(F)g(;)1690 3087 y Fn(~)1669 3110 y Fl(P)h Fn(]\()p Fl(z)25 b Fj(\000)1979 3087 y Fn(~)1959 3110 y Fl(P)13 b Fn(\))2065 3072 y Fi(\000)p Fk(1)2184 3110 y Fj(2)25 b Fn(Op)2392 3072 y Fg(w)2392 3132 y(h)2448 3110 y Fn(\()p Fl(S)2544 3072 y Fk(0)2584 3110 y Fn(\(1\)\))p Fl(:)0 3276 y Fn(Since)k(dist\(supp)626 3252 y(~)609 3276 y Fl( )668 3290 y Fk(2)706 3276 y Fn(,)40 b(supp)29 b Fl( )1048 3290 y Fk(0)1088 3276 y Fn(\))p Fl(>)c Fn(0,)31 b(it)f(follo)m(ws)f(from)h(Lemma)h(2.1)g(of)g([13)q(])g (that)895 3442 y Fj(k)959 3418 y Fn(~)940 3442 y Fl( )999 3456 y Fk(2)1039 3442 y Fn(\()p Fl(z)25 b Fj(\000)1252 3419 y Fn(~)1231 3442 y Fl(P)13 b Fn(\))1337 3405 y Fi(\000)p Fk(1)1432 3442 y Fn([)1478 3419 y(~)1457 3442 y Fl(F)g(;)1589 3419 y Fn(~)1568 3442 y Fl(P)h Fn(]\()p Fl(z)25 b Fj(\000)1878 3419 y Fn(~)1858 3442 y Fl(P)13 b Fn(\))1964 3405 y Fi(\000)p Fk(1)2059 3442 y Fl( )2118 3456 y Fk(0)2157 3442 y Fj(k)26 b Fn(=)f Fj(O)s Fn(\()p Fl(h)2486 3405 y Fg(N)2554 3442 y Fn(\))p Fl(;)56 b Fj(8)p Fl(N)35 b Fj(2)24 b Fh(N)7 b Fl(:)0 3608 y Fn(uniformly)27 b(on)k Fl(z)e Fj(2)h Fn(supp)940 3584 y(~)921 3608 y Fl(f)9 b Fn(.)150 3716 y(On)30 b(the)g(other)h(hand,)e(the)i(argumen)m(t)g(of)f(the)h(pro)s (of)e(of)i(Lemma)f(2)h(yields)1239 3875 y Fj(k)p Fn(\()p Fl(@)1367 3889 y Fg(x)1407 3898 y Ft(1)1446 3875 y Fl(V)21 b Fn(\)\()p Fl(i)g Fj(\000)f Fl(P)1791 3889 y Fk(2)1830 3875 y Fn(\()p Fl(h)p Fn(\)\))1987 3837 y Fi(\000)p Fg(m)2110 3875 y Fj(k)2155 3889 y Fk(tr)2240 3875 y Fn(=)25 b Fj(O)s Fn(\()p Fl(h)2498 3837 y Fi(\000)p Fg(n)2601 3875 y Fn(\))p Fl(:)0 4033 y Fn(Com)m(bining)j(the)j(ab)s(o)m(v)m(e)g(three)g (equalities)e(with)g(the)h(estimate)i Fj(j)p Fn(\()p Fl(z)25 b Fj(\000)20 b Fl(P)2515 4047 y Fk(2)2554 4033 y Fn(\()p Fl(h)p 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Fl(:)26 b Fn(A)f(similar)f(argumen)m(t)i (implies)d(the)i(indep)s(endence)f(of)i(the)g(asymptotic)g(expansion)e (\(7.2\))0 1905 y(on)29 b(the)h(c)m(hoice)h(of)e Fl(G;)15 b( )823 1919 y Fk(1)864 1905 y Fn(,)30 b(pro)m(vided)e(that)i Fl( )1547 1919 y Fk(1)1587 1905 y Fn(\()p Fl(x)1674 1919 y Fk(1)1713 1905 y Fn(\))c(=)f(1)30 b(for)f Fl(\016)2123 1919 y Fk(1)2182 1905 y Fj(\000)18 b Fn(1)26 b Fj(\024)f Fl(x)2490 1919 y Fk(1)2554 1905 y Fj(\024)g Fl(E)2717 1919 y Fk(1)2775 1905 y Fn(+)19 b Fj(j)p Fl(\016)2930 1919 y Fk(1)2970 1905 y Fj(j)f Fn(+)h Fj(k)p Fl(V)h Fj(k)3266 1919 y Fi(1)3360 1905 y Fn(+)e(3)p Fl(:)31 b Fn(Next,)f(w)m(e)0 2013 y(ha)m(v)m(e)752 2265 y Fl(a)800 2279 y Fk(0)864 2265 y Fn(=)25 b(\(2)p Fl(\031)s Fn(\))1130 2227 y Fi(\000)p Fg(n)1248 2150 y Ff(Z)53 b(Z)1393 2338 y Fe(R)1441 2319 y Ft(2)p Fq(n)1518 2171 y Ff(\020)1567 2265 y Fl(f)10 b Fn(\()p Fj(j)p Fl(\030)t Fj(j)1751 2227 y Fk(2)1811 2265 y Fn(+)20 b Fl(x)1954 2279 y Fk(1)2014 2265 y Fn(+)g Fl(V)g Fn(\()p Fl(x)p 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Fd(we)h(have)1224 3741 y Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))27 b(=)e(\(2)p Fl(\031)s(h)p Fn(\))1828 3704 y Fi(\000)p Fg(n)1932 3741 y Fl(c)1971 3755 y Fk(0)2010 3741 y Fn(\()p Fl(\025)p Fn(\))c(+)f Fj(O)s Fn(\()p Fl(h)2407 3704 y Fi(\000)p Fg(n)p Fk(+1)2600 3741 y Fn(\))p Fl(;)1008 b Fn(\(7.13\))0 3912 y Fd(uniformly)34 b(on)f Fl(\025)25 b Fj(2)g Fn([)p Fl(E)801 3926 y Fk(0)841 3912 y Fl(;)15 b(E)948 3926 y Fk(1)988 3912 y Fn(])p Fd(,)32 b(wher)-5 b(e)1009 4121 y Fl(c)1048 4135 y Fk(0)1088 4121 y Fn(\()p Fl(\025)p Fn(\))26 b(=)f Fj(\000)1419 4060 y Fn(1)p 1414 4100 55 4 v 1414 4184 a Fl(n)1478 4121 y(!)1535 4135 y Fg(n)1617 4006 y Ff(Z)1663 4195 y Fe(R)1711 4176 y Fq(n)1773 4121 y Fl(@)1821 4135 y Fg(x)1861 4144 y Ft(1)1900 4121 y Fl(V)20 b Fn(\()p Fl(x)p Fn(\)\()p Fl(\025)h Fj(\000)f Fl(V)g Fn(\()p Fl(x)p Fn(\))h Fj(\000)f Fl(x)2654 4135 y Fk(1)2694 4121 y Fn(\))2739 4037 y Fq(n)p 2739 4049 39 4 v 2743 4090 a Ft(2)2729 4143 y Fk(+)2791 4121 y Fl(dx)778 b Fn(\(7.14\))0 4337 y 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Fk(0)608 5143 y Fn(\()p Fl(\025)p Fn(\))p Fd(.)p eop %%Page: 24 24 24 23 bop 0 66 a Fk(24)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)150 266 y Fd(Pr)-5 b(o)g(of.)41 b Fn(F)-8 b(ollo)m(wing)28 b([8)q(],)h([12)r(],)g(the)g(pro)s(of)f(is)f(rather)i(similar)d(to)j (that)g(in)e(these)i(pap)s(ers)f(and)g(for)g(this)g(reason)0 374 y(w)m(e)k(will)d(presen)m(t)i(only)g(the)g(main)g(steps.)44 b(The)30 b(reader)i(ma)m(y)g(consult)f([8],)h([12)r(])f(for)g(more)h (details.)43 b(Let)32 b(\012)27 b Fj(\032)f Fh(C)0 482 y Fn(b)s(e)33 b(a)h(compact)h(domain)e(ha)m(ving)g(the)h(prop)s(erties) f(describ)s(ed)e(in)i(Section)h(5)g(and)f(let)h([)p Fl(E)3111 496 y Fk(0)3150 482 y Fl(;)k(E)3280 496 y Fk(1)3320 482 y Fn(])31 b Fj(\032)g Fn(\012)22 b Fj(\\)g Fh(R)s Fl(:)40 b Fn(W)-8 b(e)0 589 y(assume)38 b(that)h Fl(z)k Fj(2)38 b Fn(\012)g Fj(\))g Fn(Re)16 b Fl(z)42 b(<)c(\016)1300 603 y Fk(1)1340 589 y Fl(:)h Fn(Let)g Fl(')f Fj(2)h Fl(C)1844 556 y Fi(1)1837 613 y Fk(0)1918 589 y Fn(\()p Fh(R)s Fn(\))p Fl(;)47 b Fn(supp)24 b Fl(')39 b Fj(\032)f Fn([)p Fl(E)2632 603 y Fk(0)2697 589 y Fj(\000)25 b Fl(\017;)15 b(E)2937 603 y Fk(1)3003 589 y Fn(+)25 b Fl(\017)p Fn(])p Fl(;)41 b Fj(\032)d Fn(\012)25 b Fj(\\)g Fh(R)s Fl(;)47 b(\017)38 b(>)g Fn(0)p Fl(:)0 697 y Fn(In)m(tro)s(duce)30 b(the)g(functions)962 964 y Fl(M)1050 978 y Fg(')1101 964 y Fn(\()p Fl(\025)p Fn(\))c(=)f Fj(\000)1432 903 y Fn(1)p 1427 943 56 4 v 1427 1027 a Fl(\031)1731 884 y Ff(X)1506 1072 y Fg(w)r Fi(2)p Fk(\(Res)16 b Fg(P)1805 1081 y Ft(2)1839 1072 y Fk(\()p Fg(h)p Fk(\)\))h Fi(\\)p Fk(\012)2091 849 y Ff(Z)2174 876 y Fg(\025)2137 1038 y Fi(\0001)2358 903 y Fn(Im)e Fl(w)p 2292 943 324 4 v 2292 1027 a Fj(j)p Fl(w)23 b Fj(\000)d Fl(\026)p Fj(j)2576 1001 y Fk(2)2625 964 y Fl(')p Fn(\()p Fl(\026)p Fn(\))p Fl(d\026;)1270 1291 y(G)1341 1305 y Fg(')1391 1291 y Fn(\()p Fl(\025)p Fn(\))26 b(=)1651 1229 y(1)p 1646 1270 56 4 v 1646 1353 a Fl(\031)1726 1176 y Ff(Z)1809 1202 y Fg(\025)1773 1364 y Fi(\0001)1917 1291 y Fl(')p Fn(\()p Fl(\026)p 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2857 y Fl(\022)1731 2872 y Fg(h)1796 2857 y Fj(\003)c Fl(G)1933 2871 y Fg(')1983 2857 y Fn(\)\()p Fl(\025)p Fn(\))g(+)f Fj(O)s Fn(\()p Fl(h)2415 2819 y Fk(1)p Fi(\000)p Fg(n)2553 2857 y Fn(\))p Fl(:)0 3000 y Fn(Without)31 b(loss)f(of)h(the)g(generalit)m(y)-8 b(,)31 b(w)m(e)h(ma)m(y)f(assume)g(that)g(ev)m(ery)g Fl(\025)26 b Fj(2)g Fn(supp)19 b Fl(')31 b Fn(is)f(a)h(non-critical)f (v)-5 b(alue)30 b(of)h(the)0 3108 y(sym)m(b)s(ol)e Fj(j)p Fl(\030)t Fj(j)405 3075 y Fk(2)465 3108 y Fn(+)20 b Fl(x)608 3122 y Fk(1)668 3108 y Fn(+)g Fl(V)g Fn(\()p Fl(x)p Fn(\))p Fl(:)31 b Fn(Applying)d(Theorem)i(5,)h(w)m(e)g(deduce)359 3212 y Ff(Z)442 3239 y Fg(\025)405 3401 y Fi(\0001)550 3327 y Fl('\030)653 3290 y Fi(0)676 3327 y Fn(\()p Fl(\026)p Fn(\))p Fl(d\026)26 b Fn(=)1033 3303 y(^)1025 3327 y Fl(\022)1068 3342 y Fg(h)1133 3327 y Fj(\003)1199 3212 y Ff(Z)1282 3239 y Fg(\025)1245 3401 y Fi(\0001)1390 3327 y Fl('\030)1493 3290 y Fi(0)1516 3327 y Fn(\()p Fl(\026)p Fn(\))p Fl(d\026)21 b Fn(+)f Fj(O)s Fn(\()p Fl(h)2017 3290 y Fk(1)p Fi(\000)p Fg(n)2155 3327 y Fn(\))25 b(=)2311 3233 y Ff(\020)2361 3212 y(Z)2444 3239 y Fg(\025)2407 3401 y Fi(\0001)2552 3327 y Fl(\015)2599 3341 y Fk(0)2638 3327 y Fn(\()p Fl(\026)p Fn(\))p Fl(d\026)2865 3233 y Ff(\021)2915 3327 y Fl(h)2967 3290 y Fi(\000)p Fg(n)3090 3327 y Fn(+)20 b Fj(O)s Fn(\()p Fl(h)3343 3290 y Fk(1)p Fi(\000)p Fg(n)3481 3327 y Fn(\))p Fl(;)0 3538 y Fn(where)30 b(the)g(function)g Fl(\015)823 3552 y Fk(0)862 3538 y Fn(\()p Fl(\026)p Fn(\))h(is)e(giv)m(en)i(b)m(y)f(Theorem)g(5.)150 3754 y(No)m(w)i(w)m(e)f(c)m(ho)s(ose)i Fl(C)847 3768 y Fk(1)912 3754 y Fn(=)27 b Fl(\016)1050 3768 y Fk(1)1110 3754 y Fj(\000)21 b(k)p Fl(V)f Fj(k)1365 3768 y Fi(1)1471 3754 y Fn(and)31 b(w)m(e)h(apply)d(the)j(ab)s(o)m(v)m(e)g(argumen)m(t)g (for)e(a)i(cut-o\013)g(function)e Fl(')p Fn(\()p Fl(\025)p Fn(\))0 3862 y(for)g(whic)m(h)f(w)m(e)i(ha)m(v)m(e)1375 3986 y(supp)19 b Fl(')46 b Fj(\032)f(f)p Fl(\025)26 b Fj(2)f Fh(R)34 b Fn(:)45 b Fl(\025)26 b(<)e(C)2414 4000 y Fk(1)2454 3986 y Fj(g)p Fl(:)0 4129 y Fn(F)-8 b(or)31 b(this)e(purp)s(ose)g(w)m(e)i(c)m(ho)s(ose)g(a)g(domain)e(\012)h(as)g (ab)s(o)m(v)m(e)i(and)e(observ)m(e)h(that)f Fj(j)p Fl(\030)t Fj(j)2759 4096 y Fk(2)2819 4129 y Fn(+)20 b Fl(x)2962 4143 y Fk(1)3022 4129 y Fn(+)g Fl(V)g Fn(\()p Fl(x)p Fn(\))31 b(has)f(no)g(critical)0 4237 y(v)-5 b(alues)31 b Fl(\025)d Fj(2)f Fn(supp)20 b Fl(':)32 b Fn(It)g(is)f(easy)h(to)h (see)f(that)h Fl(\015)1667 4251 y Fg(j)1703 4237 y Fn(\()p Fl(\025)p Fn(\))c(=)e(0)p Fl(;)37 b(j)c Fn(=)28 b(0)p Fl(;)15 b(:::;)g(N)33 b Fj(\000)21 b Fn(1)32 b(for)f Fl(\025)d(<)f(C)3088 4251 y Fk(1)3128 4237 y Fl(;)32 b Fn(so)g(for)f(suc)m(h)h(cut-o\013)0 4345 y(function)d(w)m(e)i(ha)m(v) m(e)1237 4389 y Ff(Z)1320 4415 y Fg(\025)1283 4577 y Fi(\0001)1428 4504 y Fl('\030)1531 4466 y Fi(0)1555 4504 y Fn(\()p Fl(\026)p Fn(\))p Fl(d\026)26 b Fn(=)f Fj(O)s Fn(\()p Fl(h)2066 4466 y Fg(N)7 b Fi(\000)p Fg(n)2231 4504 y Fn(\))p Fl(;)36 b Fj(8)p Fl(N)f Fj(2)25 b Fh(N)6 b Fl(:)0 4696 y Fn(Consider)24 b(a)j(partition)e(of)h(unit)m(y)f Fl(')1213 4710 y Fk(1)1253 4696 y Fn(\()p Fl(\025)p Fn(\))12 b(+)g Fl(')1530 4710 y Fk(2)1570 4696 y Fn(\()p Fl(\025)p Fn(\))g(+)g Fl(')1847 4710 y Fk(3)1887 4696 y Fn(\()p Fl(\025)p Fn(\))26 b(=)f(1)p Fl(;)i(')2288 4710 y Fg(j)2325 4696 y Fn(\()p Fl(\025)p Fn(\))f Fj(2)f Fl(C)2632 4663 y Fi(1)2625 4719 y Fk(0)2706 4696 y Fn(\()p Fh(R)s Fn(\))q Fl(;)33 b(j)e Fn(=)25 b(1)p Fl(;)15 b Fn(2)p Fl(;)g Fn(3)p Fl(;)28 b Fn(on)e(the)h(in)m(terv)-5 b(al)0 4804 y([)p Fl(C)90 4818 y Fk(2)130 4804 y Fl(;)15 b(E)237 4818 y Fk(1)297 4804 y Fn(+)20 b Fl(\017)p Fn(])p Fl(;)35 b(E)577 4818 y Fk(1)637 4804 y Fn(+)20 b Fl(\017)25 b(<)g(\016)926 4818 y Fk(1)966 4804 y Fl(;)36 b(C)1092 4818 y Fk(2)1156 4804 y Fl(<)25 b(C)1317 4818 y Fk(1)1357 4804 y Fl(:)30 b Fn(W)-8 b(e)32 b(assume)e(that)1010 4965 y Fl(')1069 4979 y Fk(3)1134 4965 y Fn(=)25 b(1)c(on)f([)p Fl(E)1504 4979 y Fk(0)1544 4965 y Fl(;)35 b(E)1671 4979 y Fk(1)1711 4965 y Fn(])p Fl(;)h Fn(supp)18 b Fl(')2063 4979 y Fk(3)2128 4965 y Fj(\032)46 b Fn([)p Fl(E)2337 4979 y Fk(0)2397 4965 y Fj(\000)19 b Fl(\017;)36 b(E)2652 4979 y Fk(1)2712 4965 y Fn(+)20 b Fl(\017)p Fn(])p Fl(;)1301 5143 y Fn(supp)f Fl(')1568 5157 y Fk(1)1653 5143 y Fj(\032)h Fn(])g Fj(\000)g(1)p Fl(;)36 b(C)2097 5157 y Fk(1)2156 5143 y Fj(\000)20 b Fl(\017)p Fn([)p Fl(;)36 b(\017)25 b(>)g Fn(0)p Fl(:)p eop %%Page: 25 25 25 24 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g (RESONANCES)999 b(25)0 266 y Fn(F)-8 b(or)31 b Fl(\025)25 b Fj(2)g Fn([)p Fl(E)419 280 y Fk(0)459 266 y Fl(;)36 b(E)587 280 y Fk(1)626 266 y Fn(])p Fl(;)31 b Fn(w)m(e)g(get)753 460 y Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))22 b Fj(\000)e Fl(\030)t Fn(\()p Fl(C)1269 474 y Fk(2)1309 460 y Fl(;)15 b(h)p Fn(\))26 b(=)f Fj(\000)1644 398 y Fn(1)p 1639 439 56 4 v 1639 522 a Fl(\031)1944 379 y Ff(X)1719 567 y Fg(w)r Fi(2)p Fk(\(Res)16 b Fg(P)2018 576 y Ft(2)2052 567 y Fk(\()p Fg(h)p Fk(\)\))h Fi(\\)p Fk(\012)2304 345 y Ff(Z)2387 371 y Fi(1)2350 534 y Fg(C)2400 543 y Ft(2)2553 398 y Fn(Im)e Fl(w)p 2487 439 324 4 v 2487 522 a Fj(j)p Fl(w)23 b Fj(\000)d Fl(\026)p Fj(j)2771 496 y Fk(2)2820 460 y Fl(')2879 474 y Fk(1)2919 460 y Fn(\()p Fl(\026)p Fn(\))p Fl(d\026)1420 762 y Fn(+)1506 700 y(1)p 1501 741 56 4 v 1501 824 a Fl(\031)1580 647 y Ff(Z)1663 673 y Fi(1)1627 836 y Fg(C)1677 845 y Ft(2)1753 762 y Fl(')1812 776 y Fk(1)1852 762 y Fn(\()p Fl(\026)p Fn(\))15 b(Im)h Fl(r)s Fn(\()p Fl(\026;)f(h)p Fn(\))p Fl(d\026)1171 997 y Fn(+tr)1312 902 y Ff(h)1352 997 y Fl(')1411 1011 y Fk(2)1451 997 y Fn(\()p Fl(P)1544 1011 y Fg(j)1581 997 y Fn(\)\()p Fl(h)p Fn(\))1738 902 y Ff(i)1778 918 y Fk(2)1778 1060 y Fg(j)t Fk(=1)1925 997 y Fn(+)20 b Fl(M)2104 1011 y Fg(')2150 1020 y Ft(3)2189 997 y Fn(\()p Fl(\025)p Fn(\))h(+)f Fl(G)2495 1011 y Fg(')2541 1020 y Ft(3)2580 997 y Fn(\()p Fl(\025)p Fn(\))p Fl(:)0 1165 y Fn(F)-8 b(or)31 b(the)g(terms)f(in)m(v)m(olving)f Fl(')1021 1179 y Fk(3)1091 1165 y Fn(and)h Fl(')1327 1179 y Fk(1)1397 1165 y Fn(w)m(e)h(apply)e(the)h(ab)s(o)m(v)m(e)i(argumen)m(t)f(and)f(w) m(e)g(observ)m(e)h(that)625 1249 y Ff(Z)708 1276 y Fg(C)758 1285 y Ft(1)793 1276 y Fi(\000)p Fg(\017)671 1438 y(C)721 1447 y Ft(2)896 1364 y Fl(')955 1378 y Fk(1)995 1364 y Fl(\030)1039 1327 y Fi(0)1062 1364 y Fl(d\026)25 b Fn(=)1285 1249 y Ff(Z)1368 1276 y Fg(C)1418 1285 y Ft(1)1453 1276 y Fi(\000)p Fg(\017)1331 1438 y Fi(\0001)1556 1364 y Fl(')1615 1378 y Fk(1)1655 1364 y Fl(\030)1699 1327 y Fi(0)1722 1364 y Fl(d\026)20 b Fj(\000)1935 1249 y Ff(Z)2018 1276 y Fg(C)2068 1285 y Ft(2)1981 1438 y Fi(\0001)2126 1364 y Fl(')2185 1378 y Fk(1)2225 1364 y Fl(\030)2269 1327 y Fi(0)2292 1364 y Fl(d\026)26 b Fn(=)f Fj(O)s Fn(\()p Fl(h)2678 1327 y Fg(N)7 b Fi(\000)p Fg(n)2843 1364 y Fn(\))p Fl(;)36 b Fj(8)p Fl(N)f Fj(2)25 b Fh(N)6 b Fl(:)0 1563 y Fn(Next,)33 b(w)m(e)f(ma)m(y)g(c)m(hoice)g(the)f(v)-5 b(alue)31 b(of)h Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))33 b(at)f(one)f(p)s(oin)m(t)g(and)g(w)m(e)h(supp)s(ose)d(that)j Fl(\030)t Fn(\()p Fl(C)3158 1577 y Fk(2)3198 1563 y Fl(;)15 b(h)p Fn(\))28 b(=)f(0.)44 b(Th)m(us)30 b(w)m(e)0 1671 y(obtain)35 b(the)g(asymptotics)h(\(7.13\).)57 b(T)-8 b(o)36 b(\014nd)e(the)h(co)s(e\013cien)m(t)i Fl(c)2203 1685 y Fk(0)2243 1671 y Fn(\()p Fl(\025)p Fn(\),)g(notice)f(that)g (from)e(Theorem)h(4)h(and)f(the)0 1779 y(de\014nition)28 b(of)j Fl(\030)t Fn(\()p Fl(\025;)15 b(h)p Fn(\))31 b(w)m(e)g(ha)m(v)m (e)945 1965 y Fj(h)p Fl(c)1019 1979 y Fk(0)1059 1965 y Fl(;)15 b(f)1154 1927 y Fi(0)1177 1965 y Fj(i)26 b Fn(=)1334 1850 y Ff(Z)52 b(Z)1478 2038 y Fe(R)1526 2020 y Ft(2)p Fq(n)1603 1965 y Fn(\()p Fl(@)1686 1979 y Fg(x)1726 1988 y Ft(1)1765 1965 y Fl(V)20 b Fn(\()p Fl(x)p Fn(\)\))p Fl(f)10 b Fn(\()p Fj(j)p Fl(\030)t Fj(j)2179 1927 y Fk(2)2240 1965 y Fn(+)20 b Fl(x)2383 1979 y Fk(1)2442 1965 y Fn(+)g Fl(V)h Fn(\()p Fl(x)p Fn(\)\))p Fl(dxd\030)958 2208 y Fn(=)1064 2147 y(1)p 1064 2187 46 4 v 1064 2271 a(2)1119 2208 y Fl(!)1176 2222 y Fg(n)1258 2093 y Ff(Z)1341 2120 y Fi(1)1304 2282 y Fk(0)1431 2093 y Ff(Z)1477 2282 y Fe(R)1525 2263 y Fq(n)1587 2208 y Fl(@)1635 2222 y Fg(x)1675 2231 y Ft(1)1714 2208 y Fl(V)f Fn(\()p Fl(x)p Fn(\))p Fl(f)10 b Fn(\()p Fl(t)20 b Fn(+)g Fl(V)h Fn(\()p Fl(x)p Fn(\))f(+)g Fl(x)2502 2222 y Fk(1)2542 2208 y Fn(\))p Fl(t)2620 2144 y Fq(n)p 2620 2156 39 4 v 2624 2197 a Ft(2)2668 2171 y Fi(\000)p Fk(1)2762 2208 y Fl(dtdx)885 2433 y Fn(=)25 b Fj(\000)1062 2371 y Fn(1)p 1062 2412 46 4 v 1062 2495 a(2)1117 2433 y Fl(!)1174 2447 y Fg(n)1256 2318 y Ff(Z)1339 2344 y Fi(1)1302 2506 y Fk(0)1429 2318 y Ff(Z)1475 2506 y Fe(R)1523 2488 y Fq(n)1585 2433 y Fl(@)1633 2447 y Fg(x)1673 2456 y Ft(1)1711 2433 y Fl(V)20 b Fn(\()p Fl(x)p Fn(\))1921 2318 y Ff(Z)2005 2344 y Fi(1)1968 2506 y Fg(t)p Fk(+)p Fg(V)d Fk(\()p Fg(x)p Fk(\)+)p Fg(x)2295 2515 y Ft(1)2349 2433 y Fl(f)2404 2395 y Fi(0)2426 2433 y Fn(\()p Fl(\025)p Fn(\))p Fl(t)2592 2368 y Fq(n)p 2593 2380 39 4 v 2597 2421 a Ft(2)2641 2395 y Fi(\000)p Fk(1)2735 2433 y Fl(d\025dtdx)883 2674 y Fn(=)25 b Fj(\000)1065 2612 y Fn(1)p 1060 2653 55 4 v 1060 2736 a Fl(n)1124 2674 y(!)1181 2688 y Fg(n)1263 2559 y Ff(Z)1309 2748 y Fe(R)1357 2760 y Fq(\025)1417 2559 y Ff(Z)1463 2748 y Fe(R)1511 2729 y Fq(n)1573 2674 y Fl(@)1621 2688 y Fg(x)1661 2697 y Ft(1)1700 2674 y Fl(V)20 b Fn(\()p Fl(x)p Fn(\)\()p Fl(\025)h Fj(\000)f Fl(V)g Fn(\()p Fl(x)p Fn(\))h Fj(\000)f Fl(x)2454 2688 y Fk(1)2493 2674 y Fn(\))2538 2589 y Fq(n)p 2539 2601 39 4 v 2543 2643 a Ft(2)2528 2696 y Fk(+)2591 2674 y Fl(f)2646 2636 y Fi(0)2669 2674 y Fn(\()p Fl(\025)p Fn(\))p Fl(d\025dx;)0 2856 y Fn(whic)m(h)30 b(giv)m(es)h(\(7.14\).)46 b(The)30 b(second)h(assertion)g(of)g(Theorem) g(6)h(can)f(b)s(e)g(established)e(com)m(bining)h(Theorems)g(2,)0 2964 y(5)h(with)f(the)h(argumen)m(t)g(of)g(the)g(pro)s(of)f(of)h (Theorem)g(3)g(in)f([12)q(].)42 b(As)31 b(in)f(Theorem)g(3)i(in)d([12)q (],)j(w)m(e)f(sho)m(w)g(that)g(for)0 3072 y Fl(\025)25 b Fj(2)g Fn([)p Fl(E)256 3086 y Fk(0)296 3072 y Fl(;)36 b(E)424 3086 y Fk(1)463 3072 y Fn(])31 b(w)m(e)f(ha)m(v)m(e)876 3172 y Ff(\020)933 3242 y Fn(^)926 3266 y Fl(\022)969 3281 y Fg(h)1034 3266 y Fj(\003)1118 3205 y(\000)15 b Fn(Im)f Fl(w)p 1109 3245 294 4 v 1109 3329 a Fj(j)p Fl(:)21 b Fj(\000)f Fl(w)r Fj(j)1363 3303 y Fk(2)1413 3172 y Ff(\021)1463 3266 y Fn(\()p Fl(\025)p Fn(\))26 b(=)1740 3205 y Fj(\000)15 b Fn(Im)f Fl(w)p 1718 3245 322 4 v 1718 3329 a Fj(j)p Fl(\025)20 b Fj(\000)g Fl(w)r Fj(j)1999 3303 y Fk(2)2069 3266 y Fn(+)g Fj(O)2235 3172 y Ff(\020)2400 3205 y Fn(1)p 2295 3245 257 4 v 2295 3329 a Fj(j)15 b Fn(Im)g Fl(w)r Fj(j)2562 3266 y Fl(e)2604 3229 y Fi(\000)2678 3202 y Fq(C)p 2669 3214 67 4 v 2669 3255 a Ft(2)p Fq(h)2745 3229 y Fi(j)d Fk(Im)f Fg(w)r Fi(j)2949 3172 y Ff(\021)2998 3266 y Fl(;)723 3431 y Ff(\020)780 3501 y Fn(^)773 3525 y Fl(\022)816 3540 y Fg(h)880 3525 y Fj(\003)21 b Fn(\(1)g Fj(\000)f Fl(')1197 3539 y Fk(3)1237 3525 y Fn(\()p Fl(:)p Fn(\)\))1429 3463 y(Im)15 b Fl(w)p 1377 3504 294 4 v 1377 3587 a Fj(j)p Fl(:)22 b Fj(\000)e Fl(w)r Fj(j)1632 3561 y Fk(2)1682 3431 y Ff(\021)1731 3525 y Fn(\()p Fl(\025)p Fn(\))26 b(=)f(\(1)c Fj(\000)f Fl(')2227 3539 y Fk(3)2267 3525 y Fn(\()p Fl(\025)p Fn(\)\))2501 3463 y(Im)15 b Fl(w)p 2435 3504 322 4 v 2435 3587 a Fj(j)p Fl(\025)22 b Fj(\000)e Fl(w)r Fj(j)2718 3561 y Fk(2)2788 3525 y Fn(+)g Fj(O)s Fn(\()p Fl(h)3041 3487 y Fi(1)3116 3525 y Fn(\))p Fl(;)1449 3704 y Fn(^)1442 3728 y Fl(\022)1485 3743 y Fg(h)1550 3728 y Fj(\003)g Fl(G)1686 3742 y Fg(')1732 3751 y Ft(3)1797 3728 y Fn(=)25 b Fl(G)1964 3742 y Fg(')2010 3751 y Ft(3)2069 3728 y Fn(+)20 b Fj(O)s Fn(\()p Fl(h)2322 3690 y Fi(1)2397 3728 y Fn(\))p Fl(;)0 3862 y Fn(where)27 b(w)m(e)h(assume)g(that)g Fl(\022)f Fn(=)e(1)j(on)g([)p Fj(\000)1393 3826 y Fk(1)p 1366 3841 91 4 v 1366 3893 a(2)p Fg(C)1466 3862 y Fl(;)1558 3826 y Fk(1)p 1531 3841 V 1531 3893 a(2)p Fg(C)1631 3862 y Fn(])p Fl(:)g Fn(Since)f(for)g(all)g (resonances)h Fl(w)f Fj(2)e Fn(\012)i(w)m(e)i(ha)m(v)m(e)f(the)g(lo)m (w)m(er)g(b)s(ound)0 3970 y Fj(j)15 b Fn(Im)g Fl(w)r Fj(j)26 b(\025)f Fl(N)10 b(h)15 b Fn(log)r(\(1)p Fl(=h)p Fn(\))p Fl(;)33 b Fn(w)m(e)d(ma)m(y)h(estimate)h(the)e(exp)s(onen)m(t)h (in)m(v)m(olving)d Fj(j)15 b Fn(Im)h Fl(w)r Fj(j)p Fn(.)41 b(Th)m(us)29 b(w)m(e)i(conclude)f(that)801 4104 y(^)794 4128 y Fl(\022)837 4143 y Fg(h)901 4128 y Fj(\003)21 b Fn(\()p Fl(')1061 4142 y Fk(3)1101 4128 y Fl(\030)1145 4090 y Fi(0)1168 4128 y Fn(\)\()p Fl(\025;)15 b(h)p Fn(\))27 b(=)e Fl(')1600 4142 y Fk(3)1640 4128 y Fn(\()p Fl(\025)p Fn(\))p Fl(\030)1807 4090 y Fi(0)1831 4128 y Fn(\()p Fl(\025;)15 b(h)p Fn(\))22 b(+)e Fj(O)s Fn(\()p Fl(h)2321 4090 y Fg(N)7 b(C)s(=)p Fk(2)p Fi(\000)p Fg(n)p Fi(\000)p Fk(1)2700 4128 y Fn(\))p Fl(;)36 b Fj(8)p Fl(N)f Fj(2)25 b Fh(N)0 4275 y Fn(and)30 b(this)f(implies)f(easily)h(the)i(second)f (assertion)g(of)h(Theorem)f(6.)1531 b Fc(2)150 4491 y Fn(Let)31 b Fl(V)5 b(;)36 b Fn([)p Fl(E)524 4505 y Fk(0)563 4491 y Fl(;)15 b(E)670 4505 y Fk(1)710 4491 y Fn(])31 b(and)f Fl(\016)983 4505 y Fk(1)1053 4491 y Fn(b)s(e)g(as)g(in)f (Theorem)h(6.)41 b(F)-8 b(or)32 b Fl(f)i Fj(2)25 b Fl(C)2290 4458 y Fi(1)2283 4514 y Fk(0)2364 4491 y Fn(\()p Fh(R)s Fn(\),)37 b(in)m(tro)s(duce)29 b(the)i(measure)1200 4677 y Fj(h)p Fl(\026;)15 b(f)10 b Fj(i)26 b Fn(=)1542 4562 y Ff(Z)1625 4583 y(\020)1674 4677 y Fl(f)10 b Fn(\()p Fl(x)1816 4691 y Fk(1)1876 4677 y Fn(+)20 b Fl(V)g Fn(\()p Fl(x)p Fn(\)\))h Fj(\000)f Fl(f)10 b Fn(\()p Fl(x)2451 4691 y Fk(1)2490 4677 y Fn(\))2525 4583 y Ff(\021)2575 4677 y Fl(dx:)969 b Fn(\(7.15\))0 4857 y(Since)1372 4982 y Fj(jh)p Fl(\026;)15 b(f)10 b Fj(ij)26 b(\024)f Fn(sup)14 b Fj(j)p Fl(f)1996 4944 y Fi(0)2019 4982 y Fj(j)2059 4867 y Ff(Z)2157 4982 y Fj(j)p Fl(V)21 b Fn(\()p Fl(x)p Fn(\))p Fj(j)p Fl(dx;)0 5143 y(\026)30 b Fn(is)f(a)i(distribution)c(of) j(\014rst)g(order.)40 b(W)-8 b(e)32 b(denote)f(b)m(y)f(singsupp)2233 5165 y Fg(a)2294 5143 y Fl(\026)g Fn(the)h(analytic)f(singular)e(supp)s (ort)h(of)h Fl(\026)p Fn(.)p eop %%Page: 26 26 26 25 bop 0 66 a Fk(26)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)0 266 y Fx(Theorem)39 b(7.)44 b Fd(Supp)-5 b(ose)37 b(the)g(assumptions)h (of)f(The)-5 b(or)g(em)38 b Fn(4)f Fd(ful\014l)5 b(le)-5 b(d.)53 b(L)-5 b(et)37 b Fl(\016)2738 280 y Fk(1)2810 266 y Fl(>)31 b(E)2979 280 y Fk(1)3051 266 y Fl(>)g(\016)3193 280 y Fk(1)3256 266 y Fj(\000)22 b(k)p Fl(V)f Fj(k)3513 280 y Fi(1)3625 266 y Fd(and)37 b(let)0 374 y Fl(\025)29 b Fj(2)p Fn(])p Fl(E)235 388 y Fk(0)274 374 y Fl(;)15 b(E)381 388 y Fk(1)421 374 y Fn([)22 b Fj(\\)f Fn(singsupp)895 395 y Fg(a)958 374 y Fl(\026)p Fd(.)47 b(Then)35 b(for)g(every)g Fl(h)p Fd(-indep)-5 b(endent)36 b(c)-5 b(omplex)36 b(neighb)-5 b(orho)g(o)g(d)38 b Fn(\012)c Fd(of)h Fl(\025)f Fd(ther)-5 b(e)36 b(exist)0 482 y Fl(h)52 496 y Fk(0)117 482 y Fn(=)25 b Fl(h)p Fn(\(\012\))h Fl(>)f Fn(0)33 b Fd(su\016ciently)g(smal)5 b(l)33 b(and)h Fl(C)e Fn(=)25 b Fl(C)7 b Fn(\(\012\))25 b Fl(>)g Fn(0)33 b Fd(so)g(that)h(for)f Fl(h)26 b Fj(2)p Fn(]0)p Fl(;)15 b(h)2826 496 y Fk(0)2866 482 y Fn([)33 b Fd(we)g(have)1147 637 y Fn(#)p Fj(f)p Fl(z)c Fj(2)c Fn(\012)g(:)46 b Fl(z)29 b Fj(2)c Fn(Res)c(\()p Fl(P)2001 651 y Fk(2)2041 637 y Fn(\()p Fl(h)p Fn(\)\))p Fj(g)27 b(\025)e Fl(C)7 b Fn(\(\012\))p Fl(h)2626 599 y Fi(\000)p Fg(n)2728 637 y Fl(:)150 804 y Fd(Pr)-5 b(o)g(of.)54 b Fn(Let)34 b Fl( )i Fj(2)31 b Fl(C)867 771 y Fi(1)860 827 y Fk(0)941 804 y Fn(\()p Fh(R)s Fn(\))41 b(b)s(e)34 b(a)g(cut-o\013)i(function)d(so)h(that)h Fl( )g Fn(=)d(1)j(near)f([)p Fl(\016)2833 818 y Fk(1)2896 804 y Fj(\000)22 b(k)p Fl(V)f Fj(k)3153 818 y Fi(1)3251 804 y Fj(\000)i Fn(1)p Fl(;)38 b(E)3520 818 y Fk(1)3583 804 y Fn(+)22 b Fj(j)p Fl(\016)3741 818 y Fk(1)3781 804 y Fj(j)h Fn(+)0 912 y Fj(k)p Fl(V)e Fj(k)164 926 y Fi(1)259 912 y Fn(+)f(3].)41 b(F)-8 b(or)31 b Fl(f)k Fj(2)25 b Fl(C)887 879 y Fi(1)880 935 y Fk(0)961 912 y Fn(\(])p Fl(E)1088 926 y Fk(0)1128 912 y Fl(;)15 b(E)1235 926 y Fk(1)1275 912 y Fn([\))31 b(in)m(tro)s(duce)e(the)i (distributions)976 1105 y Fj(h)7 b Fn(~)-52 b Fl(\026;)15 b(f)10 b Fj(i)25 b Fn(=)1317 990 y Ff(Z)1400 1011 y(\020)1450 1105 y Fl(f)10 b Fn(\()p Fl( )s Fn(\()p Fl(x)1689 1119 y Fk(1)1729 1105 y Fn(\))p Fl(x)1816 1119 y Fk(1)1876 1105 y Fn(+)20 b Fl(V)g Fn(\()p Fl(x)p Fn(\)\))h Fj(\000)f Fl(f)10 b Fn(\()p Fl( )s Fn(\()p Fl(x)2548 1119 y Fk(1)2588 1105 y Fn(\))p Fl(x)2675 1119 y Fk(1)2715 1105 y Fn(\))2750 1011 y Ff(\021)2800 1105 y Fl(dx;)539 1353 y Fj(h)d Fn(~)-52 b Fl(!)s(;)15 b(f)10 b Fj(i)25 b Fn(=)g(\(2)p Fl(\031)s Fn(\))1055 1316 y Fi(\000)p Fg(n)1173 1238 y Ff(Z)1256 1259 y(\020)1306 1353 y Fl(f)10 b Fn(\()p Fj(j)p Fl(\030)t Fj(j)1490 1316 y Fk(2)1550 1353 y Fn(+)20 b Fl( )s Fn(\()p Fl(x)1790 1367 y Fk(1)1830 1353 y Fn(\))p Fl(x)1917 1367 y Fk(1)1977 1353 y Fn(+)g Fl(V)g Fn(\()p Fl(x)p Fn(\)\))h Fj(\000)f Fl(f)10 b Fn(\()p Fj(j)p Fl(\030)t Fj(j)2594 1316 y Fk(2)2654 1353 y Fn(+)20 b Fl( )s Fn(\()p Fl(x)2894 1367 y Fk(1)2934 1353 y Fn(\))p Fl(x)3021 1367 y Fk(1)3061 1353 y Fn(\))3096 1259 y Ff(\021)3146 1353 y Fl(dxd\030)t(:)150 1529 y Fn(Clearly)-8 b(,)35 b(~)-52 b Fl(\026)25 b Fj(2)g(E)704 1496 y Fi(0)727 1529 y Fn(\()p Fh(R)s Fn(\))35 b(and)g(~)-52 b Fl(\026)25 b Fn(=)g Fl(\026)p Fn(,)36 b(~)-52 b Fl(!)28 b Fn(=)d Fl(!)32 b Fn(on)c(])p Fl(E)1838 1543 y Fk(0)1878 1529 y Fl(;)15 b(E)1985 1543 y Fk(1)2025 1529 y Fn([,)29 b(where)f(the)h(distribution)c Fl(!)31 b Fn(has)e(b)s(een)f(in)m(tro)s (duced)0 1637 y(in)35 b(Theorem)i(4.)59 b(Next)38 b(the)f(pro)s(of)f (follo)m(ws)f(with)h(minor)f(mo)s(di\014cations)f(that)k(in)d([33)q(])i (and)f(w)m(e)h(will)d(presen)m(t)0 1745 y(only)29 b(the)i(main)e (steps.)150 1960 y(Denote)j(b)m(y)e Fl(W)13 b(F)745 1974 y Fg(a)787 1960 y Fn(\()p Fl(:)p Fn(\))31 b(the)g(analytic)e(w)m(a)m(v) m(e)k(fron)m(t.)41 b(As)30 b(it)g(w)m(as)h(sho)m(wn)e(in)g([33)r(],)h (w)m(e)h(ha)m(v)m(e)1554 2115 y(WF)1707 2129 y Fg(a)1749 2115 y Fn(\()7 b(~)-52 b Fl(\026)p Fn(\))25 b(=)g(WF)2148 2129 y Fg(a)2190 2115 y Fn(\()7 b(~)-52 b Fl(!)s Fn(\))p Fl(:)0 2270 y Fn(Since)33 b(~)-52 b Fl(!)30 b 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Fl(K)i Fn(+)20 b Fl(T)8 b(;)1238 b Fn(\(8.14\))0 5026 y(where)30 b Fl(K)37 b Fn(has)30 b(rank)g Fj(O)s Fn(\()p Fl(h)911 4993 y Fi(\000)p Fg(n)1013 5026 y Fn(\))h(and)f Fj(k)p Fl(T)13 b Fj(k)26 b Fn(=)f Fj(O)s Fn(\()p Fl(h)1696 4993 y Fi(1)1771 5026 y Fn(\).)150 5143 y(No)m(w)31 b(it)f(is)f(clear)i (that)g(\(6.3\))h(and)e(\(6.4\))i(are)f(ful\014lled)26 b(with)2270 5117 y Ff(b)2249 5143 y Fl(P)2307 5158 y Fk(2)p Fg(;\022)2427 5143 y Fn(=)2543 5117 y Ff(e)2523 5143 y Fl(P)2581 5158 y Fg(\022)2641 5143 y Fj(\000)20 b Fl(T)13 b Fn(.)p eop %%Page: 30 30 30 29 bop 0 66 a Fk(30)1453 b(M.)23 b(DIMASSI,)h(V.)f(PETK)n(O)n(V)1732 266 y Fv(References)38 411 y Fw([1])40 b(J.)f(Avron,)i(I.)d(W.)h (Herbst,)i Fb(Sp)l(e)l(ctr)l(al)g(and)f(sc)l(attering)h(the)l(ory)g(of) e(Schr\177)-39 b(odinger)41 b(op)l(er)l(ators)h(r)l(elate)l(d)f(to)f (the)g(Stark)g(e\013e)l(ct)p Fw(,)158 502 y(Comm)n(un.)24 b(Math.)i(Ph)n(ys.)g Fa(52)g Fw(\(1977\),)h(239-254.)38 594 y([2])40 b(J.-F.)26 b(Bon)n(y)-6 b(,)26 b Fb(R)n(\023)-37 b(esonanc)l(es)29 b(dans)g(des)f(domaines)g(de)g(tail)t(le)f Fu(h)p Fw(,)f(In)n(ter.)g(Math.)g(Res.)g(Not.)f Fa(16)h Fw(\(2001\),)i(817-847.)38 685 y([3])40 b(J.-F.)32 b(Bon)n(y)-6 b(,)32 b Fb(Minor)l(ation)i(du)f(nombr)l(e)g(de)g(r)n(\023)-37 b(esonanc)l(es)36 b(engendr)n(\023)-37 b(ees)35 b(p)l(ar)e(une)h(tr)l (aje)l(ctoir)l(e)g(ferm)n(\023)-37 b(ee)p Fw(,)34 b(Comm)n(un.)c(P)-6 b(.D.E.)32 b Fa(27)158 776 y Fw(\(2002\),)27 b(1021-1078.)38 868 y([4])40 b(J.-F.)26 b(Bon)n(y)-6 b(,)26 b(J.)g(Sj\177)-38 b(ostrand,)26 b Fb(T)-6 b(r)l(ac)l(e)29 b(formula)e(for)h(r)l(esonanc)l (es)i(in)d(smal)t(l)g(domains)p Fw(,)f(J.)h(F)-6 b(unct.)25 b(Anal.)h Fa(184)g Fw(\(2001\),)h(402-418.)38 959 y([5])40 b(P)-6 b(.)26 b(Briet,)h Fb(Gener)l(al)h(estimates)i(on)d(distor)l(de)l (d)j(r)l(esolvents)g(and)e(applic)l(ations)g(to)h(Stark)g(Hamiltonians) p Fw(,)d(Rev.)f(Math.)h(Ph)n(ys.)g Fa(8)158 1050 y Fw(\(1996\),)h (639-653.)38 1142 y([6])40 b(P)-6 b(.)25 b(Briet,)i Fb(A)n(bsolutely)h 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Fb(Sp)l(e)l(ctr)l(al)i(asymptotics)g(in)d(the)i (semi-classic)l(al)g(limit)p Fw(,)d(London)g(Mathematical)g(So)r(ciet)n (y)-6 b(,)23 b(Lecture)158 1872 y(Notes)j(Series)g Fa(268)p Fw(,)h(Cam)n(bridge)e(Univ)n(ersit)n(y)g(Press,)i(1999.)0 1963 y([11])40 b(M.)35 b(Dimassi,)j(M.)d(Zerzeri,)j Fb(A)d(lo)l(c)l(al) h(tr)l(ac)l(e)i(formula)d(for)h(r)l(esonanc)l(es)i(of)e(p)l(erturb)l(e) l(d)i(p)l(erio)l(dic)f(Schr\177)-39 b(odinger)37 b(op)l(er)l(ators)p Fw(,)j(J.)158 2055 y(F)-6 b(unct.)25 b(Anal.)h Fa(198)g Fw(\(2003\),)h(142-159.)0 2146 y([12])40 b(M.)24 b(Dimassi,)h Fb(Sp)l(e)l(ctr)l(al)j(shift)e(function)g(and)g(r)l(esonanc)l(es)j(for) d(p)l(erturb)l(aitons)j(of)c(p)l(erio)l(dic)i(Schr\177)-39 b(odinger)27 b(op)l(er)l(ators)p Fw(,)h(Preprin)n(t,)158 2237 y(2002.)0 2329 y([13])40 b(B.)34 b(Hellfer)g(and)e(J.)i(Sj\177)-38 b(ostrand)34 b Fb(A)n(nalyse)h(semi-classique)h(p)l(our)f(l')n(\023)-37 b(equation)35 b(de)f(Harp)l(er,)i Fw(M)n(\023)-36 b(emoire)34 b(de)f(la)g(So)r(ci.)h(Math.)g(de)158 2420 y(F)-6 b(rance)26 b Fa(34)p Fw(,)g(1988.)0 2511 y([14])40 b(I.)26 b(W.)f(Herbst,)h Fb(Unitary)i(e)l(quivalenc)l(e)h(of)e(Stark)i(Hamiltonians)p Fw(,)d(Math.)h(Z.)e Fa(155)i Fw(\(1977\),)g(55-70.)0 2603 y([15])40 b(I.)26 b(W.)f(Herbst,)h Fb(Dilation)h(analytic)l(al)t (ly)h(in)f(c)l(onstant)j(ele)l(ctric)e(\014eld)p Fw(,)e(Comm)n(un.)e (Math.)j(Ph)n(ys.)e Fa(64)h Fw(\(1979\),)h(279-298.)0 2694 y([16])40 b(I.)26 b(W.)f(Herbst,)h Fb(Exp)l(onential)j(de)l(c)l (ay)f(in)g(Stark)g(e\013e)l(ct)p Fw(,)g(Comm)n(un.)c(Math.)i(Ph)n(ys.)g Fa(75)g Fw(\(1980\),)h(197-205.)0 2785 y([17])40 b(P)-6 b(.)34 b(D.)g(Hislop,)k(I.)c(M.)h(Sigal,)i Fb(Intr)l(o)l(duction)h(to)e (sp)l(e)l(ctr)l(al)h(the)l(ory)g(with)f(applic)l(ations)g(to)g (Schr\177)-39 b(odinger)37 b(op)l(er)l(ators)p Fw(,)j(Applied)158 2877 y(Math.)26 b(Sciences,)h Fa(113)p Fw(,)f(Springer)g(V)-6 b(erlag,)26 b(Berlin,)h(1996.)0 2968 y([18])40 b(L.)26 b(H\177)-38 b(ormander,)25 b Fb(The)i(A)n(nalysis)h(of)g(Line)l(ar)g 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(density)f(for)f(Stark)i(hamiltonians)e(\(II\),)f(Asymptotics)j(of)e (tr)l(ac)l(e)h(for-)158 4886 y(mulae)25 b Fw(Chin.)h(Ann.)f(of)i(Math.) f Fa(12)k(B)25 b Fw(\(1991\),)j(358-384.)0 4977 y([31])40 b(I.)34 b(M.)h(Sigal,)j Fb(Sharp)e(exp)l(onential)h(b)l(ounds)g(on)f(r) l(esonanc)l(es)i(states)f(and)f(width)g(of)f(r)l(esonanc)l(es)p Fw(,)40 b(Adv.)33 b(in)h(Appl.)g(Math.)h Fa(9)158 5068 y Fw(\(1988\),)27 b(127-166.)p eop %%Page: 31 31 31 30 bop 1070 66 a Fk(SPECTRAL)31 b(SHIFT)g(FUNCTION)f(AND)g (RESONANCES)999 b(31)0 266 y Fw([32])40 b(I.)25 b(M.)g(Sigal,)i Fb(Ge)l(ometric)h(the)l(ory)h(of)d(Stark)i(r)l(esonanc)l(es)i(in)c (multi-ele)l(crin)i(systems)p Fw(,)f(Comm)n(un.)c(Math.)i(Ph)n(ysics,)h Fa(19)f Fw(\(1988\),)158 357 y(287-314.)0 448 y([33])40 b(J.)28 b(Sj\177)-38 b(ostrand,)29 b Fb(A)g(tr)l(ac)l(e)i(formula)d (for)i(r)l(esonanc)l(es)i(and)d(applic)l(ation)h(to)g(semi-classic)l (al)g(Schr\177)-39 b(odinger)31 b(op)l(er)l(ator)p Fw(,)f(Exp)r(os)n (\023)-36 b(e)28 b(I)r(I,)158 540 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