Content-Type: multipart/mixed; boundary="-------------0404290931787" This is a multi-part message in MIME format. ---------------0404290931787 Content-Type: text/plain; name="04-137.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-137.keywords" resonances, Landau levels, trace formula ---------------0404290931787 Content-Type: application/postscript; name="magne19.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="magne19.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: magne19.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o magne19.ps magne19 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.04.29:1619 %%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 N/hsize 8.5 72 mul 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Fh(\000)p Fk(1)p eop %%Page: 5 5 5 4 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1281 b(5)815 266 y Fn(+\()941 243 y(~)921 266 y Fl(P)979 280 y Fk(0)1039 266 y Fn(+)20 b Fl(i)p Fn(\))1196 229 y Fh(\000)p Fk(1)1291 266 y Fl(AV)1432 229 y Fj(!)1483 266 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fl(A)p Fn(\)\()1835 243 y(~)1814 266 y Fl(P)1872 280 y Fk(0)1933 266 y Fn(+)g Fl(i)p Fn(\))2090 229 y Fh(\000)p Fk(1)2210 266 y Fn(=)25 b Fl(L)2368 280 y Fk(1)2428 266 y Fn(+)20 b Fl(L)2581 280 y Fk(2)2640 266 y Fn(+)g Fl(L)2793 280 y Fk(3)2853 266 y Fn(+)f Fl(L)3005 280 y Fk(4)3060 266 y Fl(:)0 397 y Fn(T)-8 b(o)31 b(treat)g Fl(L)418 411 y Fk(1)458 397 y Fn(,)f(notice)h(that)g(on)f(the)h(supp)s(ort)d(of) j Fl(q)s Fn(\()p Fl(x;)15 b(y)s(;)g(\021)s Fn(\))32 b(w)m(e)e(ha)m(v)m (e)1113 553 y(\()p Fl(B)5 b Fn(\()p Fl(\021)1305 516 y Fk(2)1365 553 y Fn(+)20 b Fl(y)1504 516 y Fk(2)1543 553 y Fn(\))h(+)f Fl(x)g Fn(+)g Fl(i)p Fn(\))1919 516 y Fh(\000)p Fk(1)2039 553 y Fi(2)25 b Fl(S)2186 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Ff(Z)10 b(Z)g(Z)f(Z)1186 1612 y Fi(h)p Fl(y)s(;)15 b(\021)s Fi(i)1392 1574 y Fh(\000)p Fk(2)p Fj(k)1526 1612 y Fi(h)p Fl(x)21 b Fi(\000)f Fl(B)1799 1574 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1963 1612 y Fl(\021)s Fi(i)2046 1574 y Fh(\000)p Fk(2)p Fh(\000)p Fj(\017)2225 1612 y Fi(h)p Fl(B)2334 1574 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2498 1612 y Fl(y)j Fn(+)d Fl(B)2731 1574 y Fh(\000)p Fk(1)2825 1612 y Fl(\030)t Fi(i)2904 1574 y Fh(\000)p Fk(1)p Fh(\000)p Fj(\017)3082 1612 y Fl(dxd\030)t(dy)s(d\021)1302 1867 y Fi(\024)25 b Fl(C)1470 1830 y Fh(0)1463 1890 y Fk(0)1503 1867 y Fl(B)1591 1743 y Ff(Z)10 b(Z)1743 1867 y Fi(h)p Fl(y)s(;)15 b(\021)s Fi(i)1949 1830 y Fh(\000)p Fk(2)p Fj(k)2083 1867 y Fl(dy)s(d\021)29 b Fi(\024)c Fl(C)2467 1830 y Fh(00)2460 1890 y Fk(0)2509 1867 y Fl(B)0 2046 y Fn(with)f(constan)m(ts)j Fl(C)672 2013 y Fh(0)665 2071 y Fk(0)704 2046 y Fl(;)g(C)828 2013 y Fh(00)821 2071 y Fk(0)869 2046 y Fl(;)f Fn(indep)s(enden)m(t)e(on)h Fl(B)5 b Fn(.)39 b(T)-8 b(o)26 b(deal)f(with)f Fl(L)2270 2060 y Fj(j)2307 2046 y Fl(;)i(j)31 b Fn(=)25 b(2)p Fl(;)15 b Fn(3)p Fl(;)g Fn(4)p Fl(;)28 b Fn(w)m(e)e(will)d(sho)m(w)i(that)h(\() p Fl(I)18 b Fi(\000)11 b Fl(A)p Fn(\))p Fl(V)3850 2013 y Fj(!)0 2154 y Fn(and)22 b Fl(V)242 2121 y Fj(!)293 2154 y Fn(\()p Fl(I)12 b Fi(\000)5 b Fl(A)p Fn(\))23 b(are)g(trace)h(class)e(op)s(erators.)39 b(F)-8 b(or)23 b(our)f(analysis)g(in)f(Sections)i(4-6)g(w)m(e)h(examine)e(the)h(dep)s (endence)0 2262 y(on)30 b Fl(B)35 b Fn(of)c(the)f(trace)i(estimates.)41 b(Notice)31 b(that)g(on)g(the)f(supp)s(ort)f(of)h(the)h(sym)m(b)s(ol)e (of)i(\()p Fl(I)c Fi(\000)20 b Fl(A)p Fn(\))31 b(w)m(e)g(ha)m(v)m(e) 1268 2418 y Fi(h)p Fl(y)s(;)15 b(\021)s Fi(i)1474 2381 y Fj(k)1544 2418 y Fi(\025)25 b(j)p Fn(\()p Fl(\021)1748 2381 y Fk(2)1808 2418 y Fn(+)20 b Fl(y)1947 2381 y Fk(2)1986 2418 y Fn(\))h(+)f Fl(B)2207 2381 y Fh(\000)p Fk(1)2301 2418 y Fn(\()p Fl(x)g Fn(+)g Fl(i)p Fn(\))p Fi(j)15 b Fl(:)0 2568 y Fn(T)-8 b(aking)30 b(in)m(to)g(accoun)m(t)i(the)f (estimate)g(\(1)p 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1804 4128 a(2)1845 4105 y Fo(h)p Fr(y)r(;\021)r Fo(i)1982 4091 y Fr(k)2277 3988 y Fi(h)p Fl(B)c(u)p Fi(i)2473 3950 y Fh(\000)p Fk(2)p Fh(\000)p Fj(\017)2651 3988 y Fl(dudy)s(d\021)303 4286 y Fi(\024)25 b Fl(C)471 4249 y Fh(0)464 4309 y Fk(2)504 4286 y Fl(B)578 4249 y Fk(2)617 4186 y Ff(\020)671 4163 y(Z)10 b(Z)f(Z)843 4369 y Fh(j)p Fj(u)p Fh(j\024)p Fj(C)1029 4378 y Fs(3)1062 4369 y Fj(;)p Fh(j)p Fj(y)r Fh(j\024)p Fj(C)1264 4378 y Fs(3)1298 4369 y Fj(;)p Fh(j)p Fj(\021)r Fh(j\024)p Fj(C)1500 4378 y Fs(3)1539 4286 y Fi(h)p Fl(B)c(u)p Fi(i)1735 4249 y Fh(\000)p Fk(2)p Fh(\000)p Fj(\017)1913 4286 y Fl(dudy)s(d\021)24 b Fn(+)2314 4163 y Ff(Z)10 b(Z)f(Z)2486 4369 y Fh(j)p Fj(u)p Fh(j\025)2632 4342 y Fs(1)p 2631 4354 V 2631 4395 a(2)2671 4369 y Fh(h)p Fj(y)r(;\021)r Fh(i)2819 4350 y Fr(k)2863 4286 y Fi(h)p Fl(B)c(u)p Fi(i)3059 4249 y Fh(\000)p Fk(2)p Fh(\000)p Fj(\017)3237 4286 y Fl(dudy)s(d\021)3526 4186 y Ff(\021)230 4612 y Fi(\024)25 b Fl(C)391 4626 y Fk(4)430 4612 y Fl(B)g Fn(+)20 b 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y Fi(k)p Fn(\()1200 4990 y(~)1179 5013 y Fl(P)1237 5027 y Fk(0)1298 5013 y Fn(+)19 b Fl(i)p Fn(\))1454 4975 y Fh(\000)p Fk(1)1549 5013 y Fl(V)1623 4975 y Fj(!)1673 5013 y Fn(\()1729 4990 y(~)1708 5013 y Fl(P)1766 5027 y Fk(0)1826 5013 y Fn(+)h Fl(i)p Fn(\))1983 4975 y Fh(\000)p Fk(1)2078 5013 y Fi(k)2123 5027 y Fk(tr)2208 5013 y Fi(\024)25 b Fl(A)2372 5027 y Fk(0)2412 5013 y Fl(B)5 b(;)35 b(B)30 b Fi(\025)25 b Fn(1)0 5143 y(with)k(a)i(constan)m(t)g Fl(A)717 5157 y Fk(0)782 5143 y Fl(>)25 b Fn(0)31 b(indep)s(enden)m(t)d (on)i Fl(B)5 b(:)p eop %%Page: 6 6 6 5 bop 0 66 a Fk(6)1407 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n(V) 150 266 y Fn(T)-8 b(o)31 b(establish)e(\(2)p Fl(:)p Fn(1\),)j(w)m(e)f (write)e(the)i(left-hand)e(side)h(in)f(the)h(form)1001 346 y Ff(\020)1055 447 y Fn(\()p Fl(i)21 b Fi(\000)f Fl(P)1291 461 y Fk(1)1331 447 y Fn(\()p Fl(B)5 b Fn(\)\))1510 409 y Fh(\000)p Fk(1)1625 447 y Fi(\000)20 b Fn(\()p Fl(i)h Fi(\000)f Fl(P)1952 461 y Fk(0)1991 447 y Fn(\()p Fl(B)5 b 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Fn(=)897 280 y(~)877 303 y Fl(P)935 317 y Fk(0)975 303 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b(+)g Fl(V)1389 265 y Fj(w)1445 303 y Fn(\()p Fl(x)h Fn(+)f Fl(\022)i Fi(\000)e Fl(B)1874 265 y Fh(\000)1939 238 y Fs(1)p 1939 250 31 3 v 1939 291 a(2)1983 303 y Fl(D)2058 317 y Fj(y)2120 303 y Fi(\000)2277 241 y Fn(1)p 2221 282 159 4 v 2221 365 a(2)p Fl(B)2340 339 y Fk(2)2389 303 y Fl(;)15 b(B)2503 265 y Fh(\000)2568 238 y Fs(1)p 2568 250 31 3 v 2568 291 a(2)2613 303 y Fl(y)23 b Fn(+)c Fl(B)2845 265 y Fh(\000)p Fk(1)2939 303 y Fl(D)3014 317 y Fj(x)3058 303 y Fn(\))p Fl(;)0 481 y Fn(where)274 458 y(~)256 481 y Fi(U)313 496 y Fj(\022)378 481 y Fn(=)25 b Fl(U)10 b Fi(U)603 496 y Fj(\022)642 481 y Fn(.)38 b(Throughout)23 b(the)h(pap)s(er)f(w)m(e)h(will)e(use)h (the)h(notations)2610 458 y(~)2590 481 y Fl(P)2648 495 y Fj(j)2684 481 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))24 b(for)g(the)g(op)s(erators)g(de\014ned)0 589 y(ab)s(o)m(v)m(e)h(and)d (since)h Fl(\014)30 b Fn(=)25 b(1)f(this)e(mak)m(es)j(no)e(confusion)f (with)g(the)h(notation)h Fl(P)2597 603 y Fj(j)2634 589 y Fn(\()p Fl(B)5 b(;)15 b(\014)5 b Fn(\))24 b(giv)m(en)f(in)f(the)i(In) m(tro)s(duction.)0 764 y Fw(Lemma)g(1.)35 b Fe(Ther)-5 b(e)26 b(exists)g Fl(\022)1017 778 y Fk(0)1082 764 y Fl(>)f Fn(0)g Fe(such)h(that)h(the)f(self-adjoint)g(op)-5 b(er)g(ator)2606 741 y Fn(~)2586 764 y Fl(P)2644 778 y Fk(1)2684 764 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))p Fe(,)26 b(de\014ne)-5 b(d)27 b(for)f Fl(\022)h Fi(2)p Fn(])5 b Fi(\000)g Fl(\022)3686 778 y Fk(0)3724 764 y Fl(;)15 b(\022)3807 778 y Fk(0)3847 764 y Fn([)p Fe(,)0 872 y(extends)37 b(to)f(an)h(analytic)g(typ)-5 b(e-A)36 b(family)h(of)f(op)-5 b(er)g(ators)40 b(on)c Fl(D)s Fn(\(0)p Fl(;)15 b(\022)2377 886 y Fk(0)2417 872 y Fn(\))37 b Fe(with)f(the)h(same)g(domain)g Fi(D)i Fe(as)d(that)i(of)20 963 y Fn(~)0 986 y Fl(P)58 1000 y Fk(0)98 986 y Fn(\()p Fl(B)5 b(;)15 b Fn(0\)\))p Fe(.)43 b(Mor)-5 b(e)g(over,)682 1144 y Fl(\033)734 1158 y Fk(ess)825 1144 y Fn(\()881 1121 y(~)860 1144 y Fl(P)918 1158 y Fk(1)958 1144 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))26 b(=)f Fl(\033)1397 1158 y Fk(ess)1488 1144 y Fn(\()1543 1121 y(~)1523 1144 y Fl(P)1581 1158 y Fk(0)1621 1144 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))25 b(=)g Fl(\033)s Fn(\()2118 1121 y(~)2097 1144 y Fl(P)2155 1158 y Fk(0)2196 1144 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))25 b(=)g Fi(f)p Fl(\025)c Fn(+)f Fl(\022)s Fn(;)55 b Fl(\025)25 b Fi(2)g Fg(R)r Fi(g)q Fl(:)0 1315 y Fw(Pro)s(of.)55 b Fn(Clearly)-8 b(,)35 b(the)g(domain)f Fi(D)k Fn(of)1401 1292 y(~)1381 1315 y Fl(P)1439 1329 y Fk(0)1479 1315 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))34 b(is)g(indep)s(enden)m(t)f(of)i Fl(\022)i Fn(and)d Fl(\022)h Fi(!)2951 1292 y Fn(~)2930 1315 y Fl(P)2988 1329 y Fk(0)3028 1315 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))p Fl(u)35 b Fn(is)f(analytic)g(for)0 1423 y(all)28 b Fl(u)e Fi(2)e(D)s Fn(.)40 b(On)29 b(the)h(other)f(hand,)g (the)g(analytic)g(assumption)f(on)i Fl(V)49 b Fn(implies)27 b(that)j(there)f(exists)g Fl(\022)3457 1437 y Fk(0)3522 1423 y Fl(>)c Fn(0)k(small)0 1531 y(enough)h(suc)m(h)g(that)709 1719 y Fl(D)s Fn(\(0)p Fl(;)15 b(\022)950 1733 y Fk(0)990 1719 y Fn(\))26 b Fi(3)e Fl(\022)k Fi(!)d Fl(V)1396 1682 y Fj(w)1453 1619 y Ff(\020)1507 1719 y Fl(x)20 b Fn(+)g Fl(\022)j Fi(\000)d Fl(B)1901 1682 y Fh(\000)1966 1655 y Fs(1)p 1965 1667 V 1965 1708 a(2)2009 1719 y Fl(D)2084 1733 y Fj(y)2146 1719 y Fi(\000)2304 1658 y Fn(1)p 2247 1699 159 4 v 2247 1782 a(2)p Fl(B)2366 1756 y Fk(2)2416 1719 y Fl(;)15 b(B)2530 1682 y Fh(\000)2595 1655 y Fs(1)p 2594 1667 31 3 v 2594 1708 a(2)2639 1719 y Fl(y)23 b Fn(+)d Fl(B)2872 1682 y Fh(\000)p Fk(1)2966 1719 y Fl(D)3041 1733 y Fj(x)3085 1619 y Ff(\021)3139 1719 y Fl(u)0 1908 y Fn(is)29 b(analytic)h(for)g(an)m(y)g Fl(u)c Fi(2)f Fl(L)973 1875 y Fk(2)1012 1908 y Fn(\()p Fg(R)1107 1875 y Fk(2)1152 1908 y Fn(\).)41 b(F)-8 b(ollo)m(wing)30 b([19],)i(this)d(giv)m(es)h(the)h(\014rst)e(statemen)m(t)j(of)e(the)h (lemma.)40 b(F)-8 b(or)31 b(the)0 2016 y(second)f(one,)h(notice)g(that) 1006 2168 y Fl(\033)1058 2182 y Fk(ess)1149 2168 y Fn(\()1204 2145 y(~)1184 2168 y Fl(P)1242 2182 y Fk(0)1282 2168 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))25 b(=)g Fl(\033)s Fn(\()1779 2145 y(~)1758 2168 y Fl(P)1816 2182 y Fk(0)1857 2168 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))25 b(=)g Fi(f)p Fl(\025)c Fn(+)f Fl(\022)27 b Fn(:)46 b Fl(\025)25 b Fi(2)g Fg(R)s Fi(g)p Fl(:)0 2326 y Fn(Using)32 b(\(3)p Fl(:)p Fn(1\))j(and)d(Lemma)h(3)g(of)g([25],)i(p.)48 b(111,)35 b(w)m(e)e(deduce)g(that)g Fl(\033)2380 2340 y Fk(ess)2472 2326 y Fn(\()2527 2303 y(~)2507 2326 y Fl(P)2565 2340 y Fk(1)2605 2326 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))29 b(=)g Fl(\033)3051 2340 y Fk(ess)3143 2326 y Fn(\()3198 2303 y(~)3178 2326 y Fl(P)3236 2340 y Fk(0)3276 2326 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))33 b(and)f(this)0 2434 y(completes)f(the)f(pro)s(of.)3020 b Fd(\003)150 2649 y Fn(Belo)m(w)31 b(w)m(e)g(tak)m(e)h Fl(\022)27 b Fi(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)1147 2663 y Fk(0)1187 2649 y Fn(\))p Fl(;)41 b Fn(Im)20 b Fl(\022)27 b Fi(\024)e Fn(0)p Fl(;)31 b Fn(and)f(consider)f(the)i (domain)1377 2800 y(\012)1443 2815 y Fj(\022)1507 2800 y Fn(=)25 b Fi(f)p Fl(z)30 b Fi(2)25 b Fg(C)48 b Fn(:)e(Im)20 b Fl(z)29 b(>)c Fn(Im)20 b Fl(\022)s Fi(g)15 b Fl(:)150 2950 y Fn(It)32 b(is)f(easy)h(to)g(see)h(that)f(there)g(exist)f Fl(\022)1496 2964 y Fk(0)1563 2950 y Fl(>)c Fn(0)32 b(small)e(enough)i (suc)m(h)f(that)h(for)g Fl(\022)e Fi(2)d Fl(D)s Fn(\(0)p Fl(;)15 b(\022)3239 2964 y Fk(0)3279 2950 y Fn(\))32 b(with)e(Im)21 b Fl(\022)29 b Fi(\024)f Fn(0)0 3058 y(w)m(e)j(ha)m(v)m (e)1254 3191 y Fi(k)p Fn(\()p Fl(z)25 b Fi(\000)1512 3168 y Fn(~)1492 3191 y Fl(P)1550 3205 y Fk(0)1590 3191 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))1855 3154 y Fh(\000)p Fk(1)1949 3191 y Fi(k)26 b(\024)2358 3130 y Fn(1)p 2126 3170 511 4 v 2126 3254 a Fi(j)p Fn(Im)20 b Fl(\022)i Fi(\000)e Fn(Im)g Fl(z)t Fi(j)3713 3191 y Fn(\(3.4\))0 3371 y(for)30 b Fl(z)g Fi(2)25 b Fn(\012)363 3386 y Fj(\022)401 3371 y Fn(.)150 3479 y(No)m(w,)31 b(rep)s(eating)f(the)h(argumen)m(t)f(in)g([10],)i(w)m(e)e(will)e(sho)m (w)i(that)h(w)m(e)g(ha)m(v)m(e)h(the)e(follo)m(wing)0 3644 y Fw(Lemma)j(2.)42 b Fe(L)-5 b(et)32 b Fn(Im)20 b Fl(z)821 3658 y Fk(0)886 3644 y Fl(>)25 b Fn(Im)20 b Fl(\022)s(:)32 b Fe(Then)h(the)g(op)-5 b(er)g(ator)36 b Fl(P)2018 3658 y Fk(1)2057 3644 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))21 b Fi(\000)e Fl(z)2440 3658 y Fk(0)2513 3644 y Fe(is)32 b(a)h(F)-7 b(r)i(e)g(dholm)36 b(one)d(with)g(index)g (0.)150 3808 y(Pr)-5 b(o)g(of.)42 b Fn(Let)31 b(\010)659 3823 y Fj(\022)697 3808 y Fn(\()p Fl(x;)15 b(y)s Fn(\))27 b(=)d(\()p Fl(x)d Fn(+)f Fl(\022)s(;)15 b(y)s Fn(\))p Fl(:)30 b Fn(F)-8 b(or)31 b Fl(z)f Fi(2)25 b Fn(\012)30 b(w)m(e)h(ha)m(v)m(e)858 3987 y Fl(P)916 4001 y Fk(1)956 3987 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)30 b Fn(=)1465 3886 y Ff(\020)1519 3987 y Fl(I)d Fn(+)1677 3886 y Ff(\020)1731 3987 y Fl(V)41 b Fi(\016)21 b Fn(\010)1957 4002 y Fj(\022)1995 3886 y Ff(\021)2049 3987 y Fn(\(1)g Fi(\000)f Fl( )2300 4001 y Fj(z)2340 3987 y Fn(\()p Fl(u)p Fn(\)\))2497 3886 y Ff(h)2541 3987 y Fl(P)2599 4001 y Fk(0)2639 3987 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)864 4222 y Fn(+)935 4121 y Ff(\020)989 4222 y Fl(V)40 b Fi(\016)21 b Fn(\010)1214 4237 y Fj(\022)1252 4121 y Ff(\021)1307 4222 y Fl( )1366 4236 y Fj(z)1406 4222 y Fn(\()p Fl(u)p Fn(\))1528 4121 y Ff(i)1571 4144 y Fh(\000)p Fk(1)1666 4121 y Ff(\021h)1763 4222 y Fl(P)1821 4236 y Fk(0)1861 4222 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)k Fn(+)2359 4121 y Ff(\020)2413 4222 y Fl(V)41 b Fi(\016)20 b Fn(\010)2638 4237 y Fj(\022)2677 4121 y Ff(\021)2731 4222 y Fl( )2790 4236 y Fj(z)2830 4222 y Fn(\()p Fl(u)p Fn(\))2952 4121 y Ff(i)3011 4222 y Fl(;)0 4389 y Fn(where)g Fl(u)25 b Fn(=)g(\()p Fl(x;)15 b(y)s Fn(\))p Fl(;)i( )737 4403 y Fj(z)777 4389 y Fn(\()p Fl(u)p Fn(\))26 b Fi(2)f Fl(C)1083 4356 y Fh(1)1157 4389 y Fn(\()p Fg(R)1252 4356 y Fk(2)1297 4389 y Fn(;)15 b([0)p Fl(;)g Fn(1]\))23 b(is)d(a)g(function)g(suc)m(h)g (that)h Fl( )2510 4403 y Fj(z)2550 4389 y Fn(\()p Fl(u)p Fn(\))26 b(=)e(0)d(for)f Fi(j)p Fl(u)p Fi(j)26 b(\024)f Fl(C)3277 4403 y Fk(1)3317 4389 y Fl(;)40 b( )3441 4403 y Fj(z)3481 4389 y Fn(\()p Fl(u)p Fn(\))26 b(=)f(1)c(for)0 4525 y Fi(j)p Fl(u)p Fi(j)26 b(\025)f Fl(C)289 4539 y Fk(1)339 4525 y Fn(+)10 b(1)p Fl(:)26 b Fn(Cho)s(osing)e Fl(C)973 4539 y Fk(1)1038 4525 y Fl(>)h Fn(0)h(\(dep)s(ending)d(on)i Fl(z)t Fn(\))h(large)g(enough,)g(w)m(e)g(ma)m(y)g(assume)g(that)g Fi(j)3300 4424 y Ff(\020)3354 4525 y Fl(V)31 b Fi(\016)10 b Fn(\010)3559 4540 y Fj(\022)3599 4424 y Ff(\021)3653 4525 y Fi(j)p Fl( )3737 4539 y Fj(z)3777 4525 y Fn(\()p Fl(u)p Fn(\))0 4655 y(is)29 b(small,)h(so)g(the)h(op)s(erator)1182 4790 y Fl(A)1250 4805 y Fj(\022)1289 4790 y Fn(\()p Fl(z)t Fn(\))c(=)e Fl(P)1586 4804 y Fk(0)1625 4790 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)25 b Fn(+)2124 4689 y Ff(\020)2178 4790 y Fl(V)40 b Fi(\016)21 b Fn(\010)2403 4805 y Fj(\022)2442 4689 y Ff(\021)2496 4790 y Fl( )2555 4804 y Fj(z)2595 4790 y Fn(\()p Fl(x)p Fn(\))0 4944 y(is)29 b(in)m(v)m(ertible)g(for)h Fl(z)g Fi(2)25 b Fn(\012)856 4959 y Fj(\022)894 4944 y Fn(.)41 b(On)30 b(the)g(other)h(hand,)1203 5117 y Fl(K)1280 5132 y Fj(\022)1319 5117 y Fn(\()p Fl(z)t Fn(\))26 b(=)1557 5016 y Ff(\020)1612 5117 y Fl(V)40 b Fi(\016)21 b Fn(\010)1837 5132 y Fj(\022)1875 5016 y Ff(\021)1930 5117 y Fn(\(1)g Fi(\000)f Fl( )2181 5131 y Fj(z)2221 5117 y Fn(\()p Fl(u)p Fn(\)\))p Fl(A)2446 5132 y Fj(\022)2486 5117 y Fn(\()p Fl(z)t Fn(\))2602 5080 y Fh(\000)p Fk(1)p eop %%Page: 8 8 8 7 bop 0 66 a Fk(8)1407 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n(V)0 266 y Fn(is)29 b(compact.)42 b(Then)999 394 y(dim)13 b(Ker)20 b(\()1388 371 y(~)1367 394 y Fl(P)1425 408 y Fk(1)1466 394 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)f Fl(z)1848 408 y Fk(0)1888 394 y Fn(\))26 b(=)f(dim)13 b(Ker)20 b(\()p Fl(I)28 b Fn(+)20 b Fl(K)2649 409 y Fj(\022)2688 394 y Fn(\()p Fl(z)2765 408 y Fk(0)2805 394 y Fn(\)\))p Fl(;)0 540 y Fn(pro)m(vided)1685 667 y(Im)f Fl(z)1855 681 y Fk(0)1920 667 y Fl(>)25 b Fn(Im)20 b Fl(\022)s(:)0 813 y Fn(A)30 b(simple)f(argumen)m(t)i(sho)m(ws)f(that) h(Im)20 b(\()p Fl(P)1469 827 y Fk(1)1509 813 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)1892 827 y Fk(0)1931 813 y Fn(\))31 b(is)e(closed)i(and)1022 978 y(co)s(dim)19 b(\()p Fl(P)1375 992 y Fk(1)1415 978 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)1798 992 y Fk(0)1838 978 y Fn(\))26 b(=)e(dim)14 b(Ker)40 b(\()p Fl(I)28 b Fn(+)20 b Fl(K)2626 941 y Fh(\003)2619 1001 y Fj(\022)2665 978 y Fn(\()p Fl(z)2742 992 y Fk(0)2782 978 y Fn(\)\))p Fl(:)0 1143 y Fn(Th)m(us)29 b Fl(P)288 1157 y Fk(1)328 1143 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)35 b Fn(is)29 b(a)i(F)-8 b(redholm)29 b(op)s(erator)i(with)e(index)g(0)i(and)f(the)g(pro)s(of)g(is)f (complete.)576 b Fd(\003)0 1316 y Fw(De\014nition)35 b(1.)42 b Fe(L)-5 b(et)33 b Fn(Im)20 b Fl(\022)27 b(<)e Fn(0)p Fe(.)42 b(We)32 b(say)i(that)g Fl(z)29 b Fi(2)c Fn(\012)1913 1331 y Fj(\022)1984 1316 y Fe(is)33 b(a)g(r)-5 b(esonanc)g(e)34 b(of)f Fl(P)2745 1330 y Fk(1)2785 1316 y Fn(\()p Fl(B)5 b Fn(\))33 b Fe(if)1399 1480 y Fn(dim)14 b(Ker)40 b(\()p Fl(P)1846 1494 y Fk(1)1886 1480 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)t Fn(\))26 b Fl(>)f Fn(0)p Fl(:)150 1653 y Fn(As)45 b(in)e([10],)49 b(w)m(e)c(sho)m(w)g(that)g Fl(P)1290 1667 y Fk(1)1330 1653 y Fn(\()p Fl(B)5 b Fn(\))44 b(has)h(no)f(resonances)h Fl(z)k Fn(with)43 b(Im)29 b Fl(z)53 b(>)c Fn(0,)g(as)c(w)m(ell)e(as,)49 b(that)c(the)0 1761 y(resonances)37 b(in)f Fi(f)p Fl(z)41 b Fi(2)36 b Fg(C)60 b Fn(:)i(Im)o Fl(z)41 b(>)36 b Fn(Im)24 b Fl(\022)1458 1775 y Fk(2)1534 1761 y Fl(>)36 b Fn(Im)24 b Fl(\022)1817 1775 y Fk(1)1856 1761 y Fi(g)38 b Fn(are)f(indep)s (enden)m(t)e(of)i(the)h(c)m(hoice)g(of)f Fl(\022)i Fn(satisfying)d(the) 0 1869 y(condition)29 b(0)d Fl(>)f Fn(Im)19 b Fl(\022)737 1883 y Fk(2)802 1869 y Fi(\025)25 b Fn(Im)20 b Fl(\022)27 b Fi(\025)e Fn(Im)20 b Fl(\022)1365 1883 y Fk(1)1404 1869 y Fl(:)30 b Fn(W)-8 b(e)32 b(de\014ne)e(the)g(m)m(ultiplicit)m(y)e (of)i(a)h(resonance)g Fl(z)3159 1883 y Fk(0)3229 1869 y Fn(b)m(y)1080 2076 y Fl(m)p Fn(\()p Fl(z)1237 2090 y Fk(0)1276 2076 y Fn(\))26 b(=)f(rank)1686 2014 y(1)p 1643 2055 132 4 v 1643 2138 a(2)p Fl(\031)s(i)1800 1952 y Ff(Z)1850 2158 y Fj(\015)1886 2166 y Fr(\027)1925 2158 y Fk(\()p Fj(z)1985 2167 y Fs(0)2020 2158 y Fk(\))2052 2076 y Fn(\()p Fl(z)f Fi(\000)2265 2053 y Fn(~)2244 2076 y Fl(P)2302 2090 y Fk(1)2342 2076 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))2607 2038 y Fh(\000)p Fk(1)2702 2076 y Fl(dz)t(;)0 2317 y Fn(where)34 b Fl(\015)314 2331 y Fj(\017)347 2317 y Fn(\()p Fl(z)424 2331 y Fk(0)464 2317 y Fn(\))f(=)f Fi(f)p Fl(z)37 b Fn(=)c Fl(z)905 2331 y Fk(0)967 2317 y Fn(+)23 b Fl(\027)6 b(e)1154 2284 y Fj(i')1228 2317 y Fl(;)39 b Fn(0)33 b Fi(\024)f Fl(')h Fi(\024)f Fn(2)p Fl(\031)s Fi(g)k Fn(and)e Fl(\027)k(>)33 b Fn(0)i(is)e(small)h (enough.)53 b(In)34 b(the)h(follo)m(wing)e(w)m(e)j(\014x)0 2425 y Fl(\022)27 b Fi(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)397 2439 y Fk(0)437 2425 y Fn(\))31 b(with)e(Im)20 b Fl(\022)27 b(<)e Fn(0)31 b(and)f(w)m(e)h(denote)g(the)f(set)h(of)g(resonances)f (of)h Fl(P)2699 2439 y Fk(1)2739 2425 y Fn(\()p Fl(B)5 b Fn(\))30 b(b)m(y)h(Res)20 b Fl(P)3261 2439 y Fk(1)3301 2425 y Fn(\()p Fl(B)5 b Fn(\))p Fl(:)150 2641 y Fw(Remark.)40 b Fn(Clearly)-8 b(,)30 b(the)g(op)s(erators)622 2805 y Fl(P)680 2819 y Fk(1)720 2805 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))25 b(=)g Fi(U)1128 2820 y Fj(\022)1167 2805 y Fl(P)1225 2819 y Fk(1)1264 2805 y Fn(\()p Fl(B)5 b Fn(\))p Fi(U)1474 2767 y Fh(\000)p Fk(1)1465 2835 y Fj(\022)1594 2805 y Fn(=)25 b(\()p Fl(D)1800 2819 y Fj(x)1865 2805 y Fi(\000)20 b Fl(B)5 b(y)s Fn(\))2113 2768 y Fk(2)2172 2805 y Fn(+)20 b Fl(D)2341 2768 y Fk(2)2338 2828 y Fj(x)2402 2805 y Fn(+)g Fl(x)g Fn(+)g Fl(\022)j Fn(+)d Fl(V)g Fn(\()p Fl(x)g Fn(+)g Fl(\022)s(;)15 b(y)s Fn(\))p Fl(;)0 2984 y Fn(and)202 2961 y(~)181 2984 y Fl(P)239 2998 y Fk(1)279 2984 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))35 b(ha)m(v)m(e)h(the)f (same)g(eigen)m(v)-5 b(alues)35 b(in)e(\012)1802 2999 y Fj(\022)1876 2984 y Fn(with)h(the)h(same)g(m)m(ultiplicit)m(y)-8 b(.)52 b(In)34 b(a)h(suc)m(h)g(w)m(a)m(y)-8 b(,)37 b(w)m(e)f(can)0 3092 y(w)m(ork)30 b(directly)g(with)f Fl(P)819 3106 y Fk(1)858 3092 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\).)0 3264 y Fw(Prop)s(osition)36 b(2.)42 b Fe(L)-5 b(et)33 b Fl(V)53 b Fe(satisfy)33 b Fn(\(1)p Fl(:)p Fn(1\))i Fe(and)e(the)g(c)-5 b(ondition)1588 3429 y Fn(1)20 b(+)g Fl(@)1792 3443 y Fj(x)1836 3429 y Fl(V)h Fn(\()p Fl(x;)15 b(y)s Fn(\))26 b Fl(>)f Fn(0)p Fl(:)0 3594 y Fe(Then,)33 b(ther)-5 b(e)33 b(exists)g Fl(\022)783 3608 y Fk(0)855 3594 y Fe(,)f Fn(Im)20 b Fl(\022)1087 3608 y Fk(0)1151 3594 y Fl(<)25 b Fn(0)p Fe(,)33 b(such)g(that)h Fl(P)1804 3608 y Fk(1)1844 3594 y Fn(\()p Fl(B)5 b Fn(\))32 b Fe(has)i(no)f(r)-5 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Fn(\)\))p Fl(:)p eop %%Page: 10 10 10 9 bop 0 66 a Fk(10)1372 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n(V) 0 266 y Fw(Pro)s(of.)38 b Fn(Cho)s(ose)22 b(a)g(function)e Fl(f)34 b Fi(2)25 b Fl(C)1279 233 y Fh(1)1272 290 y Fk(0)1353 266 y Fn(\(]2)p Fl(n)r Fi(\000)r Fn(2)p Fl(;)15 b Fn(2)p Fl(n)p Fn([\))24 b(suc)m(h)e(that)g Fl(f)34 b Fn(=)25 b(1)d(near)f(2)p Fl(n)r Fi(\000)r Fn(1.)39 b(Ob)m(viously)-8 b(,)22 b Fl(f)10 b Fn(\()p Fl(D)3567 233 y Fk(2)3564 288 y Fj(y)3608 266 y Fn(+)r Fl(y)3729 233 y Fk(2)3769 266 y Fn(\))25 b(=)0 385 y(\005)30 b(and)g(the)h(pseudo)s(di\013eren)m (tial)c(calculus)i(yields)g Fl(y)s(f)1865 352 y Fh(0)1887 385 y Fn(\()p Fl(D)2000 352 y Fk(2)1997 408 y Fj(y)2060 385 y Fn(+)20 b Fl(y)2199 352 y Fk(2)2238 385 y Fn(\))26 b(=)f Fi(O)s Fn(\(1\))31 b(in)e Fi(L)p Fn(\()p Fl(L)2882 352 y Fk(2)2922 385 y Fn(\()p Fg(R)3017 352 y Fk(2)3062 385 y Fn(\)\))p Fl(:)i Fn(Th)m(us)615 512 y Ff(h)658 613 y Fl(O)s(p)776 575 y Fj(!)826 613 y Fl(G)897 512 y Ff(\020)962 551 y Fl(x)20 b 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Fl(B)1811 905 y Fj(\016)2062 768 y Ff(\021)2116 869 y Fl(y)s(f)2219 832 y Fh(0)2241 869 y Fn(\()p Fl(D)2354 832 y Fk(2)2351 892 y Fj(y)2414 869 y Fn(+)g Fl(y)2553 832 y Fk(2)2592 869 y Fn(\))2627 768 y Ff(\021)2707 869 y Fn(=)25 b Fi(O)s Fn(\()p Fl(B)2987 832 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)3241 869 y Fn(\))15 b Fl(:)3829 1022 y Fd(\003)0 1130 y Fn(T)-8 b(o)31 b(estimate)g(the)g(norms)e(of)i(the)f (comm)m(utators,)i(w)m(e)f(need)f(the)h(follo)m(wing)0 1293 y Fw(Lemma)42 b(4.)k Fe(Ther)-5 b(e)41 b(exist)f(c)-5 b(onstants)42 b Fl(C)1467 1307 y Fk(0)1545 1293 y Fl(>)c Fn(0)p Fl(;)k(C)1831 1307 y Fk(1)1909 1293 y Fl(>)c Fn(0)p Fe(,)k(indep)-5 b(endent)41 b(on)g Fl(B)5 b Fe(,)41 b(and)g Fl(B)3178 1307 y Fk(0)3256 1293 y Fi(\035)d Fn(1)j Fe(so)f(that)h(for)0 1400 y Fl(B)30 b Fi(\025)25 b Fl(B)264 1414 y Fk(0)336 1400 y Fe(we)32 b(have)1620 1607 y Fi(k)p Fl(y)24 b Fn(Op)1855 1569 y Fj(!)1905 1506 y Ff(\020)1960 1607 y Fl(G)2031 1569 y Fh(0)2031 1629 y Fj(i)2059 1506 y Ff(\020)2124 1545 y Fl(x)c Fi(\000)g Fl(B)2361 1512 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2525 1545 y Fl(D)2600 1559 y Fj(y)p 2124 1586 V 2327 1669 a Fl(B)2401 1643 y Fj(\016)2652 1506 y Ff(\021\021)2760 1607 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)579 1772 y(\024)25 b Fl(C)740 1786 y Fk(0)779 1772 y Fi(k)p Fn(\()p Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))21 b Fi(\000)f Fl(z)t Fn(\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)1628 1786 y Fk(1)1668 1772 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)36 b(i)26 b Fn(=)f(1)p Fl(;)15 b Fn(2)p Fl(;)37 b Fi(8)p Fl(z)29 b Fi(2)c Fn(\012)2787 1786 y Fj(n)2833 1772 y Fl(;)36 b(u)25 b Fi(2)g(D)s Fl(:)558 b Fn(\(4.2\))0 1935 y Fw(Pro)s(of.)41 b Fn(In)m(tro)s(duce)30 b(the)h(sym)m(b)s(ol) 1202 2200 y Fl(g)1245 2214 y Fj(i)1274 2200 y Fn(\()p Fl(x;)15 b(y)s(;)g(\021)s Fn(\))27 b(=)1705 2105 y Fl(y)s(G)1824 2072 y Fh(0)1824 2130 y Fj(i)1852 2004 y Ff(\020)1906 2105 y Fl(B)1980 2072 y Fh(\000)p Fj(\016)2073 2105 y Fn(\()p Fl(x)20 b Fi(\000)g Fl(B)2345 2072 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2509 2105 y Fl(\021)s Fn(\))2592 2004 y Ff(\021)p 1705 2179 943 4 v 1796 2262 a Fl(\021)1844 2236 y Fk(2)1904 2262 y Fn(+)g Fl(y)2043 2236 y Fk(2)2103 2262 y Fn(+)g Fl(B)2268 2236 y Fh(\000)p Fk(1)2361 2262 y Fl(x)h Fn(+)e Fl(i)2673 2200 y(:)0 2406 y Fn(W)-8 b(e)35 b(will)c(sho)m(w)i(that)h(this)f(sym)m(b)s(ol)f(is)g(in)h(the)g(class)h Fl(S)1899 2373 y Fk(0)1938 2406 y Fn(\()p Fg(R)2033 2373 y Fk(4)2079 2406 y Fn(\))p Fl(:)g Fn(In)e(fact,)k(the)e(deriv)-5 b(ativ)m(e)33 b Fl(@)3131 2373 y Fj(l)3126 2429 y(x)3170 2406 y Fl(@)3223 2362 y Fj(p)3218 2418 y(y)3263 2406 y Fl(@)3316 2373 y Fj(k)3311 2429 y(\021)3359 2406 y Fl(g)3402 2420 y Fj(i)3430 2406 y Fn(\()p Fl(x;)15 b(y)s(;)g(\021)s Fn(\))36 b(can)0 2514 y(b)s(e)30 b(written)f(as)i(a)g(sum)e(of)i(terms) 623 2707 y Fl(B)697 2670 y Fh(\000)p Fk(\()p Fj(q)r Fh(\000)p Fk(1\))p Fj(\016)1353 2646 y Fl(y)1401 2613 y Fj(p)1437 2590 y Fo(0)1459 2613 y Fk(+1)1553 2646 y Fl(\021)1601 2613 y Fj(k)1640 2590 y Fo(0)p 978 2687 1063 4 v 978 2770 a Fn(\()p Fl(\021)1061 2744 y Fk(2)1122 2770 y Fn(+)20 b Fl(y)1261 2744 y Fk(2)1320 2770 y Fn(+)g Fl(B)1485 2744 y Fh(\000)p Fk(1)1579 2770 y Fl(x)g Fn(+)g Fl(i)p Fn(\))1808 2744 y Fj(k)r Fk(+)p Fj(p)p Fk(+)p Fj(l)2015 2725 y Fo(0)2051 2707 y Fl(G)2122 2660 y Fk(\()p Fj(q)r Fk(\))2122 2735 y Fj(i)2215 2607 y Ff(\020)2270 2707 y Fl(B)2344 2670 y Fh(\000)p Fj(\016)2436 2707 y Fn(\()p Fl(x)g Fi(\000)g Fl(B)2708 2670 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2873 2707 y Fl(\021)s Fn(\))2956 2607 y Ff(\021)3011 2707 y Fl(;)15 b(q)28 b Fi(\025)d Fn(1)0 2923 y(with)k Fl(p)253 2890 y Fh(0)301 2923 y Fi(\024)c Fl(p;)36 b(k)554 2890 y Fh(0)602 2923 y Fi(\024)25 b Fl(k)s(;)36 b(l)838 2890 y Fh(0)887 2923 y Fi(\025)25 b Fl(l)d Fn(+)e(1)p Fl(:)31 b Fn(Setting)f Fl(u)25 b Fn(=)g Fl(B)1784 2890 y Fh(\000)p Fj(\016)1876 2923 y Fn(\()p Fl(x)c Fi(\000)f Fl(B)2149 2890 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2313 2923 y Fl(\021)s Fn(\))p Fl(;)31 b Fn(w)m(e)g(need)f(to)h(estimate)643 3134 y Fl(T)696 3149 y Fj(l)q(;p;k)835 3134 y Fn(\()p Fl(B)5 b(;)15 b(y)s(;)g(\021)s(;)g(u)p Fn(\))27 b(=)2028 3073 y Fl(y)2076 3040 y Fj(p)p Fk(+1)2206 3073 y Fl(\021)2254 3040 y Fj(k)p 1380 3114 1565 4 v 1380 3148 a Ff(\020)1435 3249 y Fl(\021)1483 3223 y Fk(2)1543 3249 y Fn(+)19 b Fl(y)1681 3223 y Fk(2)1741 3249 y Fn(+)h Fl(B)1906 3223 y Fh(\000)p Fk(1+)p Fj(\016)2088 3249 y Fl(u)h Fn(+)e Fl(B)2325 3222 y Fh(\000)p Fk(3)p Fj(=)p Fk(2)2490 3249 y Fl(\021)k Fn(+)d Fl(i)2680 3148 y Ff(\021)2735 3171 y Fj(k)r Fk(+)p Fj(p)p Fk(+)p Fj(l)2955 3134 y Fl(G)3026 3087 y Fk(\()p Fj(q)r Fk(\))3026 3162 y Fj(i)3119 3134 y Fn(\()p Fl(u)p Fn(\))0 3435 y(uniformly)27 b(with)i(resp)s(ect)i(to)g Fl(B)5 b(;)15 b(y)s(;)g(\021)s(;)g(u:)31 b Fn(F)-8 b(or)32 b Fl(B)1680 3402 y Fh(\000)p Fk(1+)p Fj(\016)1862 3435 y Fi(j)p Fl(u)p Fi(j)26 b(\024)2096 3399 y Fk(1)p 2096 3414 36 4 v 2096 3466 a(2)2141 3435 y Fn(\()p Fl(\021)2224 3402 y Fk(2)2284 3435 y Fn(+)20 b Fl(y)2423 3402 y Fk(2)2463 3435 y Fn(\))p Fl(;)31 b Fn(w)m(e)f(ha)m(v)m(e)991 3597 y Fi(j)p Fl(\021)1064 3560 y Fk(2)1124 3597 y Fn(+)20 b Fl(y)1263 3560 y Fk(2)1323 3597 y Fn(+)f Fl(B)1487 3560 y Fh(\000)p Fk(1+)p Fj(\016)1670 3597 y Fl(u)h Fn(+)g Fl(i)p Fi(j)26 b(\025)f Fl(c)2050 3611 y Fk(0)2090 3597 y Fn(\()p Fl(\021)2173 3560 y Fk(2)2233 3597 y Fn(+)20 b Fl(y)2372 3560 y Fk(2)2431 3597 y Fn(+)g(1\))p Fl(;)37 b(c)2703 3611 y Fk(0)2768 3597 y Fl(>)25 b Fn(0)0 3765 y(and)39 b(w)m(e)g(get)i Fl(T)542 3780 y Fj(l)q(;p;k)721 3765 y Fn(=)f Fi(O)s Fn(\(1\))g(with)e(resp)s(ect)h(to)h Fl(B)5 b(;)15 b(y)s(;)g(\021)s(;)g(u)p Fn(.)69 b(On)38 b(the)h(other)h(hand,)h(the)e(supp)s(ort)e(of)j Fl(G)3684 3717 y Fk(\()p Fj(q)r Fk(\))3684 3792 y Fj(i)3777 3765 y Fn(\()p Fl(u)p Fn(\))0 3880 y(is)c(b)s(ounded)f(and)h Fl(B)733 3848 y Fh(\000)p Fk(1+)p Fj(\016)916 3880 y Fi(j)p Fl(u)p Fi(j)g(\025)1171 3845 y Fk(1)p 1171 3860 V 1171 3912 a(2)1217 3880 y Fn(\()p Fl(\021)1300 3848 y Fk(2)1364 3880 y Fn(+)25 b Fl(y)1508 3848 y Fk(2)1547 3880 y Fn(\))37 b(leads)g(to)g(\()p Fl(\021)2054 3848 y Fk(2)2119 3880 y Fn(+)25 b Fl(y)2263 3848 y Fk(2)2302 3880 y Fn(\))36 b Fi(\024)g Fl(c)2519 3894 y Fk(1)2559 3880 y Fl(B)2633 3848 y Fh(\000)p Fk(1+)p Fj(\016)2852 3880 y Fi(\024)g Fl(c)2998 3894 y Fk(2)3075 3880 y Fn(and)g(w)m(e)h (obtain)g(again)0 3988 y Fl(T)53 4003 y Fj(l)q(;p;k)218 3988 y Fn(=)25 b Fi(O)s Fn(\(1\))p Fl(:)31 b Fn(No)m(w)g(consider)f (the)g(op)s(erator)h Fl(O)s(p)1763 3955 y Fj(!)1813 3988 y Fl(g)1856 4002 y Fj(i)1884 3988 y Fn(\()p Fl(x;)15 b(y)s(;)g(D)2174 4002 y Fj(y)2217 3988 y Fn(\))31 b(with)e(W)-8 b(eyl)31 b(sym)m(b)s(ol)e Fl(g)3074 4002 y Fj(i)3102 3988 y Fn(\()p Fl(x;)15 b(y)s(;)g(\021)s Fn(\))p Fl(:)32 b Fn(W)-8 b(e)32 b(ha)m(v)m(e)312 4209 y Fl(O)s(p)430 4171 y Fj(!)480 4209 y Fl(g)523 4223 y Fj(i)552 4209 y Fn(\()p Fl(x;)15 b(y)s(;)g(D)842 4223 y Fj(y)884 4209 y Fn(\)\()p Fl(D)1032 4171 y Fk(2)1029 4231 y Fj(y)1092 4209 y Fn(+)20 b Fl(y)1231 4171 y Fk(2)1291 4209 y Fn(+)g Fl(B)1456 4171 y Fh(\000)p Fk(1)1550 4209 y Fl(x)g Fn(+)g Fl(i)p Fn(\))26 b(=)e Fl(y)s(O)s(p)2066 4171 y Fj(!)2116 4108 y Ff(\020)2171 4209 y Fl(G)2242 4171 y Fh(0)2242 4231 y Fj(i)2270 4108 y Ff(\020)2335 4147 y Fl(x)c Fi(\000)g Fl(B)2572 4114 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2736 4147 y Fl(D)2811 4161 y Fj(y)p 2335 4188 519 4 v 2538 4271 a Fl(B)2612 4245 y Fj(\016)2863 4108 y Ff(\021\021)2991 4209 y Fn(+)g Fl(R)3151 4223 y Fj(i)3180 4209 y Fn(\()p Fl(x;)15 b(y)s(;)g(D)3470 4223 y Fj(y)3512 4209 y Fn(\))g Fl(:)0 4382 y Fn(Using)21 b(the)i(explicit)d(form)m(ulae)i(of)h Fl(R)1239 4396 y Fj(i)1289 4382 y Fn(giv)m(en)f(b)m(y)h(the)f(calculus) f(of)h(pseudo)s(di\013eren)m(tial)e(op)s(erators,)k(and)e(rep)s(eating) 0 4490 y(the)40 b(ab)s(o)m(v)m(e)h(argumen)m(ts,)i(w)m(e)e(see)f(that)h (the)f(sym)m(b)s(ol)f(of)h Fl(R)2089 4504 y Fj(i)2157 4490 y Fn(is)f(in)g(the)h(class)f Fl(S)2823 4457 y Fk(0)2863 4490 y Fn(\()p Fg(R)2958 4457 y Fk(4)3003 4490 y Fn(\).)70 b(It)40 b(follo)m(ws)f(from)h(the)0 4598 y(Calderon-V)-8 b(aillancourt's)34 b(theorem)i(\(see)g(for)f(instance,)i(Theorem)e (7.11)i(of)f([8]\))h(that)f Fl(O)s(p)3229 4565 y Fj(!)3279 4598 y Fl(g)3322 4612 y Fj(i)3350 4598 y Fn(\()p Fl(x;)15 b(y)s(;)g(D)3640 4612 y Fj(y)3683 4598 y Fn(\))36 b(and)0 4706 y Fl(R)69 4720 y Fj(i)128 4706 y Fn(are)30 b(b)s(ounded.)39 b(Th)m(us,)1177 4968 y Fi(k)p Fl(y)23 b(O)s(p)1408 4931 y Fj(!)1458 4867 y Ff(\020)1513 4968 y Fl(G)1584 4931 y Fh(0)1584 4991 y Fj(i)1612 4867 y Ff(\020)1677 4906 y Fl(x)d Fi(\000)g Fl(B)1914 4873 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2078 4906 y Fl(D)2153 4920 y Fj(y)p 1677 4947 V 1880 5030 a Fl(B)1954 5004 y Fj(\016)2205 4867 y Ff(\021\021)2313 4968 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)613 5141 y(\024)25 b(k)p Fl(O)s(p)872 5104 y Fj(!)922 5141 y Fl(g)965 5155 y Fj(i)994 5141 y Fn(\()p Fl(x;)15 b(y)s(;)g(D)1284 5155 y Fj(y)1327 5141 y Fn(\)\()p Fl(D)1475 5104 y Fk(2)1472 5164 y Fj(y)1535 5141 y Fn(+)20 b Fl(y)1674 5104 y Fk(2)1733 5141 y Fn(+)g Fl(B)1898 5104 y Fh(\000)p Fk(1)1992 5141 y Fl(x)g Fn(+)g Fl(i)p Fn(\)\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)2792 5155 y Fk(2)2832 5141 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p eop %%Page: 11 11 11 10 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(11)865 266 y Fi(\024)25 b Fl(C)1026 280 y Fk(3)1065 266 y Fi(k)p Fn(\()p Fl(D)1223 228 y Fk(2)1220 288 y Fj(y)1283 266 y Fn(+)20 b Fl(y)1422 228 y Fk(2)1482 266 y Fn(+)g Fl(B)1647 228 y Fh(\000)p Fk(1)1740 266 y Fl(x)h Fn(+)e Fl(i)p Fn(\)\()p Fl(I)29 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)2541 280 y Fk(2)2580 266 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)822 419 y(\024)25 b Fl(C)983 433 y Fk(3)1022 419 y Fi(k)p Fn(\()p Fl(D)1180 381 y Fk(2)1177 441 y Fj(y)1240 419 y Fn(+)20 b Fl(y)1379 381 y Fk(2)1439 419 y Fn(+)g Fl(B)1604 381 y Fh(\000)p Fk(1)1698 419 y Fn(\()p Fl(x)g Fi(\000)g Fl(z)t Fn(\)\)\()p Fl(I)29 b Fi(\000)19 b Fn(\005\))p Fl(u)p Fi(k)j Fn(+)e Fl(C)2591 381 y Fh(0)2584 441 y Fk(2)2623 419 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)510 612 y(\024)25 b Fl(C)671 626 y Fk(3)710 612 y Fi(k)755 511 y Ff(\020)810 612 y Fl(B)5 b Fn(\()p Fl(D)997 575 y Fk(2)994 635 y Fj(y)1056 612 y Fn(+)20 b Fl(y)1195 575 y Fk(2)1234 612 y Fn(\))h(+)f Fl(x)g Fn(+)g Fl(\022)i Fi(\000)1767 551 y Fn(1)p 1710 591 159 4 v 1710 674 a(4)p Fl(B)1829 648 y Fk(2)1899 612 y Fn(+)e Fl(V)2063 575 y Fj(!)2133 612 y Fi(\000)g Fl(z)2270 511 y Ff(\021)2325 612 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)2896 626 y Fk(4)2935 612 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)827 780 y(\024)25 b Fl(C)988 794 y Fk(3)1027 780 y Fi(k)p Fn(\()p Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))21 b Fi(\000)f Fl(z)t Fn(\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)1876 794 y Fk(1)1915 780 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)37 b Fi(8)p Fl(z)29 b Fi(2)c Fn(\012)2690 794 y Fj(n)2736 780 y Fl(;)36 b(u)25 b Fi(2)g(D)18 b Fl(:)150 912 y Fn(Here)31 b(w)m(e)f(ha)m(v)m(e)i(used)d(the)i(fact)g (that)g Fl(B)1522 879 y Fh(\000)p Fk(1)1616 912 y Fl(z)j Fn(is)29 b(b)s(ounded)f(for)i Fl(z)g Fi(2)25 b Fn(\012)2517 926 y Fj(n)2594 912 y Fn(as)30 b(w)m(ell)f(as)i(the)f(estimate)h([\005) p Fl(;)15 b(V)3728 879 y Fj(!)3779 912 y Fn(])25 b(=)0 1026 y Fi(O)s Fn(\()p Fl(B)184 993 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)349 1026 y Fn(\))p Fl(:)3395 b Fd(\003)p Fn(.)150 1242 y(T)-8 b(o)31 b(estimate)g(the)g(action)f(of)h Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))30 b(on)g Fl(A)1707 1256 y Fj(i)1735 1242 y Fn(\()p Fl(I)e Fi(\000)20 b Fn(\005\))p Fl(u;)41 b(i)26 b Fn(=)f(1)p Fl(;)15 b Fn(2)p Fl(;)32 b Fn(w)m(e)e(need)h(the)f(follo)m(wing)0 1407 y Fw(Lemma)36 b(5.)43 b Fe(Ther)-5 b(e)36 b(exists)g Fl(a)1062 1421 y Fk(1)1132 1407 y Fl(>)29 b Fn(0)p Fe(,)37 b(indep)-5 b(endent)36 b(on)g Fl(B)5 b Fe(,)36 b(and)g Fl(B)2367 1421 y Fk(0)2436 1407 y Fi(\035)30 b Fn(1)36 b Fe(so)g(that)g(for)g Fl(B)f Fi(\025)29 b Fl(B)3366 1421 y Fk(0)3441 1407 y Fe(and)36 b Fl(z)f Fi(2)29 b Fn(\012)3853 1421 y Fj(n)0 1515 y Fe(we)k(have)613 1670 y Fi(k)658 1570 y Ff(h)702 1670 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))998 1570 y Ff(\020)1053 1670 y Fl(B)5 b Fn(\()p Fl(D)1240 1633 y Fk(2)1237 1693 y Fj(y)1299 1670 y Fn(+)20 b Fl(y)1438 1633 y Fk(2)1477 1670 y Fn(\))h(+)f Fl(x)g Fn(+)g Fl(\022)i Fn(+)e Fl(V)2016 1633 y Fj(!)2067 1570 y Ff(\021)2121 1670 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))g Fi(\000)g Fl(z)2575 1570 y Ff(i)2619 1670 y Fl(A)2687 1684 y Fk(1)2726 1670 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)1026 1852 y(\025)25 b Fl(a)1170 1866 y Fk(1)1209 1852 y Fl(B)5 b Fi(k)p Fl(A)1396 1866 y Fk(1)1436 1852 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h(\000)f(O)s Fn(\()p Fl(B)2125 1815 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2324 1852 y Fn(\))p Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)37 b(u)25 b Fi(2)g(D)s Fl(:)592 b Fn(\(4.3\))0 2017 y Fw(Lemma)36 b(6.)43 b Fe(Ther)-5 b(e)36 b(exists)g Fl(a)1062 2031 y Fk(2)1132 2017 y Fl(>)29 b Fn(0)p Fe(,)37 b(indep)-5 b(endent)36 b(on)g Fl(B)5 b Fe(,)36 b(and)g Fl(B)2367 2031 y Fk(0)2436 2017 y Fi(\035)30 b Fn(1)36 b Fe(so)g(that)g(for)g Fl(B)f Fi(\025)29 b Fl(B)3366 2031 y Fk(0)3441 2017 y Fe(and)36 b Fl(z)f Fi(2)29 b Fn(\012)3853 2031 y Fj(n)0 2125 y Fe(we)k(have)613 2281 y Fi(k)658 2180 y Ff(h)702 2281 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))998 2180 y Ff(\020)1053 2281 y Fl(B)5 b Fn(\()p Fl(D)1240 2243 y Fk(2)1237 2303 y Fj(y)1299 2281 y Fn(+)20 b Fl(y)1438 2243 y Fk(2)1477 2281 y Fn(\))h(+)f Fl(x)g Fn(+)g Fl(\022)i Fn(+)e Fl(V)2016 2243 y Fj(!)2067 2180 y Ff(\021)2121 2281 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))g Fi(\000)g Fl(z)2575 2180 y Ff(i)2619 2281 y Fl(A)2687 2295 y Fk(2)2726 2281 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)820 2463 y(\025)25 b Fl(a)964 2477 y Fk(2)1003 2463 y Fi(j)p Fn(Im)20 b Fl(\022)s Fi(jk)p Fl(A)1341 2477 y Fk(2)1381 2463 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)i(\000)d(O)s Fn(\()p Fl(B)2070 2425 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2324 2463 y Fn(\))p Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)37 b(u)25 b Fi(2)g(D)s Fl(:)592 b Fn(\(4.4\))150 2627 y(Assuming)29 b(the)h(ab)s(o)m(v)m(e)i(estimates)f(established,)e(w)m(e)i(will)c (complete)k(the)g(pro)s(of)e(of)i(Prop)s(osition)d(3.)150 2843 y Fw(Pro)s(of)36 b(of)f(Prop)s(osition)h(3.)41 b Fn(W)-8 b(e)31 b(ha)m(v)m(e)1280 2994 y Fl(F)13 b Fn(\()p Fl(u)p Fn(\))26 b(=)f Fi(k)p Fn(\()p Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))21 b Fi(\000)f Fl(z)t Fn(\)\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)2564 2957 y Fk(2)742 3178 y Fn(=)838 3077 y Ff(\020)893 3178 y Fn(\()p Fl(A)996 3140 y Fk(2)996 3200 y(1)1056 3178 y Fn(+)g Fl(A)1215 3140 y Fk(2)1215 3200 y(2)1254 3178 y Fn(\)\()p Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))21 b Fi(\000)f Fl(z)t Fn(\)\()p Fl(I)28 b Fi(\000)20 b 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Fn(\)\()p Fl(I)28 b Fi(\000)20 b Fn(\005\)])h(+)f Fi(O)s Fn(\()p Fl(B)2602 4249 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2855 4287 y Fn(\))15 b Fl(;)0 4419 y Fn(since)30 b(b)m(y)g(the)h(pseudo)s(di\013eren)m(tial) c(calculus)i(w)m(e)i(get)1030 4575 y([)p Fl(A)1123 4589 y Fj(i)1152 4575 y Fl(;)15 b Fn(\005])26 b(=)f Fi(O)s Fn(\()p Fl(B)1591 4538 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)1844 4575 y Fn(\))p Fl(;)36 b Fn([)p Fl(A)2033 4589 y Fj(i)2062 4575 y Fl(;)15 b(V)2175 4538 y Fj(!)2226 4575 y Fn(])25 b(=)g Fi(O)s Fn(\()p Fl(B)2556 4538 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2809 4575 y Fn(\))p Fl(:)0 4726 y Fn(Next)363 4877 y([)p Fl(A)456 4891 y Fj(i)485 4877 y Fl(;)15 b Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(B)5 b Fn(\()p Fl(D)1009 4840 y Fk(2)1006 4900 y Fj(y)1069 4877 y Fn(+)20 b Fl(y)1208 4840 y Fk(2)1247 4877 y Fn(\)\()p Fl(I)28 b Fi(\000)20 b Fn(\005\)])26 b(=)f([)p Fl(A)1819 4891 y Fj(i)1847 4877 y Fl(;)15 b(B)5 b Fn(\()p Fl(D)2074 4840 y Fk(2)2071 4900 y Fj(y)2134 4877 y Fn(+)20 b Fl(y)2273 4840 y Fk(2)2312 4877 y Fn(\)]\()p Fl(I)29 b Fi(\000)19 b Fn(\005\))i Fi(\000)f Fl(B)5 b Fn(\()p Fl(D)2968 4840 y Fk(2)2965 4900 y Fj(y)3027 4877 y Fn(+)20 b Fl(y)3166 4840 y Fk(2)3206 4877 y Fn(\)[)p Fl(A)3334 4891 y Fj(i)3363 4877 y Fl(;)15 b Fn(\005])g Fl(:)0 5028 y Fn(Then)29 b(w)m(e)i(ha)m(v)m(e)746 5141 y Fl(B)5 b Fn(\()p Fl(D)933 5104 y Fk(2)930 5164 y Fj(y)992 5141 y Fn(+)20 b Fl(y)1131 5104 y Fk(2)1170 5141 y Fn(\)[)p Fl(A)1298 5155 y Fj(i)1327 5141 y Fl(;)15 b Fn(\005])26 b(=)f Fl(B)1656 5104 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)1854 5141 y Fn(\()p Fl(D)1967 5104 y Fk(2)1964 5164 y Fj(y)2027 5141 y Fn(+)20 b Fl(y)2166 5104 y Fk(2)2205 5141 y Fn(\))p Fl(L)2302 5155 y Fj(i)2331 5141 y Fl(f)2386 5104 y Fh(0)2408 5141 y Fn(\()p Fl(D)2521 5104 y Fk(2)2518 5164 y Fj(y)2581 5141 y Fn(+)g Fl(y)2720 5104 y Fk(2)2759 5141 y Fn(\))p Fl(;)36 b(i)26 b Fn(=)f(1)p Fl(;)15 b Fn(2)p eop %%Page: 12 12 12 11 bop 0 66 a Fk(12)1372 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n (V)0 266 y Fn(with)e(op)s(erators)h Fl(L)655 280 y Fj(i)705 266 y Fn(ha)m(ving)g(sym)m(b)s(ols)e(uniformly)g(b)s(ounded)g(with)g (resp)s(ect)i(to)h Fl(B)5 b Fn(.)38 b(The)21 b(sym)m(b)s(ol)g(of)h(the) h(op)s(erator)0 374 y(on)30 b(the)h(righ)m(t)f(hand)f(side)g(of)i(the)f (ab)s(o)m(v)m(e)i(equalit)m(y)e(is)f(b)s(ounded)g(with)g(its)g(deriv)-5 b(ativ)m(es)30 b(and)g(w)m(e)h(deduce)1315 537 y Fl(B)5 b Fn(\()p Fl(D)1502 500 y Fk(2)1499 560 y Fj(y)1562 537 y Fn(+)20 b Fl(y)1701 500 y Fk(2)1740 537 y Fn(\)[)p Fl(A)1868 551 y Fj(i)1897 537 y Fl(;)15 b Fn(\005])26 b(=)f Fi(O)s Fn(\()p Fl(B)2336 500 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2534 537 y Fn(\))0 707 y(in)h(the)i(space)g(of)g(b)s (ounded)d(op)s(erators)j Fi(L)p Fn(\()p Fl(L)1526 674 y Fk(2)1565 707 y Fn(\()p Fg(R)1660 674 y Fk(2)1706 707 y Fn(\)\))p Fl(:)g Fn(T)-8 b(o)28 b(treat)h(the)e(comm)m(utator)i(with) e Fl(B)5 b Fn(\()p Fl(D)3235 674 y Fk(2)3232 730 y Fj(y)3288 707 y Fn(+)15 b Fl(y)3422 674 y Fk(2)3461 707 y Fn(\),)28 b(w)m(e)g(apply)0 815 y(Lemma)i(4.)42 b(It)30 b(is)g(clear)g(that)133 978 y([)p Fl(A)226 992 y Fj(i)255 978 y Fl(;)15 b(B)5 b Fn(\()p Fl(D)482 941 y Fk(2)479 1001 y Fj(y)542 978 y Fn(+)20 b Fl(y)681 941 y Fk(2)720 978 y Fn(\)]\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))26 b(=)f Fi(O)s Fn(\(1\))p Fl(B)1463 941 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)1662 978 y Fl(y)e Fn(Op)1851 941 y Fj(!)1902 877 y Ff(\020)1956 978 y Fl(G)2027 941 y Fh(0)2027 1001 y Fj(i)2056 978 y Fn(\()p Fl(B)2165 941 y Fh(\000)p Fj(\016)2257 978 y Fn(\()p Fl(x)e Fi(\000)f Fl(B)2530 941 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2694 978 y Fl(D)2769 992 y Fj(y)2811 978 y Fn(\))2846 877 y Ff(\021)2900 978 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))h(+)f Fi(O)s Fn(\()p Fl(B)3493 941 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)3691 978 y Fn(\))15 b Fl(;)0 1164 y Fn(so)31 b(for)f Fl(B)35 b Fn(large)30 b(enough,)g(according)h(to)g(\(4.2\),)h(w)m(e)f(obtain)1250 1326 y Fi(k)p Fn(\()p Fl(I)d Fi(\000)20 b Fn(\005\)[)p Fl(A)1685 1340 y Fj(i)1714 1326 y Fl(;)15 b(B)5 b Fn(\()p Fl(D)1941 1288 y Fk(2)1938 1348 y Fj(y)2001 1326 y Fn(+)20 b Fl(y)2140 1288 y Fk(2)2179 1326 y Fn(\)]\()p Fl(I)29 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)431 1520 y(\024)25 b Fl(C)592 1534 y Fk(5)632 1520 y Fl(B)706 1482 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)904 1520 y Fi(k)p Fn(\()p Fl(Q)p Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)t Fn(\)\()p Fl(I)29 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h Fn(+)f Fl(C)2050 1534 y Fk(6)2089 1520 y Fl(B)2163 1482 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2361 1520 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)37 b(z)29 b Fi(2)c Fn(\012)3085 1534 y Fj(n)3132 1520 y Fl(;)36 b(u)25 b Fi(2)g(D)17 b Fl(;)0 1659 y Fn(with)27 b(constan)m(ts)i Fl(C)670 1673 y Fk(5)735 1659 y Fl(>)c Fn(0)p Fl(;)32 b(C)998 1673 y Fk(6)1062 1659 y Fl(>)25 b Fn(0)k(indep)s(enden)m(t)d(on)i Fl(z)k Fn(and)c Fl(B)5 b Fn(.)40 b(Finally)-8 b(,)27 b(taking)h(in)m(to)g(accoun)m(t)i(the)e (estimates)0 1767 y(\(4.3\),)k(\(4.4\),)h(w)m(e)e(deduce)516 1953 y(2\(1)22 b Fi(\000)e Fl(C)826 1915 y Fh(0)819 1975 y Fk(5)858 1953 y Fl(B)932 1915 y Fk(1)p Fh(\000)p Fk(2)p Fj(\016)1095 1953 y Fn(\))p Fl(F)13 b Fn(\()p Fl(u)p Fn(\))26 b Fi(\025)1445 1852 y Ff(h)1488 1953 y Fl(a)1536 1915 y Fk(2)1536 1975 y(1)1576 1953 y Fl(B)1650 1915 y Fk(2)1689 1953 y Fi(k)p Fl(A)1802 1967 y Fk(1)1842 1953 y Fn(\()p Fl(I)i Fi(\000)19 b Fn(\005\))p Fl(u)p Fi(k)2235 1915 y Fk(2)2296 1953 y Fn(+)h Fl(a)2435 1915 y Fk(2)2435 1975 y(2)2474 1953 y Fi(j)p Fn(Im)h Fl(\022)s Fi(j)2700 1915 y Fk(2)2738 1953 y Fi(k)p Fl(A)2851 1967 y Fk(2)2891 1953 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)3285 1915 y Fk(2)3326 1852 y Ff(i)1497 2167 y Fi(\000)p Fl(C)1633 2181 y Fk(7)1672 2167 y Fl(B)1746 2129 y Fk(1)p Fh(\000)p Fk(2)p Fj(\016)1909 2167 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)2348 2129 y Fk(2)921 2363 y Fi(\025)1017 2262 y Ff(h\020)1114 2363 y Fn(min)o Fi(f)p Fl(a)1359 2377 y Fk(1)1399 2363 y Fl(B)5 b(;)15 b(a)1561 2377 y Fk(2)1600 2363 y Fi(j)p Fn(Im)20 b Fl(\022)s Fi(jg)1870 2262 y Ff(\021)1924 2285 y Fk(2)1984 2363 y Fi(\000)g Fl(C)2140 2377 y Fk(7)2179 2363 y Fl(B)2253 2325 y Fk(1)p Fh(\000)p Fk(2)p Fj(\016)2416 2262 y Ff(i)2459 2363 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)2898 2325 y Fk(2)2954 2363 y Fl(:)0 2530 y Fn(F)-8 b(or)31 b Fl(B)k Fn(su\016cien)m(tly)29 b(large)h(this)g(implies)d(the)k(estimate)g(\(4.1\).)1668 b Fd(\003)150 2746 y Fw(Pro)s(of)36 b(of)f(Lemma)e(5.)40 b Fn(First)30 b(notice)h(that)g(the)f(op)s(erator)h Fl(D)2338 2713 y Fk(2)2335 2768 y Fj(y)2397 2746 y Fn(+)20 b Fl(y)2536 2713 y Fk(2)2596 2746 y Fn(+)g(1)30 b(is)g(elliptic,)e(so)358 2920 y Fi(k)p Fl(D)478 2934 y Fj(y)520 2920 y Fl(u)p Fi(k)e(\024)e(k)p Fn(\()p Fl(D)896 2882 y Fk(2)893 2942 y Fj(y)957 2920 y Fn(+)c Fl(y)1096 2882 y Fk(2)1155 2920 y Fn(+)g(1\))p Fl(u)p Fi(k)i Fn(+)d Fl(C)1600 2934 y Fk(8)1640 2920 y Fi(k)p Fl(u)p Fi(k)26 b(\024)f(k)p Fn(\()p Fl(I)j Fi(\000)20 b Fn(\005\)\()p Fl(D)2359 2882 y Fk(2)2356 2942 y Fj(y)2419 2920 y Fn(+)g Fl(y)2558 2882 y Fk(2)2597 2920 y Fn(\))p Fl(u)p Fi(k)i Fn(+)d(\()p Fl(C)2941 2934 y Fk(8)3001 2920 y Fn(+)h(2)p Fl(n)h Fn(+)e(2\))p Fi(k)p Fl(u)p Fi(k)1055 3131 y(\024)25 b Fl(B)1225 3094 y Fh(\000)p Fk(1)1318 3131 y Fi(k)p Fn(\()p Fl(I)k Fi(\000)19 b Fn(\005\))1660 3030 y Ff(h)1704 3131 y Fl(B)5 b Fn(\()p Fl(D)1891 3094 y Fk(2)1888 3154 y Fj(y)1950 3131 y Fn(+)20 b Fl(y)2089 3094 y Fk(2)2128 3131 y Fn(\))h Fi(\000)f Fl(z)2321 3030 y Ff(i)2364 3131 y Fl(u)p Fi(k)h Fn(+)f Fl(C)2645 3094 y Fh(0)2638 3154 y Fk(8)2677 3131 y Fi(k)p Fl(u)p Fi(k)p Fl(;)869 b Fn(\(4.5\))0 3306 y(where)30 b(w)m(e)i(ha)m(v)m(e)g(used)e (that)h Fl(B)1087 3273 y Fh(\000)p Fk(1)1181 3306 y Fl(z)k Fn(is)30 b(b)s(ounded)f(for)h Fl(z)h Fi(2)25 b Fn(\012)2086 3320 y Fj(n)2133 3306 y Fl(:)31 b Fn(Second,)g(applying)e(the)i (estimate)g(\(4.5\))i(for)e(the)0 3414 y(term)1383 3534 y(\()p Fl(I)d Fi(\000)19 b Fn(\005\))p Fl(B)1753 3497 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1918 3534 y Fl(D)1993 3548 y Fj(y)2035 3534 y Fl(A)2103 3548 y Fk(1)2143 3534 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u;)0 3673 y Fn(w)m(e)31 b(obtain)747 3814 y Fi(k)p Fn(\()p Fl(I)d Fi(\000)20 b Fn(\005\))1089 3713 y Ff(h)1132 3814 y Fl(B)5 b Fn(\()p Fl(D)1319 3777 y Fk(2)1316 3837 y Fj(y)1378 3814 y Fn(+)20 b Fl(y)1517 3777 y Fk(2)1557 3814 y Fn(\))g(+)g Fl(x)g Fn(+)g Fl(\022)j Fi(\000)2089 3753 y Fn(1)p 2032 3793 159 4 v 2032 3877 a(4)p Fl(B)2151 3850 y Fk(2)2221 3814 y Fn(+)d Fl(V)2385 3777 y Fj(!)2456 3814 y Fi(\000)g Fl(z)2593 3713 y Ff(i)2636 3814 y Fl(A)2704 3828 y Fk(1)2744 3814 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)191 4074 y Fn(=)25 b Fi(k)p Fn(\()p Fl(I)j Fi(\000)20 b Fn(\005\))629 3973 y Ff(h)672 4074 y Fl(B)5 b Fn(\()p Fl(D)859 4036 y Fk(2)856 4096 y Fj(y)918 4074 y Fn(+)20 b Fl(y)1057 4036 y Fk(2)1097 4074 y Fn(\))g(+)g Fl(B)1317 4036 y Fj(\016)1355 3973 y Ff(\020)1419 4012 y Fl(x)g Fi(\000)g Fl(B)1656 3979 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1820 4012 y Fl(D)1895 4026 y Fj(y)p 1419 4053 519 4 v 1622 4136 a Fl(B)1696 4110 y Fj(\016)1947 3973 y Ff(\021)2021 4074 y Fn(+)g Fl(B)2186 4036 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2351 4074 y Fl(D)2426 4088 y Fj(y)2488 4074 y Fi(\000)2645 4012 y Fn(1)p 2588 4053 159 4 v 2588 4136 a(4)p Fl(B)2707 4110 y Fk(2)2777 4074 y Fn(+)g Fl(V)2941 4036 y Fj(!)3012 4074 y Fi(\000)g Fl(z)3149 3973 y Ff(i)3192 4074 y Fl(A)3260 4088 y Fk(1)3300 4074 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)888 4293 y(\025)25 b Fn(\(1)c Fi(\000)f Fl(B)1250 4255 y Fh(\000)p Fk(3)p Fj(=)p Fk(2)1414 4293 y Fn(\))p Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))1791 4192 y Ff(h)1835 4293 y Fl(B)5 b Fn(\()p Fl(D)2022 4255 y Fk(2)2019 4316 y Fj(y)2081 4293 y Fn(+)20 b Fl(y)2220 4255 y Fk(2)2259 4293 y Fn(\))h Fi(\000)f Fl(z)2452 4192 y Ff(i)2495 4293 y Fl(A)2563 4307 y Fk(1)2603 4293 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)855 4549 y(\000)p Fl(B)1000 4512 y Fj(\016)1037 4445 y Ff(\015)1037 4499 y(\015)1037 4554 y(\015)1088 4448 y(\020)1152 4487 y Fl(x)g Fi(\000)g Fl(B)1389 4454 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1553 4487 y Fl(D)1628 4501 y Fj(y)p 1152 4528 519 4 v 1355 4612 a Fl(B)1429 4585 y Fj(\016)1680 4448 y Ff(\021)1734 4549 y Fl(A)1802 4563 y Fk(1)1842 4549 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)2190 4445 y Ff(\015)2190 4499 y(\015)2190 4554 y(\015)2262 4549 y Fi(\000)g Fl(C)2418 4563 y Fk(9)2457 4549 y Fi(k)p Fl(A)2570 4563 y Fk(1)2610 4549 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)15 b Fl(:)0 4727 y Fn(Our)29 b(assumptions)g(on)h(\012)896 4741 y Fj(n)973 4727 y Fn(and)g(the)g(sp)s(ectral)g(theorem)h(for)f Fl(D)2219 4694 y Fk(2)2216 4750 y Fj(y)2278 4727 y Fn(+)20 b Fl(y)2417 4694 y Fk(2)2487 4727 y Fn(imply)631 4926 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))973 4825 y Ff(h)1017 4926 y Fl(B)5 b Fn(\()p Fl(D)1204 4888 y Fk(2)1201 4948 y Fj(y)1263 4926 y Fn(+)20 b Fl(y)1402 4888 y Fk(2)1441 4926 y Fn(\))h Fi(\000)e Fl(z)1633 4825 y Ff(i)1677 4926 y Fl(A)1745 4940 y Fk(1)1784 4926 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)26 b(\025)f Fl(a)2348 4940 y Fk(1)2388 4926 y Fl(B)5 b Fi(k)p Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(A)2871 4940 y Fk(1)2911 4926 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fi(k)886 5143 y(\025)25 b Fl(a)1030 5157 y Fk(1)1069 5143 y Fl(B)5 b Fi(k)p Fl(A)1256 5157 y Fk(1)1296 5143 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)h(\000)f(O)s Fn(\()p Fl(B)1985 5106 y Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2184 5143 y Fn(\))p Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)p Fl(;)37 b(a)2768 5157 y Fk(1)2833 5143 y Fl(>)24 b Fn(0)15 b Fl(:)p eop %%Page: 13 13 13 12 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(13)0 266 y Fn(Next)33 b(c)m(ho)s(ose)f(a)h (function)969 243 y(~)948 266 y Fl(G)1019 280 y Fk(1)1086 266 y Fi(2)28 b Fl(C)1247 233 y Fh(1)1240 291 y Fk(0)1321 266 y Fn(\()p Fg(R)s Fn(;)15 b([0)q Fl(;)g Fn(1)q(]\))38 b(with)30 b(supp)s(ort)g(close)j(to)f(that)g(of)g Fl(G)2963 280 y Fk(1)3035 266 y Fn(and)f(suc)m(h)g(that)3638 243 y(~)3618 266 y Fl(G)3689 280 y Fk(1)3756 266 y Fn(=)d(1)0 374 y(on)i(supp)f Fl(G)415 388 y Fk(1)455 374 y Fl(:)h Fn(Then)1335 567 y Fl(B)1409 529 y Fj(\016)1446 466 y Ff(\020)1511 505 y Fl(x)20 b Fi(\000)g Fl(B)1748 472 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1912 505 y Fl(D)1987 519 y Fj(y)p 1511 546 519 4 v 1714 629 a Fl(B)1788 603 y Fj(\016)2039 466 y Ff(\021)2093 567 y Fl(A)2161 581 y Fk(1)2201 567 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)824 844 y Fn(=)25 b Fl(B)994 806 y Fj(\016)1032 743 y Ff(\020)1096 782 y Fl(x)20 b Fi(\000)g Fl(B)1333 749 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1497 782 y Fl(D)1572 796 y Fj(y)p 1096 823 V 1299 906 a Fl(B)1373 880 y Fj(\016)1624 743 y Ff(\021)1678 844 y Fl(O)s(p)1796 806 y Fj(!)1867 821 y Fn(~)1846 844 y Fl(G)1917 858 y Fk(1)1957 743 y Ff(\020)2021 782 y Fl(x)h Fi(\000)f Fl(B)2259 749 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2423 782 y Fl(D)2498 796 y Fj(y)p 2021 823 V 2225 906 a Fl(B)2299 880 y Fj(\016)2550 743 y Ff(\021)2604 844 y Fl(A)2672 858 y Fk(1)2711 844 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)0 1039 y Fn(and,)33 b(since)e Fl(u)502 1016 y Fn(~)481 1039 y Fl(G)552 1053 y Fk(1)592 1039 y Fn(\()p Fl(u)p Fn(\))i(is)f(b)s (ounded,)f(w)m(e)i(can)g(estimate)g(this)e(term)h(b)m(y)h Fl(C)2503 1053 y Fk(10)2577 1039 y Fl(B)2651 1006 y Fj(\016)2689 1039 y Fi(k)p Fl(A)2802 1053 y Fk(1)2842 1039 y Fn(\()p Fl(I)c Fi(\000)21 b Fn(\005\))p Fl(u)p Fi(k)p Fl(:)34 b Fn(T)-8 b(aking)32 b Fl(B)37 b Fn(large)0 1146 y(enough,)30 b(w)m(e)h(complete)g(the)g(pro)s(of.)2583 b Fd(\003)150 1362 y Fw(Pro)s(of)36 b(of)f(Lemma)e(6.)40 b Fn(W)-8 b(e)32 b(ha)m(v)m(e)256 1445 y Ff(\014)256 1500 y(\014)256 1555 y(\014)286 1449 y(\020)q(h)384 1550 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))680 1449 y Ff(\020)735 1550 y Fl(B)5 b Fn(\()p Fl(D)922 1512 y Fk(2)919 1572 y Fj(y)981 1550 y Fn(+)20 b Fl(y)1120 1512 y Fk(2)1160 1550 y Fn(\))g(+)g Fl(x)g Fn(+)g Fl(\022)i Fi(\000)1692 1488 y Fn(1)p 1635 1529 159 4 v 1635 1612 a(4)p Fl(B)1754 1586 y Fk(2)1824 1550 y Fn(+)e Fl(V)1988 1512 y Fj(!)2038 1449 y Ff(\021)2093 1550 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))h Fi(\000)f Fl(z)2547 1449 y Ff(i)2590 1550 y Fl(A)2658 1564 y Fk(2)2698 1550 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u;)15 b(A)3154 1564 y Fk(2)3195 1550 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)3544 1449 y Ff(\021)3598 1445 y(\014)3598 1500 y(\014)3598 1555 y(\014)139 1821 y Fi(\025)25 b(\000)p Fn(Im)434 1721 y Ff(\020h)531 1821 y Fn(\()p Fl(I)j Fi(\000)20 b Fn(\005\))828 1721 y Ff(\020)883 1821 y Fl(B)5 b Fn(\()p Fl(D)1070 1784 y Fk(2)1067 1844 y Fj(y)1129 1821 y Fn(+)20 b Fl(y)1268 1784 y Fk(2)1307 1821 y Fn(\))h(+)f Fl(x)g Fi(\000)1683 1760 y Fn(1)p 1627 1801 V 1627 1884 a(4)p Fl(B)1746 1858 y Fk(2)1815 1821 y Fn(+)g Fl(\022)j Fn(+)d Fl(V)2136 1784 y Fj(!)2186 1721 y Ff(\021)2240 1821 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))h Fi(\000)f Fl(z)2695 1721 y Ff(i)2738 1821 y Fl(A)2806 1835 y Fk(2)2846 1821 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u;)15 b(A)3302 1835 y Fk(2)3343 1821 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u)3691 1721 y Ff(\021)335 2058 y Fn(=)25 b(\(Im)c Fl(z)j Fi(\000)c Fn(Im)g Fl(\022)s Fn(\))p Fi(k)p Fl(A)1076 2072 y Fk(2)1115 2058 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)1509 2021 y Fk(2)1570 2058 y Fi(\000)f Fn(Im)1769 1958 y Ff(\020)1823 2058 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(V)2194 2021 y Fj(!)2244 2058 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(A)2609 2072 y Fk(2)2649 2058 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u;)15 b(A)3105 2072 y Fk(2)3146 2058 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)3495 1958 y Ff(\021)1004 2292 y Fi(\025)1100 2191 y Ff(\020)1155 2292 y Fn(\(1)h Fi(\000)f Fl(\026)p Fn(\))p Fi(j)p Fn(Im)g Fl(\022)s Fi(j)f(\000)h Fl(C)1837 2306 y Fk(11)1912 2292 y Fl(B)1986 2255 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2239 2191 y Ff(\021)2293 2292 y Fi(k)p Fl(A)2406 2306 y Fk(2)2446 2292 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)2840 2255 y Fk(2)570 2526 y Fi(\000)p Fn(Im)749 2425 y Ff(\020)803 2526 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(V)1174 2488 y Fj(!)1224 2526 y Fl(A)1292 2540 y Fk(2)1332 2526 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))p Fl(u;)15 b(A)1788 2540 y Fk(2)1829 2526 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)2178 2425 y Ff(\021)2253 2526 y Fi(\000)g Fl(C)2409 2540 y Fk(11)2483 2526 y Fl(B)2557 2488 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)p Fh(\000)p Fj(\016)2810 2526 y Fi(k)p Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(u)p Fi(k)3249 2488 y Fk(2)3305 2526 y Fl(:)0 2717 y Fn(No)m(w)26 b(w)m(e)f(c)m(ho)s(ose)h(a)f (function)1054 2694 y(~)1033 2717 y Fl(G)1104 2731 y Fk(2)1170 2717 y Fi(2)f Fl(C)1327 2684 y Fh(1)1320 2745 y Fj(b)1402 2717 y Fn(\()p Fg(R)s Fn(;)15 b([0)q 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y Fh(\000)p Fk(1)1449 2281 y Fi(k)p Fn(\()p Fl(B)5 b Fn(\()p Fl(D)1716 2244 y Fk(2)1713 2304 y Fj(y)1776 2281 y Fn(+)20 b Fl(y)1915 2244 y Fk(2)1955 2281 y Fn(\))g Fi(\000)g Fl(z)t Fn(\))p Fl(u)p Fn(\))p Fi(k)i Fn(+)e Fl(C)2492 2295 y Fk(9)2531 2281 y Fi(k)p Fl(u)p Fi(k)15 b Fl(:)0 2417 y Fn(Next)31 b(w)m(e)g(estimate)g(the)g (term)1680 2534 y Fl(B)1754 2496 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)1918 2534 y Fl(D)1993 2548 y Fj(y)2035 2534 y Fl(A)2103 2548 y Fk(1)2143 2534 y Fl(u;)0 2669 y Fn(exploiting)g(the)h(fact)i (that)f Fl(D)1038 2683 y Fj(y)1112 2669 y Fn(and)f Fl(A)1359 2683 y Fk(1)1431 2669 y Fn(comm)m(utes)h(and)f(apply)f(the)i(ab)s(o)m (v)m(e)g(estimate.)48 b(The)32 b(second)g(p)s(oin)m(t)g(is)0 2777 y(related)e(to)h(Lemma)g(4.)41 b(F)-8 b(or)31 b Fi(<)p Fl(z)e(<)c Fn(0)31 b(consider)e(the)i(sym)m(b)s(ol)1147 3049 y Fl(g)1190 3063 y Fj(i)1218 3049 y Fn(\()p Fl(x;)15 b(y)s(;)g(\021)s Fn(\))27 b(=)1705 2955 y Fl(y)s(G)1824 2922 y Fh(0)1824 2980 y Fj(i)1852 2854 y Ff(\020)1906 2955 y Fl(B)1980 2922 y Fh(\000)p Fj(\016)2073 2955 y Fn(\()p Fl(x)20 b Fi(\000)g Fl(B)2345 2922 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2509 2955 y Fl(\021)s Fn(\))2592 2854 y Ff(\021)p 1649 3028 1054 4 v 1649 3112 a Fl(\021)1697 3085 y Fk(2)1757 3112 y Fn(+)g Fl(y)1896 3085 y Fk(2)1956 3112 y Fn(+)g Fl(B)2121 3085 y Fh(\000)p Fk(1)2215 3112 y Fn(\()p Fl(x)g Fi(\000)g(<)p Fl(z)t Fn(\))g(+)g Fl(i)2728 3049 y(:)0 3265 y Fn(It)38 b(is)f(easy)i(to)f(sho)m(w)g(that)h(this)e (sym)m(b)s(ol)f(is)h(in)g Fl(S)1746 3232 y Fk(0)1785 3265 y Fn(\()p Fg(R)1880 3232 y Fk(4)1926 3265 y Fn(\))p Fl(:)h Fn(Then)f(the)h(pro)s(of)g(of)g(Lemma)g(4)g(go)s(es)h(without)e (an)m(y)0 3373 y(c)m(hange)32 b(and)d(w)m(e)i(obtain)f(\(4.2\).)42 b(Finally)-8 b(,)29 b(w)m(e)i(get)h(the)e(estimate)1180 3534 y Fi(k)1245 3511 y Fn(~)1225 3534 y Fl(P)1283 3548 y Fk(1)1323 3534 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)t Fn(\))p Fl(u)p Fi(k)26 b(\025)f Fl(\015)5 b Fi(j)p Fn(Im)20 b Fl(\022)s Fi(jk)p Fl(u)p Fi(k)p Fl(;)36 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a(4)p Fl(B)2524 4505 y Fk(2)2593 4469 y Fn(+)d Fl(V)2757 4431 y Fj(!)2808 4368 y Ff(\021)2862 4469 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))15 b Fl(;)1047 4715 y(L)1109 4729 y Fk(2)1148 4715 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))25 b(=)g(\005)1567 4614 y Ff(\020)1622 4715 y Fl(B)5 b Fn(\()p Fl(D)1809 4678 y Fk(2)1806 4738 y Fj(y)1868 4715 y Fn(+)20 b Fl(y)2007 4678 y Fk(2)2046 4715 y Fn(\))h(+)f Fl(x)g Fn(+)g Fl(\022)i Fi(\000)2578 4654 y Fn(1)p 2522 4694 V 2522 4777 a(4)p Fl(B)2641 4751 y Fk(2)2690 4614 y Ff(\021)2745 4715 y Fn(\005)15 b Fl(;)1065 4891 y(W)1164 4853 y Fj(!)1239 4891 y Fn(=)25 b(\()p Fl(I)i Fi(\000)20 b Fn(\005\))p Fl(V)1705 4853 y Fj(!)1756 4891 y Fn(\005)g(+)g(\005)p Fl(V)2076 4853 y Fj(!)2127 4891 y Fn(\()p Fl(I)27 b Fi(\000)20 b Fn(\005\))h(+)f(\005)p Fl(V)2676 4853 y Fj(!)2727 4891 y Fn(\005)15 b Fl(:)0 5026 y Fn(It)30 b(clear)h(that)1031 5143 y Fl(L)1093 5157 y Fk(1)1133 5143 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b(+)g Fl(L)1536 5157 y Fk(2)1575 5143 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)k Fn(+)c Fl(W)2172 5106 y Fj(!)2247 5143 y Fn(=)2364 5120 y(~)2343 5143 y Fl(P)2401 5157 y Fk(1)2441 5143 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)f(:)p eop %%Page: 15 15 15 14 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(15)0 266 y Fn(The)30 b(op)s(erator)564 243 y(~)554 266 y Fl(L)o Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)30 b Fn(=)25 b Fl(L)1186 280 y Fk(1)1225 266 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b(+)g Fl(L)1628 280 y Fk(2)1667 266 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)35 b Fn(is)29 b(in)m(v)m(ertible)g(for)h Fl(z)g Fi(2)25 b Fn(\012)2941 280 y Fj(n)2987 266 y Fl(:)31 b Fn(In)e(fact,)j(w)m(e)f(ha)m(v)m(e)538 424 y Fi(k)p Fn(\()629 401 y(~)618 424 y Fl(L)q Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)t Fn(\))p Fl(u)p Fi(k)1200 386 y Fk(2)1266 424 y Fn(=)25 b Fi(k)p Fn(\()p Fl(L)1504 438 y Fk(1)1544 424 y Fn(\()p Fl(B)5 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Fj(j)1330 3732 y Fn(\()p Fl(B)5 b Fn(\))21 b Fi(\000)f Fl(i)p Fn(\))1652 3695 y Fh(\000)p Fk(1)1747 3732 y Fn(\()p Fl(P)1840 3746 y Fj(j)1877 3732 y Fn(\()p Fl(B)5 b Fn(\))20 b(+)g Fl(i)p Fn(\))2198 3695 y Fh(\000)p Fk(1)2293 3732 y Fn(\()p Fl(z)25 b Fi(\000)20 b Fl(P)2544 3746 y Fj(j)2581 3732 y Fn(\()p Fl(B)5 b Fn(\)\))2760 3695 y Fh(\000)p Fk(1)2855 3631 y Ff(i)2898 3654 y Fk(1)2898 3796 y(0)2937 3732 y Fl(;)56 b Fi(\006)20 b Fn(Im)g Fl(z)29 b(>)c Fn(0)p Fl(;)0 3922 y Fn(where)30 b([)p Fl(a)336 3936 y Fj(j)373 3922 y Fn(])398 3889 y Fk(1)398 3947 y(0)463 3922 y Fn(=)25 b Fl(a)607 3936 y Fk(1)666 3922 y Fi(\000)20 b Fl(a)805 3936 y Fk(0)845 3922 y Fl(:)30 b Fn(It)h(follo)m(ws)e(from)h(Prop)s (osition)f(1)h(that)h Fl(\033)2326 3936 y Fh(\006)2385 3922 y Fn(\()p Fl(z)t Fn(\))h(are)e(w)m(ell)g(de\014ned)f(and)h(w)m(e)g (ha)m(v)m(e)1440 4086 y Fl(\033)1492 4100 y Fh(\000)1551 4086 y Fn(\()p Fl(z)t Fn(\))c(=)p 1789 4007 228 4 v 25 w Fl(\033)1841 4100 y Fk(+)1900 4086 y Fn(\()5 b(\026)-50 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y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))2799 4360 y Fh(\000)p Fk(1)2893 4398 y Fl(:)0 4549 y Fn(Consequen)m(tly)-8 b(,)30 b(the)h(cyclicit)m(y)f(of) g(the)h(trace)h(yields)565 4738 y Fl(\033)617 4752 y Fk(+)676 4738 y Fn(\()p Fl(z)t Fn(\))26 b(=)f(\()p Fl(z)995 4701 y Fk(2)1055 4738 y Fn(+)20 b(1\)tr)1298 4637 y Ff(h)1341 4738 y Fn(\()1397 4715 y(~)1376 4738 y Fl(P)1434 4752 y Fj(j)1471 4738 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(i)p Fn(\))1878 4701 y Fh(\000)p Fk(1)1973 4738 y Fn(\()2029 4715 y(~)2008 4738 y Fl(P)2066 4752 y Fj(j)2104 4738 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b(+)f Fl(i)p Fn(\))2510 4701 y Fh(\000)p Fk(1)2605 4738 y Fn(\()p Fl(z)25 b Fi(\000)2819 4715 y Fn(~)2798 4738 y Fl(P)2856 4752 y Fj(j)2893 4738 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\))3158 4701 y Fh(\000)p Fk(1)3253 4637 y Ff(i)3296 4660 y Fk(1)3296 4801 y(0)3713 4738 y Fn(\(5.3\))0 4921 y(for)30 b(all)f Fl(z)h Fi(2)25 b Fn(\012)489 4935 y Fk(+)573 4921 y Fn(=)g(\012)735 4935 y Fj(n)802 4921 y Fi(\\)19 b(f)p Fn(Im)i Fl(z)29 b(>)c Fn(0)p Fi(g)p Fn(,)32 b Fl(\022)27 b Fi(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)1768 4935 y Fk(0)1808 4921 y Fn(\))21 b Fi(\\)e Fg(R)s Fn(.)150 5035 y(No)m(w,)33 b(\014x)e Fl(\016)g(>)c Fn(0)32 b(and)f(let)g Fl(z)h Fi(2)27 b Fn(\012)1298 5050 y Fj(\016)1363 5035 y Fn(=)g(\012)1527 5049 y Fj(n)1594 5035 y Fi(\\)21 b(f)p Fn(Im)g Fl(z)32 b Fi(\025)27 b Fl(\016)s Fi(g)p Fn(.)45 b(Since)2440 5012 y(~)2420 5035 y Fl(P)2478 5049 y Fj(j)2515 5035 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))31 b(extends)g(to)i(an)e(analytic)g(t)m(yp)s(e-)0 5143 y(A)e(family)e(of)i(op)s(erators)g(on)g Fl(D)s Fn(\(0)p Fl(;)15 b(\022)1242 5157 y Fk(0)1282 5143 y Fn(\),)30 b(for)e(su\016cien)m(tly)g(small)f Fl(\022)2242 5157 y Fk(0)2310 5143 y Fn(and)h Fl(z)i Fi(2)25 b Fn(\012)2709 5158 y Fj(\016)2746 5143 y Fn(,)k(the)g(r.h.s)g(of)g(\(5.3\))h(extends) f(b)m(y)p eop %%Page: 16 16 16 15 bop 0 66 a Fk(16)1372 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n (V)0 266 y 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Ff(X)2533 2000 y Fj(!)r Fh(2)p Fj(\033)2666 2008 y Fr(pp)2735 2000 y Fk(\()p Fj(P)2807 2009 y Fs(1)2842 2000 y Fk(\()p Fj(B)s Fk(\)\))p Fh(\\)p Fj(J)3088 1885 y Fl(\016)s Fn(\()p Fl(\025)h Fi(\000)f Fl(!)s Fn(\))p Fl(;)262 b Fn(\(5.7\))0 2178 y Fe(wher)-5 b(e)34 b Fl(r)s Fn(\()p Fl(z)t(;)15 b(B)5 b Fn(\))33 b Fe(is)f(a)h(function)g (holomorphic)j(in)c Fn(\012)h Fe(and)1407 2328 y Fi(j)p Fl(r)s Fn(\()p Fl(z)t(;)15 b(B)5 b Fn(\))p Fi(j)26 b(\024)f Fl(C)7 b Fn(\()p Fl(W)13 b Fn(\))p Fl(B)5 b(;)34 b(z)c Fi(2)25 b Fl(W)m(:)1220 b Fn(\(5.8\))150 2493 y(Applying)26 b(the)i(ab)s(o)m(v)m(e)h(represen)m(tation)g(of)f(the)g Fl(\030)1843 2460 y Fh(0)1866 2493 y Fn(\()p Fl(\025;)15 b(B)5 b Fn(\),)30 b(w)m(e)e(obtain)g(immediately)f(a)h(lo)s(cal)f (trace)j(form)m(ula)0 2601 y(completely)g(similar)e(to)j(those)g(in)e ([28],)j([22],)g([10].)0 2766 y Fw(Theorem)i(3.)42 b Fe(Assume)32 b(that)i Fl(V)53 b Fe(satis\014es)33 b(the)g(assumption)i Fn(\(3)p Fl(:)p Fn(1\))p Fe(.)43 b(L)-5 b(et)1165 2916 y Fn(\012)25 b Fi(\032)g(f)p Fl(z)30 b Fi(2)24 b Fg(C)49 b Fn(:)d(Im)20 b Fl(z)29 b Fi(\025)c Fl(\026)p Fn(Im)20 b Fl(\022)s Fi(g)p Fl(;)35 b Fn(0)26 b Fl(<)f(\026)g(<)g Fn(1)0 3066 y Fe(b)-5 b(e)34 b(an)g(op)-5 b(en,)36 b(simply)f(c)-5 b(onne)g(cte)g(d,)36 b(r)-5 b(elatively)35 b(c)-5 b(omp)g(act)36 b(such)f(that)g Fl(I)g Fn(=)27 b(\012)21 b Fi(\\)g Fg(R)43 b Fe(is)34 b(an)g(interval.)47 b(Supp)-5 b(ose)36 b(that)0 3174 y Fl(f)42 b Fe(is)32 b(holomorphic)k(on)d(a)g(neighb)-5 b(orho)g(o)g(d)37 b(of)32 b Fn(\012)h Fe(and)g(that)h Fl( )29 b Fi(2)c Fl(C)2264 3141 y Fh(1)2257 3199 y Fk(0)2338 3174 y Fn(\()p Fg(R)s Fn(\))39 b Fe(satis\014es)1382 3381 y Fl( )s Fn(\()p Fl(\025)p Fn(\))26 b(=)1689 3253 y Ff(\032)1799 3326 y Fn(0)p Fl(;)84 b(d)p Fn(\()p Fl(I)7 b(;)15 b(\025)p Fn(\))26 b Fl(>)f Fn(2)p Fl(\021)s(;)1799 3434 y Fn(1)p Fl(;)84 b(d)p Fn(\()p Fl(I)7 b(;)15 b(\025)p Fn(\))26 b Fl(<)f(\021)s(;)0 3581 y Fe(wher)-5 b(e)34 b Fl(\021)28 b(>)d Fn(0)33 b Fe(is)g(su\016ciently)g(smal)5 b(l.)43 b(Then)32 b(for)i Fl(B)j Fe(lar)-5 b(ge)33 b(enough)g(we)g (have)954 3771 y Fn(tr)1025 3670 y Ff(h)1068 3771 y Fn(\()p Fl( )s(f)10 b Fn(\)\()p Fl(P)1348 3785 y Fj(j)1386 3771 y Fn(\()p Fl(B)5 b Fn(\)\))1565 3670 y Ff(i)1608 3693 y Fk(1)1608 3834 y Fj(j)t Fk(=0)1760 3771 y Fn(=)2063 3685 y Ff(X)1856 3886 y Fj(z)s Fh(2)15 b Fk(Res)h Fj(P)2126 3895 y Fs(1)2160 3886 y Fk(\()p Fj(B)s Fk(\))i Fh(\\)d Fk(\012)2416 3771 y Fl(f)10 b Fn(\()p Fl(z)t Fn(\))21 b(+)f Fl(E)2766 3786 y Fk(\012)p Fj(;f)s(; )3713 3771 y Fn(\(5.9\))0 4025 y Fe(with)673 4138 y Fi(j)p Fl(E)765 4153 y Fk(\012)p Fj(;f)s(; )946 4138 y Fi(j)25 b(\024)g Fl(M)10 b Fn(\()p Fl( )s(;)15 b Fn(\012\)sup)26 b Fi(fj)p Fl(f)10 b Fn(\()p Fl(z)t Fn(\))p Fi(j)52 b Fn(:)e(0)26 b Fi(\024)f Fl(d)p Fn(\(\012)p Fl(;)15 b(z)t Fn(\))27 b Fi(\024)e Fn(2)p Fl(\021)19 b(;)47 b Fn(Im)20 b Fl(z)30 b Fi(\024)25 b Fn(0)p Fi(g)p Fl(B)20 b(:)705 4335 y Fn(6.)47 b Fm(Effective)33 b(Hamil)-6 b(tonian)33 b(f)n(or)h(str)n(ong)f(ma)n (gnetic)h(fields)150 4497 y Fn(In)27 b(this)f(section)i(w)m(e)g(use)f (freely)g(the)g(notations)h(of)g(the)f(previous)f(section.)40 b(In)27 b(particular,)g(the)g(set)h(\012)3679 4511 y Fj(n)3754 4497 y Fn(and)0 4605 y(the)36 b(pro)5 b(jection)36 b(\005)h(are)g(asso)s(ciated)f(to)h(a)g(Landau)f(lev)m(el)g(\(2)p Fl(n)24 b Fi(\000)g Fn(1\))p Fl(B)41 b Fn(with)35 b(a)i(\014xed)f(in)m (teger)g Fl(n)f Fi(\025)g Fn(1.)59 b(Let)37 b(us)0 4713 y(in)m(tro)s(duce)29 b(the)i(op)s(erators)447 4905 y Fl(R)516 4919 y Fk(+)600 4905 y Fn(:)51 b Fl(L)738 4867 y Fk(2)777 4905 y Fn(\()p Fg(R)872 4867 y Fk(2)872 4927 y Fj(x;y)979 4905 y Fn(\))25 b Fi(3)g Fl(v)k Fi(\000)-16 b(!)26 b Fn(\()p Fi(h)p Fl(v)s Fn(\()p Fl(x;)15 b Fi(\001)p Fn(\))p Fl(;)g( )1773 4919 y Fj(n)1822 4905 y Fn(\()p Fi(\001)p Fn(\))p Fi(i)1952 4925 y Fj(L)2000 4906 y Fs(2)2036 4925 y Fk(\()p Fc(R)2111 4933 y Fr(y)2149 4925 y Fk(\))2206 4905 y Fn(=)2302 4781 y Ff(Z)2353 4987 y Fc(R)2401 4995 y Fr(y)2457 4905 y Fl(v)s Fn(\()p Fl(x;)g(y)s Fn(\))p 2714 4826 225 4 v Fl( )2773 4919 y Fj(n)2822 4905 y Fn(\()p Fl(y)s Fn(\))p Fl(dy)29 b Fi(2)c Fl(L)3209 4867 y Fk(2)3248 4905 y Fn(\()p Fg(R)3343 4919 y Fj(x)3393 4905 y Fn(\))p Fl(;)955 5141 y(R)1024 5155 y Fh(\000)1109 5141 y Fn(=)g Fl(R)1275 5104 y Fh(\003)1274 5164 y Fk(+)1358 5141 y Fn(:)46 b Fl(L)1491 5104 y Fk(2)1530 5141 y Fn(\()p Fg(R)1625 5155 y Fj(x)1675 5141 y Fn(\))26 b Fi(3)e Fl(u)i Fi(\000)-16 b(!)25 b Fl(u)p Fn(\()p Fl(x)p Fn(\))p Fl( )2303 5155 y Fj(n)2351 5141 y Fn(\()p Fl(y)s Fn(\))h Fi(2)f Fl(L)2643 5104 y Fk(2)2682 5141 y Fn(\()p Fg(R)2777 5104 y Fk(2)2777 5164 y Fj(x;y)2884 5141 y Fn(\))p Fl(:)p eop %%Page: 18 18 18 17 bop 0 66 a Fk(18)1372 b(M.)22 b(DIMASSI)j(AND)e(V.)g(PETK)n(O)n (V)0 266 y Fn(These)30 b(op)s(erators)h(satisfy)1281 374 y Fl(R)1350 388 y Fk(+)1409 374 y Fl(R)1478 388 y Fh(\000)1562 374 y Fn(=)25 b Fl(I)1698 393 y Fj(L)1746 374 y Fs(2)1780 393 y Fk(\()p Fc(R)1855 401 y Fr(x)1895 393 y Fk(\))1926 374 y Fl(;)198 b(R)2218 388 y Fh(\000)2277 374 y Fl(R)2346 388 y Fk(+)2430 374 y Fn(=)25 b(\005)p Fl(:)150 500 y Fn(Consider)k(the)h(follo)m(wing)f(Grushin)f(problem:) 563 696 y Fi(P)626 710 y Fk(0)665 696 y Fn(\()p Fl(z)t Fn(\))f(=)903 568 y Ff(\022)1012 641 y Fl(P)1070 655 y Fk(0)1110 641 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)87 b(R)1649 655 y Fh(\000)1190 749 y Fl(R)1259 763 y Fk(+)1621 749 y Fn(0)1750 568 y Ff(\023)1842 696 y Fn(:)50 b Fi(D)23 b(\002)d Fl(L)2163 658 y Fk(2)2203 696 y Fn(\()p Fg(R)2298 710 y Fj(x;y)2404 696 y Fn(\))26 b Fi(!)f Fl(L)2643 658 y Fk(2)2682 696 y Fn(\()p Fg(R)2777 658 y Fk(2)2777 718 y Fj(x;y)2884 696 y Fn(\))c Fi(\002)f Fl(L)3093 658 y Fk(2)3132 696 y Fn(\()p Fg(R)3227 710 y Fj(x)3277 696 y Fn(\))p Fl(:)0 892 y Fn(By)31 b(a)f(simple)f (computation,)h(w)m(e)h(ha)m(v)m(e)h(the)e(follo)m(wing)0 1052 y Fw(Lemma)k(7.)42 b Fe(The)34 b(op)-5 b(er)g(ator)37 b Fi(P)1104 1066 y Fk(0)1143 1052 y Fn(\()p Fl(z)t Fn(\))e Fe(is)e(uniformly)i(invertible)e(for)h Fl(z)d Fi(2)c Fn(\012)2580 1066 y Fj(n)2660 1052 y Fe(and)34 b Fl(\022)29 b Fi(2)e Fl(D)s Fn(\(0)p Fl(;)15 b(\022)3238 1066 y Fk(0)3278 1052 y Fn(\))p Fe(.)45 b(Its)34 b(inverse)f(is)0 1160 y(holomorphic)j(in)c Fn(\()p Fl(z)t(;)15 b(\022)s Fn(\))34 b Fe(and)f(has)h(the)f(form)1329 1361 y Fi(E)1377 1375 y Fk(0)1416 1361 y Fn(\()p Fl(z)t Fn(\))26 b(=)1654 1233 y Ff(\022)1763 1306 y Fl(E)1830 1320 y Fk(0)1869 1306 y Fn(\()p Fl(z)t Fn(\))170 b Fl(E)2222 1320 y Fk(0)p Fj(;)p Fk(+)1784 1414 y Fl(E)1851 1428 y Fk(0)p Fj(;)p Fh(\000)2069 1414 y Fl(E)2136 1428 y Fk(0)p Fj(;)p Fh(\000)p Fk(+)2305 1414 y Fn(\()p Fl(z)t Fn(\))2464 1233 y Ff(\023)2546 1361 y Fl(;)0 1556 y Fe(wher)-5 b(e)310 1723 y Fl(E)377 1737 y Fk(0)417 1723 y Fn(\()p Fl(z)t Fn(\))26 b(=)672 1700 y Ff(b)655 1723 y Fl(R)724 1737 y Fk(0)764 1723 y Fn(\()p Fl(z)t Fn(\))p Fl(;)36 b(E)1008 1737 y Fk(0)p Fj(;)p Fk(+)1148 1723 y Fn(=)25 b Fl(R)1313 1737 y Fh(\000)1372 1723 y Fl(;)35 b(E)1499 1737 y Fk(0)p Fj(;)p Fh(\000)1638 1723 y Fn(=)25 b Fl(R)1803 1737 y Fk(+)1863 1723 y Fl(;)35 b(E)1990 1737 y Fk(0)p Fj(;)p Fh(\000)p Fk(+)2159 1723 y Fn(\()p Fl(z)t Fn(\))26 b(=)f Fl(z)g Fi(\000)20 b Fn(\(2)p Fl(n)g Fi(\000)g Fn(1\))p Fl(B)26 b Fi(\000)20 b Fl(x)g Fi(\000)g Fl(\022)i Fn(+)3452 1661 y(1)p 3396 1702 159 4 v 3396 1785 a(4)p Fl(B)3515 1759 y Fk(2)3564 1723 y Fl(;)992 1929 y Ff(b)974 1952 y Fl(R)1043 1966 y Fk(0)1083 1952 y Fn(\()p Fl(z)t Fn(\))k(=)1321 1851 y Ff(\020)1375 1952 y Fn(\()p Fl(I)i Fi(\000)20 b Fn(\005\))p Fl(P)1730 1966 y Fk(0)1770 1952 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\)\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))g Fi(\000)g Fl(z)2454 1851 y Ff(\021)2509 1874 y Fh(\000)p Fk(1)2603 1952 y Fn(\()p Fl(I)28 b Fi(\000)20 b Fn(\005\))p Fl(:)150 2140 y Fn(No)m(w)31 b(consider)e(the)i(Grushin)d(problem)h(for)h(the)g (p)s(erturb)s(ed)e(Hamiltonian)h Fl(P)2873 2154 y Fk(1)2913 2140 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)29 b Fn(:)1303 2341 y Fi(P)1366 2355 y Fk(1)1406 2341 y Fn(\()p Fl(z)t Fn(\))d(=)1644 2213 y Ff(\022)1752 2286 y Fl(P)1810 2300 y Fk(1)1850 2286 y Fn(\()p Fl(B)5 b(;)15 b(\022)s Fn(\))20 b Fi(\000)g Fl(z)87 b(R)2389 2300 y Fh(\000)1931 2394 y Fl(R)2000 2408 y Fk(+)2362 2394 y Fn(0)2490 2213 y Ff(\023)2572 2341 y Fl(:)1116 b Fn(\(6.1\))0 2557 y Fw(Prop)s(osition)38 b(6.)k Fe(F)-7 b(or)35 b Fl(B)j Fe(lar)-5 b(ge)35 b(enough)f(the)g(op)-5 b(er)g(ator)37 b Fi(P)2073 2571 y Fk(1)2112 2557 y Fn(\()p Fl(z)t Fn(\))e Fe(is)f(invertible)f(for)h Fl(z)e Fi(2)27 b Fn(\012)3135 2571 y Fj(n)3215 2557 y Fe(and)35 b(its)e(inverse)h(is)0 2665 y(given)e(by)1292 2822 y Fi(E)1340 2836 y Fk(1)1379 2822 y Fn(\()p Fl(z)t Fn(\))26 b(=)1617 2694 y Ff(\022)1763 2767 y Fl(E)1830 2781 y Fk(1)1869 2767 y Fn(\()p Fl(z)t Fn(\))149 b Fl(E)2201 2781 y Fk(1)p Fj(;)p Fk(+)2315 2767 y Fn(\()p Fl(z)t Fn(\))1726 2875 y Fl(E)1793 2889 y Fk(1)p Fj(;)p Fh(\000)1907 2875 y Fn(\()p Fl(z)t Fn(\))84 b Fl(E)2174 2889 y Fk(1)p Fj(;)p Fh(\000)p Fk(+)2343 2875 y Fn(\()p Fl(z)t Fn(\))2501 2694 y Ff(\023)2583 2822 y Fl(;)0 2999 y Fe(wher)-5 b(e)2793 3145 y Fl(E)2860 3159 y Fk(1)2899 3145 y Fn(\()p Fl(z)t Fn(\))26 b(=)3157 3122 y(^)3137 3145 y Fl(R)q Fn(\()p Fl(z)t Fn(\))p Fl(a)p Fn(\()p Fl(z)t Fn(\))p Fl(;)201 b Fn(\(6.2\))2777 3285 y Fl(E)2844 3299 y Fk(1)p Fj(;)p Fh(\000)2958 3285 y Fn(\()p Fl(z)t Fn(\))26 b(=)f Fl(R)3265 3299 y Fk(+)3324 3285 y Fl(a)p Fn(\()p Fl(z)t Fn(\))p Fl(;)200 b Fn(\(6.3\))1998 3443 y Fl(E)2065 3457 y Fk(1)p Fj(;)p Fk(+)2179 3443 y Fn(\()p Fl(z)t Fn(\))26 b(=)f Fi(\000)2507 3420 y Fn(^)2488 3443 y Fl(R)p Fn(\()p Fl(z)t Fn(\))p Fl(a)p Fn(\()p Fl(z)t Fn(\)[)p Fl(V)2938 3405 y Fj(!)2988 3443 y Fl(;)15 b Fn(\005])p Fl(R)3190 3457 y Fh(\000)3270 3443 y Fn(+)20 b Fl(R)3430 3457 y Fh(\000)3489 3443 y Fl(;)199 b Fn(\(6.4\))386 3635 y Fl(E)453 3649 y Fk(1)p Fj(;)p Fh(\000)p Fk(+)621 3635 y Fn(\()p Fl(z)t Fn(\))27 b(=)e Fi(\000)p Fl(R)1000 3649 y Fk(+)1058 3635 y Fl(a)p Fn(\()p Fl(z)t Fn(\)[)p Fl(V)1322 3597 y Fj(!)1372 3635 y Fl(;)15 b Fn(\005])p Fl(R)1574 3649 y Fh(\000)1655 3635 y Fn(+)1745 3534 y Ff(h)1788 3635 y Fl(z)25 b Fi(\000)20 b Fn(\(2)p Fl(n)g Fi(\000)g Fn(1\))p Fl(B)26 b Fi(\000)20 b Fl(x)g Fi(\000)g Fl(\022)i Fn(+)2843 3573 y(1)p 2787 3614 V 2787 3697 a(4)p Fl(B)2906 3671 y Fk(2)2976 3635 y Fi(\000)d Fl(R)3135 3649 y Fk(+)3195 3635 y Fl(V)3268 3597 y Fj(!)3318 3635 y Fl(R)3387 3649 y Fh(\000)3446 3534 y Ff(i)3489 3635 y Fl(:)199 b Fn(\(6.5\))0 3864 y Fe(Her)-5 b(e)32 b Fl(a)p Fn(\()p Fl(z)t Fn(\))27 b(=)504 3763 y Ff(\020)558 3864 y Fl(I)g Fn(+)20 b([\005)p Fl(;)15 b(V)923 3831 y Fj(!)974 3864 y Fn(])1019 3841 y(^)999 3864 y Fl(R)q Fn(\()p Fl(z)t Fn(\))1185 3763 y Ff(\021)1240 3786 y Fh(\000)p Fk(1)1367 3864 y Fe(and)1563 3841 y Fn(^)1543 3864 y Fl(R)q Fn(\()p Fl(z)t Fn(\))34 b Fe(is)e(given)g(by)h Fn(\(4)p Fl(:)p Fn(6\))p Fl(:)0 4048 y Fw(Pro)s(of.)41 b Fn(Set)694 4197 y(~)678 4220 y Fi(E)9 b Fn(\()p Fl(z)t Fn(\))26 b(=)973 4064 y Ff( )1106 4119 y Fn(^)1086 4142 y Fl(R)q Fn(\()p Fl(z)t Fn(\))876 b Fl(R)2217 4156 y Fh(\000)1116 4279 y Fl(R)1185 4293 y Fk(+)1356 4279 y Fl(z)24 b 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Fn(\()p Fl(B)3214 4536 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)3378 4574 y Fn(\))15 b Fl(:)0 4720 y Fn(A)30 b(simple)f(calculus)g(implies)973 4919 y Fi(P)1036 4933 y Fk(1)1076 4919 y Fn(\()p Fl(z)t Fn(\))21 b Fi(\016)1294 4896 y Fn(~)1279 4919 y Fi(E)8 b Fn(\()p Fl(z)t Fn(\))26 b(=)1573 4791 y Ff(\022)1682 4870 y Fl(I)h Fn(+)20 b([\005)p Fl(;)15 b(V)2047 4837 y Fj(!)2097 4870 y Fn(])2142 4847 y(^)2122 4870 y Fl(R)q Fn(\()p Fl(z)t Fn(\))84 b([)p Fl(V)2491 4837 y Fj(!)2541 4870 y Fl(;)15 b Fn(\005])p Fl(R)2743 4884 y Fh(\000)1973 4977 y Fn(0)556 b Fl(I)2845 4791 y Ff(\023)1611 5143 y Fn(=)25 b Fl(I)i Fn(+)20 b Fi(O)s Fn(\()p Fl(B)2049 5106 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)2214 5143 y Fn(\))15 b Fl(;)p eop %%Page: 19 19 19 18 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(19)0 266 y Fn(since)24 b([)p Fl(V)316 233 y Fj(!)366 266 y Fl(;)15 b Fn(\005])26 b(=)f Fi(O)s Fn(\()p Fl(B)805 233 y Fh(\000)p Fk(1)p Fj(=)p Fk(2)970 266 y Fn(\).)39 b(Consequen)m(tly)-8 b(,)26 b(for)f 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1967 4939 507 4 v 1967 5023 a(\(1)21 b Fi(\000)f Fl(\026)p Fn(\))p Fi(j)p Fn(Im)g Fl(\022)s Fi(j)2483 4960 y Fl(;)1160 b Fn(\(6.11\))0 5143 y(uniformly)27 b(with)i(resp)s(ect)i(to)g Fl(z)e Fi(2)c Fn(\012)1267 5157 y Fj(n)1314 5143 y Fn(.)p eop %%Page: 21 21 21 20 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(21)150 266 y Fn(Let)31 b Fl(R)26 b(>)f Fn(0)31 b(b)s(e)e(a)i(large)g(constan)m(t)g(suc)m(h)f (that)914 460 y(sup)1051 482 y Fh(j)p Fj(x)p Fh(j)p Fj(>R;)14 b(\030)s Fh(2)p Fc(R)1406 460 y Fi(j)p Fl(@)1484 423 y Fj(\013)1479 483 y(x;\030)1577 460 y Fl(V)20 b Fn(\()p Fl(x)h Fn(+)e Fl(\022)s(;)c(\030)t Fn(\))p Fi(j)26 b Fl(<)2170 399 y Fn(\(1)21 b Fi(\000)f Fl(\026)p Fn(\))p Fi(j)p Fl(\022)s Fi(j)p 2170 439 378 4 v 2336 523 a Fn(2)2558 460 y Fl(;)35 b Fi(j)p Fl(\013)p Fi(j)26 b(\024)f Fl(N)2921 474 y Fk(0)2961 460 y Fl(;)0 633 y Fn(where)40 b Fl(N)346 647 y Fk(0)426 633 y Fn(is)f(an)h(in)m(teger)h(indep)s(enden)m(t)d(on)i Fl(B)45 b Fn(and)40 b Fl(n:)g 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b(Ph)m(ys.)0 4691 y(Sci.,)30 b(490,)i(Dordrec)m(h)m(t,)g (Klu)m(w)m(er)d(Acad.)41 b(Publ.1997.)0 4799 y([28])32 b(J.)e(Sj\177)-45 b(ostrand,)30 b Fe(R)-5 b(esonanc)g(es)34 b(for)f(b)-5 b(ottles)34 b(and)g(tr)-5 b(ac)g(e)33 b(formulae)p Fn(,)f(Math.)42 b(Nac)m(hr.)f Fw(221)31 b Fn(\(2001\),)i(95-149.)0 4907 y([29])g(X.)e(P)-8 b(.)32 b(W)-8 b(ang,)33 b Fe(Bounds)h(on)g (widths)h(of)f(r)-5 b(esonanc)g(es)36 b(for)e(Stark)g(hamiltonians)p Fn(,)g(Acta)f(Math.)44 b(Sinica,)30 b(Ser.)0 5015 y(B,)h Fw(6)g Fn(\(1990\),)i(100-119.)p eop %%Page: 23 23 23 22 bop 1316 66 a Fk(MA)n(GNETIC)31 b(ST)-6 b(ARK)31 b(HAMIL)-6 b(TONIANS)1245 b(23)0 266 y Fn([30])34 b(X.)f(P)-8 b(.)34 b(W)-8 b(ang,)34 b Fe(On)h(the)g(Magnetic)f(Stark)i(R)-5 b(esonanc)g(es)37 b(in)e(Two)g(Dimensional)h(Case)p Fn(,)e(Lecture)f (Notes)h(in)0 374 y(Ph)m(ysics,)c Fw(403)p Fn(,)h(Springer,)e(Berlin,)f (1992,)33 b(pp.)40 b(211-233.)0 482 y([31])c(X.)f(P)-8 b(.)35 b(W)-8 b(ang,)37 b Fe(Barrier)h(r)-5 b(esonanc)g(es)38 b(in)f(str)-5 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Fv(:)37 b Fa(petkov@math.u-bordeaux.fr)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0404290931787--