Content-Type: multipart/mixed; boundary="-------------0107260559170" This is a multi-part message in MIME format. ---------------0107260559170 Content-Type: text/plain; name="01-289.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-289.keywords" Schrodinger operator, magnetic Schrodinger operator, classical wave operator, acoustic operator, Maxwell operator, Combes-Thomas estimate ---------------0107260559170 Content-Type: application/postscript; name="DecayKernelAMS.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="DecayKernelAMS.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: DecayKernelAMS.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips DecayKernelAMS.dvi -o %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2001.07.26:1248 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def 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Fo( )6 b(;)28 b(B)8 b( )e Fn(i)67 b Fq(for)758 8051 y Fo( )52 b Fn(2)46 b(D)5 b Fq(\()p Fo(B)j Fq(\).\))957 8251 y(A)55 b Fj(gener)-8 b(alize)g(d)60 b(Schr\177)-85 b(odinger)61 b(op)-8 b(er)g(ator)72 b Fq(is)56 b(of)f(the)g(form)2896 8564 y Fo(H)60 b Fq(=)46 b Fo(W)60 b Fq(+)37 b Fo(V)147 b(on)111 b Fn(H)4329 8495 y Fm(\()p Fl(n)p Fm(\))4523 8564 y Fo(;)-3811 b Fq(\(1\))758 8872 y(with)2522 9180 y Fo(W)69 b Fq(=)2923 9031 y Fn(p)p 3062 9031 129 7 v 3062 9180 a(K)r Fq(\()p Ft(D)36 b Fn(\000)i(A)p Fq(\))3803 9111 y Fi(\003)3878 9180 y Fn(R)p Fq(\()p Ft(D)f Fn(\000)g(A)p Fq(\))4631 9031 y Fn(p)p 4768 9031 V 4768 9180 a(K)r Fo(;)-4185 b Fq(\(2\))758 9489 y(where:)1003 9742 y Fn(\017)83 b Ft(D)62 b Fq(is)h(a)f(\014rst)g(order)h(constan)-5 b(t)62 b(co)5 b(e\016cien)-5 b(t)61 b(partial)h(di\013eren)-5 b(tial)63 b(op)5 b(erator)61 b(from)h Fn(H)6541 9682 y Fm(\()p Fl(n)p Fm(\))1169 9941 y Fq(to)k Fn(H)1525 9881 y Fm(\()p Fl(m)p Fm(\))1755 9941 y Fq(;)72 b(it)66 b(is)g(giv)-5 b(en)67 b(b)-5 b(y)66 b(m)-5 b(ultiplication)68 b(b)-5 b(y)66 b(an)h Fo(m)44 b Fn(\002)g Fo(n)66 b Fq(matrix)g Fo(D)5 b Fq(\()p Fn(\000)p Fo(i)p Fn(r)p Fq(\),)69 b(where,)p 758 10159 997 7 v 957 10347 a Fs(1991)49 b Fh(Mathematics)h(Subje)-7 b(ct)49 b(Classi\014c)-7 b(ation.)71 b Fs(Primary)46 b(81Q10,)f(47F05;)h(Secondary)h(35P05.)957 10508 y Fh(Key)54 b(wor)-7 b(ds)54 b(and)g(phr)-7 b(ases.)70 b Fs(Sc)l(hr\177)-71 b(odinger)50 b(op)t(erator,)g(magnetic)e(Sc)l(hr\177)-71 b(odinger)49 b(op)t(erator,)h(classical)c(w)l(a)l(v)l(e)758 10674 y(op)t(erator,)g(acoustic)e(op)t(erator,)i(Maxw)l(ell)g(op)t (erator,)g(Com)l(b)t(es-Thomas)f(estimate.)957 10840 y(A.K.)i(w)l(as)f(supp)t(orted)h(in)g(part)g(b)l(y)h(NSF)f(Gran)l(ts)g (DMS-9800883)d(and)k(DMS-9800860.)3713 11039 y Fm(1)p eop %%Page: 2 2 2 1 bop 758 861 a Fm(2)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)1169 1259 y Fq(for)50 b(a)g(d-comp)5 b(onen)-5 b(t)51 b(v)-5 b(ector)49 b Fo(k)5 b Fq(,)52 b Fo(D)5 b Fq(\()p Fo(k)g Fq(\))49 b(is)i(an)f Fo(m)26 b Fn(\002)g Fo(n)50 b Fq(matrix)g(with)f(en)-5 b(tries)51 b(of)f(the)f(form)1169 1458 y Fo(D)5 b Fq(\()p Fo(k)g Fq(\))1532 1483 y Fl(r)l(;s)1747 1458 y Fq(=)46 b Fo(a)2010 1483 y Fl(r)l(;s)2215 1458 y Fn(\001)37 b Fo(k)61 b Fq(with)55 b Fo(a)2911 1483 y Fl(r)l(;s)3126 1458 y Fn(2)45 b Fk(C)3393 1398 y Fl(d)3480 1458 y Fq(,)55 b(for)g Fo(r)c Fq(=)46 b(1)p Fo(;)28 b(:)g(:)g(:)55 b(;)28 b(m;)83 b(s)46 b Fq(=)g(1)p Fo(;)28 b(:)g(:)g(:)55 b(;)28 b(n)p Fq(.)73 b(W)-14 b(e)55 b(set)2162 1773 y Fo(D)2299 1798 y Fm(+)2456 1773 y Fq(=)46 b(sup)q Fn(fk)p Fo(D)5 b Fq(\()p Fo(k)g Fq(\))p Fn(k)p Fq(;)120 b Fo(k)52 b Fn(2)46 b Fk(C)4064 1705 y Fl(d)4151 1773 y Fo(;)74 b Fn(j)p Fo(k)5 b Fn(j)47 b Fq(=)f(1)p Fn(g)g Fo(<)h Fn(1)28 b Fo(;)-4546 b Fq(\(3\))1169 2101 y(note)55 b Fn(k)p Fo(D)5 b Fq(\()p Fo(k)g Fq(\))p Fn(k)46 b(\024)g Fo(D)2425 2126 y Fm(+)2536 2101 y Fn(j)p Fo(k)5 b Fn(j)56 b Fq(for)g(all)f Fo(k)d Fn(2)46 b Fk(C)3666 2040 y Fl(d)3752 2101 y Fq(.)1003 2306 y Fn(\017)83 b(A)65 b Fq(is)i(the)e(op)5 b(erator)65 b(from)g Fn(H)3065 2246 y Fm(\()p Fl(n)p Fm(\))3325 2306 y Fq(to)g Fn(H)3680 2246 y Fm(\()p Fl(m)p Fm(\))3976 2306 y Fq(giv)-5 b(en)65 b(b)-5 b(y)66 b(m)-5 b(ultiplication)67 b(b)-5 b(y)66 b(the)f Fo(m)44 b Fn(\002)g Fo(n)1169 2505 y Fq(matrix-v)-9 b(alued)69 b(measurable)g(function)g Fn(A)p Fq(\()p Fo(x)p Fq(\).)112 b(W)-14 b(e)69 b(require)f Fn(A)p Fq(\()p Fo(x)p Fq(\))f(to)i(b)5 b(e)68 b(lo)5 b(cally)68 b(in)1169 2705 y Fo(L)1282 2644 y Fm(2)1357 2705 y Fq(.)1003 2904 y Fn(\017)83 b(K)59 b Fq(and)f Fn(R)f Fq(are)g(co)5 b(e\016cien)-5 b(ts)57 b(op)5 b(erators)58 b(on)f Fn(H)4111 2844 y Fm(\()p Fl(n)p Fm(\))4362 2904 y Fq(and)h Fn(H)4829 2844 y Fm(\()p Fl(m)p Fm(\))5059 2904 y Fq(,)g(resp)5 b(ectiv)-5 b(ely)-14 b(.)79 b(A)56 b(co)5 b(e\016-)1169 3103 y(cien)-5 b(t)55 b(op)5 b(erator)54 b Fn(S)67 b Fq(on)55 b Fn(H)2779 3043 y Fm(\()p Fl(k)s Fm(\))3019 3103 y Fq(is)g(a)g(b)5 b(ounded,)56 b(in)-5 b(v)g(ertible)55 b(op)5 b(erator)54 b(giv)-5 b(en)55 b(b)-5 b(y)56 b(m)-5 b(ultipli-)1169 3302 y(cation)61 b(b)-5 b(y)61 b(a)g(co)5 b(e\016cien)-5 b(t)60 b(matrix:)84 b(an)61 b Fo(k)47 b Fn(\002)40 b Fo(k)67 b Fq(matrix)60 b(-v)-9 b(alued)61 b(measurable)h(function)1169 3502 y Fn(S)12 b Fq(\()p Fo(x)p Fq(\))54 b(on)i Fk(R)1903 3441 y Fl(d)1989 3502 y Fq(,)g(satisfying)1950 3803 y Fo(S)2052 3828 y Fi(\000)2164 3803 y Fo(I)2237 3828 y Fl(k)2365 3803 y Fn(\024)46 b(S)12 b Fq(\()p Fo(x)p Fq(\))45 b Fn(\024)i Fo(S)3201 3828 y Fm(+)3311 3803 y Fo(I)3384 3828 y Fl(k)3521 3803 y Fo(;)120 b Fq(with)92 b(0)46 b Fo(<)g(S)4508 3828 y Fi(\000)4666 3803 y Fn(\024)h Fo(S)4944 3828 y Fm(+)5100 3803 y Fo(<)f Fn(1)55 b Fo(:)-4784 b Fq(\(4\))1169 4113 y(\()p Fo(I)1307 4138 y Fl(k)1444 4113 y Fq(is)56 b(the)f Fo(k)42 b Fn(\002)37 b Fo(k)61 b Fq(iden)-5 b(tit)g(y)56 b(matrix.\))1003 4313 y Fn(\017)83 b Fo(V)h Fq(is)49 b(a)e(p)5 b(oten)-5 b(tial,)49 b(an)f(op)5 b(erator)47 b(giv)-5 b(en)47 b(b)-5 b(y)48 b(m)-5 b(ultiplication)49 b(b)-5 b(y)48 b(a)f(real-v)-9 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Fn(h)p Fo( )6 b(;)28 b(W)23 b( )6 b Fn(i)36 b Fq(+)h(\002)3807 5236 y Fm(2)3881 5211 y Fn(k)p Fo( )6 b Fn(k)4161 5143 y Fm(2)4328 5211 y Fq(for)55 b(all)139 b Fo( )52 b Fn(2)46 b(Q)p Fq(\()p Fo(W)23 b Fq(\))28 b Fo(:)-4975 b Fq(\(5\))957 5513 y(A)49 b(generalized)h(Sc)-5 b(hr\177)-83 b(odinger)50 b(op)5 b(erator)49 b(is)i(w)-5 b(ell)49 b(de\014ned)i(as)f(a)f(semib)5 b(ounded)51 b(self-adjoin)-5 b(t)758 5712 y(op)5 b(erator.)72 b(T)-14 b(o)52 b(see)h(that,)f(note)g (that)g Ft(D)32 b Fn(\000)f(A)p Fq(,)52 b(initially)h(de\014ned)g(on)f (in\014nitely)h(di\013eren)-5 b(tiable)758 5912 y(functions)66 b(with)g(compact)f(supp)5 b(ort)67 b(from)f Fn(H)3718 5851 y Fm(\()p Fl(n)p Fm(\))3977 5912 y Fq(to)g Fn(H)4333 5851 y Fm(\()p Fl(m)p Fm(\))4562 5912 y Fq(,)j(extends)d(to)f(a)h (closed,)j(densely)758 6111 y(de\014ned)56 b(op)5 b(erator.)73 b(W)-14 b(e)55 b(set)g(\(as)g(in)h([13,)g(15]\))1936 6432 y Fo(A)46 b Fq(=)2282 6284 y Fn(p)p 2421 6284 141 7 v 2421 6432 a(R)o Fq(\()p Ft(D)37 b Fn(\000)g(A)p Fq(\))3173 6284 y Fn(p)p 3311 6284 129 7 v 148 x(K)112 b Fq(on)f Fn(D)5 b Fq(\()p Fo(A)p Fq(\))45 b(=)4444 6284 y Fn(p)p 4582 6284 V 148 x(K)30 b(D)5 b Fq(\()p Ft(D)37 b Fn(\000)g(A)p Fq(\))p Fo(;)-4772 b Fq(\(6\))758 6766 y(a)55 b(closed,)h(densely)g (de\014ned)g(op)5 b(erator)54 b(from)i Fn(H)3795 6706 y Fm(\()p Fl(n)p Fm(\))4044 6766 y Fq(to)f Fn(H)4389 6706 y Fm(\()p Fl(m)p Fm(\))4619 6766 y Fq(,)g(and)h(de\014ne)g Fo(W)78 b Fq(b)-5 b(y)3346 7067 y Fo(W)69 b Fq(=)46 b Fo(A)3872 6999 y Fi(\003)3948 7067 y Fo(A;)-3361 b Fq(\(7\))758 7378 y(a)56 b(p)5 b(ositiv)-5 b(e)55 b(self-adjoin)-5 b(t)57 b(op)5 b(erator)55 b(with)h Fn(Q)p Fq(\()p Fo(W)23 b Fq(\))46 b(=)h Fn(D)5 b Fq(\()p Fo(A)p Fq(\).)74 b(It)55 b(follo)-5 b(ws)57 b(from)f(the)f(conditions)758 7584 y(on)g Fo(V)1122 7523 y Fi(\006)1289 7584 y Fq(that)g Fo(H)60 b Fq(=)46 b Fo(W)60 b Fq(+)37 b Fo(V)92 b Fq(is)56 b(a)f(semib)5 b(ounded)57 b(self-adjoin)-5 b(t)56 b(op)5 b(erator)55 b(on)h Fn(H)5798 7523 y Fm(\()p Fl(n)p Fm(\))5992 7584 y Fq(,)f(with)1774 7884 y Fo(H)60 b Fn(\025)46 b Fq(\(1)37 b Fn(\000)g Fq(\002)2627 7909 y Fm(1)2702 7884 y Fq(\))p Fo(W)59 b Fn(\000)37 b Fq(\002)3278 7909 y Fm(2)3491 7884 y Fq(and)139 b Fn(Q)p Fq(\()p Fo(H)13 b Fq(\))45 b(=)h Fn(Q)p Fq(\()p Fo(W)23 b Fq(\))36 b Fn(\\)h(Q)p Fq(\()p Fo(V)5498 7816 y Fm(+)5608 7884 y Fq(\))p Fo(:)-4961 b Fq(\(8\))957 8186 y(Our)56 b(estimates)f(will)h(b) 5 b(e)55 b(stated)g(in)h(terms)f(of)g(the)g(constan)-5 b(ts)57 b(\002)5064 8211 y Fm(1)5138 8186 y Fo(;)28 b Fq(\002)5341 8211 y Fm(2)5416 8186 y Fq(,)55 b(and)3093 8496 y(\004)46 b(=)g Fo(D)3562 8521 y Fm(+)3673 8350 y Fg(p)p 3839 8350 488 7 v 146 x Fo(R)3965 8521 y Fm(+)4075 8496 y Fo(K)4216 8521 y Fm(+)4354 8496 y Fo(:)-3642 b Fq(\(9\))758 8797 y(Note)54 b(that)h(while)g(\004)g(dep)5 b(ends)56 b(only)g(on)f Fo(W)23 b Fq(,)56 b(\002)3766 8822 y Fm(1)3896 8797 y Fq(and)g(\002)4348 8822 y Fm(2)4478 8797 y Fq(dep)5 b(end)55 b(on)h(b)5 b(oth)55 b Fo(W)78 b Fq(and)56 b Fo(V)37 b Fq(.)957 8997 y(Examples)56 b(include)g(the)f Fj(Schr\177)-85 b(odinger)61 b(op)-8 b(er)g(ator)72 b Fq(\(e.g.,)55 b([16]\))2072 9306 y Fo(H)60 b Fq(=)46 b Fn(\000)p Fq(\001)38 b(+)f Fo(V)147 b Fq(on)111 b Fo(L)3559 9237 y Fm(2)3634 9306 y Fq(\()p Fk(R)3809 9237 y Fl(d)3895 9306 y Fo(;)28 b Fq(d)p Fo(x)p Fq(\))p Fo(;)120 b Fq(where)92 b(\004)46 b(=)g(1)28 b Fo(;)-4635 b Fq(\(10\))758 9607 y(where)65 b(\001)i(is)g(the)e Fo(d)p Fq(-dimensional)k(Laplacian)d(op) 5 b(erator,)68 b(the)e Fj(magnetic)71 b(\014eld)f(Schr\177)-85 b(odinger)758 9807 y(op)-8 b(er)g(ator)1396 10108 y Fo(H)60 b Fq(=)46 b(\()p Fn(\000)p Fo(i)p Fn(r)38 b(\000)f Ft(a)p Fq(\))f Fn(\001)h Fq(\()p Fn(\000)p Fo(i)p Fn(r)g(\000)g Ft(a)p Fq(\))f(+)h Fo(V)148 b Fq(on)111 b Fo(L)4235 10039 y Fm(2)4309 10108 y Fq(\()p Fk(R)4485 10039 y Fl(d)4571 10108 y Fo(;)28 b Fq(d)p Fo(x)p Fq(\))p Fo(;)120 b Fq(where)92 b(\004)46 b(=)g(1)28 b Fo(;)-5311 b Fq(\(11\))758 10441 y(where)46 b Ft(a)g Fq(is)h(a)f(v)-5 b(ector)45 b(p)5 b(oten)-5 b(tial,)48 b(an)f(op)5 b(erator)45 b(from)i Fn(H)4292 10381 y Fm(\(1\))4516 10441 y Fq(to)f Fn(H)4852 10381 y Fm(\()p Fl(d)p Fm(\))5079 10441 y Fq(giv)-5 b(en)47 b(b)-5 b(y)46 b(m)-5 b(ultiplication)758 10641 y(b)g(y)46 b(the)f(v)-5 b(ector-v)c(alued)46 b(function)g Ft(a)p Fq(\()p Fo(x)p Fq(\))f Fn(2)h Fo(L)3540 10580 y Fm(2)3540 10688 y(lo)t(c)3714 10641 y Fq(\()p Fk(R)3890 10580 y Fl(d)3976 10641 y Fo(;)28 b Fq(d)p Fo(x)p Fq(;)g Fk(R)4422 10580 y Fl(d)4508 10641 y Fq(\),)48 b(and)e(the)g Fj(classic)-8 b(al)51 b(wave)h(op)-8 b(er)g(a-)758 10840 y(tor)69 b Fo(W)74 b Fq(giv)-5 b(en)51 b(b)-5 b(y)52 b(\(2\))e(with)h Fn(A)46 b Fq(=)g(0)52 b(\(see)f([13,)g(15]\).)72 b(Examples)52 b(of)f(classical)i(w)-5 b(a)g(v)g(e)51 b(op)5 b(erators)p eop %%Page: 3 3 3 2 bop 758 886 a Fm(OPERA)-11 b(TOR)59 b(KERNEL)g(ESTIMA)-11 b(TES)59 b(F)l(OR)g(FUNCTIONS)f(OF)f(GENERALIZED)j(SCHR)5862 857 y(\177)5844 886 y(ODINGER)f(OPERA)-11 b(TORS)-66 b(3)758 1259 y Fq(include)56 b(the)f Fj(ac)-8 b(oustic)59 b(op)-8 b(er)g(ator)2321 1613 y Fo(W)69 b Fq(=)46 b Fn(\000)2947 1501 y Fq(1)p 2871 1575 235 7 v 2871 1608 a Fn(p)p 3010 1608 96 7 v 3010 1728 a Fo(\024)3125 1613 y Fn(r)38 b(\001)3405 1501 y Fq(1)p 3404 1575 86 7 v 3404 1727 a Fo(\032)3509 1613 y Fn(r)3743 1501 y Fq(1)p 3667 1575 235 7 v 3667 1608 a Fn(p)p 3806 1608 96 7 v 3806 1728 a Fo(\024)4032 1613 y(on)111 b(L)4436 1545 y Fm(2)4511 1613 y Fq(\()p Fk(R)4686 1545 y Fl(d)4773 1613 y Fo(;)28 b Fq(d)p Fo(x)p Fq(\))p Fo(;)-4387 b Fq(\(12\))758 1995 y(where)56 b Fn(r)h Fq(is)g(the)f Fo(d)p Fq(-dimensional)j(gradien)-5 b(t,)57 b Fo(\024)p Fq(\()p Fo(x)p Fq(\))e(is)j(the)e(compressibilit)-5 b(y)-14 b(,)58 b(and)f Fo(\045)p Fq(\()p Fo(x)p Fq(\))f(is)h(the)758 2194 y(mass)f(densit)-5 b(y)-14 b(,)57 b(and)e(the)g Fj(Maxwel)8 b(l)62 b(op)-8 b(er)g(ator)2107 2549 y Fo(W)69 b Fq(=)2606 2436 y(1)p 2528 2511 239 7 v 2528 2559 a Fn(p)p 2667 2559 101 7 v 2667 2663 a Fo(\026)2787 2549 y Fn(r)37 b(\002)3148 2436 y Fq(1)p 3148 2511 84 7 v 3151 2663 a Fo(")3251 2549 y Fn(r)g(\002)3690 2436 y Fq(1)p 3612 2511 239 7 v 3612 2559 a Fn(p)p 3750 2559 101 7 v 104 x Fo(\026)3981 2549 y(on)111 b(L)4385 2480 y Fm(2)4459 2549 y Fq(\()p Fk(R)4635 2480 y Fm(3)4718 2549 y Fo(;)28 b Fq(d)p Fo(x)p Fq(;)g Fk(C)5164 2480 y Fm(3)5248 2549 y Fq(\))p Fo(;)-4601 b Fq(\(13\))758 2946 y(where)56 b Fn(r\002)h Fq(is)g(the)f(curl,)h Fo(\026)p Fq(\()p Fo(x)p Fq(\))e(is)i(the)f(magnetic)h(p)5 b(ermeabilit)-5 b(y)-14 b(,)57 b(and)g Fo(")p Fq(\()p Fo(x)p Fq(\))f(is)h(the)f (dielectric)758 3145 y(constan)-5 b(t.)758 3453 y Ft(De\014nition)62 b(1.)84 b Fq(The)51 b Fj(op)-8 b(er)g(ator)55 b(kernel)70 b Fq(of)51 b(a)g(b)5 b(ounded)52 b(op)5 b(erator)51 b Fo(B)60 b Fq(on)51 b Fn(H)5584 3393 y Fm(\()p Fl(n)p Fm(\))5829 3453 y Fq(is)h(de\014ned)g(b)-5 b(y)2351 3739 y Fo(K)2492 3764 y Fl(B)2607 3739 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))45 b(=)h Fo(\037)3317 3764 y Fl(x)3401 3739 y Fo(B)8 b(\037)3639 3764 y Fl(y)3811 3739 y Fo(f)18 b(or)61 b(al)s(l)94 b(x;)28 b(y)52 b Fn(2)46 b Fk(R)4981 3670 y Fl(d)5068 3739 y Fo(;)-4356 b Fq(\(14\))758 4025 y(where)40 b Fo(\037)1327 4050 y Fl(x)1451 4025 y Fq(denotes)g(the)h (op)5 b(erator)39 b(giv)-5 b(en)41 b(b)-5 b(y)41 b(m)-5 b(ultiplication)41 b(b)-5 b(y)41 b(the)f(c)-5 b(haracteristic)40 b(function)758 4224 y(of)55 b(a)g(cub)5 b(e)55 b(of)g(side)h(1)g(cen)-5 b(tered)55 b(at)g Fo(x)p Fq(.)957 4532 y(A)65 b(b)5 b(ounded)66 b(op)5 b(erator)65 b Fo(B)73 b Fq(ma)-5 b(y)66 b(or)f(ma)-5 b(y)66 b(not)f(ha)-5 b(v)g(e)66 b(an)g(in)-5 b(tegral)66 b(k)-5 b(ernel,)68 b(but)d(it)g(alw)-5 b(a)g(ys)758 4732 y(has)59 b(an)g(op)5 b(erator)57 b(k)-5 b(ernel.)83 b(\(The)58 b(c)-5 b(hoice)59 b(of)f(a)h(cub)5 b(e)58 b(of)g(side)h(1)f(is)h (arbitrary)-14 b(,)60 b(one)e(ma)-5 b(y)59 b(\014x)f(a)758 4931 y(scale)h Fo(L)h Fq(and)g(replace)f(the)h(c)-5 b(haracteristic)59 b(functions)h(in)g(\(14\))f(b)-5 b(y)59 b(c)-5 b(haracteristic)60 b(functions)758 5130 y(of)55 b(cub)5 b(es)55 b(of)g(side)h Fo(L)p Fq(.\))74 b(Note)54 b(that)h Fn(k)p Fo(K)3209 5155 y Fl(B)3323 5130 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)46 b(\024)g(k)p Fo(B)8 b Fn(k)p Fq(,)56 b(and)1951 5449 y(\()p Fo(B)8 b( )e Fq(\))2329 5474 y Fl(x)2458 5449 y Fq(=)2757 5291 y Fg(X)2634 5659 y Fl(y)t Fi(2)p Fl(x)p Fm(+)p Ff(Z)3056 5625 y Fe(d)3147 5449 y Fo(K)3288 5474 y Fl(B)3402 5449 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fo( )3896 5474 y Fl(y)4114 5449 y Fq(in)138 b(L)4494 5380 y Fm(2)4569 5449 y Fq(\()p Fk(R)4744 5380 y Fl(d)4831 5449 y Fo(;)28 b Fq(d)p Fo(x)p Fq(;)g Fk(C)5276 5380 y Fl(n)5376 5449 y Fq(\))g Fo(;)-4757 b Fq(\(15\))758 5942 y(for)59 b(all)h Fo( )g Fn(2)53 b Fq(L)1687 5881 y Fm(2)1761 5942 y Fq(\()p Fk(R)1937 5881 y Fl(d)2023 5942 y Fo(;)28 b Fq(d)p Fo(x)p Fq(;)g Fk(C)2469 5881 y Fl(n)2568 5942 y Fq(\),)61 b(where)f Fo(\036)3324 5967 y Fl(x)3461 5942 y Fq(=)53 b Fo(\037)3747 5967 y Fl(x)3831 5942 y Fo(\036)p Fq(.)87 b(Op)5 b(erator)59 b(k)-5 b(ernels)61 b(a)-5 b(v)g(oid)60 b(man)-5 b(y)61 b(tec)-5 b(hni-)758 6141 y(cal)61 b(problems)h(asso)5 b(ciated)61 b(with)g(in)-5 b(tegral)62 b(k)-5 b(ernels;)64 b(they)d(ha)-5 b(v)g(e)61 b(pro)-5 b(v)g(ed)62 b(v)-5 b(ery)60 b(useful)i(in)g(the)758 6340 y(study)50 b(of)h(Anderson)g(lo)5 b(calization)50 b(of)g(random)h(Sc)-5 b(hr\177)-83 b(odinger)52 b(op)5 b(erators)50 b(and)h(classical)h(w)-5 b(a)g(v)g(e)758 6539 y(op)5 b(erators)55 b([2,)g(7)q(,)g(8,)g(9,)h(13,)f(14)q(].)957 6739 y(Sc)-5 b(hr\177)-83 b(odinger)47 b(op)5 b(erators)47 b(and)g(classical)g(w)-5 b(a)g(v)g(e)47 b(op)5 b(erators)46 b(satisfy)g(a)h(resolv)-5 b(en)g(t)47 b(deca)-5 b(y)46 b(esti-)758 6938 y(mate,)51 b(commonly)g(called)f(the)g(Com)-5 b(b)5 b(es-Thomas)53 b(estimate)d(in)h(the)f(Mathematical)g(Ph)-5 b(ysics)758 7137 y(literature.)76 b(The)57 b(pro)5 b(ofs)56 b(are)h(based)g(on)g(an)f(argumen)-5 b(t)57 b(of)g(Com)-5 b(b)5 b(es)57 b(and)g(Thomas)h([3].)77 b(\(F)-14 b(or)758 7336 y(Sc)-5 b(hr\177)-83 b(odinger)54 b(op)5 b(erators)53 b(see)g([16,)g(1];)h(for)e(classical)i(w)-5 b(a)g(v)g(e)54 b(op)5 b(erators)52 b(see)h([7)q(,)g(8,)g(13].\))73 b(In)53 b(the)758 7536 y(follo)-5 b(wing)64 b(theorem)f(w)-5 b(e)64 b(state)f(and)h(pro)-5 b(v)g(e)64 b(a)g(Com)-5 b(b)5 b(es-Thomas)66 b(estimate)d(for)g(generalized)758 7735 y(Sc)-5 b(hr\177)-83 b(odinger)52 b(op)5 b(erators,)51 b(using)h(ideas)f(from)g([7,)h(Lemma)f(12])g(and)g([13)q(,)g(Theorem)g (3.5].)73 b(Our)758 7934 y(pro)5 b(of)63 b(do)5 b(es)64 b(not)g(use)h(analyticit)-5 b(y)-14 b(.)99 b(W)-14 b(e)64 b(giv)-5 b(e)64 b(explicit)f(estimates)h(on)h(the)e(constan)-5 b(ts,)67 b(v)-9 b(alid)758 8133 y(for)55 b(all)g(energies)h Fo(z)7 b Fq(.)75 b(This)56 b(will)f(b)5 b(e)55 b(of)g(imp)5 b(ortance)56 b(in)f(Theorem)h(2.)758 8442 y Ft(Theorem)63 b(1.)83 b Fj(L)-8 b(et)62 b Fo(H)75 b Fj(b)-8 b(e)63 b(a)e(gener)-8 b(alize)g(d)63 b(Schr\177)-85 b(odinger)63 b(op)-8 b(er)g(ator,)62 b(let)h Fo(R)q Fq(\()p Fo(z)7 b Fq(\))50 b(=)h(\()p Fo(H)f Fn(\000)37 b Fo(z)7 b Fq(\))6506 8381 y Fi(\000)p Fm(1)6684 8442 y Fj(,)758 8641 y Fo(z)114 b(=)-102 b Fn(2)88 b Fo(\033)6 b Fq(\()p Fo(H)13 b Fq(\))p Fj(,)87 b(b)-8 b(e)83 b(its)g(r)-8 b(esolvent,)89 b(and)83 b(let)g Fo(K)3649 8666 y Fl(n;z)3847 8641 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))82 b Fj(b)-8 b(e)83 b(the)g(op)-8 b(er)g(ator)81 b(kernel)k(of)d Fo(R)q Fq(\()p Fo(z)7 b Fq(\))6594 8581 y Fl(n)6684 8641 y Fj(,)758 8840 y Fo(n)45 b Fq(=)i(1)p Fo(;)28 b Fq(2)p Fo(;)g(:)g(:)g(:)f Fj(.)76 b(Then)60 b(for)f(any)g Fq(0)46 b Fo(<)h(\015)55 b(<)46 b Fq(1)60 b Fj(we)g(have)1976 9225 y Fn(k)p Fo(K)2200 9250 y Fl(n;z)2398 9225 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)46 b(\024)3088 8991 y Fg(\022)3210 9041 y(\020)3330 9151 y Fm(1+)p Fl(\015)p 3329 9187 247 7 v 3329 9282 a Fm(1)p Fi(\000)p Fl(\015)3596 9041 y Fg(\021)3695 9075 y Fm(2)3853 9160 y(1)p 3817 9187 137 7 v 3817 9282 a Fl(\021)3885 9299 y Fe(z)3974 8991 y Fg(\023)4096 9025 y Fl(n)4187 9225 y Fq(e)4261 9156 y Fl(\015)7 b(m)4456 9173 y Fe(z)4524 9057 y Fi(p)p 4633 9057 70 6 v 99 x Fl(d)4738 9225 y Fq(e)4812 9156 y Fi(\000)p Fl(\015)g(m)5111 9173 y Fe(z)5179 9156 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)758 9225 y Fq(\(16\))758 9626 y Fj(for)59 b(al)8 b(l)60 b Fo(x;)28 b(y)53 b Fn(2)45 b Fk(R)1830 9566 y Fl(d)1917 9626 y Fj(,)59 b(wher)-8 b(e)60 b Fo(\021)2578 9651 y Fl(z)2701 9626 y Fq(=)46 b(dist)q(\()p Fo(z)7 b(;)28 b(\033)6 b Fq(\()p Fo(H)13 b Fq(\)\))p Fj(,)58 b(and)2831 9954 y Fo(m)2977 9979 y Fl(z)3099 9954 y Fq(=)46 b(\000)3874 9842 y Fo(\021)3956 9867 y Fl(z)p 3398 9916 1112 7 v 3398 9950 a Fg(p)p 3564 9950 946 7 v 141 x Fq(\002)3693 10116 y Fm(2)3805 10091 y Fq(+)37 b Fo(\021)4053 10116 y Fl(z)4166 10091 y Fq(+)g Fn(j)p Fo(z)7 b Fn(j)4589 9954 y Fo(;)-3877 b Fq(\(17\))758 10360 y Fj(with)3047 10726 y Fq(\000)46 b(=)3392 10614 y(3)3475 10476 y Fn(p)p 3614 10476 84 7 v 3614 10614 a Fq(7)3724 10485 y Fn(p)p 3863 10485 490 7 v 3863 10614 a Fq(1)36 b Fn(\000)i Fq(\002)4278 10639 y Fm(1)p 3392 10688 961 7 v 3720 10840 a Fq(32)28 b(\004)4400 10726 y Fo(:)-3688 b Fq(\(18\))p eop %%Page: 4 4 4 3 bop 758 861 a Fm(4)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fj(If)h Fq(Re)28 b Fo(z)54 b(<)46 b Fq(\006)g(=)g(inf)41 b Fo(\033)6 b Fq(\()p Fo(H)13 b Fq(\))59 b Fj(we)h(also)f(have)1530 1588 y Fn(k)p Fo(K)1754 1613 y Fl(n;z)1952 1588 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)45 b(\024)2642 1404 y Fg(\020)3214 1523 y Fm(1)p 2761 1550 973 7 v 2761 1645 a(\(1)p Fi(\000)p Fl(\015)3060 1612 y Fd(2)3124 1645 y Fm(\)\(\006)p Fi(\000)p Fl(R)q(e)22 b(z)5 b Fm(\))3753 1404 y Fg(\021)3852 1438 y Fl(n)3943 1588 y Fq(e)4017 1519 y Fl(\015)4113 1404 y Fc(p)p 4204 1404 392 5 v 63 x Fd(\006)p Fc(\000)p Fe(Re)20 b(z)4617 1396 y Fc(p)p 4708 1396 62 5 v 71 x Fe(d)p 4113 1494 656 6 v 4404 1559 a Fd(\004)4825 1588 y Fq(e)4899 1519 y Fi(\000)p Fl(\015)5099 1404 y Fc(p)p 5191 1404 392 5 v 5191 1467 a Fd(\006)p Fc(\000)p Fe(Re)f(z)p 5099 1494 484 6 v 5304 1559 a Fd(\004)5625 1519 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)758 1588 y Fq(\(19\))758 1923 y Fj(for)59 b(al)8 b(l)60 b Fo(x;)28 b(y)53 b Fn(2)45 b Fk(R)1830 1863 y Fl(d)1917 1923 y Fj(.)957 2219 y Fq(Theorem)52 b(1)g(is)h(the)f(basic)h(to)5 b(ol)51 b(in)i(our)f(deriv)-9 b(ation)52 b(of)g(estimates)g(on)g(op)5 b(erator)52 b(k)-5 b(ernels)52 b(of)758 2418 y(functions)k(of)f(generalized)g(Sc)-5 b(hr\177)-83 b(odinger)57 b(op)5 b(erators.)73 b(The)56 b(pro)5 b(of)55 b(is)h(giv)-5 b(en)55 b(in)h(Section)f(2.)957 2617 y(If)44 b(the)g(energy)g Fo(z)52 b Fq(is)46 b(real)e(and)h(sits)g (in)g(a)g(sp)5 b(ectral)44 b(gap)h(of)f(the)h(op)5 b(erator)43 b Fo(H)13 b Fq(,)48 b(one)c(can)h(obtain)758 2816 y(estimates)51 b(where)g(the)f(rate)h(of)g(the)f(exp)5 b(onen)-5 b(tial)51 b(deca)-5 b(y)51 b(is)h(prop)5 b(ortional)51 b(to)g(the)g(pro)5 b(duct)50 b(of)758 3016 y(the)57 b(square)h(ro)5 b(ots)57 b(of)h(the)f(distances)h(from)g Fo(z)65 b Fq(to)57 b(the)h(edges)g(of)f (the)g(gap,)i(similarly)g(to)e(\(19\);)758 3215 y(see)e([1,)h(13].)957 3414 y(Note)66 b(that)h Fo(m)1888 3439 y Fl(z)2031 3414 y Fq(gro)-5 b(ws)69 b(as)2726 3311 y Fn(p)p 2865 3311 159 7 v 2865 3414 a Fo(\021)2947 3439 y Fl(z)3091 3414 y Fq(as)f Fo(\021)3389 3439 y Fl(z)3532 3414 y Fn(!)e(1)p Fq(,)k(so)e(the)f(the)g(constan)-5 b(t)67 b(in)h(\(16\))f(gro)-5 b(ws)68 b(as)758 3631 y(e)832 3571 y Fl(c)891 3498 y Fi(p)p 1000 3498 137 6 v 73 x Fl(\021)1068 3588 y Fe(z)1145 3631 y Fq(.)97 b(This)64 b(can)g(b)5 b(e)62 b(a)-5 b(v)g(oided)64 b(b)-5 b(y)64 b(pic)-5 b(king)64 b Fo(\015)k Fq(=)59 b(\(2)42 b(+)4355 3527 y Fn(p)p 4493 3527 159 7 v 104 x Fo(\021)4575 3656 y Fl(z)4652 3631 y Fq(\))4717 3571 y Fi(\000)p Fm(1)4895 3631 y Fq(.)97 b(The)64 b(resulting)f(estimate,) 758 3830 y(whic)-5 b(h)56 b(will)f(b)5 b(e)55 b(used)i(in)e(the)g(pro)5 b(of)55 b(of)g(Theorem)h(2,)f(is)h(giv)-5 b(en)56 b(in)g(the)f(follo)-5 b(wing)56 b(corollary)-14 b(.)758 4126 y Ft(Corollary)64 b(1.)83 b Fj(L)-8 b(et)65 b Fo(H)79 b Fj(b)-8 b(e)65 b(a)f(gener)-8 b(alize)g(d)66 b(Schr\177)-85 b(odinger)66 b(op)-8 b(er)g(ator.)92 b(Then,)66 b(with)f(the)h(nota-)758 4325 y(tion)59 b(of)h(The)-8 b(or)g(em)58 b(1,)2314 4643 y Fn(k)p Fo(K)2538 4668 y Fl(n;z)2736 4643 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)45 b(\024)3425 4459 y Fg(\020)3580 4578 y Fm(9)p 3544 4605 137 7 v 3544 4701 a Fl(\021)3612 4718 y Fe(z)3701 4459 y Fg(\021)3800 4494 y Fl(n)3891 4643 y Fq(e)3965 4575 y Fm(\000)4047 4476 y Fi(p)p 4156 4476 70 6 v 99 x Fl(d)4261 4643 y Fq(e)4335 4558 y Fi(\000)4555 4504 y Fe(m)4657 4521 y(z)p 4458 4532 364 6 v 4458 4598 a Fd(2+)4600 4547 y Fc(p)p 4691 4547 131 5 v 51 x Fe(\021)4753 4615 y(z)4842 4558 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)758 4643 y Fq(\(20\))758 4979 y Fj(for)59 b(al)8 b(l)60 b Fo(x;)28 b(y)53 b Fn(2)45 b Fk(R)1830 4918 y Fl(d)1917 4979 y Fj(.)957 5274 y Fq(If)73 b Fo(f)95 b Fn(2)76 b Fo(C)1623 5299 y Fm(0)1697 5274 y Fq(\()p Fk(R)21 b Fq(\))83 b(\(the)72 b(con)-5 b(tin)g(uous)76 b(functions)e(v)-9 b(anishing)74 b(at)f(in\014nit)-5 b(y\),)79 b(it)73 b(follo)-5 b(ws)74 b(from)758 5473 y(the)64 b(Stone-W)-14 b(eierstrass)65 b(theorem)g(and)g(Corollary)f(1)h(that)e(the)i(op)5 b(erator)63 b(k)-5 b(ernel)65 b Fo(K)6264 5498 y Fl(f)6350 5473 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))758 5673 y(of)52 b Fo(f)18 b Fq(\()p Fo(H)13 b Fq(\))52 b(go)5 b(es)52 b(to)f(0)i(as)f Fn(j)p Fo(x)31 b Fn(\000)g Fo(y)6 b Fn(j)46 b(!)g(1)p Fq(.)73 b(But)52 b(this)g(soft)g(argumen)-5 b(t)53 b(giv)-5 b(es)53 b(no)f(estimate)g(on)h(the)758 5872 y(rate)65 b(of)h(deca)-5 b(y)-14 b(.)106 b(In)67 b(fact,)h(it)d(is)i(kno)-5 b(wn)67 b(that)e(for)h Fo(H)78 b Fq(=)64 b Fn(\000)p Fq(\001)i(the)g(k)-5 b(ernel)66 b(of)g Fo(f)18 b Fq(\()p Fo(H)13 b Fq(\))66 b(will)g(not)758 6071 y(ha)-5 b(v)g(e)55 b(rapid)i(deca)-5 b(y)55 b(as)h Fn(j)p Fo(x)37 b Fn(\000)g Fo(y)6 b Fn(j)46 b(!)g(1)56 b Fq(if)f Fo(f)73 b Fq(is)56 b(not)g(smo)5 b(oth)55 b([16,)h(Remark)f(3)h(on)f(p.)75 b(476].)957 6270 y(In)70 b(this)g(article)g(w)-5 b(e)70 b(com)-5 b(bine)71 b(Corollary)f(1)g(with)f(the)h(Hel\013er-Sj\177)-83 b(ostrand)70 b(form)-5 b(ula)71 b([11])758 6470 y(to)77 b(deriv)-5 b(e)77 b(estimates)h(on)g(the)f(deca)-5 b(y)77 b(of)h(op)5 b(erator)77 b(k)-5 b(ernels)78 b(of)f(functions)h(of)f (generalized)758 6669 y(Sc)-5 b(hr\177)-83 b(odinger)52 b(op)5 b(erators.)72 b(The)51 b(existing)g(literature)f(deriv)-5 b(es)51 b(estimates)g(on)h(in)-5 b(tegral)51 b(k)-5 b(ernels)758 6868 y(from)38 b(the)g(heat)h(k)-5 b(ernel)38 b(estimate)g([16)q(,)g (4,)h(5].)68 b(But)38 b(heat)g(k)-5 b(ernels)39 b(estimates)g(are)f (only)h(a)-5 b(v)c(ailable)758 7067 y(for)48 b(Sc)-5 b(hr\177)-83 b(odinger)50 b(op)5 b(erators)49 b([16])g(and)g(acoustic)g (op)5 b(erators)48 b(with)h(constan)-5 b(t)49 b(compressibilit)-5 b(y)758 7267 y([5];)97 b(they)82 b(are)h(not)f(a)-5 b(v)c(ailable)84 b(for)e(Maxw)-5 b(ell)83 b(op)5 b(erators,)90 b(for)83 b(example.)156 b(W)-14 b(e)83 b(sho)-5 b(w)84 b(that)758 7466 y(a)75 b(Com)-5 b(b)5 b(es-Thomas)78 b(estimate)d(as)h(in)g (Corollary)f(1)h(is)g(a)f(natural)h(to)5 b(ol)75 b(to)g(obtain)g(k)-5 b(ernel)758 7665 y(estimates)51 b(for)f(a)h(large)f(class)i(of)e(op)5 b(erators.)72 b(Ev)-5 b(en)51 b(for)f(Sc)-5 b(hr\177)-83 b(odinger)52 b(op)5 b(erators)51 b(p)5 b(olynomial)758 7864 y(deca)-5 b(y)55 b(for)g(the)f(\(in)-5 b(tegral)55 b(or)g(op)5 b(erator\))54 b(k)-5 b(ernel)56 b(of)e Fo(f)18 b Fq(\()p Fo(H)13 b Fq(\))55 b(with)g Fo(f)74 b Fq(an)55 b(arbitrary)g(function)g(in)758 8064 y Fo(C)889 8003 y Fi(1)877 8105 y Fm(0)1029 8064 y Fq(\()p Fk(R)21 b Fq(\))75 b(is)66 b(an)g(op)5 b(en)66 b(question)g(\(see)g([16,)137 b(p.)106 b(476]\).)f(In)66 b(the)g(case)g(when)g(there)f(is)i(a)f(heat) 758 8263 y(k)-5 b(ernel)59 b(estimate,)h(Da)-5 b(vies)59 b([5)q(,)h(Theorem)f(3.4.10])h(pro)-5 b(v)g(ed)60 b(p)5 b(olynomial)60 b(deca)-5 b(y)59 b(of)g(an)-5 b(y)59 b(order)758 8462 y(for)67 b Fo(f)85 b Fn(2)66 b(S)12 b Fq(\()p Fk(R)21 b Fq(\).)120 b(In)67 b(Theorem)h(2)g(w)-5 b(e)67 b(allo)-5 b(w)69 b(an)-5 b(y)67 b(generalized)h(Sc)-5 b(hr\177)-83 b(odinger)69 b(op)5 b(erator)67 b(and)758 8661 y(pro)-5 b(v)g(e)52 b(p)5 b(olynomial)52 b(deca)-5 b(y)52 b(of)f(an)-5 b(y)52 b(order)g(if)g Fo(f)70 b Fq(b)5 b(elongs)52 b(to)f(the)h(m)-5 b(uc)g(h)53 b(larger)e(class)i(of)f(slo)-5 b(wly)758 8861 y(decreasing)70 b(smo)5 b(oth)70 b(functions)g(\(see)f (De\014nition)h(2\).)116 b(A)69 b(particular)h(example)g(of)f(in)-5 b(terest)758 9086 y([10])55 b(is)h Fo(f)18 b Fq(\()p Fo(u)p Fq(\))46 b(=)1803 9006 y Fm(1)p Fi(\000X)2068 9023 y Fe(I)2135 9006 y Fm(\()p Fl(u)p Fm(\))p 1803 9048 515 7 v 1917 9144 a Fl(u)p Fi(\000)p Fl(E)2337 9086 y Fq(,)56 b(where)f Fo(I)k Fn(\032)47 b Fo(\033)6 b Fq(\()p Fo(H)13 b Fq(\))55 b(is)h(a)f(b)5 b(ounded)56 b(op)5 b(en)56 b(in)-5 b(terv)c(al,)55 b Fo(E)g Fn(2)46 b Fo(I)13 b Fq(,)56 b(and)g Fn(X)6659 9111 y Fl(I)758 9289 y Fq(is)51 b(a)h(smo)5 b(oth)51 b(c)-5 b(harateristic)51 b(function)g(for)g(the)g (in)-5 b(terv)c(al)51 b Fo(I)13 b Fq(.)73 b(Note)50 b(that)g(in)i(this) f(case)h(w)-5 b(e)51 b(ha)-5 b(v)g(e)758 9489 y Fo(f)18 b Fq(\()p Fo(H)13 b Fq(\))46 b(=)g(\(1)36 b Fn(\000)h(X)1827 9514 y Fl(I)1904 9489 y Fq(\()p Fo(H)13 b Fq(\)\))p Fo(R)q Fq(\()p Fo(E)d Fq(\))53 b(with)i Fo(E)h Fn(2)46 b Fo(\033)6 b Fq(\()p Fo(H)13 b Fq(\).)957 9688 y(If)55 b Fo(w)50 b Fn(2)c Fk(R)1560 9628 y Fl(n)1715 9688 y Fq(w)-5 b(e)55 b(write)3102 9887 y Fn(h)p Fo(w)t Fn(i)45 b Fq(=)3575 9737 y Fg(p)p 3741 9737 577 7 v 150 x Fq(1)37 b(+)g Fn(j)p Fo(w)t Fn(j)4242 9839 y Fm(2)4345 9887 y Fo(:)758 10183 y Ft(De\014nition)62 b(2.)84 b Fo(f)36 b Fq(:)65 b Fk(R)122 b Fn(!)91 b Fk(C)128 b Fq(is)83 b(called)g(a)f Fj(slow)8 b(ly)85 b(de)-8 b(cr)g(e)g(asing)84 b(smo)-8 b(oth)83 b(function)96 b Fq(if)83 b(it)f(is)758 10382 y(in\014nitely)55 b(di\013eren)-5 b(tiable)56 b(and)g(there)f(exists)g Fo(\015)g(>)47 b Fq(0)55 b(suc)-5 b(h)56 b(that)3014 10684 y Fn(j)p Fo(f)3159 10616 y Fm(\()p Fl(n)p Fm(\))3353 10684 y Fq(\()p Fo(u)p Fq(\))p Fn(j)45 b(\024)4030 10572 y Fo(c)4102 10597 y Fl(n)p 3865 10646 494 7 v 3865 10798 a Fn(h)p Fo(u)p Fn(i)4090 10750 y Fl(\015)7 b Fm(+)p Fl(n)4406 10684 y Fo(;)-3694 b Fq(\(21\))p eop %%Page: 5 5 5 4 bop 758 886 a Fm(OPERA)-11 b(TOR)59 b(KERNEL)g(ESTIMA)-11 b(TES)59 b(F)l(OR)g(FUNCTIONS)f(OF)f(GENERALIZED)j(SCHR)5862 857 y(\177)5844 886 y(ODINGER)f(OPERA)-11 b(TORS)-66 b(5)758 1259 y Fq(for)55 b(some)h(constan)-5 b(ts)56 b Fo(c)2235 1284 y Fl(n)2371 1259 y Fo(<)47 b Fn(1)p Fq(,)55 b(all)h Fo(u)46 b Fn(2)g Fk(R)22 b Fq(,)64 b(and)56 b(all)g Fo(n)45 b Fq(=)i(0)p Fo(;)28 b Fq(1)p Fo(;)g Fq(2)p Fo(;)g(:)g(:)g(:)f Fq(.)957 1585 y(If)54 b Fo(f)72 b Fq(is)54 b(a)g(slo)-5 b(wly)55 b(decreasing)f(smo)5 b(oth)55 b(function,)f(w)-5 b(e)54 b(de\014ne)g(the)g(norms)h([6,)g (eq.)73 b(\(2.1.3\)]:)1811 2036 y Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)2074 2061 y Fl(n)2213 2036 y Fq(=)2467 1828 y Fl(n)2388 1878 y Fg(X)2391 2230 y Fl(r)t Fm(=0)2656 1810 y Fg(Z)2748 2187 y Ff(R)2868 2036 y Fn(j)p Fo(f)3013 1967 y Fm(\()p Fl(r)t Fm(\))3190 2036 y Fq(\()p Fo(u)p Fq(\))p Fn(jh)p Fo(u)p Fn(i)3686 1967 y Fl(r)t Fi(\000)p Fm(1)3928 2036 y Fq(d)p Fo(x)93 b(;)166 b(n)46 b Fq(=)g(1)p Fo(;)28 b Fq(2)p Fo(;)g(:)g(:)g(:)55 b(:)-4592 b Fq(\(22\))758 2505 y Ft(Theorem)63 b(2.)83 b Fj(L)-8 b(et)74 b Fo(H)88 b Fj(b)-8 b(e)74 b(a)f(gener)-8 b(alize)g(d)75 b(Schr\177)-85 b(odinger)75 b(op)-8 b(er)g(ator)73 b(and)h(let)h Fo(f)92 b Fj(b)-8 b(e)74 b(a)g(slow)8 b(ly)758 2704 y(de)-8 b(cr)g(e)g(asing)83 b(smo)-8 b(oth)82 b(function.)148 b(Then)84 b(the)f(op)-8 b(er)g(ator)82 b(kernel)j Fo(K)5045 2729 y Fl(f)5131 2704 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))82 b Fj(of)h Fo(f)18 b Fq(\()p Fo(H)13 b Fq(\))83 b Fj(de)-8 b(c)g(ays)758 2904 y(faster)48 b(than)g(any)g(p)-8 b(olynomial)47 b(as)h Fn(j)p Fo(x)12 b Fn(\000)g Fo(y)6 b Fn(j)47 b(!)f(1)p Fj(.)72 b(Mor)-8 b(e)48 b(pr)-8 b(e)g(cisely,)50 b(ther)-8 b(e)48 b(is)g(a)g(\014nite)h(c)-8 b(onstant)791 3082 y Fg(e)758 3124 y Fo(C)11 b Fj(,)60 b(indep)-8 b(endent)60 b(of)g Fo(f)77 b Fj(and)60 b Fo(H)13 b Fj(,)59 b(such)h(that)g(for)f (al)8 b(l)60 b Fo(k)52 b Fq(=)46 b(1)p Fo(;)28 b Fq(2)p Fo(;)g(:)g(:)g(:)87 b Fj(we)60 b(have)2048 3594 y Fn(k)p Fo(K)2272 3619 y Fl(f)2358 3594 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)45 b(\024)3081 3552 y Fg(e)3048 3594 y Fo(C)39 b Fq(e)3280 3526 y Fm(\000)3362 3426 y Fi(p)p 3471 3426 70 6 v 100 x Fl(d)3548 3594 y Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)3811 3619 y Fl(k)s Fm(+2)4111 3337 y Fg(\012)4189 3471 y Fq(3)p Fo(k)5 b Fq(\(\002)4557 3496 y Fm(2)4669 3471 y Fq(+)37 b(8\)\000)5087 3411 y Fi(\000)p Fm(1)5265 3337 y Fg(\013)5343 3371 y Fl(k)p 4111 3556 1314 7 v 4470 3740 a Fn(h)p Fo(x)f Fn(\000)h Fo(y)6 b Fn(i)4984 3657 y Fl(k)758 3594 y Fq(\(23\))758 4057 y Fj(for)59 b(al)8 b(l)60 b Fo(x;)28 b(y)53 b Fn(2)45 b Fk(R)1830 3997 y Fl(d)1917 4057 y Fj(.)957 4383 y Fq(Theorem)40 b(2)g(is)g(pro)-5 b(v)g(ed)41 b(in)f(Section)g(3.)69 b(The)40 b(pro)5 b(of)39 b(is)i(based)g(on)f(the)f(estimate)h(\(20\))f (and)i(the)758 4583 y(Hel\013er-Sj\177)-83 b(ostrand)60 b(form)-5 b(ula.)90 b(Theorem)61 b(2)g(pro)-5 b(vides)61 b(a)f(crucial)h(ingredien)-5 b(t)61 b(for)f(the)g(results)758 4782 y(in)55 b([10)q(].)957 5040 y Fj(A)-8 b(cknow)8 b(le)-8 b(dgement.)148 b Fq(The)79 b(authors)h(w)-5 b(an)g(t)79 b(to)g(thank)g(V)-14 b(o)9 b(jk)-9 b(an)78 b(Jaksic)h(for)g(calling)h (the)758 5239 y(Hel\013er-Sj\177)-83 b(ostrand)55 b(form)-5 b(ula)56 b(to)f(their)g(atten)-5 b(tion.)2386 5716 y(2.)92 b Fp(The)61 b(Combes-Thomas)g(estima)-11 b(te)957 6014 y Fq(In)50 b(this)g(section)g(w)-5 b(e)50 b(pro)-5 b(v)g(e)51 b(Theorem)f(1.)72 b(W)-14 b(e)49 b(start)h(b)-5 b(y)50 b(de\014ning)h(the)e(op)5 b(erators)50 b(formally)758 6214 y(giv)-5 b(en)55 b(b)-5 b(y)1562 6531 y Fo(H)1700 6556 y Fl(\013)1840 6531 y Fq(=)47 b(e)2090 6462 y Fl(\013)p Fi(\001)p Fl(x)2299 6531 y Fo(H)13 b Fq(e)2524 6462 y Fi(\000)p Fl(\013)p Fi(\001)p Fl(x)2929 6531 y Fq(and)93 b Fo(W)3446 6556 y Fl(\013)3587 6531 y Fq(=)46 b(e)3836 6462 y Fl(\013)p Fi(\001)p Fl(x)4045 6531 y Fo(W)23 b Fq(e)4299 6462 y Fi(\000)p Fl(\013)p Fi(\001)p Fl(x)4640 6531 y Fo(;)120 b Fq(where)92 b Fo(\013)47 b Fn(2)f Fk(R)5743 6462 y Fl(d)5858 6531 y Fq(;)-5146 b(\(24\))758 6849 y(note)55 b(that)3216 7166 y Fo(H)3354 7191 y Fl(\013)3495 7166 y Fq(=)46 b Fo(W)3827 7191 y Fl(\013)3959 7166 y Fq(+)37 b Fo(V)9 b(:)-3519 b Fq(\(25\))957 7484 y(T)-14 b(o)55 b(do)h(so,)g(let)f(us)h(consider)g(the)f(b)5 b(ounded)56 b(op)5 b(erator)55 b Fo(G)4547 7509 y Fl(\013)4641 7484 y Fq(,)g(de\014ned)h(as)2318 7812 y Fo(G)2449 7837 y Fl(\013)2589 7812 y Fq(=)2765 7664 y Fn(p)p 2903 7664 141 7 v 148 x(R)p Fo(D)5 b Fq(\()p Fo(\013)q Fq(\))3423 7664 y Fn(p)p 3560 7664 129 7 v 3560 7812 a(K)30 b Fo(;)120 b Fq(so)92 b Fn(k)p Fo(G)4337 7837 y Fl(\013)4432 7812 y Fn(k)46 b(\024)g(j)p Fo(\013)q Fn(j)28 b Fq(\004)g Fo(;)-4390 b Fq(\(26\))758 8139 y(as)56 b(in)f([13,)h(Theorem)f(3.5].) 75 b(Then)2172 8456 y Fo(A)2297 8481 y Fl(\013)2557 8456 y Fq(=)166 b(e)2926 8388 y Fl(\013)p Fi(\001)p Fl(x)3135 8456 y Fo(A)p Fq(e)3334 8388 y Fi(\000)p Fl(\013)p Fi(\001)p Fl(x)3693 8456 y Fq(=)46 b Fo(A)36 b Fq(+)h Fo(iG)4383 8481 y Fl(\013)4570 8456 y Fq(on)93 b Fn(D)5 b Fq(\()p Fo(A)p Fq(\))28 b Fo(;)-4542 b Fq(\(27\))1966 8706 y(\()p Fo(A)2156 8637 y Fi(\003)2232 8706 y Fq(\))2297 8731 y Fl(\013)2557 8706 y Fq(=)166 b(e)2926 8637 y Fl(\013)p Fi(\001)p Fl(x)3135 8706 y Fo(A)3260 8637 y Fi(\003)3336 8706 y Fq(e)3410 8637 y Fi(\000)p Fl(\013)p Fi(\001)p Fl(x)3769 8706 y Fq(=)46 b Fo(A)4069 8637 y Fi(\003)4182 8706 y Fq(+)37 b Fo(iG)4536 8637 y Fi(\003)4536 8747 y Fl(\013)4723 8706 y Fq(on)92 b Fn(D)5 b Fq(\()p Fo(A)5313 8637 y Fi(\003)5388 8706 y Fq(\))28 b Fo(;)-4769 b Fq(\(28\))758 9023 y(are)81 b(closed,)88 b(densely)82 b(de\014ned)f(op)5 b(erators.)152 b(\(Note)79 b(\()p Fo(A)4398 8962 y Fi(\003)4474 9023 y Fq(\))4539 9048 y Fl(\013)4722 9023 y Fn(6)p Fq(=)90 b(\()p Fo(A)5131 9048 y Fl(\013)5224 9023 y Fq(\))5289 8962 y Fi(\003)5365 9023 y Fq(.\))151 b(W)-14 b(e)81 b(no)-5 b(w)82 b(de\014ne)758 9222 y Fo(W)915 9247 y Fl(\013)1078 9222 y Fq(=)69 b(\()p Fo(A)1466 9162 y Fi(\003)1542 9222 y Fq(\))1607 9247 y Fl(\013)1701 9222 y Fo(A)1826 9247 y Fl(\013)1989 9222 y Fq(as)h(a)f(quadratric)g(form)g(with)g(form) g(domain)h Fn(D)5 b Fq(\()p Fo(A)p Fq(\))67 b(=)i Fn(Q)p Fq(\()p Fo(W)23 b Fq(\),)72 b(i.e.,)g(for)758 9421 y(eac)-5 b(h)55 b Fo(\013)47 b Fn(2)f Fk(R)1552 9361 y Fl(d)1638 9421 y Fq(,)2361 9738 y Fo(W)2518 9763 y Fl(\013)2640 9738 y Fq([)p Fo( )6 b Fq(])46 b(=)h Fn(h)o Fq(\()p Fo(A)3322 9670 y Fi(\003)3398 9738 y Fq(\))3463 9670 y Fi(\003)3463 9779 y Fl(\013)3557 9738 y Fo( )6 b(;)28 b(A)3870 9763 y Fl(\013)3964 9738 y Fo( )6 b Fn(i)27 b Fo(;)167 b( )52 b Fn(2)46 b(D)5 b Fq(\()p Fo(A)p Fq(\))p Fo(:)-4376 b Fq(\(29\))758 10056 y(If)65 b Fo(\013)f Fq(=)g(0,)k Fo(W)1651 10081 y Fm(0)1791 10056 y Fq(is)e(the)g(closed,)j(nonegativ)-5 b(e)65 b(quadratic)h(form)f(asso)5 b(ciated)66 b(to)g(the)f(op)5 b(erator)758 10255 y Fo(W)23 b Fq(.)73 b(W)-14 b(e)55 b(also)h(de\014ne)g(the)f(quadratic)g(form)h Fo(H)3716 10280 y Fl(\013)3856 10255 y Fq(=)47 b Fo(W)4189 10280 y Fl(\013)4320 10255 y Fq(+)37 b Fo(V)92 b Fq(on)56 b Fn(Q)p Fq(\()p Fo(W)23 b Fq(\))35 b Fn(\\)i(Q)p Fq(\()p Fo(V)5869 10195 y Fm(+)5979 10255 y Fq(\).)957 10455 y(It)55 b(follo)-5 b(ws)56 b(from)f(\(27\))g(and)h(\(28\))e(that)1415 10772 y Fo(H)1553 10797 y Fl(\013)1675 10772 y Fq([)p Fo( )6 b Fq(])37 b Fn(\000)g Fo(H)42 b Fq([)p Fo( )6 b Fq(])46 b(=)g Fo(W)2848 10797 y Fl(\013)2970 10772 y Fq([)p Fo( )6 b Fq(])38 b Fn(\000)f Fo(W)50 b Fq([)p Fo( )6 b Fq(])47 b(=)f(2)p Fo(i)28 b Fq(Re)g Fn(h)p Fo(A )6 b(;)28 b(G)4916 10797 y Fl(\013)5009 10772 y Fo( )6 b Fn(i)36 b(\000)h(h)p Fo(G)5586 10797 y Fl(\013)5680 10772 y Fo( )6 b(;)28 b(G)5999 10797 y Fl(\013)6093 10772 y Fo( )6 b Fn(i)28 b Fo(;)-5588 b Fq(\(30\))p eop %%Page: 6 6 6 5 bop 758 861 a Fm(6)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fq(so,)e(using)g(\(26\))f (and)h(\(8\),)e(w)-5 b(e)56 b(get)1208 1664 y Fn(j)p Fo(H)1392 1689 y Fl(\013)1515 1664 y Fq([)p Fo( )6 b Fq(])37 b Fn(\000)g Fo(H)k Fq([)p Fo( )6 b Fq(])p Fn(j)47 b Fq(=)2577 1479 y Fg(\020)2676 1664 y Fq(4)28 b(\()o(Re)g Fn(h)p Fo(A )6 b(;)28 b(G)3584 1689 y Fl(\013)3677 1664 y Fo( )6 b Fn(i)p Fq(\))3921 1580 y Fm(2)4032 1664 y Fq(+)37 b Fn(k)p Fo(G)4412 1689 y Fl(\013)4506 1664 y Fo( )6 b Fn(k)4703 1595 y Fm(4)4778 1479 y Fg(\021)4897 1469 y Fd(1)p 4897 1487 57 6 v 4897 1554 a(2)758 1664 y Fq(\(31\))1540 2039 y Fn(\024)46 b(k)p Fo(G)1929 2064 y Fl(\013)2024 2039 y Fo( )6 b Fn(k)2249 1904 y Fg(\000)2325 2039 y Fq(4)p Fn(k)p Fo(A )g Fn(k)2813 1970 y Fm(2)2924 2039 y Fq(+)37 b Fn(k)p Fo(G)3304 2064 y Fl(\013)3398 2039 y Fo( )6 b Fn(k)3595 1970 y Fm(2)3670 1904 y Fg(\001)3766 1891 y Fd(1)p 3766 1909 V 3766 1976 a(2)3896 2039 y Fn(\024)47 b Fq(2)p Fo(sW)j Fq([)p Fo( )6 b Fq(])37 b(+)4869 1973 y Fm(1)p 4869 2000 67 7 v 4869 2096 a(2)4955 2039 y Fq(\()5040 1973 y Fm(1)p 5040 2000 V 5042 2096 a Fl(s)5163 2039 y Fq(+)g Fo(s)p Fq(\))p Fn(j)p Fo(\013)q Fn(j)5671 1978 y Fm(2)5745 2039 y Fq(\004)5856 1978 y Fm(2)5930 2039 y Fn(k)p Fo( )6 b Fn(k)6210 1978 y Fm(2)1540 2354 y Fn(\024)1839 2289 y Fm(2)p Fl(s)p 1735 2316 337 7 v 1735 2411 a Fm(1)p Fi(\000)p Fm(\002)2007 2428 y Fd(1)2092 2354 y Fo(H)41 b Fq([)p Fo( )6 b Fq(])37 b(+)2681 2170 y Fg(\020)2820 2286 y Fm(2)p Fl(s)p Fm(\002)3051 2303 y Fd(2)p 2800 2316 V 2800 2411 a Fm(1)p Fi(\000)p Fm(\002)3072 2428 y Fd(1)3193 2354 y Fq(+)3379 2289 y Fm(1)p 3379 2316 67 7 v 3379 2411 a(2)3465 2354 y Fq(\()3550 2289 y Fm(1)p 3550 2316 V 3552 2411 a Fl(s)3673 2354 y Fq(+)g Fo(s)p Fq(\))p Fn(j)p Fo(\013)q Fn(j)4181 2294 y Fm(2)4255 2354 y Fq(\004)4366 2294 y Fm(2)4440 2170 y Fg(\021)4567 2354 y Fn(k)p Fo( )6 b Fn(k)4847 2294 y Fm(2)4949 2354 y Fo(;)758 2701 y Fq(for)46 b(an)-5 b(y)46 b Fo(s)g(>)h Fq(0.)71 b(It)46 b(follo)-5 b(ws)47 b([12,)h(Theorem)f(VI.1.33])f(that)g Fo(H)4620 2726 y Fl(\013)4761 2701 y Fq(is)h(a)f(closed)h(sectorial)f (form)h(on)758 2901 y(the)g(form)g(domain)i(of)e Fo(H)13 b Fq(.)72 b(W)-14 b(e)47 b(de\014ne)h Fo(H)3341 2926 y Fl(\013)3483 2901 y Fq(as)g(the)f(unique)h Fo(m)p Fq(-sectorial)f(op) 5 b(erator)47 b(asso)5 b(ciated)758 3100 y(with)55 b(it)g([12,)h (Theorem)f(VI.2.1].)957 3299 y(Let)f Fo(z)73 b(=)-102 b Fn(2)46 b Fo(\033)6 b Fq(\()p Fo(H)13 b Fq(\),)55 b(if)g(0)46 b Fo(<)g(\015)56 b(<)46 b Fq(1)55 b(and)2750 3597 y(\012)46 b(=)h(2)p Fn(k)p Fq(\()p Fo(a)36 b Fq(+)h Fo(bH)13 b Fq(\))p Fo(R)q Fq(\()p Fo(z)7 b Fq(\))p Fn(k)47 b(\024)f Fo(\015)37 b(;)-3957 b Fq(\(32\))758 3895 y(where)2256 4193 y Fo(a)46 b Fq(=)2606 4125 y Fm(2)p Fl(s)p Fm(\002)2837 4142 y Fd(2)p 2586 4155 337 7 v 2586 4250 a Fm(1)p Fi(\000)p Fm(\002)2858 4267 y Fd(1)2979 4193 y Fq(+)3165 4128 y Fm(1)p 3165 4155 67 7 v 3165 4250 a(2)3251 4193 y Fq(\()3336 4128 y Fm(1)p 3336 4155 V 3338 4250 a Fl(s)3459 4193 y Fq(+)37 b Fo(s)p Fq(\))p Fn(j)p Fo(\013)q Fn(j)3967 4133 y Fm(2)4041 4193 y Fq(\004)4152 4133 y Fm(2)4254 4193 y Fo(;)166 b(b)46 b Fq(=)4882 4128 y Fm(2)p Fl(s)p 4778 4155 337 7 v 4778 4250 a Fm(1)p Fi(\000)p Fm(\002)5050 4267 y Fd(1)5163 4193 y Fo(;)-4451 b Fq(\(33\))758 4506 y(it)55 b(follo)-5 b(ws)56 b(from)f([12)q(,)g(Theorem)h(VI.3.9])f(that) g Fo(z)72 b(=)-102 b Fn(2)46 b Fo(\033)6 b Fq(\()p Fo(H)4372 4531 y Fl(\013)4466 4506 y Fq(\))55 b(and)1918 4889 y Fn(k)p Fo(R)q Fq(\()p Fo(z)7 b Fq(\))37 b Fn(\000)g Fo(R)2671 4914 y Fl(\013)2766 4889 y Fq(\()p Fo(z)7 b Fq(\))p Fn(k)46 b(\024)3507 4776 y Fq(4\012)p 3304 4851 610 7 v 3304 5003 a(\(1)36 b Fn(\000)i Fq(\012\))3840 4955 y Fm(2)3933 4889 y Fn(k)p Fo(R)q Fq(\()p Fo(z)7 b Fq(\))p Fn(k)47 b(\024)4978 4776 y Fq(4)p Fo(\015)p 4682 4851 772 7 v 4682 5003 a Fq(\(1)36 b Fn(\000)h Fo(\015)9 b Fq(\))5192 4955 y Fm(2)5294 5003 y Fo(\021)5376 5028 y Fl(z)5501 4889 y Fo(;)-4789 b Fq(\(34\))758 5294 y(where)55 b Fo(R)1364 5319 y Fl(\013)1459 5294 y Fq(\()p Fo(z)7 b Fq(\))45 b(=)i(\()p Fo(H)2097 5319 y Fl(\013)2228 5294 y Fn(\000)37 b Fo(z)7 b Fq(\))2543 5234 y Fi(\000)p Fm(1)2776 5294 y Fq(and)56 b Fo(\021)3181 5319 y Fl(z)3304 5294 y Fq(=)47 b(dist\()p Fo(z)7 b(;)28 b(\033)6 b Fq(\()p Fo(H)13 b Fq(\)\),)55 b(so)2815 5726 y Fn(k)p Fo(R)3024 5751 y Fl(\013)3119 5726 y Fq(\()p Fo(z)7 b Fq(\))p Fn(k)46 b(\024)3637 5491 y Fg(\022)3779 5613 y Fq(1)37 b(+)g Fo(\015)p 3779 5687 382 7 v 3779 5839 a Fq(1)g Fn(\000)g Fo(\015)4180 5491 y Fg(\023)4303 5526 y Fm(2)4463 5613 y Fq(1)p 4425 5687 159 7 v 4425 5839 a Fo(\021)4507 5864 y Fl(z)4631 5726 y Fo(:)-3919 b Fq(\(35\))957 6112 y(Since)2832 6446 y(\012)46 b Fn(\024)3232 6334 y Fq(2)p 3194 6408 V 3194 6560 a Fo(\021)3276 6585 y Fl(z)3400 6446 y Fq(\()p Fo(a)36 b Fq(+)h Fo(b)p Fq(\()p Fo(\021)3973 6471 y Fl(z)4087 6446 y Fq(+)g Fn(j)p Fo(z)7 b Fn(j)p Fq(\)\))28 b Fo(;)-3875 b Fq(\(36\))758 6832 y(\(32\))54 b(will)i(b)5 b(e)55 b(satis\014ed)h(if)2934 7130 y Fo(As)37 b Fq(+)g(\()3425 7064 y Fm(1)p 3424 7091 67 7 v 3426 7187 a Fl(s)3547 7130 y Fq(+)g Fo(s)p Fq(\))p Fn(j)p Fo(\013)q Fn(j)4055 7069 y Fm(2)4175 7130 y Fn(\024)47 b Fo(B)8 b(;)-3773 b Fq(\(37\))758 7428 y(with)2133 7762 y Fo(A)45 b Fq(=)2859 7650 y(4)p 2499 7724 804 7 v 2499 7876 a(\(1)36 b Fn(\000)h Fq(\002)2978 7901 y Fm(1)3053 7876 y Fq(\)\004)3229 7819 y Fm(2)3229 7921 y(2)3322 7762 y Fq(\(\002)3516 7787 y Fm(2)3628 7762 y Fq(+)g Fo(\021)3876 7787 y Fl(z)3989 7762 y Fq(+)h Fn(j)p Fo(z)7 b Fn(j)p Fq(\))28 b Fo(;)166 b(B)54 b Fq(=)5012 7650 y Fo(\015)9 b(\021)5189 7675 y Fl(z)p 5012 7724 255 7 v 5047 7876 a Fq(\004)5158 7828 y Fm(2)5314 7762 y Fo(:)-4602 b Fq(\(38\))758 8160 y(The)55 b(left)g(hand)h(side)g(of)f(\(37\))f(is)i(minimized)h(b)-5 b(y)55 b(the)g(c)-5 b(hoice)3143 8553 y Fo(s)46 b Fq(=)3746 8441 y Fn(j)p Fo(\013)q Fn(j)p 3462 8515 768 7 v 3462 8548 a Fg(p)p 3628 8548 602 7 v 142 x Fo(A)36 b Fq(+)h Fn(j)p Fo(\013)q Fn(j)4154 8642 y Fm(2)4276 8553 y Fo(;)-3564 b Fq(\(39\))758 8981 y(in)55 b(whic)-5 b(h)56 b(case)g(\(37\))e(b)5 b(ecomes)2993 9301 y(2)p Fn(j)p Fo(\013)q Fn(j)3275 9150 y Fg(p)p 3441 9150 V 151 x Fo(A)37 b Fq(+)g Fn(j)p Fo(\013)q Fn(j)3968 9253 y Fm(2)4089 9301 y Fn(\024)46 b Fo(B)36 b(;)-3714 b Fq(\(40\))758 9598 y(i.e.,)2817 9896 y(4)p Fn(j)p Fo(\013)q Fn(j)3099 9827 y Fm(4)3211 9896 y Fq(+)37 b(4)p Fo(A)p Fn(j)p Fo(\013)q Fn(j)3784 9827 y Fm(2)3895 9896 y Fn(\000)g Fo(B)4195 9827 y Fm(2)4316 9896 y Fn(\024)46 b Fq(0)28 b Fo(;)-3890 b Fq(\(41\))758 10194 y(so)56 b(it)f(su\016ces)h(to)f(require)1700 10632 y Fn(j)p Fo(\013)q Fn(j)1899 10564 y Fm(2)2019 10632 y Fn(\024)2215 10567 y Fm(1)p 2215 10594 67 7 v 2215 10689 a(2)2328 10448 y Fg(\020)2428 10632 y Fn(\000)p Fo(A)36 b Fq(+)2884 10469 y Fg(p)p 3050 10469 611 7 v 163 x Fo(A)3175 10584 y Fm(2)3286 10632 y Fq(+)h Fo(B)3586 10584 y Fm(2)3661 10448 y Fg(\021)3806 10632 y Fq(=)4001 10520 y Fo(A)p 4001 10594 125 7 v 4022 10746 a Fq(2)4174 10348 y Fg( )4305 10363 y(r)p 4471 10363 535 7 v 269 x Fq(1)g(+)4777 10520 y Fo(B)4911 10472 y Fm(2)p 4777 10594 209 7 v 4782 10746 a Fo(A)4907 10698 y Fm(2)5043 10632 y Fn(\000)g Fq(1)5292 10348 y Fg(!)5469 10632 y Fn(\021)46 b Fo(\014)9 b(:)-5035 b Fq(\(42\))p eop %%Page: 7 7 7 6 bop 758 886 a Fm(OPERA)-11 b(TOR)59 b(KERNEL)g(ESTIMA)-11 b(TES)59 b(F)l(OR)g(FUNCTIONS)f(OF)f(GENERALIZED)j(SCHR)5862 857 y(\177)5844 886 y(ODINGER)f(OPERA)-11 b(TORS)-66 b(7)758 1262 y Fq(Since)1191 1132 y Fn(p)p 1330 1132 381 7 v 1330 1262 a Fq(1)37 b(+)g Fo(x)46 b Fn(\025)g Fq(1)37 b(+)2238 1197 y Fl(x)p 2238 1224 76 7 v 2243 1319 a Fm(2)2370 1262 y Fn(\000)2556 1197 y Fl(x)2631 1147 y Fd(2)p 2556 1224 141 7 v 2593 1319 a Fm(8)2771 1262 y Fq(for)55 b Fo(x)46 b Fn(\025)h Fq(0,)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)1921 1688 y Fo(\014)175 b Fn(\025)2505 1575 y Fo(B)2639 1515 y Fm(2)p 2505 1650 209 7 v 2506 1802 a Fq(4)p Fo(A)2761 1454 y Fg(\022)2883 1688 y Fq(1)37 b Fn(\000)3226 1575 y Fo(B)3360 1515 y Fm(2)p 3189 1650 283 7 v 3189 1802 a Fq(4)p Fo(A)3397 1754 y Fm(2)3491 1454 y Fg(\023)3660 1688 y Fn(\025)3855 1575 y Fo(B)3989 1515 y Fm(2)p 3855 1650 209 7 v 3856 1802 a Fq(4)p Fo(A)4111 1454 y Fg(\022)4233 1688 y Fq(1)g Fn(\000)4539 1575 y Fo(\015)4634 1515 y Fm(2)4709 1575 y Fq(\(1)f Fn(\000)h Fq(\002)5188 1600 y Fm(1)5263 1575 y Fq(\))5328 1515 y Fm(2)p 4539 1650 863 7 v 4888 1802 a Fq(64)5422 1454 y Fg(\023)758 1688 y Fq(\(43\))2190 2159 y Fn(\025)2505 2047 y Fq(63)p Fo(B)2805 1987 y Fm(2)p 2505 2121 375 7 v 2506 2273 a Fq(256)p Fo(A)2946 2159 y Fq(=)3378 2047 y(63)28 b(\(1)36 b Fn(\000)h Fq(\002)4051 2072 y Fm(1)4126 2047 y Fq(\))p Fo(\015)4286 1987 y Fm(2)4360 2047 y Fo(\021)4448 1987 y Fm(2)4442 2088 y Fl(z)p 3141 2121 1620 7 v 3141 2273 a Fq(1024)28 b(\004)3612 2225 y Fm(2)3686 2273 y Fq(\(\002)3880 2298 y Fm(2)3991 2273 y Fq(+)37 b Fo(\021)4239 2298 y Fl(z)4353 2273 y Fq(+)g Fn(j)p Fo(z)7 b Fn(j)p Fq(\))4826 2159 y(=)46 b Fo(\015)5096 2091 y Fm(2)5171 2159 y Fo(m)5317 2091 y Fm(2)5317 2201 y Fl(z)5421 2159 y Fo(;)758 2560 y Fq(with)55 b Fo(m)1282 2585 y Fl(z)1414 2560 y Fq(as)g(in)h(\(17\).)957 2760 y(Th)-5 b(us,)57 b(if)e(w)-5 b(e)55 b(tak)-5 b(e)55 b Fn(j)p Fo(\013)q Fn(j)46 b(\024)h Fo(\015)9 b(m)2842 2785 y Fl(z)2918 2760 y Fq(,)56 b(w)-5 b(e)55 b(satisfy)g(\(32\))g(and)h(obtain)f (\(35\).)957 2959 y(No)-5 b(w)55 b(let)g Fo(x)1670 2984 y Fm(0)1744 2959 y Fo(;)28 b(y)1899 2984 y Fm(0)2020 2959 y Fn(2)46 b Fk(R)2288 2899 y Fl(d)2374 2959 y Fq(,)56 b Fo(n)46 b Fn(2)f Fk(N)26 b Fq(,)65 b(and)56 b(tak)-5 b(e)54 b Fo(\013)47 b Fq(=)g Fo(\015)4260 2891 y Fl(m)4378 2908 y Fe(z)p 4126 2921 455 7 v 4126 3016 a Fi(j)p Fl(x)4240 3033 y Fd(0)4305 3016 y Fi(\000)p Fl(y)4476 3033 y Fd(0)4541 3016 y Fi(j)4601 2959 y Fq(\()p Fo(x)4761 2984 y Fm(0)4871 2959 y Fn(\000)37 b Fo(y)5118 2984 y Fm(0)5193 2959 y Fq(\).)74 b(W)-14 b(e)55 b(ha)-5 b(v)g(e)1851 3307 y Fo(K)1992 3332 y Fl(n;z)2190 3307 y Fq(\()p Fo(x)2350 3332 y Fm(0)2424 3307 y Fo(;)28 b(y)2579 3332 y Fm(0)2654 3307 y Fq(\))46 b(=)g Fo(\037)3044 3332 y Fl(x)3119 3349 y Fd(0)3192 3307 y Fo(R)q Fq(\()p Fo(z)7 b Fq(\))3533 3238 y Fl(n)3624 3307 y Fo(\037)3728 3332 y Fl(y)3795 3349 y Fd(0)3914 3307 y Fq(=)46 b Fo(\037)4193 3332 y Fl(x)4268 3349 y Fd(0)4342 3307 y Fq(e)4416 3238 y Fi(\000)p Fl(\013)p Fi(\001)p Fl(x)4729 3307 y Fo(R)4855 3332 y Fl(\013)4949 3307 y Fq(\()p Fo(z)7 b Fq(\))5163 3238 y Fl(n)5253 3307 y Fq(e)5327 3238 y Fl(\013)p Fi(\001)p Fl(x)5537 3307 y Fo(\037)5641 3332 y Fl(y)5708 3349 y Fd(0)758 3307 y Fq(\(44\))2017 3576 y(=)167 b(e)2387 3508 y Fi(\000)p Fl(\015)7 b(m)2686 3525 y Fe(z)2753 3508 y Fi(j)p Fl(x)2867 3525 y Fd(0)2933 3508 y Fi(\000)p Fl(y)3104 3525 y Fd(0)3169 3508 y Fi(j)3217 3576 y Fo(\037)3321 3601 y Fl(x)3396 3618 y Fd(0)3469 3576 y Fq(e)3543 3508 y Fi(\000)p Fl(\013)p Fi(\001)p Fm(\()p Fl(x)p Fi(\000)p Fl(x)4078 3525 y Fd(0)4143 3508 y Fm(\))4204 3576 y Fo(R)4330 3601 y Fl(\013)4424 3576 y Fq(\()p Fo(z)g Fq(\))4638 3508 y Fl(n)4728 3576 y Fq(e)4802 3508 y Fl(\013)p Fi(\001)p Fm(\()p Fl(x)p Fi(\000)p Fl(y)5225 3525 y Fd(0)5291 3508 y Fm(\))5351 3576 y Fo(\037)5455 3601 y Fl(y)5522 3618 y Fd(0)5595 3576 y Fo(;)758 3880 y Fq(so)1294 4184 y Fn(k)p Fo(K)1518 4209 y Fl(n;z)1717 4184 y Fq(\()p Fo(x)1877 4209 y Fm(0)1950 4184 y Fo(;)28 b(y)2105 4209 y Fm(0)2180 4184 y Fq(\))p Fn(k)46 b(\024)-1745 b Fq(\(45\))1627 4313 y Fg(\022)1749 4363 y(\020)1869 4474 y Fm(1+)p Fl(\015)p 1868 4509 247 7 v 1868 4605 a Fm(1)p Fi(\000)p Fl(\015)2135 4363 y Fg(\021)2234 4398 y Fm(2)2391 4482 y(1)p 2356 4509 137 7 v 2356 4605 a Fl(\021)2424 4622 y Fe(z)2513 4313 y Fg(\023)2635 4348 y Fl(n)2753 4548 y Fn(k)p Fo(\037)2940 4573 y Fl(x)3015 4590 y Fd(0)3088 4548 y Fq(e)3162 4479 y Fi(\000)p Fl(\013)p Fi(\001)p Fm(\()p Fl(x)p Fi(\000)p Fl(x)3697 4496 y Fd(0)3763 4479 y Fm(\))3823 4548 y Fn(k)3906 4573 y Fi(1)4047 4548 y Fn(k)p Fo(\037)4234 4573 y Fl(y)4301 4590 y Fd(0)4374 4548 y Fq(e)4448 4479 y Fl(\013)p Fi(\001)p Fm(\()p Fl(x)p Fi(\000)p Fl(y)4871 4496 y Fd(0)4937 4479 y Fm(\))4997 4548 y Fn(k)5080 4573 y Fi(1)5221 4548 y Fq(e)5295 4479 y Fi(\000)p Fl(\015)7 b(m)5594 4496 y Fe(z)5661 4479 y Fi(j)p Fl(x)5775 4496 y Fd(0)5841 4479 y Fi(\000)p Fl(y)6012 4496 y Fd(0)6077 4479 y Fi(j)6152 4548 y Fo(:)758 4940 y Fq(Since)2353 5255 y Fn(k)p Fo(\037)2540 5280 y Fl(x)2615 5297 y Fd(0)2688 5255 y Fq(e)2762 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b(o)51 b(estimate)f Fn(k)p Fo(K)3035 8631 y Fl(f)3121 8606 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)51 b Fq(w)-5 b(e)51 b(use)g(\(53\))f(and)i(\(20\),)f(and)g(mak)-5 b(e)51 b(use)h(in)f(\(56\))758 8806 y(and)44 b(\(57\))f(of)g(the)g (fact)g(that)g(the)g(function)h Fo(g)6 b Fq(\()p Fo(s)p Fq(\))45 b(=)4329 8740 y Fl(s)p 3970 8767 782 7 v 3970 8878 a Fm(\()p Fl(a)p Fm(+)4196 8794 y Fi(p)p 4305 8794 63 6 v 84 x Fl(s)o Fm(\))4419 8785 y Fi(p)p 4528 8785 223 6 v 93 x Fl(b)p Fm(+)p Fl(s)4771 8806 y Fq(,)h(with)e Fo(a;)28 b(b)46 b Fn(\025)g Fq(0,)g(is)e(increasing)758 9048 y(for)55 b Fo(s)46 b Fn(\025)g Fq(0.)74 b(Note)54 b(that)h Fo(\021)2357 9073 y Fl(z)2480 9048 y Fn(\025)46 b(j)p Fo(v)6 b Fn(j)56 b Fq(and)g(hence)2308 9445 y Fo(m)2454 9470 y Fl(z)p 2127 9519 584 7 v 2127 9671 a Fq(2)37 b(+)2413 9568 y Fn(p)p 2552 9568 159 7 v 2552 9671 a Fo(\021)2634 9696 y Fl(z)2777 9558 y Fn(\025)3811 9445 y Fq(\000)28 b Fn(j)p Fo(v)6 b Fn(j)p 2972 9519 1988 7 v 2972 9546 a Fg(\020)3071 9730 y Fq(2)37 b(+)3357 9588 y Fg(p)p 3523 9588 179 7 v 142 x Fn(j)p Fo(v)6 b Fn(j)3702 9546 y Fg(\021)3829 9588 y(p)p 3995 9588 965 7 v 142 x Fq(\002)4124 9755 y Fm(2)4235 9730 y Fq(+)37 b Fn(j)p Fo(v)6 b Fn(j)38 b Fq(+)f Fn(j)p Fo(z)7 b Fn(j)5007 9558 y Fo(:)-4295 b Fq(\(56\))957 10220 y(Th)-5 b(us,)57 b(if)e(\()p Fo(u;)28 b(v)6 b Fq(\))45 b Fn(2)h Fo(A)p Fq(,)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)56 b Fn(j)p Fo(v)6 b Fn(j)46 b(\025)h Fq(1)55 b(and)h Fn(j)p Fo(z)7 b Fn(j)47 b(\024)4276 10083 y(p)p 4414 10083 84 7 v 137 x Fq(2)p Fn(j)p Fo(v)6 b Fn(j)p Fq(,)57 b(hence)2234 10606 y Fo(m)2380 10631 y Fl(z)p 2054 10680 584 7 v 2054 10832 a Fq(2)37 b(+)2340 10729 y Fn(p)p 2478 10729 159 7 v 103 x Fo(\021)2560 10857 y Fl(z)2703 10719 y Fn(\025)3428 10606 y Fq(\000)p 2898 10680 1163 7 v 2898 10877 a(3)2981 10714 y Fg(p)p 3147 10714 914 7 v 163 x Fq(\002)3276 10902 y Fm(2)3388 10877 y Fq(+)g(1)g(+)3840 10740 y Fn(p)p 3978 10740 84 7 v 137 x Fq(2)4127 10719 y Fn(\025)4626 10606 y Fq(\000)p 4322 10680 711 7 v 4322 10842 a(3)4405 10714 y Fn(p)p 4544 10714 490 7 v 4544 10842 a Fq(\002)4673 10867 y Fm(2)4784 10842 y Fq(+)g(3)5081 10719 y Fo(;)-4369 b Fq(\(57\))p eop %%Page: 9 9 9 8 bop 758 886 a Fm(OPERA)-11 b(TOR)59 b(KERNEL)g(ESTIMA)-11 b(TES)59 b(F)l(OR)g(FUNCTIONS)f(OF)f(GENERALIZED)j(SCHR)5862 857 y(\177)5844 886 y(ODINGER)f(OPERA)-11 b(TORS)-66 b(9)758 1259 y Fq(so)56 b(for)f(an)-5 b(y)55 b Fo(r)c Fq(=)46 b(0)p Fo(;)28 b Fq(1)p Fo(;)g Fq(2)p Fo(;)g(:)g(:)g(:)82 b Fq(it)55 b(follo)-5 b(ws)57 b(that)1511 1395 y Fg(Z)1603 1772 y Ff(R)1680 1739 y Fd(2)1788 1621 y Fn(j)p Fo(f)1933 1552 y Fm(\()p Fl(r)t Fm(\))2111 1621 y Fq(\()p Fo(u)p Fq(\))p Fn(j)2402 1508 y(j)p Fo(v)6 b Fn(j)2580 1448 y Fl(r)p 2401 1582 253 7 v 2415 1734 a Fn(h)p Fo(u)p Fn(i)2673 1621 y Fo(\037)2777 1646 y Fl(A)2885 1621 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\))p Fn(k)p Fo(K)3494 1646 y Fl(z)3570 1621 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)p Fq(d)p Fo(u)p Fq(d)p Fo(v)-3640 b Fq(\(58\))1843 2090 y Fn(\024)47 b Fq(9e)2176 2021 y Fm(\000)2258 1922 y Fi(p)p 2367 1922 70 6 v 99 x Fl(d)2472 1856 y Fg(\024)2559 1864 y(Z)2652 2241 y Ff(R)2729 2208 y Fd(2)2836 2090 y Fn(j)p Fo(f)2981 2021 y Fm(\()p Fl(r)t Fm(\))3159 2090 y Fq(\()p Fo(u)p Fq(\))p Fn(j)3450 1978 y(j)p Fo(v)6 b Fn(j)3628 1917 y Fl(r)t Fi(\000)p Fm(1)p 3449 2052 423 7 v 3548 2204 a Fn(h)p Fo(u)p Fn(i)3891 2090 y Fo(\037)3995 2115 y Fl(A)4103 2090 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\)d)p Fo(u)p Fq(d)p Fo(v)4853 1856 y Fg(\025)4969 2090 y Fq(e)5043 1971 y Fi(\000)5361 1926 y Fd(\000)p 5166 1944 458 6 v 5166 2035 a(3)5223 1961 y Fi(p)p 5332 1961 292 6 v 74 x Fd(\002)5417 2060 y(2)5482 2035 y(+3)5644 1971 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)1843 2555 y Fn(\024)47 b Fq(9e)2176 2486 y Fm(\000)2258 2387 y Fi(p)p 2367 2387 70 6 v 99 x Fl(d)2472 2321 y Fg(\024)2559 2555 y Fo(t)2619 2580 y Fl(r)2720 2329 y Fg(Z)2813 2706 y Ff(R)2933 2555 y Fn(j)p Fo(f)3078 2486 y Fm(\()p Fl(r)t Fm(\))3255 2555 y Fq(\()p Fo(u)p Fq(\))p Fn(jh)p Fo(u)p Fn(i)3751 2486 y Fl(r)t Fi(\000)p Fm(1)3993 2555 y Fq(d)p Fo(u)4180 2321 y Fg(\025)4296 2555 y Fq(e)4370 2436 y Fi(\000)4688 2391 y Fd(\000)p 4493 2409 458 6 v 4493 2500 a(3)4550 2426 y Fi(p)p 4659 2426 292 6 v 74 x Fd(\002)4744 2525 y(2)4809 2500 y(+3)4971 2436 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)5336 2555 y Fo(;)758 2976 y Fq(where)55 b Fo(t)1298 3001 y Fm(0)1418 2976 y Fq(=)47 b(2)28 b(log)h(2,)56 b Fo(t)2191 3001 y Fl(r)2311 2976 y Fq(=)2506 2910 y Fm(2)2572 2860 y Fe(r)s Fd(+1)p 2506 2937 275 7 v 2611 3033 a Fl(r)2856 2976 y Fq(for)f Fo(r)c Fq(=)46 b(1)p Fo(;)28 b Fq(2)g Fo(:)g(:)g(:)55 b(;)28 b(n)p Fq(.)73 b(W)-14 b(e)55 b(can)g(th)-5 b(us)57 b(conclude)e(that)1775 3416 y Fo(C)1933 3190 y Fg(Z)2025 3567 y Ff(R)2102 3534 y Fd(2)2210 3132 y Fg(\()2423 3208 y Fl(n)2344 3258 y Fg(X)2347 3610 y Fl(r)t Fm(=0)2652 3304 y Fq(1)p 2631 3378 126 7 v 2631 3530 a Fo(r)5 b Fq(!)2777 3416 y Fn(j)p Fo(f)2922 3347 y Fm(\()p Fl(r)t Fm(\))3099 3416 y Fq(\()p Fo(u)p Fq(\))p Fn(j)3390 3304 y(j)p Fo(v)h Fn(j)3568 3243 y Fl(r)p 3390 3378 253 7 v 3404 3530 a Fn(h)p Fo(u)p Fn(i)3662 3132 y Fg(\))3823 3416 y Fo(\037)3927 3441 y Fl(A)4035 3416 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\))p Fn(k)p Fo(K)4644 3441 y Fl(z)4720 3416 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)p Fq(d)p Fo(u)p Fq(d)p Fo(v)-4790 b Fq(\(59\))3427 3913 y Fn(\024)3690 3871 y Fg(b)3657 3913 y Fo(C)3776 3938 y Fm(1)3878 3913 y Fq(e)3952 3844 y Fm(\000)4034 3745 y Fi(p)p 4143 3745 70 6 v 99 x Fl(d)4248 3913 y Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)4511 3938 y Fl(n)4632 3913 y Fq(e)4706 3794 y Fi(\000)5023 3749 y Fd(\000)p 4829 3767 458 6 v 4829 3858 a(3)4886 3785 y Fi(p)p 4995 3785 292 6 v 73 x Fd(\002)5080 3883 y(2)5145 3858 y(+3)5307 3794 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)5672 3913 y Fo(;)758 4209 y Fq(where)1271 4167 y Fg(b)1238 4209 y Fo(C)1357 4234 y Fm(1)1477 4209 y Fq(=)46 b Fo(C)67 b Fq(max)2147 4234 y Fl(r)2268 4141 y(t)2318 4158 y Fe(r)p 2268 4171 118 7 v 2275 4266 a Fl(r)t Fm(!)2452 4209 y Fo(<)46 b Fn(1)p Fq(.)957 4433 y(When)73 b(\()p Fo(u;)28 b(v)6 b Fq(\))76 b Fn(2)g Fo(B)8 b Fq(,)79 b(w)-5 b(e)74 b(ha)-5 b(v)g(e)74 b Fn(j)p Fo(v)6 b Fn(j)77 b(\024)f Fq(2)p Fn(h)p Fo(u)p Fn(i)d Fq(and)h Fn(j)p Fo(z)7 b Fn(j)77 b(\024)4671 4296 y(p)p 4809 4296 84 7 v 137 x Fq(5)p Fn(h)p Fo(u)p Fn(i)f(\024)h Fq(3)p Fn(h)p Fo(u)p Fn(i)p Fq(.)127 b(Using)74 b(these)758 4632 y(inequalities)55 b(in)h(\(56\),)f(w)-5 b(e)55 b(get)g(\(using)h(2)p Fo(ab)46 b Fn(\024)g Fo(a)3792 4572 y Fm(2)3904 4632 y Fq(+)37 b Fo(b)4141 4572 y Fm(2)4271 4632 y Fq(in)55 b(the)g(third)h(inequalit) -5 b(y\))1442 4891 y Fo(m)1588 4916 y Fl(z)p 1262 4965 584 7 v 1262 5117 a Fq(2)37 b(+)1548 5014 y Fn(p)p 1686 5014 159 7 v 103 x Fo(\021)1768 5142 y Fl(z)2031 5003 y Fn(\025)3123 4891 y Fq(\000)28 b Fn(j)p Fo(v)6 b Fn(j)p 2346 4965 1866 7 v 2346 4992 a Fg(\020)2445 5176 y Fq(2)37 b(+)2731 5034 y Fg(p)p 2897 5034 308 7 v 142 x Fq(2)p Fn(h)p Fo(u)p Fn(i)3204 4992 y Fg(\021)3331 5034 y(p)p 3497 5034 714 7 v 142 x Fq(\002)3626 5201 y Fm(2)3738 5176 y Fq(+)g(5)p Fn(h)p Fo(u)p Fn(i)4277 5003 y(\025)5197 4891 y Fq(\000)28 b Fn(j)p Fo(v)6 b Fn(j)p 4472 4965 1759 7 v 4472 5140 a Fq(2)4555 4998 y Fg(p)p 4721 4998 1510 7 v 142 x Fq(\(2)37 b(+)g Fn(h)p Fo(u)p Fn(i)o Fq(\))27 b(\(\002)5582 5165 y Fm(2)5693 5140 y Fq(+)38 b(5)p Fn(h)p Fo(u)p Fn(i)p Fq(\))758 5003 y(\(60\))2031 5587 y Fn(\025)2691 5475 y Fq(\000)28 b Fn(j)p Fo(v)6 b Fn(j)p 2346 5549 1000 7 v 2346 5701 a Fq(\002)2475 5726 y Fm(2)2587 5701 y Fq(+)37 b(2)f(+)i(6)p Fn(h)p Fo(u)p Fn(i)3412 5587 y(\025)3873 5475 y Fq(\000)28 b Fn(j)p Fo(v)6 b Fn(j)p 3607 5549 843 7 v 3607 5701 a Fq(\(\002)3801 5726 y Fm(2)3912 5701 y Fq(+)37 b(8\))p Fn(h)p Fo(u)p Fn(i)4498 5587 y Fo(:)758 5950 y Fq(If)55 b Fo(k)61 b Fq(is)56 b(a)f(p)5 b(ositiv)-5 b(e)55 b(in)-5 b(teger,)56 b(w)-5 b(e)55 b(recall)h(that)e(for)h(all)h Fo(t)46 b Fn(\025)g Fq(0)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)2593 6302 y(e)2667 6234 y Fi(\000)p Fl(t)2875 6302 y Fn(\024)3071 6190 y Fo(b)3142 6215 y Fl(k)p 3071 6264 153 7 v 3077 6416 a Fo(t)3137 6368 y Fl(k)3290 6302 y Fo(;)120 b Fq(where)138 b Fo(b)4090 6327 y Fl(k)4218 6302 y Fq(=)46 b(e)4467 6234 y Fi(\000)p Fl(k)4652 6302 y Fo(k)4743 6234 y Fl(k)4854 6302 y Fo(:)-4142 b Fq(\(61\))758 6624 y(Th)-5 b(us,)56 b(if)g Fo(n)45 b Fn(\025)i Fo(k)42 b Fq(+)37 b(1,)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)1426 6862 y(1)p 1353 6936 229 7 v 1353 7088 a(2)p Fo(n)p Fq(!)1629 6749 y Fg(Z)1722 7126 y Ff(R)1799 7093 y Fd(2)1906 6975 y Fn(j)p Fo(f)2051 6906 y Fm(\()p Fl(n)p Fm(+1\))2414 6975 y Fq(\()p Fo(u)p Fq(\))p Fn(jj)p Fo(v)6 b Fn(j)2863 6906 y Fl(n)2953 6975 y Fo(\037)3057 7000 y Fl(B)3171 6975 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\))p Fn(k)p Fo(K)3780 7000 y Fl(z)3856 6975 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)p Fq(d)p Fo(u)p Fq(d)p Fo(v)-3926 b Fq(\(62\))1665 7475 y Fn(\024)1860 7363 y Fq(9)28 b(e)2045 7303 y Fm(\000)2127 7204 y Fi(p)p 2236 7204 70 6 v 99 x Fl(d)2313 7363 y Fq(\(\002)2507 7388 y Fm(2)2618 7363 y Fq(+)38 b(8\))2933 7303 y Fl(k)3014 7363 y Fo(b)3085 7388 y Fl(k)p 1860 7437 1307 7 v 2013 7589 a Fq(2)p Fo(n)p Fq(!)27 b(\000)2373 7541 y Fl(k)2455 7589 y Fn(j)p Fo(x)37 b Fn(\000)g Fo(y)6 b Fn(j)2932 7541 y Fl(k)3214 7249 y Fg(Z)3307 7627 y Ff(R)3384 7593 y Fd(2)3491 7475 y Fn(j)p Fo(f)3636 7407 y Fm(\()p Fl(n)p Fm(+1\))3999 7475 y Fq(\()p Fo(u)p Fq(\))p Fn(jj)p Fo(v)g Fn(j)4448 7407 y Fl(n)p Fi(\000)p Fl(k)s Fi(\000)p Fm(1)4885 7475 y Fn(h)p Fo(u)p Fn(i)5110 7407 y Fl(k)5191 7475 y Fo(\037)5295 7500 y Fl(B)5410 7475 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\)d)p Fo(u)p Fq(d)p Fo(v)1665 7980 y Fn(\024)1860 7868 y Fq(9e)2017 7808 y Fm(\000)2099 7708 y Fi(p)p 2208 7708 70 6 v 100 x Fl(d)2286 7868 y Fq(\(\002)2480 7893 y Fm(2)2591 7868 y Fq(+)37 b(8\))2905 7808 y Fl(k)2986 7868 y Fo(b)3057 7893 y Fl(k)p 1860 7942 1279 7 v 2054 8094 a Fo(n)p Fq(!\000)2304 8046 y Fl(k)2386 8094 y Fn(j)p Fo(x)g Fn(\000)g Fo(y)6 b Fn(j)2863 8046 y Fl(k)3201 7868 y Fq(2)3284 7808 y Fl(n)p Fi(\000)p Fl(k)p 3179 7942 395 7 v 3179 8094 a Fo(n)36 b Fn(\000)i Fo(k)3621 7754 y Fg(Z)3713 8131 y Ff(R)3833 7980 y Fn(j)p Fo(f)3978 7912 y Fm(\()p Fl(n)p Fm(+1\))4341 7980 y Fq(\()p Fo(u)p Fq(\))p Fn(jh)p Fo(u)p Fn(i)4837 7912 y Fl(n)4925 7980 y Fq(d)p Fo(u)28 b(:)758 8343 y Fq(If)55 b(w)-5 b(e)55 b(no)-5 b(w)56 b(pic)-5 b(k)56 b Fo(n)45 b Fq(=)i Fo(k)42 b Fq(+)37 b(1,)56 b(it)f(follo)-5 b(ws)56 b(that)f(for)g(an)-5 b(y)56 b Fo(k)51 b Fq(=)c(1)p Fo(;)28 b Fq(2)p Fo(;)g(:)g(:)g(:)82 b Fq(w)-5 b(e)55 b(ha)-5 b(v)g(e)2279 8582 y(1)p 2003 8656 636 7 v 2003 8808 a(2\()p Fo(k)42 b Fq(+)37 b(1\)!)2686 8468 y Fg(Z)2778 8845 y Ff(R)2855 8812 y Fd(2)2963 8694 y Fn(j)p Fo(f)3108 8625 y Fm(\()p Fl(k)s Fm(+2\))3462 8694 y Fq(\()p Fo(u)p Fq(\))p Fn(jj)p Fo(v)6 b Fn(j)3911 8625 y Fl(k)s Fm(+1)4161 8694 y Fo(\037)4265 8719 y Fl(B)4379 8694 y Fq(\()p Fo(u;)28 b(v)6 b Fq(\))p Fn(k)p Fo(K)4988 8719 y Fl(z)5064 8694 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)p Fq(d)p Fo(u)p Fq(d)p Fo(v)-5134 b Fq(\(63\))2315 9199 y Fn(\024)2523 9157 y Fg(b)2490 9199 y Fo(C)2609 9224 y Fm(2)2703 9086 y Fq(e)2777 9026 y Fm(\000)2859 8927 y Fi(p)p 2968 8927 70 6 v 99 x Fl(d)3046 9086 y Fq(\(\002)3240 9111 y Fm(2)3351 9086 y Fq(+)37 b(8\))3665 9026 y Fl(k)p 2703 9161 1043 7 v 2852 9313 a Fq(\000)2956 9265 y Fl(k)3038 9313 y Fn(j)p Fo(x)g Fn(\000)g Fo(y)6 b Fn(j)3515 9265 y Fl(k)3794 8973 y Fg(Z)3886 9350 y Ff(R)4006 9199 y Fn(j)p Fo(f)4151 9130 y Fm(\()p Fl(k)s Fm(+2\))4505 9199 y Fq(\()p Fo(u)p Fq(\))p Fn(jh)p Fo(u)p Fn(i)5001 9130 y Fl(k)s Fm(+1)5249 9199 y Fq(d)p Fo(u)28 b(;)758 9608 y Fq(with)1169 9566 y Fg(b)1136 9608 y Fo(C)1255 9633 y Fm(2)1375 9608 y Fq(=)46 b(18)28 b(max)2053 9633 y Fl(k)2308 9539 y(b)2366 9557 y Fe(k)p 2182 9570 383 7 v 2182 9666 a Fm(\()p Fl(k)s Fm(+1\)!)2631 9608 y Fo(<)47 b Fn(1)p Fq(.)957 9836 y(It)55 b(no)-5 b(w)56 b(follo)-5 b(ws)56 b(from)f(\(55\),)g(\(53\),)g(\(59\),)f(\(63\))h(,)g(\(22\),)g (and)h(\(61\),)f(that)f(w)-5 b(e)56 b(ha)-5 b(v)g(e)1455 10223 y Fn(k)p Fo(K)1679 10248 y Fl(f)1765 10223 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)165 b(\024)2728 10181 y Fg(e)2695 10223 y Fo(C)39 b Fq(e)2927 10154 y Fm(\000)3009 10055 y Fi(p)p 3118 10055 70 6 v 99 x Fl(d)3195 10223 y Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)3458 10248 y Fl(k)s Fm(+2)3766 9989 y Fg(\022)3888 10223 y Fq(e)3962 10104 y Fi(\000)4280 10059 y Fd(\000)p 4085 10077 458 6 v 4085 10168 a(3)4142 10095 y Fi(p)p 4251 10095 292 6 v 73 x Fd(\002)4336 10193 y(2)4401 10168 y(+3)4563 10104 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)t Fi(j)4937 10223 y Fq(+)5146 10111 y(\(\002)5340 10136 y Fm(2)5451 10111 y Fq(+)37 b(8\))5765 10050 y Fl(k)p 5123 10185 745 7 v 5123 10337 a Fq(\000)5227 10289 y Fl(k)5309 10337 y Fn(j)p Fo(x)g Fn(\000)g Fo(y)6 b Fn(j)5786 10289 y Fl(k)5888 9989 y Fg(\023)758 10223 y Fq(\(64\))2399 10751 y Fn(\024)167 b Fq(2)2838 10709 y Fg(e)2806 10751 y Fo(C)39 b Fq(e)3038 10682 y Fm(\000)3120 10583 y Fi(p)p 3229 10583 70 6 v 99 x Fl(d)3306 10751 y Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)3569 10776 y Fl(k)s Fm(+2)3869 10493 y Fg(\000)3945 10628 y Fq(3)p Fo(k)5 b Fq(\(\002)4313 10653 y Fm(2)4424 10628 y Fq(+)38 b(8\))o(\000)4842 10568 y Fi(\000)p Fm(1)5020 10493 y Fg(\001)5096 10528 y Fl(k)p 3869 10713 1310 7 v 4244 10865 a Fn(j)p Fo(x)f Fn(\000)g Fo(y)6 b Fn(j)4721 10817 y Fl(k)758 10751 y Fq(\(65\))p eop %%Page: 10 10 10 9 bop 758 861 a Fm(10)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fq(for)i(an)-5 b(y)61 b Fo(k)g Fq(=)55 b(1)p Fo(;)28 b Fq(2)p Fo(;)g(:)g(:)g(:)f Fq(,)62 b(where)2829 1217 y Fg(e)2796 1259 y Fo(C)67 b(<)55 b Fn(1)61 b Fq(is)g(some)g(constan)-5 b(t)61 b(indep)5 b(enden)-5 b(t)62 b(of)e Fo(k)5 b Fq(,)63 b Fo(f)18 b Fq(,)62 b(and)g Fo(H)13 b Fq(.)758 1476 y(Since)56 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)56 b Fn(k)p Fo(K)2044 1501 y Fl(f)2130 1476 y Fq(\()p Fo(x;)28 b(y)6 b Fq(\))p Fn(k)47 b(\024)g Fo(c)p Fn(j)-28 b(j)g(j)p Fo(f)18 b Fn(j)-28 b(j)g(j)3157 1501 y Fl(k)s Fm(+2)3465 1476 y Fq(b)-5 b(y)56 b(\(54\),)f(and)i(min) 4721 1341 y Fg(\010)4818 1476 y Fq(1)p Fo(;)4975 1335 y Fg(\014)4975 1434 y(\014)5050 1410 y Fl(a)p 5050 1437 73 7 v 5057 1533 a(b)5142 1335 y Fg(\014)5142 1434 y(\014)5197 1341 y(\011)5341 1476 y Fn(\024)5537 1395 y Fi(h)p Fl(a)p Fi(i)p 5537 1437 180 7 v 5544 1533 a(h)p Fl(b)p Fi(i)5737 1476 y Fq(,)f(the)f(estimate)758 1740 y(\(23\))f(follo)-5 b(ws)56 b(with)g(some)g(di\013eren)-5 b(t)55 b(constan)-5 b(t)3806 1698 y Fg(e)3773 1740 y Fo(C)11 b Fq(.)957 1940 y(Theorem)55 b(2)h(is)g(pro)-5 b(v)g(en.)3262 2278 y Fp(References)828 2544 y Fs([1])71 b(Barbaroux,)h(J.M.,)i(Com)l(b)t (es,)d(J.M.,)j(Hislop,)c(P)-12 b(.D.:)104 b(Lo)t(calization)63 b(near)j(band)i(edges)f(for)g(random)1048 2710 y(Sc)l(hr\177)-71 b(odinger)46 b(op)t(erators.)g Fh(Helv.)k(Phys.)h(A)-7 b(cta)57 b Fb(70)p Fs(,)48 b(16-43)e(\(1997\))828 2876 y([2])71 b(Com)l(b)t(es,)62 b(J.M.,)j(Hislop,)c(P)-12 b(.D.:)89 b(Lo)t(calization)56 b(for)k(some)e(con)l(tin)l(uous,)63 b(random)d(Hamiltonian)c(in)k(d-)1048 3042 y(dimension.)45 b(J.)j(F)-12 b(unct.)48 b(Anal.)e Fb(124)p Fs(,)i(149-180)d(\(1994\)) 828 3208 y([3])71 b(Com)l(b)t(es,)100 b(J.M.,)j(Thomas,)d(L.:)150 b(Asymptotic)89 b(b)t(eha)l(vior)g(of)h(eigenfunctions)f(for)h(m)l (ulti-particle)1048 3374 y(Sc)l(hr\177)-71 b(odinger)46 b(op)t(erators.)g(Comm)l(un.)h(Math.)g(Ph)l(ys.)h Fb(34)p Fs(,)g(251-270)d(\(1973\))828 3540 y([4])71 b(Da)l(vies,)d(E.B.:)100 b(Kernel)64 b(estimates)e(for)j(functions)f(of)h(second)g(order)g (elliptic)d(op)t(erators.)i(Quart.)h(J.)1048 3706 y(Math.)47 b(Oxford)h(\(2\))e Fb(39)p Fs(,)i(37-46)d(\(1988\))828 3872 y([5])71 b(Da)l(vies,)46 b(E.B.:)63 b Fh(He)-7 b(at)50 b(kernels)h(and)g(sp)-7 b(e)g(ctr)g(al)50 b(the)-7 b(ory)p Fs(.)47 b(Cam)l(bridge)f(Univ)l(ersit)l(y)g(Press,)g(1989)828 4039 y([6])71 b(Da)l(vies,)46 b(E.B.:)63 b Fh(Sp)-7 b(e)g(ctr)g(al)50 b(the)-7 b(ory)51 b(and)g(di\013er)-7 b(ential)50 b(op)-7 b(er)g(ators)p Fs(.)47 b(Cam)l(bridge)f(Univ)l(ersit)l(y)g(Press,)g (1995)828 4205 y([7])71 b(Figotin,)62 b(A.,)i(Klein,)f(A.:)90 b(Lo)t(calization)56 b(of)61 b(classical)56 b(w)l(a)l(v)l(es)k(I:)h 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y(Harp)t(er.)65 b(In)g Fh(Schr\177)-73 b(odinger)67 b(Op)-7 b(er)g(ators)p Fs(,)70 b(H.)65 b(Holden)g(and)h(A.)f(Jensen,)70 b(eds.,)h(pp.)66 b(118-197.)d(Lectures)1048 5865 y(Notes)45 b(in)i(Ph)l(ysics)f Fb(345)p Fs(,)j(Springer-V)-12 b(erlag,)45 b(1989)758 6031 y([12])70 b(Kato,)46 b(T.:)63 b Fh(Perturb)-7 b(ation)49 b(The)-7 b(ory)51 b(for)g(Line)-7 b(ar)50 b(Op)-7 b(er)g(ators)p Fs(.)47 b(Springer-V)-12 b(erlag,)45 b(1976)758 6197 y([13])70 b(Klein,)49 b(A.,)i(Koines,)f(A.:)69 b(A)50 b(general)f(framew)l(ork)g(for)h(lo)t(calization)45 b(of)50 b(classical)c(w)l(a)l(v)l(es:)69 b(I.)50 b(Inhomoge-)1048 6363 y(neous)d(media)e(and)j(defect)e(eigenmo)t(des.)f(Math.)i(Ph)l (ys.)h(Anal.)f(Geom.)e(T)-12 b(o)47 b(app)t(ear)758 6529 y([14])70 b(Klein,)55 b(A.,)i(Koines,)e(A.:)79 b(A)54 b(general)g(framew)l(ork)g(for)g(lo)t(calization)c(of)k(classical)d(w)l (a)l(v)l(es:)78 b(I)t(I.)55 b(Random)1048 6695 y(media.)46 b(In)h(preparation)758 6861 y([15])70 b(Klein,)59 b(A.,)i(Koines,)f (A.,)h(Seifert,)f(M.:)86 b(Generalized)56 b(eigenfunctions)g(for)i(w)l (a)l(v)l(es)g(in)f(inhomogeneous)1048 7027 y(media.)46 b(J.)h(F)-12 b(unct.)48 b(Anal.)e(T)-12 b(o)47 b(app)t(ear)758 7193 y([16])70 b(Simon,)46 b(B.:)63 b(Sc)l(hr\177)-71 b(odinger)47 b(semi-groups.)e(Bull.)h(Amer.)g(Math.)i(So)t(c.)f Fb(7)p Fs(,)g(447-526)e(\(1982\))957 7505 y Fr(UMR)52 b(8524)h(CNRS,)f(UFR)f(de)g(Ma)-9 b(th)3123 7494 y(\023)3123 7505 y(ema)g(tiques,)54 b(Universit)4527 7494 y(\023)4527 7505 y(e)e(de)f(Lille)h(1,)g(F-59655)h(Villeneuve)758 7671 y(d'Ascq)d(C)1369 7660 y(\023)1369 7671 y(edex,)i(France)957 7837 y Fh(E-mail)e(addr)-7 b(ess)11 b Fs(:)62 b Fa (germinet@agat.univ-lille)o(1.)o(fr)957 8119 y Fr(Dep)-9 b(ar)g(tment)44 b(of)g(Ma)-9 b(thema)g(tics,)46 b(University)e(of)g (Calif)m(ornia,)k(Ir)-12 b(vine,)47 b(Ir)-12 b(vine,)46 b(CA)e(92697-3875,)758 8285 y(USA)957 8451 y Fh(E-mail)50 b(addr)-7 b(ess)11 b Fs(:)62 b Fa(aklein@uci.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if 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