Content-Type: multipart/mixed; boundary="-------------0403080948370" This is a multi-part message in MIME format. ---------------0403080948370 Content-Type: text/plain; name="04-68.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-68.comments" E-mail: hajo.leschke@physik.uni-erlangen.de, peter.mueller@physik.uni-goettingen.de ---------------0403080948370 Content-Type: text/plain; name="04-68.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-68.keywords" Schroedinger operators, semigroups, Feynman Kac formula, integral kernels, random Schroedinger operators ---------------0403080948370 Content-Type: application/postscript; name="jfa.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="jfa.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: jfa.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Helvetica Times-Italic Times-Bold RMTMI %%+ MTSY MTEX %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -y1200 jfa %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.03.08:1630 %%BeginProcSet: texc.pro %! 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cleartomark %%EndFont %%BeginFont: MTSY %!PS-AdobeFont-1.1: MTSY 1.1 %%CreationDate: 1993 May 30 16:26:28 % Copyright (c) 1992, 1993 The TeXplorators Corporation % Hinting Copyright (c) 1992, 1993 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.1) readonly def /Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def /FullName (MTSY) readonly def /FamilyName (MathTime) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /MTSY def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 10 /circlemultiply put dup 15 /bullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 66 /openbullet1 put dup 67 /plus put dup 68 /equal put dup 73 /semicolon put dup 79 /circumflex put dup 81 /tilde put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{0 -954 1043 796}readonly def /UniqueID 5018947 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MTEX %!PS-AdobeFont-1.1: MTEX 1.1 %%CreationDate: 1993 Jun 26 10:30:36 % Copyright (c) 1992, 1993 The TeXplorators Corporation % Hinting Copyright (c) 1992, 1993 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.1) readonly def /Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def /FullName (MTEX) readonly def /FamilyName (MathTime) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /MTEX def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 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Fw(x)h Fs(D)31 b Fx(.)s Fw(x)1599 2332 y Fz(1)1639 2317 y Fx(;)14 b(:)g(:)g(:)g(;)j Fw(x)1906 2332 y Ft(d)1951 2317 y Fx(/)32 b Fs(2)f Fv(R)2180 2281 y Ft(d)2231 2317 y FD(.)67 b(The)36 b(Euclidean)h(scalar)g(product)297 2437 y Fw(x)29 b Fs(\001)c Fw(y)31 b FD(:)p Fs(D)628 2362 y Fq(P)733 2389 y Ft(d)744 2465 y(j)8 b Fr(D)p Fz(1)882 2437 y Fw(x)940 2452 y Ft(j)978 2437 y Fw(y)1033 2452 y Ft(j)1089 2437 y FD(of)28 b Fw(x)9 b Fx(;)20 b Fw(y)30 b Fs(2)25 b Fv(R)1534 2401 y Ft(d)1610 2437 y FD(induces)f(the)h(Euclidean)f(norm)h Fp(j)s Fw(x)9 b Fp(j)24 b FD(:)p Fs(D)g Fx(.)s Fw(x)29 b Fs(\001)22 b Fw(x)9 b Fx(/)3264 2401 y Fz(1)p Fu(=)p Fz(2)3372 2437 y FD(.)394 2572 y(W)-8 b(e)48 b(denote)f(the)g(v)n(olume)g(of)h(a)f (Borel)h(subset)f Fx(3)38 b Fs(\022)f Fv(R)2485 2536 y Ft(d)2584 2572 y FD(with)47 b(respect)h(to)f(the)g Fw(d)7 b FD(-)294 2692 y(dimensional)18 b(Lebesgue)i(measure)g(as)g Fp(j)p Fx(3)p Fp(j)g FD(:)p Fs(D)1939 2612 y Fo(R)1985 2726 y Fu(3)2059 2692 y FD(d)13 b Fw(x)29 b Fs(D)2293 2612 y Fo(R)2338 2726 y Fn(R)2391 2706 y Fl(d)2427 2692 y FD(d)13 b Fw(x)2557 2674 y Fx(\037)2625 2707 y Fu(3)2686 2692 y Fx(.)s Fw(x)c Fx(/)p FD(,)21 b(where)3125 2674 y Fx(\037)3194 2707 y Fu(3)3274 2692 y FD(stands)294 2811 y(for)k(the)g(indicator)f(function)g(of)g Fx(3)p FD(.)31 b(In)25 b(particular)l(,)g(if)f Fx(3)h FD(is)f(the)h(strictly)f (positi)n(v)o(e)e(half-line,)294 2931 y Fx(2)j FD(:)p Fs(D)529 2913 y Fx(\037)598 2946 y Fz(])10 b(0)p Fu(;)p Fr(1)p Fz([)813 2931 y FD(denotes)25 b(the)f(left-continuous)g(Hea)n (viside)g(unit-step)g(function.)394 3051 y(The)51 b(Banach)i(space)e(L) 1292 3014 y Ft(p)1334 3051 y Fx(.)p Fv(R)1437 3014 y Ft(d)1488 3051 y Fx(/)p FD(,)70 b Fw(p)43 b Fs(2)c FD([1)p Fx(;)14 b Fs(1)p FD(],)58 b(consists)50 b(of)h(all)g(Borel-measurable) 294 3170 y(comple)o(x-v)n(alued)27 b(functions)50 b Fw(f)d FD(:)27 b Fv(R)1579 3134 y Ft(d)1657 3170 y Fs(!)g Fv(C)55 b FD(which)28 b(are)h(identi\002ed)g(if)f(their)h(v)n(alues)f(dif)n (fer)294 3290 y(only)23 b(on)h(a)g(set)f(of)h(Lebesgue)g(measure)g (zero)g(and)g(which)f(possess)g(a)h(\002nite)g(norm)f Fp(k)f Fw(f)e Fp(k)3352 3305 y Ft(p)3418 3290 y FD(:)p Fs(D)294 3353 y Fq(\000)340 3354 y Fo(R)386 3468 y Fn(R)439 3448 y Fl(d)475 3434 y FD(d)13 b Fw(x)22 b Fp(j)g Fw(f)e Fx(.)s Fw(x)9 b Fx(/)p Fp(j)868 3398 y Ft(p)910 3353 y Fq(\001)956 3376 y Fz(1)p Fu(=)f Ft(p)1105 3434 y Fx(<)31 b Fs(1)p FD(,)38 b(if)47 b Fw(p)35 b Fx(<)c Fs(1)p FD(,)39 b(and)d Fp(k)21 b Fw(f)g Fp(k)2197 3449 y Fr(1)2304 3434 y FD(:)p Fs(D)31 b FD(ess)14 b(sup)2717 3459 y Ft(x)6 b Fr(2)p Fn(R)2851 3439 y Fl(d)2901 3434 y Fp(j)21 b Fw(f)g Fx(.)s Fw(x)9 b Fx(/)p Fp(j)31 b Fx(<)g Fs(1)p FD(,)39 b(if)305 3554 y Fw(p)d Fs(D)c(1)p FD(.)69 b(W)-8 b(e)38 b(recall)h(that)e(L)1373 3517 y Fz(2)1413 3554 y Fx(.)p Fv(R)1516 3517 y Ft(d)1567 3554 y Fx(/)h FD(is)f(a)i (separable)f(Hilbert)f(space)h(with)f(scalar)i(prod-)294 3673 y(uct)d Fp(h)p Fs(\001)p Fx(;)14 b Fs(\001)p Fp(i)36 b FD(gi)n(v)o(en)f(by)h Fp(h)21 b Fw(f)6 b Fx(;)16 b Fw(g)t Fp(i)31 b FD(:)p Fs(D)1462 3593 y Fo(R)1508 3707 y Fn(R)1561 3687 y Fl(d)1611 3673 y FD(d)13 b Fw(x)44 b(f)1811 3637 y Fr(\003)1852 3673 y Fx(.)s Fw(x)9 b Fx(/)16 b Fw(g)t Fx(.)s Fw(x)9 b Fx(/)p FD(.)65 b(Here)37 b(the)g(star)f (denotes)g(comple)o(x)294 3793 y(conjugation)29 b(and)i(the)g(function) 52 b Fw(f)1556 3757 y Fr(\003)1629 3793 y FD(is)30 b(de\002ned)h (pointwise)f(by)52 b Fw(f)2670 3757 y Fr(\003)2711 3793 y Fx(.)s Fw(x)9 b Fx(/)28 b FD(:)p Fs(D)g Fx(.)22 b Fw(f)f Fx(.)s Fw(x)9 b Fx(//)3278 3757 y Fr(\003)3319 3793 y FD(.)49 b(W)-8 b(e)294 3912 y(write)54 b Fw(f)49 b Fs(2)28 b FD(L)790 3862 y Ft(p)782 3939 y Fz(loc)872 3912 y Fx(.)p Fv(R)975 3876 y Ft(d)1027 3912 y Fx(/)p FD(,)33 b(if)53 b Fw(f)1285 3894 y Fx(\037)1354 3927 y Fu(3)1443 3912 y Fs(2)28 b FD(L)1603 3876 y Ft(p)1645 3912 y Fx(.)p Fv(R)1748 3876 y Ft(d)1799 3912 y Fx(/)k FD(for)g(an)o(y)f(bounded)g (Borel)h(set)g Fx(3)d Fs(\032)f Fv(R)3241 3876 y Ft(d)3292 3912 y FD(.)52 b(The)294 4032 y(uniform)27 b(local)h(Lebesgue)g(spaces) g(L)1632 3981 y Ft(p)1624 4059 y Fz(unif)p Fu(;)p Fz(loc)1850 4032 y Fx(.)p Fv(R)1953 3996 y Ft(d)2004 4032 y Fx(/)g FD(consist)f(of)h(all)f(those)49 b Fw(f)f Fs(2)26 b FD(L)3106 3981 y Ft(p)3098 4059 y Fz(loc)3188 4032 y Fx(.)p Fv(R)3291 3996 y Ft(d)3342 4032 y Fx(/)i FD(for)294 4152 y(which)36 b(sup)715 4177 y Ft(x)6 b Fr(2)p Fn(Z)848 4156 y Fl(d)895 4152 y Fp(k)22 b Fw(f)1015 4134 y Fx(\037)1084 4167 y Fu(3)1140 4177 y Fk(1)1169 4167 y Fu(.)r Ft(x)6 b Fu(/)1265 4152 y Fp(k)1323 4167 y Ft(p)1396 4152 y Fx(<)32 b Fs(1)p FD(.)65 b(The)37 b(Kato)f(class)h([28,)f(2,)h(48)o(,)g(23])f(o)o(v)o (er)g Fv(R)3264 4115 y Ft(d)3352 4152 y FD(may)294 4271 y(be)i(de\002ned)g(as)f(the)g(v)o(ector)g(space)h Fj(K)31 b Fx(.)p Fv(R)1806 4235 y Ft(d)1857 4271 y Fx(/)h FD(:)p Fs(D)2063 4190 y Fq(\010)2143 4271 y Fw(f)52 b Fs(2)32 b FD(L)2379 4235 y Fz(1)2379 4298 y(loc)2469 4271 y Fx(.)p Fv(R)2572 4235 y Ft(d)2623 4271 y Fx(/)g FD(:)g(lim)2885 4286 y Ft(t)6 b Fr(#)p Fz(0)3007 4271 y Fv({)3073 4286 y Ft(t)3104 4271 y Fx(.)22 b Fw(f)f Fx(/)32 b Fs(D)g FD(0)3441 4190 y Fq(\011)3499 4271 y FD(,)294 4411 y(where)48 b Fv({)651 4426 y Ft(t)682 4411 y Fx(.)22 b Fw(f)f Fx(/)37 b FD(:)p Fs(D)g FD(sup)1148 4436 y Ft(x)6 b Fr(2)p Fn(R)1282 4416 y Fl(d)1332 4331 y Fo(R)1397 4358 y Ft(t)1377 4445 y Fz(0)1427 4411 y FD(d)k Fw(s)1559 4331 y Fo(R)1605 4445 y Fn(R)1658 4425 y Fl(d)1694 4411 y FD(d)g Fx(\030)25 b FD(e)1865 4375 y Fr(\000)p Fi(j)p Fu(\030)8 b Fi(j)2005 4350 y Fk(2)2039 4411 y Fp(j)22 b Fw(f)f Fx(.)s Fw(x)36 b Fs(C)28 b Fx(\030)2417 4339 y Fs(p)p 2502 4339 45 5 v 2502 4411 a Fw(s)6 b Fx(/)p Fp(j)p FD(.)98 b(It)47 b(obe)o(ys)g(the)g(inclu-)294 4531 y(sion)36 b Fj(K)31 b Fx(.)p Fv(R)722 4495 y Ft(d)773 4531 y Fx(/)h Fs(\022)f FD(L)1012 4494 y Fz(1)1012 4558 y(unif)p Fu(;)p Fz(loc)1237 4531 y Fx(.)p Fv(R)1340 4495 y Ft(d)1392 4531 y Fx(/)36 b FD(with)g(equality)f(if)h Fw(d)j Fs(D)31 b FD(1.)65 b(W)-8 b(e)37 b(say)f(that)58 b Fw(f)f FD(belongs)35 b(to)294 4651 y Fj(K)385 4666 y Fz(loc)476 4651 y Fx(.)p Fv(R)579 4614 y Ft(d)630 4651 y Fx(/)p FD(,)g(if)56 b Fw(f)892 4633 y Fx(\037)961 4666 y Fu(3)1051 4651 y Fs(2)29 b Fj(K)j Fx(.)p Fv(R)1369 4614 y Ft(d)1420 4651 y Fx(/)h FD(for)h(an)o(y)f(bounded)g(Borel)h(set)f Fx(3)d Fs(\032)f Fv(R)2874 4614 y Ft(d)2925 4651 y FD(.)57 b(Moreo)o(v)o(er)l(,)f Fw(f)294 4770 y FD(is)29 b(called)h(Kato)g(decomposable,)g(in)f (symbols)50 b Fw(f)e Fs(2)27 b Fj(K)2256 4785 y Fr(\006)2316 4770 y Fx(.)p Fv(R)2419 4734 y Ft(d)2470 4770 y Fx(/)p FD(,)k(if)f(sup)o Fs(f)p FD(0)p Fx(;)35 b Fw(f)21 b Fs(g)27 b(2)h Fj(K)3242 4785 y Fz(loc)3332 4770 y Fx(.)p Fv(R)3435 4734 y Ft(d)3486 4770 y Fx(/)294 4890 y FD(and)i(sup)o Fs(f)p FD(0)p Fx(;)14 b Fs(\000)21 b Fw(f)g Fs(g)27 b(2)g Fj(K)k Fx(.)p Fv(R)1267 4854 y Ft(d)1318 4890 y Fx(/)p FD(.)45 b(Finally)-6 b(,)29 b Fj(C)1839 4853 y Fr(1)1822 4916 y Fz(0)1916 4890 y Fx(.)p Fv(R)2019 4854 y Ft(d)2070 4890 y Fx(/)g FD(is)g(the)g(v)o(ector)g(space)h(of)f(all)h(functions) 316 5009 y Fw(f)46 b FD(:)24 b Fv(R)508 4973 y Ft(d)584 5009 y Fs(!)h Fv(C)51 b FD(which)25 b(are)g(arbitrarily)g(often)g(dif)n (ferentiable)f(and)h(ha)n(v)o(e)g(compact)g(supports)294 5129 y(supp)c Fw(f)g FD(.)394 5249 y(The)36 b(absolute)f(v)n(alue)g(of) h(a)h(closed)e(operator)43 b Fw(F)d FD(:)31 b(dom)o Fx(.)7 b Fw(F)i Fx(/)32 b Fs(!)e FD(L)2788 5212 y Fz(2)2828 5249 y Fx(.)p Fv(R)2931 5212 y Ft(d)2982 5249 y Fx(/)p FD(,)39 b(with)d(dense)294 5368 y(domain)21 b(of)h(de\002nition)g(dom)o Fx(.)7 b Fw(F)i Fx(/)22 b Fs(\022)g FD(L)1636 5332 y Fz(2)1676 5368 y Fx(.)p Fv(R)1779 5332 y Ft(d)1830 5368 y Fx(/)h FD(and)e(Hilbert)h(adjoint)28 b Fw(F)2736 5332 y Fr(\003)2777 5368 y FD(,)23 b(is)e(the)h(positi)n(v)o(e)e(op-)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)259 b FD(5)294 384 y(erator)21 b Fp(j)7 b Fw(F)i Fp(j)20 b FD(:)p Fs(D)g Fx(.)7 b Fw(F)940 348 y Fr(\003)989 384 y Fw(F)i Fx(/)1096 348 y Fz(1)p Fu(=)p Fz(2)1203 384 y FD(.)29 b(The)21 b(\(uniform\))f(norm)f(of)i(a)f(bounded)g(operator)27 b Fw(F)j FD(:)20 b(L)3162 348 y Fz(2)3202 384 y Fx(.)p Fv(R)3305 348 y Ft(d)3356 384 y Fx(/)h Fs(!)294 503 y FD(L)355 467 y Fz(2)395 503 y Fx(.)p Fv(R)498 467 y Ft(d)549 503 y Fx(/)k FD(is)g(de\002ned)g(as)g Fp(k)7 b Fw(F)i Fp(k)25 b FD(:)p Fs(D)f FD(sup)1613 423 y Fq(\010)1671 503 y Fp(k)7 b Fw(F)19 b(f)i Fp(k)1907 518 y Fz(2)1985 503 y FD(:)60 b Fw(f)46 b Fs(2)24 b FD(L)2295 467 y Fz(2)2335 503 y Fx(.)p Fv(R)2438 467 y Ft(d)2489 503 y Fx(/)14 b(;)g Fp(k)22 b Fw(f)e Fp(k)2757 518 y Fz(2)2822 503 y Fs(D)25 b FD(1)2975 423 y Fq(\011)3033 503 y FD(.)396 757 y(D)t FG(E)t(FI)t(N)t(I)t(T)t(I)t(O)t(N)34 b FD(1)t(.)t(1)t(.)125 b(Let)35 b Fw(d)i Fs(2)30 b Fv(N)9 b FD(.)66 b(A)35 b Fw(vector)g(potential)43 b(A)36 b FD(is)e(a)h(Borel-measurable,)294 877 y Fv(R)360 841 y Ft(d)411 877 y FD(-v)n(alued)i(function)f(on)h Fv(R)1317 841 y Ft(d)1405 877 y FD(and)g(a)g Fw(scalar)f(potential)i(V) 52 b FD(is)37 b(a)g(Borel-measurable,)j Fv(R)5 b FD(-)294 997 y(v)n(alued)24 b(function)g(on)h Fv(R)1130 960 y Ft(d)1181 997 y FD(.)31 b(Furthermore,)357 1161 y(\()p Fh(A)p FD(\))60 b(a)33 b(v)o(ector)f(potential)41 b Fw(A)33 b FD(is)f(said)g(to)f(satisfy)h(property)g(\()p Fh(A)p FD(\))q(,)h(if)g(both)e(its)g(square)h Fp(j)10 b Fw(A)r Fp(j)3484 1125 y Fz(2)543 1281 y FD(and)28 b(its)f(di)n(v)o(er)n(gence) g Fs(r)h(\001)i Fw(A)g FD(lie)d(in)g(the)h(intersection)e(L)2478 1244 y Fz(2)2478 1307 y(loc)2569 1281 y Fx(.)p Fv(R)2672 1245 y Ft(d)2723 1281 y Fx(/)21 b Fs(\\)f Fj(K)2959 1296 y Fz(loc)3049 1281 y Fx(.)p Fv(R)3152 1245 y Ft(d)3203 1281 y Fx(/)p FD(.)40 b(Here,)543 1400 y Fs(r)e(D)30 b Fx(.@)842 1415 y Fz(1)881 1400 y Fx(;)14 b(:)g(:)g(:)h(;)f(@)1145 1415 y Ft(d)1190 1400 y Fx(/)34 b FD(stands)g(for)h(the)g(gradient,)h (which)f(is)f(supposed)f(to)i(act)f(in)h(the)543 1520 y(sense)25 b(of)g(distrib)n(utions)e(on)h Fj(C)1621 1483 y Fr(1)1604 1547 y Fz(0)1698 1520 y Fx(.)p Fv(R)1801 1484 y Ft(d)1852 1520 y Fx(/)p FD(.)352 1684 y(\()p Fh(C)p FD(\))60 b(a)32 b(v)o(ector)e(potential)40 b Fw(A)32 b FD(is)f(said)f(to)h(satisfy)f(property)g(\()p Fh(C)p FD(\))q(,)j(if)d(there)h(e)o(xist)f(real)h(con-)543 1804 y(stants)h Fw(B)875 1819 y Ft(j)8 b(k)966 1804 y Fs(D)25 b(\000)8 b Fw(B)1216 1819 y Ft(k)f(j)1285 1804 y FD(,)25 b(where)41 b Fw(j)5 b Fx(;)14 b Fw(k)30 b Fs(2)25 b(f)p FD(1)p Fx(;)14 b(:)g(:)g(:)g(;)g Fw(d)7 b Fs(g)p FD(,)25 b(such)g(that)1603 2129 y Fw(A)1666 2144 y Ft(k)1705 2129 y Fx(.)s Fw(x)9 b Fx(/)26 b Fs(D)1975 2059 y FD(1)p 1975 2106 50 4 v 1975 2199 a(2)2117 2004 y Ft(d)2065 2034 y Fq(X)2084 2247 y Ft(j)8 b Fr(D)p Fz(1)2225 2129 y Fw(x)2283 2144 y Ft(j)2337 2129 y Fw(B)2409 2144 y Ft(j)g(k)3333 2129 y FD(\(1.1\))543 2469 y(for)23 b(all)h Fw(x)32 b Fs(2)22 b Fv(R)1032 2432 y Ft(d)1106 2469 y FD(and)g(all)f Fw(k)29 b Fs(2)22 b(f)p FD(1)p Fx(;)14 b(:)g(:)g(:)g(;)g Fw(d)7 b Fs(g)p FD(.)30 b(In)22 b(other)g(w)o(ords,) 32 b Fw(A)24 b FD(generates)e(a)h(spatially)543 2588 y(constant)37 b(magnetic)g(\002eld)g(gi)n(v)o(en)f(by)h(the)h(sk)o(e)n (w-symmetric)d Fw(d)d Fs(\002)24 b Fw(d)7 b FD(-matrix)37 b(with)543 2708 y(entries)c Fw(B)914 2723 y Ft(j)8 b(k)1005 2708 y Fs(D)24 b Fx(@)1164 2723 y Ft(j)1206 2708 y Fw(A)1269 2723 y Ft(k)1328 2708 y Fs(\000)19 b Fx(@)1471 2723 y Ft(k)1521 2708 y Fw(A)1595 2723 y Ft(j)1627 2708 y FD(.)357 2872 y(\()p Fh(V)p FD(\))60 b(a)30 b(scalar)f(potential)i Fw(V)43 b FD(is)29 b(said)f(to)h(satisfy)f(property)g(\()p Fh(V)p FD(\))q(,)i(if)f(it)f(can)h(be)g(written)f(as)h(a)543 2992 y(sum)1774 3216 y Fw(V)40 b Fs(D)27 b Fw(V)2038 3231 y Fz(1)2098 3216 y Fs(C)22 b Fw(V)2256 3231 y Fz(2)3333 3216 y FD(\(1.2\))543 3440 y(with)28 b Fw(V)807 3455 y Fz(1)871 3440 y FD(being)d(locally)f(square-inte)o(grable)g(and)h (Kato)g(decomposable,)1511 3664 y Fw(V)1569 3679 y Fz(1)1634 3664 y Fs(2)g FD(L)1783 3623 y Fz(2)1783 3691 y(loc)1873 3664 y Fx(.)p Fv(R)1976 3623 y 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y(whose)e(proof)g(is)f(an)h(application)f(of)h(Thm.)f(2.5)g (in)h([24].)396 1891 y(P)t FG(R)q(O)t(P)t(O)t(S)t(I)5 b(T)t(I)g(O)t(N)35 b FD(1)t(.)t(3)t(.)125 b Fw(Let)40 b(A)31 b(be)e(a)h(vector)f(potential)f(with)h(pr)l(operty)f FD(\()p Fh(A)p FD(\))i Fw(and)f(let)j(V)294 2011 y(be)25 b(a)g(scalar)f(potential)f(with)i(pr)l(operty)f FD(\()p Fh(V)p FD(\))q Fw(.)30 b(Then)c(the)e(dif)n(fer)l(ential)g(oper)o(ator) 1047 2322 y Fj(C)1130 2281 y Fr(1)1113 2349 y Fz(0)1207 2322 y Fx(.)p Fv(R)1310 2281 y Ft(d)1361 2322 y Fx(/)h Fs(3)g Fx(')k Fs(7!)1743 2252 y FD(1)p 1743 2299 50 4 v 1743 2393 a(2)1870 2197 y Ft(d)1818 2227 y Fq(X)1838 2441 y Ft(j)8 b Fr(D)p Fz(1)1962 2322 y Fx(.)f FD(i)g Fx(@)2098 2337 y Ft(j)2149 2322 y Fs(C)2278 2301 y(O)2257 2322 y Fw(A)2331 2337 y Ft(j)2362 2322 y Fx(/)2399 2281 y Fz(2)2453 2322 y Fx(')24 b Fs(C)2659 2301 y(O)2634 2322 y Fw(V)15 b Fx(')567 b FD(\(1.5\))294 2663 y Fw(is)28 b(essentially)g(self-adjoint)f(on)h FD(L)1511 2627 y Fz(2)1551 2663 y Fx(.)p Fv(R)1654 2627 y Ft(d)1705 2663 y Fx(/)p Fw(.)42 b(Her)l(e)36 b FD(i)e Fs(D)2207 2584 y(p)p 2291 2584 128 5 v 79 x(\000)p FD(1)28 b Fw(denotes)h(the)f(ima)o (ginary)f(unit)294 2783 y(and)35 b(a)g(superposed)g(hat)f(on)h(a)h (function)e(indicates)g(the)h(corr)l(esponding)f(multiplication)294 2902 y(oper)o(ator)-11 b(.)396 3157 y FD(D)t FG(E)t(FI)t(N)t(I)t(T)t(I) t(O)t(N)34 b FD(1)t(.)t(4)t(.)125 b(The)35 b(self-adjoint)g(closure)g (of)g(\(1.5\))g(on)g(L)2761 3120 y Fz(2)2800 3157 y Fx(.)p Fv(R)2904 3120 y Ft(d)2955 3157 y Fx(/)g FD(is)g(called)g(the)294 3276 y Fw(\(ma)o(gnetic\))25 b(Sc)o(hr\366ding)o(er)e(oper)o(ator)j FD(and)e(denoted)h(by)33 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)p FD(.)394 3530 y(As)31 b(suggested)f(in)h([43],) i(we)f(introduce)e(v)o(ector)h(spaces)h(of)f(L)2602 3494 y Ft(p)2644 3530 y Fx(.)p Fv(R)2747 3494 y Ft(d)2798 3530 y Fx(/)p FD(-functions)g(with)f(a)294 3650 y(decay)21 b(at)f(in\002nity)f(which)h(is)g(f)o(aster)g(than)g(that)g(of)g(some)f 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FD(by)g Fx(\026)2761 2396 y Fz(0)p Fu(;)p Ft(t)2763 2459 y(x)6 b Fu(;)t Ft(y)2868 2432 y FD(.)43 b(Gi)n(v)o(en)28 b Fw(t)36 b Fx(>)27 b FD(0,)j(a)294 2552 y(v)o(ector)h(potential)40 b Fw(A)33 b FD(with)d(property)g(\()p Fh(A)p FD(\))i(and)f(a)h(scalar)f(potential)i Fw(V)46 b FD(with)30 b(property)h(\()p Fh(V)p FD(\))q(,)294 2671 y(then)25 b(the)g Fw(Euclidean)f(action)g(functional)350 2961 y(S)398 2976 y Ft(t)429 2961 y Fx(.)10 b Fw(A)5 b Fx(;)17 b Fw(V)d Fs(I)g Fw(b)r Fx(/)24 b FD(:)p Fs(D)32 b FD(i)1016 2825 y Fo(Z)1114 2852 y Ft(t)1069 3050 y Fz(0)1145 2961 y FD(d)10 b Fw(b)r Fx(.)p Fw(s)c Fx(/)19 b Fs(\001)30 b Fw(A)r Fx(.)p Fw(b)r Fx(.)p Fw(s)6 b Fx(//)19 b Fs(C)1901 2891 y FD(i)p 1890 2938 50 4 v 1890 3031 a(2)1979 2825 y Fo(Z)2077 2852 y Ft(t)2032 3050 y Fz(0)2108 2961 y FD(d)10 b Fw(s)31 b Fx(.)p Fs(r)26 b(\001)j Fw(A)r Fx(/.)p Fw(b)r Fx(.)p Fw(s)6 b Fx(//)20 b Fs(C)2891 2825 y Fo(Z)2989 2852 y Ft(t)2943 3050 y Fz(0)3020 2961 y FD(d)10 b Fw(s)33 b(V)15 b Fx(.)p Fw(b)r Fx(.)p 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Fx(.)p FD(d)k Fw(b)r Fx(/)974 4113 y Fq(\014)974 4172 y(\014)974 4232 y(\014)974 4292 y(\014)1008 4122 y Fo(Z)1106 4148 y Ft(t)1060 4346 y Fz(0)1137 4257 y FD(d)g Fw(s)33 b(V)1327 4272 y Fz(2)1367 4257 y Fx(.)p Fw(b)r Fx(.)p Fw(s)6 b Fx(//)1612 4113 y Fq(\014)1612 4172 y(\014)1612 4232 y(\014)1612 4292 y(\014)1670 4257 y Fs(\024)25 b Fw(t)9 b Fx(v)1856 4272 y Fu(")1914 4257 y Fs(C)19 b Fx(")2073 4122 y Fo(Z)2171 4148 y Ft(t)2126 4346 y Fz(0)2202 4257 y FD(d)10 b Fw(s)2320 4122 y Fo(Z)2419 4257 y Fx(\026)2487 4216 y Fz(0)p Fu(;)p Ft(t)2489 4284 y(x)c Fu(;)t Ft(y)2593 4257 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b Fp(j)p Fw(b)r Fx(.)p Fw(s)6 b Fx(/)p Fp(j)3031 4216 y Fz(2)3095 4257 y Fx(<)25 b Fs(1)14 b Fx(:)3283 4459 y FD(\(1.10\))294 4690 y(The)21 b(latter)f(is)g(v)n(alid)f(for)i(all)f Fx(")k(>)d FD(0)f(and)h(relies)f(on)g(\(1.4\),)h(Fubini')-5 b(s)20 b(theorem)g(and)g(an)h(e)o(xplicit)294 4810 y(computation.)51 b(As)32 b(to)f(the)h(applicability)f(of)h(\(1.4\))g(in)g(this)f (estimate,)i(we)f(ha)n(v)o(e)g(used)g(the)294 4930 y(basic)g(f)o(act)f (that)g(for)h Fx(\026)1107 4893 y Fz(0)p Fu(;)p Ft(t)1109 4956 y(x)6 b Fu(;)t Ft(y)1214 4930 y FD(-almost)30 b(e)n(v)o(ery)h (path)g Fw(b)i FD(of)f(the)f(Bro)n(wnian)g(bridge)g(the)g(set)h Fs(f)p Fw(s)i Fs(2)294 5049 y FD([0)p Fx(;)14 b Fw(t)9 b FD(])35 b(:)h Fw(b)r Fx(.)p Fw(s)6 b Fx(/)21 b Fs(2)g Fx(3)p Fs(g)h FD(of)f(time)g(instances,)g(for)g(which)g Fw(b)i FD(stays)e(in)g(a)g(gi)n(v)o(en)f(Lebesgue-null)h(set)294 5169 y Fx(3)k Fs(\032)g Fv(R)567 5133 y Ft(d)618 5169 y FD(,)f(is)h(itself)f(of)g(Lebesgue)h(measure)g(zero)g(in)f([0)p Fx(;)14 b Fw(t)9 b FD(],)24 b(that)g(is,)2692 5089 y Fo(R)2757 5115 y Ft(t)2737 5203 y Fz(0)2801 5169 y FD(d)10 b Fw(s)2930 5151 y Fx(\037)2999 5184 y Fu(3)3059 5088 y Fq(\000)3105 5169 y Fw(b)r Fx(.)p Fw(s)c Fx(/)3276 5088 y Fq(\001)3346 5169 y Fs(D)25 b FD(0.)294 5288 y(W)-8 b(e)26 b(will)e(mak)o(e)g(use)h(of)g(this)f(f)o(act)h(in)g(the)f(follo) n(wing)f(without)h(further)h(notice.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 294 90 a FD(8)893 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)396 384 y FD(L)t FG(E)t(M)t(M)t(A)33 b FD(1)t(.)t(7)t(.)125 b Fw(Let)43 b(A)34 b(be)f(a)f(vector)h (potential)e(with)i(pr)l(operty)e FD(\()p Fh(A)p FD(\))j Fw(and)e(let)j(V)48 b(be)33 b(a)294 503 y(scalar)24 b(potential)g(with) g(pr)l(operty)g FD(\()p Fh(V)p FD(\))q Fw(.)31 b(F)l(inally)-5 b(,)22 b(let)j(t)34 b Fx(>)24 b FD(0)p Fw(.)31 b(Then)549 645 y FD(\(i\))100 b Fw(the)24 b(function)g(k)1287 660 y Ft(t)1343 645 y FD(:)h Fv(R)1462 609 y Ft(d)1532 645 y Fs(\002)20 b Fv(R)1696 609 y Ft(d)1772 645 y Fs(!)k Fv(C)c Fw(,)31 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b Fs(7!)f Fw(k)2466 660 y Ft(t)2497 645 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fw(,)25 b(wher)l(e)958 945 y(k)1004 960 y Ft(t)1035 945 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b FD(:)p Fs(D)1435 875 y FD(e)1479 839 y Fr(\000)p Fi(j)r Ft(x)6 b Fr(\000)t Ft(y)t Fi(j)1713 813 y Fk(2)1744 839 y Fu(=.)p Fz(2)p 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Fx(/)24 b Fs(D)1710 1588 y Fo(Z)1762 1813 y Fn(R)1815 1793 y Fl(d)1851 1724 y FD(d)16 b Fw(y)31 b(k)2038 1739 y Ft(t)2069 1724 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)14 b Fw(k)2362 1739 y Ft(t)2387 1719 y Fg(0)2411 1724 y Fx(.)6 b Fw(y)g Fx(;)16 b Fw(z)5 b Fx(/)649 b FD(\(1.12\))294 2010 y Fw(for)24 b(all)k(x)9 b Fx(;)16 b Fw(z)29 b Fs(2)24 b Fv(R)892 1974 y Ft(d)968 2010 y Fw(and)h(all)f(t)1310 1974 y Fr(0)1358 2010 y Fx(>)h FD(0)p Fw(.)521 2174 y FD(\(ii\))100 b Fw(for)29 b(e)o(very)g Fx(\016)j(>)27 b FD(0)i Fw(ther)l(e)h(e)n(xists)f(a)h(\002nite)f(constant)f(a)2601 2125 y Fu(.\016)s(/)2597 2192 y Ft(t)2718 2174 y Fx(>)g FD(0)p Fw(,)i(independent)f(of)297 2294 y(x)9 b Fx(;)20 b Fw(y)31 b Fs(2)24 b Fv(R)631 2257 y Ft(d)682 2294 y Fw(,)h(suc)o(h)f(that)g(the)h(estimate)911 2581 y Fp(j)p Fw(k)985 2596 y Ft(t)1016 2581 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)24 b Fs(\024)h Fw(a)1455 2540 y Fu(.\016)s(/)1451 2608 y Ft(t)1559 2581 y FD(e)o(xp)1702 2441 y Fq(\032)1777 2581 y Fs(\000)1867 2511 y Fp(j)s Fw(x)i Fs(\000)f Fw(y)6 b Fp(j)2151 2475 y Fz(2)p 1866 2558 324 4 v 1985 2652 a FD(4)p Fw(t)2222 2581 y Fs(C)19 b Fx(\016)t Fp(j)s Fw(x)9 b Fp(j)2478 2540 y Fz(2)2536 2581 y Fs(C)20 b Fx(\016)t Fp(j)6 b Fw(y)g Fp(j)2793 2540 y Fz(2)2832 2441 y Fq(\033)3283 2581 y FD(\(1.13\))294 2873 y Fw(holds)24 b(for)g(all)k(x)9 b Fx(;)20 b Fw(y)30 b Fs(2)25 b Fv(R)1144 2837 y Ft(d)1195 2873 y Fw(.)494 3037 y FD(\(iii\))99 b Fw(the)24 b(function)g(k)1287 3052 y Ft(t)1343 3037 y Fw(obe)m(ys)1157 3260 y(k)1203 3275 y Ft(t)1234 3260 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)25 b Fs(2)f FD(L)1612 3219 y Fr(1)1612 3287 y Fz(G)1688 3260 y Fx(.)p Fv(R)1791 3219 y Ft(d)1843 3260 y Fx(/)199 b Fw(for)24 b(all)52 b(x)34 b Fs(2)25 b Fv(R)2610 3219 y Ft(d)3283 3260 y FD(\(1.14\))294 3484 y Fw(and)h(thus)f(has)g(the)h (Carleman)g(pr)l(operty)f FD(\(1.15\))g Fw(below)-7 b(.)34 b(Mor)l(eo)o(ver)-11 b(,)27 b(the)f(mapping)e Fv(R)3337 3448 y Ft(d)3414 3484 y Fs(!)294 3604 y FD(L)355 3568 y Fz(2)395 3604 y Fx(.)p Fv(R)498 3568 y Ft(d)549 3604 y Fx(/)p Fw(,)k(x)34 b Fs(7!)24 b Fw(k)897 3619 y Ft(t)928 3604 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)25 b Fw(is)f(str)l(ongly)g(continuous.)394 3858 y(Remarks)h FD(1.8.)154 b(\(i\))100 b(The)25 b(lemma)f(is)g(pro)o(v)o(en)g(in)h (Section)f(2.)521 4022 y(\(ii\))100 b(Concerning)22 b(the)h(asserted)g (continuity)d(of)j Fw(k)2297 4037 y Ft(t)2328 4022 y FD(,)g(the)f(proof)h(will)f(e)n(v)o(en)f(sho)n(w)h(that)294 4142 y(the)j(function)f(]0)p Fx(;)14 b Fs(1)p FD([)p Fs(\002)p Fv(R)1207 4106 y Ft(d)1278 4142 y Fs(\002)19 b Fv(R)1441 4106 y Ft(d)1517 4142 y Fs(3)25 b Fx(.)p Fw(t)8 b Fx(;)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(7!)h Fw(k)2126 4157 y Ft(t)2157 4142 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(is)h(continuous.)494 4306 y(\(iii\))99 b(The)25 b(estimate)f(\(1.13\))g(corresponds)h(to)f (Thm.)h(2.1)f(in)h([43)o(].)502 4471 y(\(i)n(v\))99 b(P)o(art)39 b(\(iii\))f(of)h(Lemma)f(1.7)h(continues)e(to)i(hold)f(with)g 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b(the)h(Hermiticity)e(and)i(the)f(con-)294 5442 y(tinuity)e(of)i Fw(k)733 5457 y Ft(t)764 5442 y FD(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)259 b FD(9)396 384 y(D)t FG(E)t(FI)t(N)t(I)t(T)t(I)t(O)t(N)34 b FD(1)t(.)t(9)t(.)125 b(Let)28 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)20 b FD(be)g(the)g(Schr\366dinger)g(operator)g(of)g(De\002nition)f (1.4)294 503 y(and)32 b(let)f Fw(t)37 b Fs(2)28 b Fv(R)5 b FD(.)57 b(Then)31 b(the)h(operator)f(e)o(xponential)g(e)2198 467 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2561 503 y FD(is)31 b(densely)g(de\002ned,)i(self-)294 623 y(adjoint)f(and)h(positi)n(v)o(e)d(by)i(the)h(spectral)g(theorem)f (and)h(the)f(functional)g(calculus)h(for)g(un-)294 743 y(bounded)24 b(functions)g(of)h(unbounded)f(self-adjoint)g(operators)h (\(see)g(e.g.)g(Chap.)g(5)g(in)f([7]\).)394 997 y(W)-8 b(e)28 b(are)g(no)n(w)e(in)h(a)g(position)f(to)h(gi)n(v)o(e)e(a)j (probabilistic)e(representation)h(of)g(e)3065 960 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)3424 997 y FD(by)294 1116 y(a)25 b(Fe)o(ynman-Kac-It\364)g(formula.)396 1368 y(T)t FG(H)t(E)t(O)t(R)t(E)t(M)34 b FD(1)t(.)t(1)t(0)t(.)126 b Fw(Let)31 b(A)23 b(be)e(a)f(vector)h(potential)f(with)g(pr)l(operty)g FD(\()p Fh(A)p FD(\))i Fw(and)e(let)k(V)36 b(be)21 b(a)294 1487 y(scalar)k(potential)f(with)h(pr)l(operty)f FD(\()p Fh(V)p FD(\))q Fw(.)32 b(Mor)l(eo)o(ver)-11 b(,)26 b(let)f(t)34 b Fx(>)25 b FD(0)g Fw(and)g(let)g FD(e)2831 1451 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)3188 1487 y Fw(be)26 b(given)294 1607 y(by)f(De\002nition)f FD(1.9)p Fw(.)30 b(Then)549 1749 y FD(\(i\))100 b Fw(the)24 b(domain)g(of)h FD(e)1360 1712 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1717 1749 y Fw(is)24 b(given)h(by)573 2022 y FD(dom)750 1941 y Fq(\000)796 2022 y FD(e)840 1981 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1172 1941 y Fq(\001)1242 2022 y Fs(D)1345 1911 y Fq(n)1411 2022 y Fx( )34 b Fs(2)25 b FD(L)1664 1981 y Fz(2)1704 2022 y Fx(.)p Fv(R)1807 1981 y Ft(d)1858 2022 y Fx(/)g FD(:)1973 1886 y Fo(Z)2025 2111 y Fn(R)2078 2091 y Fl(d)2128 2022 y FD(d)16 b Fw(y)31 b(k)2315 2037 y Ft(t)2346 2022 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)24 b Fs(2)h FD(L)2947 1981 y Fz(2)2987 2022 y Fx(.)p Fv(R)3090 1981 y Ft(d)3141 2022 y Fx(/)3178 1911 y Fq(o)3283 2022 y FD(\(1.16\))294 2313 y Fw(with)35 b(k)547 2328 y Ft(t)613 2313 y Fw(de\002ned)g(in)f FD(\(1.11\))p Fw(.)61 b(Mor)l(eo)o(ver)-11 b(,)37 b FD(L)1875 2276 y Fz(2)1875 2339 y(G)1931 2313 y Fx(.)p Fv(R)2034 2277 y Ft(d)2085 2313 y Fx(/)31 b Fs(\022)f FD(dom)2438 2232 y Fq(\000)2484 2313 y FD(e)2528 2277 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2859 2232 y Fq(\001)2940 2313 y Fw(is)35 b(an)f(oper)o(ator)294 2432 y(cor)l(e)26 b(for)e FD(e)678 2396 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1010 2432 y Fw(.)521 2596 y FD(\(ii\))100 b(e)787 2560 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1148 2596 y Fw(is)29 b(the)g(maximal)f(Carleman)h(oper)o(ator)f(induced)h(by)g (the)g(continu-)294 2716 y(ous)c(inte)l(gr)o(al)e(k)o(ernel)i FD(\(1.11\))f Fw(in)h(the)g(sense)f(that)1336 2989 y FD(e)1380 2948 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1712 2989 y Fx( )34 b Fs(D)1919 2853 y Fo(Z)1971 3078 y Fn(R)2024 3058 y Fl(d)2060 2989 y FD(d)16 b Fw(y)31 b(k)2247 3004 y Ft(t)2278 2989 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)577 b FD(\(1.17\))294 3280 y Fw(for)24 b(all)h Fx( )34 b Fs(2)24 b FD(dom)935 3199 y Fq(\000)980 3280 y FD(e)1024 3244 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1356 3199 y Fq(\001)1427 3280 y Fw(and)24 b(that)g(k)1827 3295 y Ft(t)1883 3280 y Fw(has)g(the)h (Carleman)f(pr)l(operty)g FD(\(1.15\))p Fw(.)494 3453 y FD(\(iii\))99 b Fw(the)45 b(ima)o(g)o(e)g FD(e)1241 3417 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1572 3453 y Fx( )55 b Fw(of)45 b(any)g Fx( )g Fs(2)36 b FD(dom)2400 3373 y Fq(\000)2446 3453 y FD(e)2490 3417 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2822 3373 y Fq(\001)2913 3453 y Fw(has)44 b(a)h(continu-)294 3573 y(ous)d(r)l(epr)l(esentative)h(in)f FD(L)1262 3537 y Fz(2)1302 3573 y Fx(.)p Fv(R)1405 3537 y Ft(d)1456 3573 y Fx(/)h Fw(given)g(by)g(the)f(right-hand)f(side)h(of)h FD(\(1.17\))p Fw(.)84 b(If)42 b(e)o(ven)294 3693 y Fx( )34 b Fs(2)25 b FD(L)547 3656 y Fz(2)547 3719 y(G)602 3693 y Fx(.)p Fv(R)705 3657 y Ft(d)756 3693 y Fx(/)p Fw(,)g(then,)g(in)f (addition,)f FD(e)1593 3657 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1925 3693 y Fx( )34 b Fs(2)25 b FD(L)2178 3656 y Fr(1)2178 3719 y Fz(G)2254 3693 y Fx(.)p Fv(R)2357 3657 y Ft(d)2408 3693 y Fx(/)p Fw(.)394 3947 y(Remarks)g FD(1.11.)154 b(\(i\))100 b(The)25 b(proof)f(of)h(Theorem)g(1.10)f(is)h(deferred)h(to)e(Section)h(3.)521 4111 y(\(ii\))100 b(F)o(or)19 b(the)h(theory)f(of)h(Carleman)g (operators)g(we)g(refer)h(to)e([45,)g(3,)h(49)o(].)30 b(W)-8 b(e)20 b(follo)n(w)294 4231 y(mostly)k(the)g(terminology)f(and)i (con)l(v)o(entions)e(of)i([49].)494 4395 y(\(iii\))99 b(The)27 b(right-hand)g(side)h(of)f(\(1.17\))h(maps)f(e)n(v)o(en)g(an)o (y)g Fx( )36 b Fs(2)26 b FD(L)2812 4359 y Fz(1)2812 4422 y(G)2867 4395 y Fx(.)p Fv(R)2970 4359 y Ft(d)3021 4395 y Fx(/)i FD(\(and)g(hence)294 4515 y(an)o(y)h Fx( )37 b Fs(2)27 b FD(L)732 4464 y Ft(p)724 4542 y Fz(G)779 4515 y Fx(.)p Fv(R)882 4479 y Ft(d)934 4515 y Fx(/)i FD(for)h(all)40 b Fw(p)31 b Fs(2)c FD([1)p Fx(;)14 b Fs(1)p FD(]\))30 b(to)f(an)g(element)g(of)h(L)2534 4478 y Fr(1)2534 4542 y Fz(G)2610 4515 y Fx(.)p Fv(R)2713 4479 y Ft(d)2764 4515 y Fx(/)p FD(.)45 b(This)29 b(f)o(act)h(is)f(well) 294 4635 y(kno)n(wn)g(for)g(the)g(free)i(case)40 b Fw(A)29 b Fs(D)e FD(0)j(and)i Fw(V)42 b Fs(D)27 b FD(0.)45 b(It)29 b(e)o(xtends)g(to)g(the)g(general)h(situation)d(of)294 4754 y(Theorem)e(1.10)f(simply)g(by)g(the)h(basic)g(estimate)f (\(1.13\).)502 4919 y(\(i)n(v\))99 b(Theorem)35 b(1.10)g(e)o(xtends)g (the)g(main)g(result)g(of)h([43],)i(where)e(the)g(Fe)o(ynman-)294 5038 y(Kac-It\364)30 b(formula)f(\(1.17\))g(w)o(as)g(pro)o(v)o(en)f (for)40 b Fw(A)29 b Fs(D)e FD(0)i(and)g Fx( )37 b Fs(2)27 b FD(L)2590 5002 y Fz(2)2590 5065 y(G)2645 5038 y Fx(.)p Fv(R)2748 5002 y Ft(d)2799 5038 y Fx(/)j FD(under)f(some)n(what)294 5158 y(more)c(restricti)n(v)o(e)e(assumptions)g(on)i(the)g(scalar)g (potential)i Fw(V)14 b FD(,)25 b(see)g(Remark)h(1.2\(v\).)527 5322 y(\(v\))100 b(If)29 b Fw(V)896 5337 y Fz(2)960 5322 y Fs(D)c FD(0,)h(then)f(the)g(scalar)h(potential)h Fw(V)40 b Fs(D)28 b Fw(V)2408 5337 y Fz(1)2473 5322 y FD(is)d(Kato)g (decomposable)g(and)302 5442 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)605 5457 y Fz(1)645 5442 y Fx(/)27 b FD(therefore)h(bounded)e(from)h(belo)n(w)-6 b(.)35 b(Re)o(gularity)27 b(properties)f(of)h(the)g(associated)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 294 90 a FD(10)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y Fw(bounded)37 b FD(Schr\366dinger)e(semigroup)e Fs(f)p FD(e)1733 348 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2020 358 y Fk(1)2051 348 y Fu(/)2082 384 y Fs(g)2119 399 y Ft(t)f Fr(\025)p Fz(0)2272 384 y FD(are)35 b(well)f(kno)n(wn)g(and)g (ha)n(v)o(e)g(been)294 503 y(studied)28 b(in)g(great)h(detail,)h(see)f (the)f(seminal)g(paper)h([42])g(and)g([23])g(for)g(the)f(non-magnetic) 294 623 y(case)39 b Fw(A)29 b Fs(D)e FD(0.)42 b(P)o(art)28 b(of)h(these)f(results)g(were)h(e)o(xtended)f(to)h(situations)d(with)i (rather)h(general)294 743 y(v)o(ector)c(potentials)e(in)i([10].)394 997 y(So)k(f)o(ar)h(we)f(ha)n(v)o(e)f(been)h(concerned)h(with)e(the)h (\(possibly)e(unbounded\))h(operator)h(e)o(xpo-)294 1116 y(nential)38 b(e)648 1080 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1017 1116 y FD(for)38 b(a)h(\002x)o(ed)f(b)n (ut)f(arbitrary)h(time)f(parameter)i Fw(t)i Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([.)69 b(Ne)o(xt)37 b(we)294 1236 y(compile)24 b(some)h(semigroup)e(properties)i(of)g(the)f(f)o(amily)h Fs(f)p FD(e)2365 1200 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2696 1236 y Fs(g)2733 1251 y Ft(t)6 b Fr(\025)p Fz(0)2851 1236 y FD(.)396 1487 y(T)t FG(H)t(E)t(O)t(R)t(E)t(M)34 b FD(1)t(.)t(1)t(2)t(.)126 b Fw(Assume)39 b(the)h(situation)e(of)h(Theor)l(em)i FD(1.10)p Fw(.)75 b(Then)41 b(the)e(family)294 1607 y Fs(f)p FD(e)375 1571 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)707 1607 y Fs(g)744 1622 y Ft(t)6 b Fr(\025)p Fz(0)884 1607 y Fw(is)22 b(a)h(str)l(ongly)e (continuous)g(\(one-par)o(ameter\))g(semigr)l(oup)g(of)h(self-adjoint) 294 1727 y(oper)o(ator)o(s)d(g)o(ener)o(ated)g(by)i(the)f(Sc)o (hr\366ding)o(er)e(oper)o(ator)27 b(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)20 b Fw(in)g(the)g(following)f(sense:)549 1868 y FD(\(i\))100 b Fw(the)24 b(semigr)l(oup)g(law)1111 2102 y FD(e)1155 2060 y Fr(\000)p Fu(.)p Ft(t)6 b Fr(C)p Ft(t)1339 2035 y Fg(0)1360 2060 y Fu(/)f Ft(H)k Fu(.)e Ft(A)r Fu(;)r Ft(V)j Fu(/)1637 2102 y Fx( )34 b Fs(D)25 b FD(e)1888 2060 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2234 2102 y FD(e)2278 2060 y Fr(\000)p Ft(t)2357 2035 y Fg(0)2381 2060 y Ft(H)d Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2628 2102 y Fx( )585 b FD(\(1.18\))294 2335 y Fw(holds)24 b(for)g(all)h(t)8 b Fx(;)14 b Fw(t)927 2299 y Fr(0)974 2335 y Fs(2)25 b FD([0)p Fx(;)14 b Fs(1)p FD([)25 b Fw(and)f(all)g Fx( )34 b Fs(2)25 b FD(L)1909 2298 y Fz(2)1909 2362 y(G)1964 2335 y Fx(.)p Fv(R)2067 2299 y Ft(d)2119 2335 y Fx(/)p Fw(.)521 2499 y FD(\(ii\))100 b Fw(the)23 b(orbit)f(mapping)g(u)1532 2514 y Fu( )1617 2499 y FD(:)h([0)p Fx(;)14 b Fs(1)p FD([)p Fs(!)23 b FD(L)2127 2463 y Fz(2)2166 2499 y Fx(.)p Fv(R)2270 2463 y Ft(d)2321 2499 y Fx(/)p Fw(,)g(t)33 b Fs(7!)23 b Fw(u)2655 2514 y Fu( )2715 2499 y Fx(.)p Fw(t)9 b Fx(/)24 b FD(:)p Fs(D)f FD(e)3023 2463 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)3354 2499 y Fx( )33 b Fw(is)294 2618 y(str)l(ongly)23 b(continuous)h(\(at)g(t)34 b Fs(D)25 b FD(0)f Fw(only)h(fr)l(om)f(the)g(right\))g(for)h(all)f Fx( )34 b Fs(2)24 b FD(L)2806 2582 y Fz(2)2806 2645 y(G)2862 2618 y Fx(.)p Fv(R)2965 2582 y Ft(d)3016 2618 y Fx(/)p Fw(.)494 2782 y FD(\(iii\))99 b Fw(for)25 b(e)o(very)i Fx(')k Fs(2)25 b Fj(C)1384 2745 y Fr(1)1367 2808 y Fz(0)1460 2782 y Fx(.)p Fv(R)1564 2746 y Ft(d)1615 2782 y Fx(/)h Fw(the)g(orbit)f(mapping)h(u)2477 2797 y Fu(')2551 2782 y Fw(is)f(str)l(ongly)g(dif)n(fer)l(entiable)294 2901 y(\(at)31 b(t)37 b Fs(D)28 b FD(0)j Fw(only)g(fr)l(om)f(the)h(right\))f (and)h(the)g(unique)f(solution)f(of)i(the)g(linear)f(initial-value)294 3021 y(pr)l(oblem)1000 3233 y FD(d)p 981 3280 97 4 v 981 3374 a(d)10 b Fw(t)1090 3303 y Fx(8)s(.)p Fw(t)f Fx(/)25 b Fs(D)f(\000)8 b Fw(H)i Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/8)s(.)p Fw(t)9 b Fx(/)14 b(;)313 b(8)s(.)p FD(0)p Fx(/)25 b Fs(D)f Fx(')19 b(;)435 b FD(\(1.19\))294 3567 y Fw(for)26 b(a)h(str)l(ongly)e(dif)n(fer)l(entiable)h(\(at)g(t)35 b Fs(D)26 b FD(0)h Fw(only)f(fr)l(om)g(the)h(right\))f(mapping)f Fx(8)h FD(:)g([0)p Fx(;)14 b Fs(1)p FD([)p Fs(!)294 3686 y FD(dom)471 3605 y Fq(\000)525 3686 y Fw(H)c Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)883 3605 y Fq(\001)929 3686 y Fw(,)25 b(t)33 b Fs(7!)25 b Fx(8)s(.)p Fw(t)9 b Fx(/)p Fw(.)394 3940 y(Remarks)25 b FD(1.13.)154 b(\(i\))100 b(The)25 b(proof)f(of)h(Theorem)g(1.12)f(is)h(gi)n(v)o(en)e(in)i (Section)g(3.)521 4105 y(\(ii\))100 b(Interesting)27 b(analytic)g(results)g(on)h(semigroups)e(of)i(unbounded)f(operators)h (on)294 4224 y(abstract)h(Hilbert)g(and)g(Banach)h(spaces)g(were)f(pre) n(viously)f(obtained)g(in)h(e.g.)g([35,)g(25,)g(18,)294 4344 y(31].)394 4598 y(In)24 b(man)o(y)f(situations)f(it)i(is)g(useful) f(to)h(kno)n(w)f(that)h(not)f(only)h(e)2505 4562 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2861 4598 y FD(has)24 b(a)g(continuous)294 4718 y(inte)o(gral)g(k)o(ernel)h(b)n (ut)f(also)h(certain)g(bounded)f(functions)g(of)33 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)p FD(.)396 4969 y(T)t FG(H)t(E)t(O)t(R)t(E)t(M)34 b FD(1)t(.)t(1)t(4)t(.)126 b Fw(Assume)27 b(the)h(situation)e(of)i(Theor)l(em)h FD(1.10)e Fw(and)h(let)35 b(F)h Fs(2)26 b FD(L)3301 4933 y Fr(1)3377 4969 y Fx(.)p Fv(R)5 b Fx(/)294 5089 y Fw(be)27 b(a)f(bounded)f(function)g(with)i(an)f(at)f(least)h(e)n(xponentially)f (fast)h(decay)g(at)g(plus)g(in\002nity)f(in)294 5208 y(the)g(sense)g(that)f(the)g(inequality)1383 5442 y Fp(j)7 b Fw(F)i Fx(.)e Fw(E)i Fx(/)p Fp(j)24 b Fs(\024)h Fx(\015)30 b FD(min)2026 5361 y Fq(\010)2084 5442 y FD(1)p Fx(;)14 b FD(e)2225 5401 y Fr(\000)p Fu(\034)f Ft(E)2377 5361 y Fq(\011)3283 5442 y FD(\(1.20\))p eop end %%Page: 11 11 TeXDict begin 11 10 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(11)294 384 y Fw(holds)36 b(for)g(Lebesgue-almost)h(all)43 b(E)d Fs(2)31 b Fv(R)48 b Fw(with)37 b(some)f(constants)g Fx(\015)10 b(;)k(\034)44 b Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fw(.)66 b(Fur)n(-)294 503 y(thermor)l(e)o(,)27 b(let)33 b(F)910 423 y Fq(\000)964 503 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1322 423 y Fq(\001)1394 503 y Fw(be)27 b(de\002ned)f(by)h(the)g(spectr)o(al)e(theor)l(em)i(and)f(the) g(functional)294 623 y(calculus.)k(Then)549 765 y FD(\(i\))107 b Fw(F)820 684 y Fq(\000)873 765 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1231 684 y Fq(\001)1302 765 y Fw(is)25 b(a)g(bounded)f(Carleman)h(oper)o(ator)f(induced)h(by)g(the)g (contin-)294 884 y(uous)f(inte)l(gr)o(al)g(k)o(ernel)46 b(f)g FD(:)24 b Fv(R)1329 848 y Ft(d)1400 884 y Fs(\002)c Fv(R)1564 848 y Ft(d)1640 884 y Fs(!)k Fv(C)c Fw(,)31 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b Fs(7!)46 b Fw(f)21 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fw(,)24 b(wher)l(e)904 1123 y(f)d Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(:)p Fs(D)1341 1042 y Fq(\012)1388 1123 y Fw(k)1434 1138 y Ft(t)1465 1123 y Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b FD(e)1761 1082 y Fz(2)p Ft(t)d(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2080 1123 y Fw(F)2150 1042 y Fq(\000)2203 1123 y Fw(H)f Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)2561 1042 y Fq(\001)2607 1123 y Fw(k)2653 1138 y Ft(t)2684 1123 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)2889 1042 y Fq(\013)3283 1123 y FD(\(1.21\))294 1362 y Fw(with)25 b(arbitr)o(ary)e(t)33 b Fs(2)p FD(]0)p Fx(;)14 b(\034)9 b(=)p FD(2[)p Fw(,)24 b(in)h(the)g(sense)f(that)1155 1648 y(F)1225 1567 y Fq(\000)1279 1648 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1637 1567 y Fq(\001)1682 1648 y Fx( )34 b Fs(D)1889 1512 y Fo(Z)1942 1737 y Fn(R)1995 1717 y Fl(d)2031 1648 y FD(d)16 b Fw(y)52 b(f)21 b Fx(.)p Fs(\001)p Fx(;)f Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)613 b FD(\(1.22\))294 1950 y Fw(for)24 b(all)h Fx( )34 b Fs(2)24 b FD(L)818 1913 y Fz(2)858 1950 y Fx(.)p Fv(R)961 1913 y Ft(d)1012 1950 y Fx(/)h Fw(and)g(that)46 b(f)f(has)24 b(the)h(Carleman)g(pr)l(operty)e FD(\(1.15\))p Fw(.)521 2113 y FD(\(ii\))100 b Fw(the)23 b(left-hand)g(side)g(of)g FD(\(1.22\))h Fw(has)f(a)g(continuous)g(r)l (epr)l(esentative)g(in)g FD(L)3267 2077 y Fz(2)3307 2113 y Fx(.)p Fv(R)3410 2077 y Ft(d)3461 2113 y Fx(/)p Fw(,)294 2233 y(whic)o(h)i(is)f(given)h(by)g(the)g(right-hand)e(side)h(of)h FD(\(1.22\))p Fw(.)494 2396 y FD(\(iii\))99 b Fw(for)52 b(e)o(very)i Fx(w)44 b Fs(2)c FD(L)1461 2360 y Fr(1)1461 2423 y Fz(G)1537 2396 y Fx(.)p Fv(R)1640 2360 y Ft(d)1692 2396 y Fx(/)53 b Fw(the)g(pr)l(oduct)59 b(F)2393 2316 y Fq(\000)2446 2396 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)2804 2316 y Fq(\001)2876 2396 y Fs(O)-59 b Fx(w)57 b Fw(is)52 b(a)h(Hilbert-)294 2516 y(Sc)o(hmidt)24 b(oper)o(ator)f(with)i(squar)l(ed)f(norm)g(given)h(by)555 2801 y FD(T)m(race)779 2721 y Fq(\010)862 2801 y Fs(O)-58 b Fx(w)913 2760 y Fr(\003)955 2717 y Fq(\014)955 2776 y(\014)995 2801 y Fw(F)1065 2721 y Fq(\000)1118 2801 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1476 2721 y Fq(\001)1522 2717 y(\014)1522 2776 y(\014)1555 2744 y Fz(2)1621 2801 y Fs(O)-59 b Fx(w)1671 2721 y Fq(\011)1754 2801 y Fs(D)1857 2666 y Fo(Z)1910 2891 y Fn(R)1963 2871 y Fl(d)1999 2801 y FD(d)13 b Fw(x)33 b Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2401 2760 y Fz(2)2441 2666 y Fo(Z)2493 2891 y Fn(R)2546 2871 y Fl(d)2582 2801 y FD(d)16 b Fw(y)31 b Fp(j)21 b Fw(f)g Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)3082 2760 y Fz(2)3135 2801 y Fx(:)121 b FD(\(1.23\))294 3095 y Fw(Her)l(e)58 b Fs(O)-59 b Fx(w)35 b Fw(denotes)c(the)g(bounded) g(multiplication)d(oper)o(ator)i(uniquely)h(corr)l(esponding)f(to)294 3214 y Fx(w)s Fw(,)25 b(and)50 b Fs(O)-58 b Fx(w)671 3178 y Fr(\003)737 3214 y Fw(denotes)25 b(its)f(Hilbert)g(adjoint.)394 3468 y(Remarks)h FD(1.15.)154 b(\(i\))100 b(The)33 b(right-hand)g(side) g(of)g(\(1.21\))g(is)g(well)g(de\002ned)h(and)f(con-)294 3588 y(tinuous)23 b(in)h Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(2)g Fv(R)1124 3552 y Ft(d)1194 3588 y Fs(\002)19 b Fv(R)1357 3552 y Ft(d)1432 3588 y FD(by)24 b(Lemma)f(1.7\(iii\))o(,)i(Remark)f(1.8\(i)n(v\))o(,)h(the)f (boundedness)294 3708 y(of)35 b(e)456 3671 y Fz(2)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)775 3708 y Fw(F)845 3627 y Fq(\000)899 3708 y Fw(H)f Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1257 3627 y Fq(\001)1337 3708 y FD(and)35 b(the)g(continuity)e(of)i(the)g(L)2448 3671 y Fz(2)2487 3708 y Fx(.)p Fv(R)2591 3671 y Ft(d)2642 3708 y Fx(/)p FD(-scalar)g(product)g Fp(h)p Fs(\001)p Fx(;)14 b Fs(\001)p Fp(i)p FD(.)294 3827 y(Moreo)o(v)o(er)l(,)24 b(\(1.21\))h(is)f (independent)g(of)h(the)g(chosen)g Fw(t)33 b Fs(2)p FD(]0)p Fx(;)14 b(\034)9 b(=)p FD(2[.)521 3992 y(\(ii\))100 b(The)29 b(proof)g(of)g(Theorem)g(1.14)f(is)h(gi)n(v)o(en)e(in)i(Section)g(4)g (and)g(rests)g(on)f(a)i(more)294 4111 y(general)36 b(result,)h(which)e (is)f(formulated)h(as)g(Lemma)f(4.1.)62 b(This)34 b(lemma)g(is)h(in)g (the)g(spirit)294 4231 y(of)i(Thm.)g(B.7.8)g(in)f([42],)k(b)n(ut,)f (among)e(others,)i(we)f(ha)n(v)o(e)e(relax)o(ed)h(a)g(boundedness)f (as-)294 4351 y(sumption)f(in)h(a)h(suitable)e(w)o(ay)-6 b(.)66 b(Theorem)36 b(1.14)g(itself)g(may)g(be)h(vie)n(wed)f(as)g(a)h (general-)294 4470 y(ization)i(of)h(Thm.)f(B.7.1\(d\))h(in)f([42])h (from)g(Kato-decomposable)e(scalar)j(potentials)d(to)294 4590 y(ones)31 b(with)e(property)i(\()p Fh(V)p FD(\))g(and)g(to)f(v)o (ector)g(potentials)f(with)h(property)g(\()p Fh(A)p FD(\))q(.)48 b(But,)32 b(whereas)294 4709 y(Thm.)d(B.7.1\(d\))h(in)g([42)o(])g (relies)g(on)f(resolv)o(ent)g(techniques)g(and)h(requires)g(the)f(po)n (wer)n(-la)o(w)294 4829 y(decay)g Fp(j)7 b Fw(F)i Fx(.)e Fw(E)i Fx(/)p Fp(j)27 b Fs(\024)g FD(const)p Fx(:)o(.)p FD(1)21 b Fs(C)g Fp(j)7 b Fw(E)i Fp(j)p Fx(/)1582 4793 y Fr(\000)p Fu(\013)1712 4829 y FD(with)28 b Fx(\013)j(>)26 b Fw(d)7 b Fx(=)p FD(2)29 b(for)g(ener)n(gies)35 b Fw(E)j FD(in)28 b(the)g(spectrum)294 4949 y(of)33 b Fw(H)10 b FD(,)25 b(we)h(w)o(ork)e(with)g(the)h(semigroup)f(and)h(thus)f(need)h (the)g(decay)g(property)f(\(1.20\).)396 5203 y(C)t FG(O)t(R)q(O)t(L)t (L)t(A)t(R)o(Y)35 b FD(1)t(.)t(1)t(6)t(.)126 b Fw(Assume)27 b(the)g(situation)f(of)h(Theor)l(em)h FD(1.14)f Fw(and)g(let)34 b(I)40 b Fs(\032)26 b Fv(R)38 b Fw(be)294 5322 y(a)28 b(Bor)l(el)g(set)g(in)g(the)g(r)l(eal)g(line)g(whic)o(h)g(is)f(bounded) h(fr)l(om)f(abo)o(ve)o(,)i FD(sup)20 b Fw(I)40 b Fx(<)27 b Fs(1)p Fw(.)40 b(Then)29 b(The-)294 5442 y(or)l(em)i FD(1.14)f Fw(holds)g(with)38 b(F)f Fs(D)1392 5424 y Fx(\037)1465 5457 y Ft(I)1503 5442 y Fw(,)32 b(that)e(is,)h(for)f(the)h(spectr)o(al) e(pr)l(ojection)2955 5424 y Fx(\037)3029 5457 y Ft(I)3066 5361 y Fq(\000)3120 5442 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)3478 5361 y Fq(\001)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 294 90 a FD(12)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y Fw(associated)29 b(with)h(the)g(ener)l(gy)h(r)l(e)l(gime)37 b(I)43 b(of)30 b(the)g(Sc)o(hr\366ding)o(er)e(oper)o(ator)37 b(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)p Fw(.)47 b(De-)294 503 y(noting)20 b(the)h(corr)l(esponding)g(continuous)e(inte)l(gr)o(al) h(k)o(ernel)h FD(\(1.21\))g Fw(by)33 b(p)2816 518 y Ft(I)2854 503 y Fw(,)22 b(Eq.)f FD(\(1.23\))g Fw(tak)o(es)294 623 y(the)k(form)844 897 y FD(T)m(race)1067 816 y Fq(\002)1134 897 y Fs(O)-58 b Fx(w)1185 855 y Fr(\003)1226 879 y Fx(\037)1300 912 y Ft(I)1337 816 y Fq(\000)1391 897 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1749 816 y Fq(\001)1820 897 y Fs(O)-59 b Fx(w)1871 816 y Fq(\003)1937 897 y Fs(D)2040 761 y Fo(Z)2092 986 y Fn(R)2145 966 y Fl(d)2181 897 y FD(d)13 b Fw(x)34 b Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2584 855 y Fz(2)2648 897 y Fw(p)2703 912 y Ft(I)2741 897 y Fx(.)s Fw(x)g Fx(;)17 b Fw(x)9 b Fx(/)309 b FD(\(1.24\))294 1187 y Fw(for)24 b(all)h Fx(w)j Fs(2)c FD(L)815 1150 y Fr(1)815 1213 y Fz(G)892 1187 y Fx(.)p Fv(R)995 1150 y Ft(d)1046 1187 y Fx(/)p Fw(.)394 1441 y(Remark)h FD(1.17.)119 b(The)25 b(proof)f(of)h(Corollary)g(1.16)g(is)f(gi)n(v)o(en)f(in)i (Section)g(4.)394 1695 y(Finally)-6 b(,)24 b(we)h(note)f(that)h(the)g (functional)f(calculus)g(e)o(xtends)g(to)g(inte)o(gral)g(k)o(ernels.) 396 1946 y(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)35 b FD(1)t(.)t(1)t(8)t(.)126 b Fw(Assume)25 b(the)f(situation)f(of)i(Theor) l(em)g FD(1.14)p Fw(.)30 b(Then)1293 2220 y(f)21 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b Fs(D)1702 2084 y Fo(Z)1755 2309 y Fn(R)1810 2220 y FD(d)c Fw(p)s Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)31 b Fw(F)9 b Fx(.)e Fw(E)i Fx(/)737 b FD(\(1.25\))294 2510 y Fw(holds)33 b(for)h(all)i(x)9 b Fx(;)20 b Fw(y)36 b Fs(2)29 b Fv(R)1181 2474 y Ft(d)1267 2510 y Fw(and)k(all)41 b(F)i(obe)m(ying)33 b FD(\(1.20\))p Fw(.)58 b(In)35 b(addition,)f FD(\(1.25\))g Fw(holds)f(for)294 2629 y(the)h(function)39 b(F)k(given)33 b(by)41 b(F)9 b Fx(.)e Fw(E)i Fx(/)30 b Fs(D)f FD(e)1707 2593 y Fr(\000)p Ft(t)11 b(E)1880 2629 y Fw(with)33 b(some)g(arbitr)o(ary)f(t)39 b Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fw(,)35 b(in)e(whic)o(h)294 2749 y(case)c(one)g(has)g(to)f(set)51 b(f)c Fs(D)27 b Fw(k)1335 2764 y Ft(t)1366 2749 y Fw(.)43 b(The)29 b(right-hand)f(side) g(of)h FD(\(1.25\))f Fw(is)h(to)f(be)h(under)o(stood)f(as)294 2869 y(a)h(Lebesgue-Stieltjes)f(inte)l(gr)o(al)e(with)j(r)l(espect)g (to)f(the)g(comple)n(x)h(\223distrib)n(ution\224)e(function)294 2988 y Fv(R)36 b Fs(3)c Fw(E)h Fs(7!)j Fw(p)s Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(:)p Fs(D)36 b Fw(p)1355 3003 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)1609 2988 y Fx(.)s Fw(x)k Fx(;)20 b Fw(y)6 b Fx(/)p Fw(.)394 3242 y(Remark)25 b FD(1.19.)119 b(The)25 b(proof)f(of)h(Corollary)g(1.18)g(is)f(gi)n(v)o(en)f(in)i (Section)g(4.)294 3576 y(1.3.)99 b Fw(Applications)23 b(to)i(r)o(andom)e(Sc)o(hr\366ding)o(er)g(oper)o(ator)o(s)394 3766 y FD(The)28 b(results)g(of)h(the)f(pre)n(vious)g(subsection)f(are) i(nicely)f(illustrated)g(by)g(random)g(Schr\366-)294 3885 y(dinger)38 b(operators.)71 b(In)38 b(f)o(act,)k(certain)d(random) e(potentials)g(of)h(wide-spread)h(use)f(in)g(the)294 4005 y(physics)26 b(literature)g(on)h(disordered)f(systems)g(lead)g(to) h(Schr\366dinger)g(operators)f(which)h(are)294 4124 y(almost)19 b(surely)g(unbounded)g(from)g(belo)n(w)g(and)h(hence)g(to)g (Schr\366dinger)g(semigroups)e(which)294 4244 y(are)26 b(almost)e(surely)g(unbounded)g(from)h(abo)o(v)o(e.)396 4498 y(D)t FG(E)t(FI)t(N)t(I)t(T)t(I)t(O)t(N)34 b FD(1)t(.)t(2)t(0)t(.) 126 b(A)38 b Fw(r)o(andom)e(scalar)h(potential)j(V)52 b FD(on)38 b Fv(R)2729 4462 y Ft(d)2818 4498 y FD(is)f(a)h(random)f (\002eld)297 4618 y Fw(V)j FD(:)25 b Fx(\177)19 b Fs(\002)h Fv(R)712 4582 y Ft(d)788 4618 y Fs(!)25 b Fv(R)5 b FD(,)31 b Fx(.!)5 b(;)17 b Fw(x)9 b Fx(/)25 b Fs(7!)j Fw(V)1531 4582 y Fu(.!)q(/)1635 4618 y Fx(.)s Fw(x)9 b Fx(/)p FD(,)25 b(on)g(a)g(complete)g(probability)e(space)j Fx(.\177)5 b(;)14 b Fj(A)23 b Fx(;)14 b Fv(P)p Fx(/)294 4737 y FD(which)26 b(is)f(measurable)g(with)g(respect)h(to)g(the)f(product)h(of)f(the)h (sigma-algebra)f Fj(A)49 b FD(of)26 b(e)n(v)o(ent)294 4857 y(sets)c(in)f Fx(\177)h FD(and)g(the)g(sigma-algebra)f(of)h(Borel) h(sets)e(in)h Fv(R)2247 4821 y Ft(d)2298 4857 y FD(.)30 b(Furthermore,)22 b(a)h(random)e(scalar)294 4977 y(potential)27 b Fw(V)40 b FD(is)24 b(said)h(to)f(satisfy)g(property)357 5141 y(\()p Fh(S)p FD(\))60 b(if)25 b(there)g(e)o(xist)f(tw)o(o)g (reals)37 b Fw(p)1513 5156 y Fz(1)1578 5141 y Fx(>)e Fw(p)s Fx(.)p Fw(d)7 b Fx(/)26 b FD(and)36 b Fw(p)2131 5156 y Fz(2)2196 5141 y Fx(>)g Fw(p)2360 5156 y Fz(1)2400 5141 y Fw(d)7 b Fx(=)2518 5060 y Fq(\002)2560 5141 y FD(2)p Fx(.)k Fw(p)2708 5156 y Fz(1)2768 5141 y Fs(\000)30 b Fw(p)s Fx(.)p Fw(d)7 b Fx(//)3097 5060 y Fq(\003)3165 5141 y FD(such)24 b(that)1494 5378 y(sup)1484 5470 y Ft(x)6 b Fr(2)p Fn(Z)1617 5450 y Fl(d)1672 5378 y Fv(E)1733 5298 y Fq(\002)1780 5378 y Fp(k)s Fw(V)1909 5360 y Fx(\037)1978 5393 y Fu(3)2034 5403 y Fk(1)2063 5393 y Fu(.)r Ft(x)g Fu(/)2159 5378 y Fp(k)2217 5337 y Ft(p)2252 5347 y Fk(2)2217 5405 y Ft(p)2252 5415 y Fk(1)2287 5298 y Fq(\003)2353 5378 y Fx(<)25 b Fs(1)p Fx(:)698 b FD(\(1.26\))p eop end %%Page: 13 13 TeXDict begin 13 12 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(13)543 384 y(Here,)59 b Fv(E)13 b FD([)8 b Fw(X)i FD(])45 b(:)p Fs(D)1217 303 y Fo(R)1262 418 y Fu(\177)1335 384 y Fv(P)p Fx(.)p FD(d)10 b Fx(!)r(/)22 b Fw(X)1688 348 y Fu(.!)q(/)1846 384 y FD(denotes)51 b(the)g(e)o(xpectation)g(of)h(a)f(\(comple)o(x-)543 503 y(v)n(alued\))31 b(random)f(v)n(ariable)38 b Fw(X)j FD(on)30 b Fx(\177)p FD(,)i(and)e(the)h(real)42 b Fw(p)s Fx(.)p Fw(d)7 b Fx(/)32 b FD(is)e(de\002ned)h(as)g(follo)n(ws:)554 623 y Fw(p)s Fx(.)p Fw(d)7 b Fx(/)24 b FD(:)p Fs(D)e FD(2)g(if)g Fw(d)30 b Fs(\024)23 b FD(3,)34 b Fw(p)s Fx(.)p Fw(d)7 b Fx(/)23 b FD(:)p Fs(D)f Fw(d)7 b Fx(=)p FD(2)23 b(if)f Fw(d)30 b Fs(\025)22 b 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b(sup)1754 1969 y Ft(x)6 b Fr(2)p Fn(R)1888 1949 y Fl(d)1987 1878 y Fv(E)2047 1797 y Fq(\002)2095 1878 y FD(e)2139 1837 y Fr(\000)p Ft(t)i(V)j Fu(.)r Ft(x)6 b Fu(/)2370 1797 y Fq(\003)2437 1878 y Fx(<)24 b Fs(1)642 b FD(\(1.28\))543 2172 y(holds)24 b(for)h(all)g Fw(t)34 b Fx(>)24 b FD(0.)347 2336 y(\()p Fh(G)p FD(\))59 b(if)f Fw(V)70 b FD(is)54 b(a)h(Gaussian)f(random)h(\002eld)g([1,)f(34])h (which)f(is)h Fv(R)2861 2300 y Ft(d)2912 2336 y FD(-homogeneous,)543 2456 y(has)40 b(zero)h(mean,)j Fv(E)26 b FD([)d Fw(V)15 b Fx(.)p FD(0)p Fx(/)p FD(])34 b Fs(D)f FD(0,)43 b(and)d(a)h(co)o(v)n (ariance)e(function)k Fw(x)f Fs(7!)33 b Fw(C)9 b Fx(.)s Fw(x)g Fx(/)33 b FD(:)p Fs(D)543 2576 y Fv(E)27 b FD([)c Fw(V)15 b Fx(.)s Fw(x)9 b Fx(/)s Fw(V)15 b Fx(.)p FD(0)p Fx(/)p FD(])30 b(that)g(is)g(continuous)f(at)h(the)h(origin)e(where)i (it)f(obe)o(ys)f(0)f Fx(<)g Fw(C)9 b Fx(.)p FD(0)p Fx(/)28 b(<)543 2695 y Fs(1)p FD(.)394 2949 y Fw(Remarks)d FD(1.21.)154 b(\(i\))100 b(While)33 b(property)h(\()p Fh(S)p 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b(V)j Fu(.)p Fz(0)p Fu(/)1192 3641 y Fq(\003)1264 3722 y Fx(<)30 b Fs(1)p FD(.)59 b(Property)34 b(\()p Fh(L)p FD(\))h(implies)e(neither)h(\()p Fh(S)p FD(\))h(nor)f(\()p Fh(I)p FD(\))h(and)f(vice)294 3841 y(v)o(ersa.)55 b(Moreo)o(v)o(er)l(,) 34 b(if)f Fw(d)k Fs(6D)29 b FD(4,)35 b(property)e(\()p Fh(I)p FD(\))g(in)g(general)g(does)g(not)f(imply)g(property)h(\()p Fh(S)p FD(\))q(,)294 3961 y(e)n(v)o(en)26 b(if)h(property)f(\()p Fh(E)p FD(\))h(is)g(supposed.)35 b(Gi)n(v)o(en)25 b(\()p Fh(E)p FD(\))q(,)i(a)g(simple)e(suf)n(\002cient)h(criterion)g(for)h (both)294 4081 y(\()p Fh(S)p FD(\))f(and)f(\()p Fh(I)p FD(\))g(to)g(hold)f(is)g(the)h(\002niteness)1565 4314 y Fv(E)1626 4233 y Fq(\002)1673 4314 y Fp(j)s Fw(V)15 b Fx(.)p FD(0)p Fx(/)p Fp(j)1940 4272 y Ft(p)1982 4233 y Fq(\003)2048 4314 y Fx(<)25 b Fs(1)1030 b FD(\(1.29\))294 4547 y(of)21 b(the)31 b Fw(p)s FD(-th)21 b(absolute)f(moment)f(for)i (some)e(real)32 b Fw(p)25 b Fx(>)20 b FD(max)p Fs(f)p FD(3)p Fx(;)14 b Fw(d)22 b Fs(C)16 b FD(1)p 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y(ble)h(de\002nition)f(of)h(the)g(inte) o(grated)f(density)f(of)i(states)g(in)396 4285 y(P)t FG(R)q(O)t(P)t(O)t(S)t(I)5 b(T)t(I)g(O)t(N)35 b FD(1)t(.)t(2)t(5)t(.) 126 b Fw(Let)45 b(A)35 b(be)g(a)f(vector)g(potential)e(with)i(pr)l (operty)f FD(\()p Fh(C)p FD(\))i Fw(and)f(let)297 4405 y(V)52 b(be)38 b(a)f(r)o(andom)f(scalar)g(potential)g(with)h(pr)l (operties)f FD(\()p Fh(S)p FD(\))p Fw(,)41 b FD(\()p Fh(E)p FD(\))d Fw(and)f FD(\()p Fh(I)p FD(\))p Fw(.)68 b(Let)38 b Fx(0)e Fs(\032)31 b Fv(R)3472 4368 y Ft(d)294 4524 y Fw(be)f(a)f(bounded)g(open)g(cube)g(and)g(let)1636 4509 y Fs(O)1618 4507 y Fx(\037)1687 4539 y Fu(0)1769 4524 y Fw(denote)g(the)g(bounded)f(multiplication)f(oper)o(ator)294 4644 y(associated)d(with)h(the)f(indicator)g(function)f(of)i Fx(0)t Fw(.)31 b(Then)25 b(the)g(e)n(xpectation)f(value)858 4920 y(N)12 b Fx(.)7 b Fw(E)i Fx(/)25 b FD(:)p Fs(D)1291 4850 y FD(1)p 1254 4897 124 4 v 1254 4991 a Fp(j)p Fx(0)t Fp(j)1414 4920 y Fv(E)1475 4809 y Fq(n)1547 4920 y FD(T)m(race)1770 4809 y Fq(h)1835 4904 y Fs(O)1818 4902 y Fx(\037)1886 4935 y Fu(0)1953 4902 y Fx(\037)2021 4935 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)2275 4839 y Fq(\000)2329 4920 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)2686 4839 y Fq(\001)2750 4904 y Fs(O)2732 4902 y Fx(\037)2801 4935 y Fu(0)2853 4809 y Fq(io)3283 4920 y FD(\(1.31\))294 5203 y Fw(is)25 b(well)g(de\002ned)g(for)g(e)o(very)g(ener)l(gy)33 b(E)h Fs(2)25 b Fv(R)36 b Fw(in)25 b(terms)f(of)h(the)g(spatially)f (localized)g(spectr)o(al)294 5322 y(pr)l(ojection)38 b(associated)g(with)h(the)g(half-line)f FD(])26 b Fs(\000)e(1)p Fx(;)d Fw(E)9 b FD([)39 b Fw(of)g(the)g(r)o(andom)f(Sc)o(hr\366ding)o (er)294 5442 y(oper)o(ator)h(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)p Fw(.)53 b(Furthermor)l(e)32 b(it)g(is)f(independent)h (of)g Fx(0)t Fw(.)53 b(The)33 b FD(inte)o(grated)f(density)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(15)294 384 y(of)30 b(states)37 b Fw(E)f Fs(7!)f Fw(N)12 b Fx(.)7 b Fw(E)i Fx(/)30 b Fw(is)f(the)h(unbounded)f(left-continuous)f(distrib) n(ution)f(function)i(of)h(a)294 503 y(positive)24 b(Bor)l(el)h(measur)l (e)g(on)f(the)h(r)l(eal)g(line)f Fv(R)19 b Fw(.)399 755 y(Pr)l(oof)k(.)120 b FD(W)-8 b(e)28 b(refer)i(to)e(Thm.)f(3.1)h(in)g ([26)o(])h(for)f(the)g(case)h Fw(d)34 b Fs(\025)27 b FD(2)h(and)g(to)g(Thm.)f(5.20)h(in)294 874 y([36])d(for)g(the)g(case)g Fw(d)33 b Fs(D)24 b FD(1.)p 1349 883 41 90 v 394 1129 a Fw(Remark)h FD(1.26.)119 b(Mostly)-6 b(,)57 b Fw(N)12 b Fx(.)7 b Fw(E)i Fx(/)46 b FD(is)g(de\002ned)h(as)g(the)f(almost)f (surely)h(non-random)294 1248 y(quantity)30 b(arising)f(in)i(the)f (in\002nite-v)n(olume)f(limit)g(from)i(the)f(number)g(of)h(eigen)l(v)n (alues)e(per)294 1368 y(v)n(olume)d(\(counting)h(multiplicities\))d(of) j(a)g(\002nite-v)n(olume)g(restriction)f(of)35 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)3228 1332 y Fu(.!)q(/)3332 1368 y Fx(/)27 b FD(be-)294 1487 y(lo)n(w)41 b Fw(E)9 b FD(.)60 b(This)34 b(de\002nition)g(coincides)h(with)f(the)g(one)h(in) f(Proposition)g(1.25)g(abo)o(v)o(e,)i(as)f(is)294 1607 y(sho)n(wn)24 b(in)g(Cor)-5 b(.)25 b(3.3)g(of)g([26])g(under)f(the)h (present)g(assumptions)e(on)34 b Fw(A)27 b FD(and)h Fw(V)15 b FD(.)394 1861 y(On)25 b(account)g(of)f(Corollary)h(1.24)g(and)g (\(1.31\))f(we)h(conclude)396 2113 y(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o (Y)35 b FD(1)t(.)t(2)t(7)t(.)126 b Fw(Let)38 b(A)30 b(be)e(a)g(vector)g (potential)e(with)h(pr)l(operty)g FD(\()p Fh(C)p FD(\))i Fw(and)e(let)k(V)294 2232 y(be)25 b(a)g(r)o(andom)e(scalar)i(potential) e(with)i(pr)l(operties)e FD(\()p Fh(S)p FD(\))q Fw(,)h FD(\()p Fh(E)p FD(\))i Fw(and)f FD(\()p Fh(I)p FD(\))p Fw(.)31 b(Then)25 b(the)g(equality)1454 2521 y(N)12 b Fx(.)7 b Fw(E)i Fx(/)25 b Fs(D)f Fv(E)1871 2441 y Fq(\002)1929 2521 y Fw(p)s Fx(.)7 b Fw(E)i Fs(I)14 b FD(0)p Fx(;)g FD(0)p Fx(/)2330 2441 y Fq(\003)3283 2521 y FD(\(1.32\))294 2833 y Fw(holds)36 b(for)h(all)43 b(E)d Fs(2)32 b Fv(R)19 b Fw(,)45 b(wher)l(e)k(p)1537 2797 y Fu(.!)q(/)1642 2833 y Fx(.)7 b Fw(E)i Fs(I)14 b(\001)p Fx(;)g Fs(\001)p Fx(/)30 b Fs(D)43 b Fw(p)2150 2784 y Fu(.!)q(/)2147 2859 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)2438 2833 y Fw(denotes)36 b(the)h(continuous)f(in-)294 2965 y(te)l(gr)o(al)k(k)o(ernel)i(of)f (the)h(spectr)o(al)e(pr)l(ojection)1949 2947 y Fx(\037)2017 2980 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)2271 2884 y Fq(\000)2325 2965 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)2646 2928 y Fu(.!)q(/)2750 2965 y Fx(/)2787 2884 y Fq(\001)2832 2965 y Fw(.)81 b(W)-9 b(e)42 b(r)l(ecall)f(that)305 3084 y(p)358 3048 y Fu(.!)q(/)463 3084 y Fx(.)7 b Fw(E)i Fs(I)14 b(\001)p Fx(;)g Fs(\001)p Fx(/)23 b Fw(e)n(xists)i(for)f Fv(P)p Fw(-almost)i(all)e Fx(!)j Fs(2)e Fx(\177)f Fw(accor)l(ding)h(to) f(Cor)l(ollary)g FD(1.24)p Fw(.)394 3338 y(Remarks)h FD(1.28.)154 b(\(i\))100 b(The)25 b(corollary)f(is)h(pro)o(v)o(en)e(in) i(Section)g(5.)521 3553 y(\(ii\))100 b(The)27 b(representation)g (\(1.32\))g(for)g(the)g(inte)o(grated)f(density)g(of)i(states)e(has)h (been)294 3673 y(kno)n(wn)32 b(pre)n(viously)e(from)j(a)f(rigorous)g (point)g(of)g(vie)n(w)g(only)g(under)g(additional)f(assump-)294 3792 y(tions)j(on)h(the)f(random)h(scalar)g(potential.)60 b(F)o(or)35 b(e)o(xample,)h(Remark)g(VI.1.5)e(in)h([12)o(])h(and)294 3912 y(Remark)e(3.4)f(in)g([26])g(require)h(from)f(the)g(outset)f(the)h Fv(P)p FD(-almost)h(sure)f(e)o(xistence)g(of)g(con-)294 4031 y(tinuous)25 b(inte)o(gral)f(k)o(ernels)i(for)g(the)g(spectral)f (projections.)33 b(A)26 b(suf)n(\002cient)f(criterion)g(for)h(this)294 4151 y(requirement)36 b(is)h(that)i Fw(V)51 b FD(is)36 b Fv(P)p FD(-almost)i(surely)e(Kato)g(decomposable)g([42,)h(10)o(].)66 b(Earlier)294 4271 y(deri)n(v)n(ations)24 b(of)h(the)g(representation)g (\(1.32\))h(by)f(dif)n(ferent)g(authors)g(require)g(e)n(v)o(en)g (stronger)294 4390 y(conditions)j(on)k Fw(V)15 b FD(,)31 b(see)e(Thms.)g(5.18)g(and)g(5.23)g(in)g([36].)45 b(The)29 b(latter)h(theorem,)g(ho)n(we)n(v)o(er)l(,)294 4510 y(co)o(v)o(ers)24 b(dif)n(ferential)h(operators)f(more)h(general)g(than)g(Schr\366dinger) g(operators.)494 4724 y(\(iii\))99 b(T)-8 b(o)33 b(our)g(kno)n(wledge,) h(Corollary)f(1.27)g(pro)o(vides)f(the)h(\002rst)g(rigorous)f(deri)n(v) n(a-)294 4844 y(tion)27 b(of)h(the)f(representation)h(\(1.32\))f(for)h (a)g(wide)g(class)g(of)f(random)h(scalar)g(potentials.)38 b(As)294 4963 y(we)32 b(ha)n(v)o(e)f(seen,)i(this)d(class)h(includes)f (also)h(random)g(potentials)f(leading)h(to)f(Schr\366dinger)294 5083 y(operators)j(which)f(are)i Fv(P)p FD(-almost)f(surely)g (unbounded)e(from)i(belo)n(w)-6 b(.)53 b(F)o(or)33 b(e)o(xample,)g (this)294 5203 y(is)j(the)g(case)h(if)i Fw(V)51 b FD(has)36 b(properties)f(\()p Fh(G)p FD(\))i(and)f(\()p Fh(E)p FD(\))h([29)o(,)g(12)o(,)f(36].)65 b(F)o(or)36 b(such)g(a)g(choice)g (of)297 5322 y Fw(V)47 b FD(the)32 b(relation)g(\(1.32\))g(is)g (frequently)g(tak)o(en)g(for)h(granted)f(in)g(the)g(physics)f (literature)h(on)294 5442 y(disordered)25 b(systems,)e(see)j(e.g.)e ([39,)h(33,)g(17)o(].)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 294 90 a FD(16)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)502 384 y FD(\(i)n(v\))99 b(Corollary)28 b(1.27)g(strengthens)g(Cor)-5 b(.)28 b(3.3)g(in)g([26])h (in)f(the)g(sense)g(that)g(Eq.)h(\(3.6\))294 503 y(in)c([26])g(may)f (be)h(replaced)h(by)e(Eq.)h(\(3.7\))g(in)f([26])h(without)f(an)h (additional)e(assumption.)294 757 y Fw(1.3.2.)99 b(Disor)l(der)n(-aver) o(a)o(g)o(ed)67 b(semigr)l(oup.)98 b FD(The)68 b(second)f(application,) 78 b(for)68 b(which)294 877 y(Corollary)32 b(1.24)f(pro)o(vides)g(a)h (rigorous)f(justi\002cation,)h(concerns,)h(loosely)e(speaking,)i(the) 294 997 y(e)o(xpectation)24 b(v)n(alue)g(of)h(the)g(random)f(operator)i (e)o(xponential)d(e)2489 960 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2821 997 y FD(.)396 1248 y(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)35 b FD(1)t(.)t(2)t(9)t(.) 126 b Fw(Let)39 b(A)30 b(be)e(a)g(vector)h(potential)e(with)h(pr)l (operty)f FD(\()p Fh(A)p FD(\))i Fw(and)f(let)j(V)294 1368 y(be)24 b(a)g(r)o(andom)e(scalar)h(potential)f(with)h(pr)l (operties)f FD(\()p Fh(S)p FD(\))j Fw(and)e FD(\()p Fh(L)p FD(\))q Fw(.)30 b(Mor)l(eo)o(ver)-11 b(,)24 b(let)f(t)33 b Fx(>)23 b FD(0)h Fw(and)294 1487 y(let)30 b(k)474 1439 y Fu(.!)q(/)470 1505 y Ft(t)609 1487 y Fw(denote)g(the)g(continuous)f (inte)l(gr)o(al)g(k)o(ernel)h(of)g FD(e)2296 1451 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2597 1426 y Fm(.!)q(/)2673 1451 y Fu(/)2704 1487 y Fw(.)47 b(W)-9 b(e)30 b(r)l(ecall)h(that)e(k) 3419 1439 y Fu(.!)q(/)3415 1505 y Ft(t)294 1607 y Fw(e)n(xists)c(for)f Fv(P)p Fw(-almost)i(all)e Fx(!)j Fs(2)e Fx(\177)f Fw(accor)l(ding)g(to) h(Cor)l(ollary)f FD(1.24)p Fw(.)30 b(Then)549 1749 y FD(\(i\))100 b Fw(the)36 b(disor)l(der)n(-aver)o(a)o(g)o(ed)f(inte)l (gr)o(al)f(k)o(ernel)p 2301 1670 77 4 v 37 w(k)2347 1764 y Ft(t)2409 1749 y FD(:)d Fv(R)2534 1712 y Ft(d)2609 1749 y Fs(\002)24 b Fv(R)2777 1712 y Ft(d)2859 1749 y Fs(!)31 b Fv(C)j Fw(,)45 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)31 b Fs(7!)p 294 1789 V 294 1868 a Fw(k)340 1883 y Ft(t)371 1868 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)33 b FD(:)p Fs(D)g Fv(E)12 b FD([)p Fw(k)915 1883 y Ft(t)952 1868 y Fx(.)s Fw(x)d Fx(;)20 b Fw(y)6 b Fx(/)p FD(])40 b Fw(is)f(well)g(de\002ned,)k(Hermitian)c(in)g(the)g(sense)g (that)p 3103 1789 V 39 w(k)3149 1883 y Ft(t)3180 1868 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)33 b Fs(D)p 294 1909 V 294 1988 a Fw(k)340 2003 y Ft(t)389 1928 y Fr(\003)430 1988 y Fx(.)6 b Fw(y)g Fx(;)17 b Fw(x)9 b Fx(/)37 b Fw(for)g(all)j(x)9 b Fx(;)20 b Fw(y)38 b Fs(2)31 b Fv(R)1348 1952 y Ft(d)1400 1988 y Fw(,)40 b(continuous)c(and)i (dominated)e(by)i(the)f(fr)l(ee)h(heat)f(k)o(ernel)294 2107 y(accor)l(ding)25 b(to)1351 2443 y Fp(j)p 1379 2364 V Fw(k)1425 2458 y Ft(t)1456 2443 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)24 b Fs(\024)h Fj(L)1928 2458 y Ft(t)1996 2373 y FD(e)2040 2337 y Fr(\000)p Fi(j)r Ft(x)6 b Fr(\000)t Ft(y)t Fi(j)2274 2311 y Fk(2)2304 2337 y Fu(=.)p Fz(2)p Ft(t)g Fu(/)p 1996 2420 459 4 v 2054 2513 a Fx(.)p FD(2)p Fx(\031)k Fw(t)f Fx(/)2285 2485 y Ft(d)c Fu(=)p Fz(2)3283 2443 y FD(\(1.33\))294 2767 y Fw(for)29 b(all)j(x)9 b Fx(;)20 b Fw(y)33 b Fs(2)27 b Fv(R)917 2731 y Ft(d)968 2767 y Fw(.)44 b(In)30 b(particular)-11 b(,)p 1598 2689 77 4 v 28 w(k)1644 2782 y Ft(t)1675 2767 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)27 b Fs(2)g FD(L)2058 2731 y Fr(1)2058 2794 y Fz(G)2135 2767 y Fx(.)p Fv(R)2238 2731 y Ft(d)2289 2767 y Fx(/)i Fw(for)g(all)j(x)k Fs(2)27 b Fv(R)2875 2731 y Ft(d)2926 2767 y Fw(.)45 b(The)30 b(mapping)294 2887 y Fv(R)360 2851 y Ft(d)436 2887 y Fs(!)25 b FD(L)632 2851 y Fz(2)671 2887 y Fx(.)p Fv(R)775 2851 y Ft(d)826 2887 y Fx(/)p Fw(,)j(x)33 b Fs(7!)p 1128 2808 V 25 w Fw(k)1174 2902 y Ft(t)1205 2887 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)24 b Fw(is)h(str)l(ongly)e (continuous.)521 3071 y FD(\(ii\))100 b Fw(the)24 b(function)p 1240 2992 V 23 w(k)1286 3086 y Ft(t)1342 3071 y Fw(induces)g(a)g (bounded,)g(self-adjoint)e(and)i(positive)g(Carleman)294 3190 y(oper)o(ator)i(T)721 3205 y Ft(t)776 3190 y Fw(on)e FD(L)961 3154 y Fz(2)1001 3190 y Fx(.)p Fv(R)1104 3154 y Ft(d)1156 3190 y Fx(/)h Fw(in)f(the)h(sense)g(that)1358 3496 y(T)1410 3511 y Ft(t)1440 3496 y Fx( )34 b FD(:)p Fs(D)1675 3360 y Fo(Z)1727 3585 y Fn(R)1780 3565 y Fl(d)1830 3496 y FD(d)r Fw(y)p 1957 3417 V 31 w(k)2003 3511 y Ft(t)2034 3496 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)821 b FD(\(1.34\))294 3822 y Fw(for)24 b(all)h Fx( )34 b Fs(2)24 b FD(L)818 3786 y Fz(2)858 3822 y Fx(.)p Fv(R)961 3786 y Ft(d)1012 3822 y Fx(/)h Fw(and)g(that)p 1429 3743 V 24 w(k)1475 3837 y Ft(t)1531 3822 y Fw(has)f(the)h(Carleman)f(pr)l(operty)g FD(\(1.15\))p Fw(.)494 4006 y FD(\(iii\))99 b Fw(the)27 b(ima)o(g)o(e)j(T)1216 4021 y Ft(t)1246 4006 y Fx( )37 b Fw(of)28 b(any)g Fx( )35 b Fs(2)27 b FD(L)1887 3970 y Fz(2)1926 4006 y Fx(.)p Fv(R)2030 3970 y Ft(d)2081 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5209 y Fk(1)1440 5199 y Fr(I)p Ft(b)q Fu(/)1533 5099 y Fq(\025)1601 5094 y Fk(1)p 1597 5107 33 4 v 1603 5147 a Fl(p)1660 5099 y Fq(\024)1713 5104 y Fo(Z)1825 5240 y Fx(\026)1893 5199 y Fz(0)p Fu(;)p Ft(t)1895 5267 y(x)e Fu(;)t Ft(y)2000 5240 y Fx(.)p FD(d)k Fw(b)r Fx(/)2213 5155 y Fq(\014)2213 5215 y(\014)2246 5240 y FD(e)2290 5199 y Fr(\000)t Ft(S)2381 5209 y Fl(t)2406 5199 y Fu(.)p Fz(0)p Fu(;)r Ft(V)2533 5209 y Fk(2)2562 5199 y Fr(I)p Ft(b)q Fu(/)2674 5240 y Fs(\000)19 b FD(e)2815 5199 y Fr(\000)t Ft(S)2906 5209 y Fl(t)2931 5199 y Fu(.)p Fz(0)p Fu(;)r Ft(V)3058 5209 y Fk(2)p Fm(;)t Fl(R)3141 5199 y Fr(I)p Ft(b)q Fu(/)3233 5155 y Fq(\014)3233 5215 y(\014)3274 5178 y Ft(p)3311 5152 y Fg(0)3335 5099 y Fq(\025)3412 5094 y Fk(1)p 3399 5107 51 4 v 3405 5152 a Fl(p)3431 5138 y Fg(0)3481 5240 y Fx(:)3333 5442 y FD(\(2.8\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 294 90 a FD(20)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(The)25 b(\002rst)g(e)o (xpectation)f(in)g(the)h(last)g(line)f(of)h(\(2.8\))g(is)f(bounded)g (according)h(to)704 559 y Fq(\024)756 564 y Fo(Z)868 699 y Fx(\026)936 658 y Fz(0)p Fu(;)p Ft(t)938 726 y(x)6 b Fu(;)t Ft(y)1043 699 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b FD(e)1298 658 y Fr(\000)t Ft(S)1389 668 y Fl(t)1413 658 y Fu(.)p Fz(0)p Fu(;)8 b Ft(p)t(V)1585 668 y Fk(1)1615 658 y Fr(I)p Ft(b)q Fu(/)1707 559 y Fq(\025)1760 582 y Fz(1)p Fu(=)g Ft(p)1902 699 y Fs(\024)25 b Fw(C)2071 714 y Fz(1)2139 699 y FD(e)o(xp)2295 619 y Fq(\010)2353 699 y Fp(j)s Fw(x)j Fs(\000)d Fw(y)6 b Fp(j)2637 658 y Fz(2)2677 699 y Fx(=.)p FD(4)p Fx(\034)2854 714 y Fz(1)2905 699 y Fw(p)s Fx(/)2995 619 y Fq(\011)3081 699 y Fx(;)219 b FD(\(2.9\))294 991 y(confer)31 b(\(2.2\).)48 b(Here)31 b Fw(C)1136 1006 y Fz(1)1204 991 y Fs(\021)d Fw(C)1379 1006 y Fz(1)1418 991 y Fx(.)11 b Fw(p)s Fx(;)j(\034)1609 1006 y Fz(1)1650 991 y Fx(;)g(\034)1740 1006 y Fz(2)1780 991 y Fx(/)30 b FD(is)g(a)h(\002nite)g(constant.)46 b(In)31 b(order)g(to)f(bound)g(the)294 1111 y(second)36 b(e)o(xpectation)e(in)h (the)h(last)f(line)g(of)h(\(2.8\))f(we)h(emplo)o(y)f(the)g(elementary)h (inequal-)294 1231 y(ity)31 b Fp(j)p FD(e)502 1194 y Ft(r)563 1231 y Fs(\000)22 b FD(e)706 1194 y Ft(r)739 1169 y Fg(0)763 1231 y Fp(j)28 b Fs(\024)h Fp(j)n Fw(r)j Fs(\000)19 b Fw(r)1166 1194 y Fr(0)1190 1231 y Fp(j)14 b FD(e)1276 1194 y Fz(max)p Fr(f)o Ft(r)n Fu(;)o Ft(r)1501 1169 y Fg(0)1521 1194 y Fr(g)1583 1231 y FD(for)30 b Fw(r)o Fx(;)12 b Fw(r)1861 1194 y Fr(0)1913 1231 y Fs(2)28 b Fv(R)43 b FD(together)31 b(with)g Fp(j)s Fw(V)2764 1246 y Fz(2)p Fu(;)6 b Ft(R)2878 1231 y Fp(j)29 b Fs(\024)f Fp(j)s Fw(V)3127 1246 y Fz(2)3167 1231 y Fp(j)j FD(and)h(the)294 1350 y(Cauchy-Schw)o(arz)27 b(inequality)-6 b(.)29 b(This)24 b(gi)n(v)o(es)352 1501 y Fo(Z)465 1637 y Fx(\026)533 1595 y Fz(0)p Fu(;)p Ft(t)535 1663 y(x)6 b Fu(;)t Ft(y)639 1637 y Fx(.)p FD(d)k Fw(b)r Fx(/)853 1552 y Fq(\014)853 1612 y(\014)886 1637 y FD(e)930 1595 y Fr(\000)t Ft(S)1021 1605 y Fl(t)1045 1595 y Fu(.)p Fz(0)p Fu(;)r Ft(V)1172 1605 y Fk(2)1202 1595 y Fr(I)p Ft(b)q Fu(/)1314 1637 y Fs(\000)19 b FD(e)1455 1595 y Fr(\000)t Ft(S)1546 1605 y Fl(t)1570 1595 y Fu(.)p Fz(0)p Fu(;)r Ft(V)1697 1605 y Fk(2)p Fm(;)t Fl(R)1780 1595 y Fr(I)p Ft(b)q Fu(/)1873 1552 y Fq(\014)1873 1612 y(\014)1914 1574 y Ft(p)1951 1549 y Fg(0)472 1898 y Fs(\024)572 1762 y Fo(Z)684 1898 y Fx(\026)752 1857 y Fz(0)p Fu(;)p Ft(t)754 1925 y(x)6 b Fu(;)t Ft(y)859 1898 y Fx(.)p FD(d)k Fw(b)r Fx(/)k FD(e)1107 1857 y Ft(S)1140 1867 y Fl(t)1164 1857 y Fu(.)p Fz(0)p Fu(;)8 b Ft(p)1293 1831 y Fg(0)1311 1857 y Fi(j)r Ft(V)1378 1867 y Fk(2)1407 1857 y Fi(j)i Fr(I)p Ft(b)q Fu(/)1547 1813 y Fq(\014)1547 1873 y(\014)1585 1898 y Fw(S)1633 1913 y Ft(t)1664 1898 y Fx(.)p FD(0)p Fx(;)17 b Fw(V)1859 1913 y Fz(2)1918 1898 y Fs(\000)23 b Fw(V)2077 1913 y Fz(2)p Fu(;)6 b Ft(R)2191 1898 y Fs(I)14 b Fw(b)r Fx(/)2330 1813 y Fq(\014)2330 1873 y(\014)2371 1840 y Ft(p)2408 1815 y Fg(0)472 2188 y Fs(\024)572 2048 y Fq(\024)625 2053 y Fo(Z)737 2188 y Fx(\026)805 2147 y Fz(0)p Fu(;)p Ft(t)807 2215 y(x)6 b Fu(;)t Ft(y)911 2188 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b FD(e)1170 2147 y Ft(S)1203 2157 y Fl(t)1227 2147 y Fu(.)p Fz(0)p Fu(;)p Fz(2)8 b Ft(p)1391 2122 y Fg(0)1410 2147 y Fi(j)r Ft(V)1477 2157 y Fk(2)1506 2147 y Fi(j)i Fr(I)p Ft(b)q Fu(/)1631 2048 y Fq(\025)1684 2071 y Fz(1)p Fu(=)p Fz(2)1805 2048 y Fq(\024)1858 2053 y Fo(Z)1970 2188 y Fx(\026)2038 2147 y Fz(0)p Fu(;)p Ft(t)2040 2215 y(x)c Fu(;)t Ft(y)2145 2188 y Fx(.)p FD(d)k Fw(b)r Fx(/)2345 2104 y Fq(\014)2345 2164 y(\014)2383 2188 y Fw(S)2431 2203 y Ft(t)2462 2188 y Fx(.)p FD(0)p Fx(;)17 b Fw(V)2657 2203 y Fz(2)2716 2188 y Fs(\000)22 b Fw(V)2874 2203 y Fz(2)p Fu(;)6 b Ft(R)2989 2188 y Fs(I)14 b Fw(b)r Fx(/)3128 2104 y Fq(\014)3128 2164 y(\014)3161 2131 y Fz(2)8 b Ft(p)3241 2105 y Fg(0)3264 2048 y Fq(\025)3317 2071 y Fz(1)p Fu(=)p Fz(2)3438 2188 y Fx(:)3283 2391 y FD(\(2.10\))294 2629 y(The)25 b(\002rst)g(e)o(xpectation)f(in)g(the)h (last)g(line)f(of)h(\(2.10\))g(can)g(be)g(estimated)f(as)h(in)f (\(2.3\),)598 2780 y Fo(Z)710 2916 y Fx(\026)778 2874 y Fz(0)p Fu(;)p Ft(t)780 2942 y(x)6 b Fu(;)t Ft(y)885 2916 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b FD(e)1144 2874 y Ft(S)1177 2884 y Fl(t)1200 2874 y Fu(.)p Fz(0)p Fu(;)p Fz(2)8 b Ft(p)1364 2849 y Fg(0)1383 2874 y Fi(j)r Ft(V)1450 2884 y Fk(2)1479 2874 y Fi(j)i Fr(I)p Ft(b)q Fu(/)1630 2916 y Fs(\024)25 b Fw(C)1805 2865 y Fz(2)8 b Ft(p)1885 2839 y Fg(0)1799 2942 y Fz(2)1935 2916 y FD(e)o(xp)2078 2835 y Fq(\010)2136 2916 y FD(4)j Fw(p)2250 2874 y Fr(0)2274 2916 y Fx(")s(\034)2365 2931 y Fz(2)2405 2835 y Fq(\000)2451 2916 y Fp(j)s Fw(x)e Fp(j)2563 2874 y Fz(2)2621 2916 y Fs(C)20 b Fp(j)6 b Fw(y)g Fp(j)2831 2874 y Fz(2)2870 2835 y Fq(\001\011)2987 2916 y Fx(;)263 b FD(\(2.11\))294 3214 y(where)31 b Fx(")g Fs(2)p FD(]0)p Fx(;)14 b(.)p FD(2)d Fw(p)988 3178 y Fr(0)1011 3214 y Fx(\034)1066 3178 y Fz(2)1054 3241 y(2)1106 3214 y Fx(/)1143 3178 y Fr(\000)p Fz(1)1238 3214 y FD([)30 b(is)g(arbitrary)g(and)g Fw(C)2014 3229 y Fz(2)2081 3214 y Fs(\021)e Fw(C)2256 3229 y Fz(2)2295 3214 y Fx(.)11 b Fw(p)s Fx(;)j(";)g(\034)2578 3229 y Fz(2)2619 3214 y Fx(/)30 b FD(is)g(another)f(\002nite)h(con-)294 3334 y(stant.)h(Here)26 b(we)f(ha)n(v)o(e)g(used)f(the)h(monotonicity)e (of)i(the)g(right-hand)f(side)h(of)g(\(2.3\))g(in)g Fw(t)9 b FD(.)31 b(T)-8 b(o)294 3454 y(bound)24 b(the)h(second)g(e)o (xpectation)f(in)g(the)h(last)f(line)h(of)g(\(2.10\))f(we)h(observ)o(e) g(that)691 3758 y Fp(j)s Fw(V)780 3773 y Fz(2)819 3758 y Fx(.)s Fw(x)9 b Fx(/)20 b Fs(\000)j Fw(V)1128 3773 y Fz(2)p Fu(;)6 b Ft(R)1242 3758 y Fx(.)s Fw(x)j Fx(/)p Fp(j)25 b Fs(\024)g Fx(.")s Fp(j)s Fw(x)9 b Fp(j)1722 3716 y Fz(2)1781 3758 y Fs(C)19 b Fx(v)1927 3773 y Fu(")1965 3758 y Fx(/)14 b(2)s(.)p Fp(j)s Fw(x)9 b Fp(j)19 b Fs(\000)27 b Fw(R)t Fx(/)f Fs(\024)f Fx(.")e Fs(C)d Fx(v)2852 3773 y Fu(")2890 3758 y Fx(/)2964 3688 y Fp(j)s Fw(x)9 b Fp(j)3076 3651 y Fz(4)p 2964 3735 152 4 v 2990 3828 a Fw(R)3055 3799 y Fz(2)3283 3758 y FD(\(2.12\))294 4040 y(for)35 b(all)f Fx(")g(>)c FD(0)k(and)h(Lebesgue-almost)e(all)k Fw(x)i Fs(2)30 b Fv(R)2123 4004 y Ft(d)2174 4040 y FD(.)60 b(Here)35 b(we)g(ha)n(v)o(e)f(e)o(xploited)42 b Fw(R)35 b Fx(>)30 b FD(1)294 4160 y(and)k(the)f(\223Chebyshe)n(v\224)g (inequality)f Fx(2)s(.\030)i Fs(\000)22 b FD(1)p Fx(/)30 b Fs(\024)g Fx(\030)2192 4124 y Fz(2)2232 4160 y FD(,)35 b Fx(\030)41 b Fs(2)30 b Fv(R)5 b FD(.)62 b(By)33 b(the)g(Jensen)g(and) h(the)294 4279 y(triangle)23 b(inequality)-6 b(,)21 b(Fubini')-5 b(s)22 b(theorem)g(and)h(upon)g(standardizing)f(the)g(Bro)n(wnian)h (bridge)294 4399 y(according)i(to)g Fw(b)r Fx(.)p Fw(s)6 b Fx(/)24 b Fs(D)p FD(:)h Fw(t)1178 4363 y Fz(1)p Fu(=)p Fz(2)1313 4378 y Fs(Q)1299 4399 y Fw(b)r Fx(.)p Fw(s)6 b Fx(=)f Fw(t)k Fx(/)19 b Fs(C)k Fw(x)28 b Fs(C)20 b Fx(.)6 b Fw(y)25 b Fs(\000)d Fw(x)9 b Fx(/)p Fw(s)d Fx(=)f Fw(t)k FD(,)25 b(the)g(estimate)f(\(2.12\))h(yields)492 4550 y Fo(Z)604 4685 y Fx(\026)672 4644 y Fz(0)p Fu(;)p Ft(t)674 4712 y(x)6 b Fu(;)t Ft(y)779 4685 y Fx(.)p FD(d)k Fw(b)r Fx(/)979 4601 y Fq(\014)979 4660 y(\014)1017 4685 y Fw(S)1065 4700 y Ft(t)1096 4685 y Fx(.)p FD(0)p Fx(;)17 b Fw(V)1291 4700 y Fz(2)1350 4685 y Fs(\000)23 b Fw(V)1509 4700 y Fz(2)p Fu(;)6 b Ft(R)1623 4685 y Fs(I)14 b Fw(b)r Fx(/)1762 4601 y Fq(\014)1762 4660 y(\014)1795 4628 y Fz(2)8 b Ft(p)1875 4602 y Fg(0)612 4959 y Fs(\024)712 4848 y Fq(\020)783 4889 y Fx(.")23 b Fs(C)d Fx(v)1035 4904 y Fu(")1073 4889 y Fx(/)p Fw(t)p 783 4936 364 4 v 916 5029 a(R)981 5001 y Fz(2)1159 4848 y Fq(\021)1218 4871 y Fz(2)8 b Ft(p)1298 4846 y Fg(0)1335 4823 y Fo(Z)1434 4850 y Ft(t)1388 5048 y Fz(0)1490 4889 y FD(d)i Fw(s)p 1490 4936 105 4 v 1524 5029 a(t)1620 4823 y Fo(Z)1733 4959 y Fx(\026)1801 4917 y Fz(0)p Fu(;)p Ft(t)1803 4985 y(x)c Fu(;)t Ft(y)1907 4959 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b Fp(j)p Fw(b)r Fx(.)p Fw(s)6 b Fx(/)p Fp(j)2345 4917 y Fz(8)i Ft(p)2425 4892 y Fg(0)612 5244 y Fs(D)715 5134 y Fq(\020)786 5175 y Fx(.")23 b Fs(C)c Fx(v)1037 5190 y Fu(")1075 5175 y Fx(/)p Fw(t)p 786 5222 364 4 v 919 5315 a(R)984 5286 y Fz(2)1161 5134 y Fq(\021)1221 5157 y Fz(2)8 b Ft(p)1301 5131 y Fg(0)1338 5109 y Fo(Z)1436 5136 y Fz(1)1391 5334 y(0)1490 5244 y FD(d)i Fx(\033)1655 5109 y Fo(Z)1767 5244 y Fx(\026)1835 5198 y Fz(0)p Fu(;)p Fz(1)1835 5271 y(0)p Fu(;)p Fz(0)1933 5244 y Fx(.)p FD(d)2044 5223 y Fs(Q)2030 5244 y Fw(b)r Fx(/)25 b Fp(j)p Fw(t)2209 5203 y Fz(1)p Fu(=)p Fz(2)2330 5223 y Fs(Q)2316 5244 y Fw(b)r Fx(.\033)13 b(/)20 b Fs(C)i Fw(x)29 b Fs(C)19 b Fx(.)6 b Fw(y)25 b Fs(\000)e Fw(x)9 b Fx(/\033)k Fp(j)3195 5203 y Fz(8)8 b Ft(p)3275 5178 y Fg(0)3298 5244 y Fx(:)3283 5442 y FD(\(2.13\))p eop end %%Page: 21 21 TeXDict begin 21 20 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(21)294 384 y(This)24 b(result)h(and)g(se)n(v)o(eral)f(applications)f(of)i(the) g(elementary)g(inequality)1363 630 y Fp(j)n Fw(r)k Fs(C)18 b Fw(r)1602 589 y Fr(0)1625 630 y Fp(j)1653 589 y Fu(\013)1725 630 y Fs(\024)25 b FD(2)1875 589 y Fu(\013)1923 549 y Fq(\000)1969 630 y Fp(j)n Fw(r)10 b Fp(j)2072 589 y Fu(\013)2138 630 y Fs(C)19 b Fp(j)n Fw(r)2310 589 y Fr(0)2334 630 y Fp(j)2362 589 y Fu(\013)2409 549 y Fq(\001)3283 630 y FD(\(2.14\))294 876 y(for)32 b Fx(\013)h(>)28 b FD(0)j(and)f Fw(r)o Fx(;)12 b Fw(r)1025 840 y Fr(0)1077 876 y Fs(2)28 b Fv(R)1234 840 y Ft(d)1317 876 y FD(sho)n(w)i(that)h(there)h(e)o(xist) f(tw)o(o)g(further)g(\002nite)h(constants)e Fw(C)3377 891 y Fz(3)3446 876 y Fs(\021)294 996 y Fw(C)363 1011 y Fz(3)403 996 y Fx(.)11 b Fw(p)s Fx(;)j(")s(/)25 b FD(and)g Fw(C)899 1011 y Fz(4)964 996 y Fs(\021)f Fw(C)1135 1011 y Fz(4)1175 996 y Fx(.)11 b Fw(p)s Fx(;)j(")s(/)26 b FD(such)e(that)366 1182 y Fq(\024)419 1187 y Fo(Z)531 1322 y Fx(\026)599 1281 y Fz(0)p Fu(;)p Ft(t)601 1349 y(x)6 b Fu(;)t Ft(y)705 1322 y Fx(.)p FD(d)k Fw(b)r Fx(/)905 1238 y Fq(\014)905 1297 y(\014)943 1322 y Fw(S)991 1337 y Ft(t)1022 1322 y Fx(.)p FD(0)p Fx(;)17 b Fw(V)1217 1337 y Fz(2)1277 1322 y Fs(\000)22 b Fw(V)1435 1337 y Fz(2)p Fu(;)6 b Ft(R)1550 1322 y Fs(I)14 b Fw(b)r Fx(/)1689 1238 y Fq(\014)1689 1297 y(\014)1721 1265 y Fz(2)8 b Ft(p)1801 1239 y Fg(0)1825 1182 y Fq(\025)1877 1205 y Fz(1)p Fu(=.)p Fz(2)g Ft(p)2051 1179 y Fg(0)2069 1205 y Fu(/)2125 1322 y Fs(\024)2253 1253 y Fx(\034)2296 1268 y Fz(2)p 2237 1299 114 4 v 2245 1393 a Fw(R)2310 1364 y Fz(2)2388 1242 y Fq(\002)2429 1322 y Fw(C)2498 1337 y Fz(3)2538 1322 y Fx(\034)2593 1281 y Fz(2)2581 1349 y(2)2653 1322 y Fs(C)19 b Fw(C)2819 1337 y Fz(4)2859 1242 y Fq(\000)2904 1322 y Fp(j)s Fw(x)9 b Fp(j)3016 1281 y Fz(4)3075 1322 y Fs(C)19 b Fp(j)6 b Fw(y)g Fp(j)3284 1281 y Fz(4)3323 1242 y Fq(\001\003)3424 1322 y Fx(:)3283 1525 y FD(\(2.15\))294 1771 y(Combining)24 b(\(2.8\),)h(\(2.9\),)f (\(2.10\),)h(\(2.11\))g(and)g(\(2.15\),)f(we)h(obtain)368 1929 y Fq(\014)368 1989 y(\014)368 2049 y(\014)368 2109 y(\014)402 1938 y Fo(Z)514 2074 y Fx(\026)582 2033 y Fz(0)p Fu(;)p Ft(t)584 2101 y(x)6 b Fu(;)t Ft(y)688 2074 y Fx(.)p FD(d)k Fw(b)r Fx(/)913 1993 y Fq(\002)955 2074 y FD(e)999 2033 y Fr(\000)t Ft(S)1090 2043 y Fl(t)1114 2033 y Fu(.)d Ft(A)r Fu(;)r Ft(V)k Fr(I)p Ft(b)q Fu(/)1383 2074 y Fs(\000)19 b FD(e)1524 2033 y Fr(\000)t Ft(S)1615 2043 y Fl(t)1640 2033 y Fu(.)7 b Ft(A)r Fu(;)r Ft(V)1788 2043 y Fl(R)1826 2033 y Fr(I)p Ft(b)q Fu(/)1918 1993 y Fq(\003)1960 1929 y(\014)1960 1989 y(\014)1960 2049 y(\014)1960 2109 y(\014)488 2362 y Fs(\024)600 2292 y Fw(C)669 2307 y Fz(1)709 2292 y Fw(C)778 2307 y Fz(2)817 2292 y Fx(\034)860 2307 y Fz(2)p 600 2339 301 4 v 701 2432 a Fw(R)766 2404 y Fz(2)937 2281 y Fq(\002)979 2362 y Fw(C)1048 2377 y Fz(3)1087 2362 y Fx(\034)1142 2320 y Fz(2)1130 2388 y(2)1202 2362 y Fs(C)19 b Fw(C)1368 2377 y Fz(4)1408 2281 y Fq(\000)1454 2362 y Fp(j)s Fw(x)9 b Fp(j)1566 2320 y Fz(4)1624 2362 y Fs(C)20 b Fp(j)6 b Fw(y)g Fp(j)1834 2320 y Fz(4)1873 2281 y Fq(\001)o(\003)1974 2362 y FD(e)o(xp)2130 2221 y Fq(\032)2217 2292 y Fp(j)s Fw(x)28 b Fs(\000)d Fw(y)6 b Fp(j)2501 2256 y Fz(2)p 2217 2339 324 4 v 2280 2432 a FD(4)p Fx(\034)2373 2447 y Fz(1)2424 2432 y Fw(p)2572 2362 y Fs(C)19 b FD(2)p Fx(")s(\034)2810 2377 y Fz(2)2850 2281 y Fq(\000)2896 2362 y Fp(j)s Fw(x)9 b Fp(j)3008 2320 y Fz(2)3067 2362 y Fs(C)19 b Fp(j)6 b Fw(y)g Fp(j)3276 2320 y Fz(2)3315 2281 y Fq(\001)3361 2221 y(\033)3283 2564 y FD(\(2.16\))294 2810 y(for)25 b(all)f Fw(t)34 b Fs(2)24 b FD([)p Fx(\034)784 2825 y Fz(1)824 2810 y Fx(;)14 b(\034)914 2825 y Fz(2)954 2810 y FD(],)25 b(all)f Fx(")k Fs(2)p FD(]0)p Fx(;)14 b(.)p FD(2)d Fw(p)1578 2774 y Fr(0)1602 2810 y Fx(\034)1657 2773 y Fz(2)1645 2837 y(2)1696 2810 y Fx(/)1733 2774 y Fr(\000)p Fz(1)1828 2810 y FD([)25 b(and)f(all)k Fw(x)9 b Fx(;)20 b Fw(y)30 b Fs(2)24 b Fv(R)2515 2774 y Ft(d)2566 2810 y FD(.)31 b(Another)24 b(application)f(of)294 2930 y(\(2.14\))i(and)g(choosing)35 b Fw(p)28 b Fs(D)d FD(2)p Fx(\034)1399 2945 y Fz(2)1438 2930 y Fx(=\034)1528 2945 y Fz(1)1593 2930 y Fs(\025)g FD(2)g(then)g(yields)505 3203 y(sup)445 3286 y Ft(t)6 b Fr(2)p Fz([)p Fu(\034)567 3296 y Fk(1)598 3286 y Fu(;\034)651 3296 y Fk(2)681 3286 y Fz(])718 3092 y Fq(h)765 3203 y FD(e)809 3161 y Fu(\032)t Fi(j)r Ft(x)g Fi(j)938 3136 y Fk(2)968 3161 y Fr(\000)15 b(Q)-38 b Fu(\032)t Fi(j)t Ft(y)t Fi(j)1151 3136 y Fk(2)1199 3203 y Fp(j)p Fw(k)1273 3218 y Ft(t)1304 3203 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)19 b Fs(\000)h Fw(k)1704 3161 y Fu(.)6 b Ft(R)s Fu(/)1700 3229 y Ft(t)1813 3203 y Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)2074 3092 y Fq(i)564 3484 y Fs(\024)778 3414 y Fw(C)847 3429 y Fz(1)886 3414 y Fw(C)955 3429 y Fz(2)995 3414 y Fx(\034)1038 3429 y Fz(2)p 676 3461 504 4 v 684 3555 a Fw(R)749 3526 y Fz(2)790 3555 y Fx(.)p FD(2)p Fx(\031)k(\034)990 3570 y Fz(1)1030 3555 y Fx(/)1067 3526 y Ft(d)5 b Fu(=)p Fz(2)1217 3403 y Fq(\002)1258 3484 y Fw(C)1327 3499 y Fz(3)1367 3484 y Fx(\034)1422 3443 y Fz(2)1410 3511 y(2)1481 3484 y Fs(C)20 b Fw(C)1648 3499 y Fz(4)1687 3403 y Fq(\000)1733 3484 y Fp(j)s Fw(x)9 b Fp(j)1845 3443 y Fz(4)1904 3484 y Fs(C)19 b Fp(j)6 b Fw(y)g Fp(j)2113 3443 y Fz(4)2152 3403 y Fq(\001\003)659 3700 y Fs(\002)20 b FD(e)o(xp)913 3620 y Fq(\010)991 3700 y Fs(\000)1088 3620 y Fq(\002)1130 3700 y FD(1)p Fx(=.)p FD(4)p Fx(\034)1357 3715 y Fz(2)1396 3700 y Fx(/)g Fs(\000)f FD(4)p Fx(\032)26 b Fs(\000)20 b FD(8)p Fx(")s(\034)1919 3715 y Fz(2)1958 3620 y Fq(\003)2000 3700 y Fp(j)s Fw(x)28 b Fs(\000)d Fw(y)6 b Fp(j)2284 3659 y Fz(2)2343 3700 y Fs(\000)20 b Fx(.)h Fs(Q)-54 b Fx(\032)26 b Fs(\000)19 b FD(4)p Fx(\032)26 b Fs(\000)19 b FD(10)14 b Fx(")s(\034)3087 3715 y Fz(2)3127 3700 y Fx(/)p Fp(j)6 b Fw(y)g Fp(j)3276 3659 y Fz(2)3315 3620 y Fq(\011)3283 3850 y FD(\(2.17\))294 4096 y(for)37 b(all)f Fx(\032)9 b(;)36 b Fs(Q)-55 b Fx(\032)38 b(>)31 b FD(0,)39 b(all)e Fx(")d Fs(2)p FD(]0)p Fx(;)14 b(.)p FD(2)p Fx(\034)1547 4111 y Fz(2)1611 4096 y Fs(\000)23 b Fx(\034)1755 4111 y Fz(1)1796 4096 y Fx(/=.)p FD(4)p Fx(\034)2022 4059 y Fz(3)2010 4123 y(2)2062 4096 y Fx(/)p FD([)37 b(and)f(all)j Fw(x)9 b Fx(;)20 b Fw(y)37 b Fs(2)31 b Fv(R)2835 4060 y Ft(d)2887 4096 y FD(.)66 b(The)36 b(assertion)294 4216 y(of)41 b(the)f(lemma)g(no)n(w)g(follo)n(ws)f(by)h(choosing)f Fx(\032)47 b FD(and)41 b Fx(")i FD(so)d(small)g(that)g(4)p Fx(\032)31 b Fs(C)26 b FD(10)14 b Fx(")s(\034)3373 4231 y Fz(2)3446 4216 y Fx(<)294 4335 y FD(min)o Fs(f)22 b(Q)-55 b Fx(\032)7 b(;)14 b(.)p FD(4)p Fx(\034)724 4350 y Fz(2)763 4335 y Fx(/)800 4299 y Fr(\000)p Fz(1)895 4335 y Fs(g)p FD(.)p 1076 4344 41 90 v 394 4558 a(Lemma)35 b(2.2)g(possesses)f(an)i (immediate)e(corollary)-6 b(,)37 b(which)e(completes)g(the)g(proof)g (of)294 4677 y(Lemma)25 b(1.7.)396 4929 y(C)t FG(O)t(R)q(O)t(L)t(L)t(A) t(R)o(Y)35 b FD(2)t(.)t(3)t(.)125 b Fw(The)26 b(function)931 5175 y FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fs(\002)p Fv(R)1340 5134 y Ft(d)1410 5175 y Fs(\002)20 b Fv(R)1574 5134 y Ft(d)1650 5175 y Fs(!)25 b Fv(C)19 b Fx(;)220 b(.)p Fw(t)8 b Fx(;)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(7!)g Fw(k)2623 5190 y Ft(t)2654 5175 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)396 b FD(\(2.18\))294 5421 y Fw(is)25 b(continuous)e(under)h(the)h(assumptions)e(of)h(Lemma)i FD(1.7)p Fw(.)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 294 90 a FD(22)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)399 384 y Fw(Pr)l(oof)k(.)120 b FD(Since)28 b(by)f(assumption)i Fw(V)1716 399 y Ft(R)1794 384 y FD(lies)e(in)g Fj(K)2156 399 y Fr(\006)2215 384 y Fx(.)p Fv(R)2319 348 y Ft(d)2370 384 y Fx(/)g FD(and)h(both)f Fp(j)10 b Fw(A)r Fp(j)2940 348 y Fz(2)3006 384 y FD(and)28 b Fs(r)f(\001)k Fw(A)e FD(lie)294 503 y(in)d Fj(K)489 518 y Fz(loc)580 503 y Fx(.)p Fv(R)683 467 y Ft(d)734 503 y Fx(/)p FD(,)h(Thm.)e(6.1)h(in)g([10])h(for)f(the)h(case)g Fw(d)33 b Fs(\025)26 b FD(2,)g(respecti)n(v)o(ely)f(Prop.)i(1.3.5)e(in) h([46])294 623 y(for)f(the)g(case)h Fw(d)32 b Fs(D)24 b FD(1,)h(guarantee)g(the)g(continuity)e(of)i(the)g(function)890 841 y(]0)p Fx(;)14 b Fs(1)p FD([)p Fs(\002)p Fv(R)1299 800 y Ft(d)1369 841 y Fs(\002)20 b Fv(R)1533 800 y Ft(d)1609 841 y Fs(!)k Fv(C)c Fx(;)220 b(.)p Fw(t)8 b Fx(;)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(7!)g Fw(k)2586 800 y Fu(.)6 b Ft(R)s Fu(/)2582 868 y Ft(t)2695 841 y Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)355 b FD(\(2.19\))294 1059 y(for)32 b(all)39 b Fw(R)33 b Fx(>)c FD(0.)50 b(But)32 b(according)f(to)g(Lemma)g(2.2)g(the)g(k)o(ernel)h Fw(k)2587 1074 y Fg(\017)2648 1059 y FD(is)f(the)h(locally)e(uniform)294 1179 y(limit)24 b(of)g Fw(k)665 1143 y Fu(.)6 b Ft(R)s Fu(/)661 1206 y Fg(\017)799 1179 y FD(as)33 b Fw(R)d Fs(!)25 b(1)p FD(.)30 b(Hence,)25 b Fw(k)1648 1194 y Fg(\017)1704 1179 y FD(inherits)e(the)i(continuity)e(properties)i(of)g Fw(k)3185 1143 y Fu(.)6 b Ft(R)s Fu(/)3181 1206 y Fg(\017)3294 1179 y FD(.)p 3438 1188 41 90 v 818 1498 a(3)t(.)106 b(P)t FG(R)q(O)t(O)t(F)t(S)35 b(O)t(F)30 b FD(T)t FG(H)t(E)t(O)t(R)t(E) t(M)i FD(1.10)26 b FG(A)t(N)t(D)31 b FD(T)t FG(H)t(E)t(O)t(R)t(E)t(M)h FD(1.12)394 1717 y(Gi)n(v)o(en)37 b(the)i(tw)o(o)g(probabilistic)e (Lemmata)h(1.7)h(and)g(2.2,)j(the)c(additional)g(ar)n(guments)294 1837 y(needed)32 b(to)e(pro)o(v)o(e)g(Theorem)h(1.10)f(and)h(Theorem)g (1.12)g(are)h(purely)e(analytic.)49 b(First,)32 b(we)294 1956 y(e)o(xploit)20 b(the)h(f)o(act)h(that)f(the)h(function)e Fw(k)1618 1971 y Ft(t)1649 1956 y FD(,)i(as)g(de\002ned)g(in)f(Lemma)g (1.7,)h(is)f(a)g(Carleman)h(k)o(ernel)294 2076 y([49].)396 2327 y(L)t FG(E)t(M)t(M)t(A)33 b FD(3)t(.)t(1)t(.)125 b Fw(Let)50 b(A)41 b(be)f(a)f(vector)h(potential)e(with)h(pr)l(operty)f FD(\()p Fh(A)p FD(\))j Fw(and)e(let)j(V)54 b(be)294 2447 y(a)39 b(scalar)e(potential)g(with)h(pr)l(operty)g FD(\()p Fh(V)p FD(\))q Fw(.)71 b(F)-10 b(or)38 b(t)j Fx(>)32 b FD(0)39 b Fw(we)g(denote)f(by)45 b(K)2984 2462 y Ft(t)3053 2447 y Fw(the)38 b(inte)l(gr)o(al)294 2567 y(oper)o(ator)24 b(induced)g(by)h(the)g(k)o(ernel)g(k)1592 2582 y Ft(t)1647 2567 y Fw(with)g(domain)701 2831 y FD(dom)o Fx(.)6 b Fw(K)993 2846 y Ft(t)1024 2831 y Fx(/)25 b FD(:)p Fs(D)1217 2721 y Fq(n)1283 2831 y Fx( )34 b Fs(2)25 b FD(L)1536 2790 y Fz(2)1576 2831 y Fx(.)p Fv(R)1679 2790 y Ft(d)1730 2831 y Fx(/)g FD(:)1845 2696 y Fo(Z)1897 2921 y Fn(R)1950 2900 y Fl(d)2000 2831 y FD(d)16 b Fw(y)31 b(k)2187 2846 y Ft(t)2218 2831 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)24 b Fs(2)h FD(L)2819 2790 y Fz(2)2859 2831 y Fx(.)p Fv(R)2962 2790 y Ft(d)3013 2831 y Fx(/)3050 2721 y Fq(o)3333 2831 y FD(\(3.1\))294 3097 y Fw(and)g(action)1343 3342 y(K)1415 3357 y Ft(t)1446 3342 y Fx( )34 b FD(:)p Fs(D)1680 3206 y Fo(Z)1733 3431 y Fn(R)1786 3411 y Fl(d)1836 3342 y FD(d)16 b Fw(y)30 b(k)2022 3357 y Ft(t)2053 3342 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)852 b FD(\(3.2\))294 3608 y Fw(for)32 b(all)g Fx( )38 b Fs(2)29 b FD(dom)p Fx(.)6 b Fw(K)1074 3623 y Ft(t)1105 3608 y Fx(/)p Fw(.)54 b(Then)39 b(K)1531 3623 y Ft(t)1594 3608 y Fw(is)32 b(a)h(maximal)f (Carleman)g(oper)o(ator)-11 b(,)33 b(hence)g(closed,)294 3728 y(and)25 b(its)f(domain)g(is)g(dense)h(thanks)f(to)h(the)f (inclusion)1490 3946 y FD(L)1551 3905 y Fz(2)1551 3973 y(G)1607 3946 y Fx(.)p Fv(R)1710 3905 y Ft(d)1761 3946 y Fx(/)h Fs(\022)g FD(dom)o Fx(.)6 b Fw(K)2218 3961 y Ft(t)2249 3946 y Fx(/)14 b(:)1006 b FD(\(3.3\))294 4165 y Fw(Mor)l(eo)o(ver)-11 b(,)32 b(the)f(ima)o(g)o(e)36 b(K)1231 4180 y Ft(t)1262 4165 y Fx( )k Fw(of)30 b(any)g Fx( )37 b Fs(2)28 b FD(dom)o Fx(.)6 b Fw(K)2144 4180 y Ft(t)2175 4165 y Fx(/)31 b Fw(has)f(a)h(continuous)e(r)l(epr)l (esentative)294 4284 y(in)35 b FD(L)468 4248 y Fz(2)508 4284 y Fx(.)p Fv(R)611 4248 y Ft(d)662 4284 y Fx(/)h Fw(given)g(by)f(the)h(right-hand)e(side)h(of)g FD(\(3.2\))p Fw(.)63 b(If)35 b(e)o(ven)h Fx( )k Fs(2)30 b FD(L)2901 4247 y Fz(2)2901 4311 y(G)2957 4284 y Fx(.)p Fv(R)3060 4248 y Ft(d)3111 4284 y Fx(/)p Fw(,)38 b(then,)g(in)294 4404 y(addition,)29 b(K)754 4419 y Ft(t)785 4404 y Fx( )34 b Fs(2)25 b FD(L)1038 4367 y Fr(1)1038 4430 y Fz(G)1114 4404 y Fx(.)p Fv(R)1217 4368 y Ft(d)1268 4404 y Fx(/)p Fw(.)399 4655 y(Pr)l(oof)48 b FD(\(of)25 b(Lemma)g(3.1\))p Fw(.)119 b FD(By)27 b(Lemma)f(1.7\(i\))h(and)f(\(iii\))g(we)h(kno)n(w)f (that)g Fw(k)3121 4670 y Ft(t)3179 4655 y FD(is)g(a)h(Her)n(-)294 4775 y(mitian)h(Carleman)h(k)o(ernel.)41 b(Thus,)29 b(Thm.)f(6.13\(a\)) h(in)f([49])g(yields)g(the)g(closedness)g(of)h(the)294 4894 y(induced)k(maximal)f(Carleman)h(operator)39 b Fw(K)1888 4909 y Ft(t)1919 4894 y FD(.)55 b(The)33 b(inclusion)f(\(3.3\))h(is)f (implied)g(by)h(Re-)294 5014 y(mark)26 b(1.6\(i\))g(and)g(the)f (inclusion)31 b Fw(K)1557 5029 y Ft(t)1588 5014 y FD(L)1649 4977 y Fz(2)1649 5041 y(G)1704 5014 y Fx(.)p Fv(R)1807 4978 y Ft(d)1858 5014 y Fx(/)26 b Fs(\022)f FD(L)2085 4977 y Fr(1)2085 5041 y Fz(G)2161 5014 y Fx(.)p Fv(R)2265 4978 y Ft(d)2316 5014 y Fx(/)p FD(,)h(which)g(we)g(pro)o(v)o(e)f(ne)o (xt.)33 b(T)-8 b(o)26 b(do)294 5134 y(so,)f(we)g(note)f(that)h (\(1.13\))f(implies)1151 5378 y(sup)1138 5470 y Ft(x)6 b Fr(2)p Fn(R)1272 5450 y Fl(d)1303 5268 y Fq(h)1351 5378 y FD(e)1395 5337 y Fu(\032)t Fi(j)r Ft(x)g Fi(j)1524 5312 y Fk(2)1558 5378 y Fp(j)p Fw(k)1632 5393 y Ft(t)1663 5378 y Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)1924 5268 y Fq(i)1995 5378 y Fs(\024)25 b Fw(a)2149 5337 y Fu(.\016)s(/)2145 5405 y Ft(t)2264 5378 y FD(e)2308 5337 y Fu(.)p Fz(4)p Fu(\032)t Fr(C)p Fz(5)p Fu(\016)s(/)p Fi(j)t Ft(y)t Fi(j)2646 5312 y Fk(2)3333 5378 y FD(\(3.4\))p eop end %%Page: 23 23 TeXDict begin 23 22 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(23)294 384 y(for)30 b(all)f Fx(\032)9 b(;)14 b(\016)31 b(>)c FD(0)i(with)g Fx(\032)e Fs(C)21 b Fx(\016)31 b(<)c FD(1)p Fx(=.)p FD(16)p Fw(t)9 b Fx(/)29 b FD(and)g(all)35 b Fw(y)e Fs(2)28 b Fv(R)2382 348 y Ft(d)2433 384 y FD(.)44 b(In)29 b(deri)n(ving)f(\(3.4\))h(we)h(ha)n(v)o(e)294 503 y(also)25 b(used)f(the)h(elementary)g(inequality)e(\(2.14\))i(with) d Fw(r)35 b Fs(D)28 b Fw(x)g Fs(\000)d Fw(y)6 b FD(,)23 b Fw(r)2693 467 y Fr(0)2741 503 y Fs(D)31 b Fw(y)g FD(and)25 b Fx(\013)j Fs(D)d FD(2.)394 623 y(Consequently)-6 b(,)23 b(gi)n(v)o(en)h(an)o(y)g Fx( )34 b Fs(2)25 b FD(L)1648 586 y Fz(2)1648 649 y(G)1703 623 y Fx(.)p Fv(R)1806 587 y Ft(d)1857 623 y Fx(/)p FD(,)g(we)g(get)788 944 y(ess)14 b(sup)843 1036 y Ft(x)6 b Fr(2)p Fn(R)977 1015 y Fl(d)1062 829 y Fq(\014)1062 889 y(\014)1062 949 y(\014)1095 944 y FD(e)1139 903 y Fu(\032)t Fi(j)r Ft(x)g Fi(j)1268 877 y Fk(2)1303 944 y Fx(.)g Fw(K)1418 959 y Ft(t)1449 944 y Fx( )j(/.)s Fw(x)g Fx(/)1695 829 y Fq(\014)1695 889 y(\014)1695 949 y(\014)1753 944 y Fs(\024)25 b Fw(a)1907 903 y Fu(.\016)s(/)1903 971 y Ft(t)2011 808 y Fo(Z)2064 1033 y Fn(R)2117 1013 y Fl(d)2153 944 y FD(d)16 b Fw(y)31 b FD(e)2338 903 y Fu(.)p Fz(4)p Fu(\032)t Fr(C)p Fz(5)p Fu(\016)s(/)p Fi(j)t Ft(y)t Fi(j)2676 877 y Fk(2)2724 944 y Fp(j)p Fx( )9 b(.)d Fw(y)g Fx(/)p Fp(j)14 b Fx(:)303 b FD(\(3.5\))294 1273 y(No)n(w)-6 b(,)31 b(choosing)e Fx(\032)36 b FD(and)31 b Fx(\016)j FD(small)29 b(enough,)i(the)f (right-hand)f(side)h(of)h(\(3.5\))f(is)f(\002nite)i(since)294 1392 y(L)355 1356 y Fz(2)355 1419 y(G)411 1392 y Fx(.)p Fv(R)514 1356 y Ft(d)565 1392 y Fx(/)25 b Fs(\022)g FD(L)791 1356 y Fz(1)791 1419 y(G)846 1392 y Fx(.)p Fv(R)949 1356 y Ft(d)1000 1392 y Fx(/)g FD(by)g(Remark)g(1.6\(i\).)394 1512 y(In)h(order)f(to)h(complete)f(the)g(proof)g(of)h(the)g(lemma)e (we)i(ha)n(v)o(e)g(to)f(sho)n(w)f(the)i(continuity)d(of)300 1632 y Fw(K)372 1647 y Ft(t)403 1632 y Fx( )34 b FD(for)25 b(all)g Fx( )34 b Fs(2)25 b FD(dom)o Fx(.)6 b Fw(K)1257 1647 y Ft(t)1288 1632 y Fx(/)p FD(.)31 b(T)-8 b(o)25 b(this)e(end)i(we)g(observ)o(e)807 1815 y Fq(\014)807 1874 y(\014)840 1899 y Fx(.)6 b Fw(K)955 1914 y Ft(t)986 1899 y Fx( )j(/.)s Fw(x)g Fx(/)20 b Fs(\000)f Fx(.)6 b Fw(K)1464 1914 y Ft(t)1495 1899 y Fx( )j(/.)s Fw(x)1704 1858 y Fr(0)1729 1899 y Fx(/)1766 1815 y Fq(\014)1766 1874 y(\014)1824 1899 y Fs(\024)25 b Fp(k)p Fx( )9 b Fp(k)2103 1914 y Fz(2)2168 1899 y Fp(k)p Fw(k)2264 1914 y Ft(t)2294 1899 y Fx(.)s Fw(x)g Fx(;)14 b Fs(\001)p Fx(/)20 b Fs(\000)f Fw(k)2662 1914 y Ft(t)2693 1899 y Fx(.)s Fw(x)2786 1858 y Fr(0)2810 1899 y Fx(;)14 b Fs(\001)p Fx(/)p Fp(k)2972 1914 y Fz(2)3333 1899 y FD(\(3.6\))294 2167 y(by)24 b(the)f(triangle)g(and)h(the)f(Cauchy-Schw)o(arz)i (inequality)d(for)i(all)i Fw(x)9 b Fx(;)17 b Fw(x)2751 2131 y Fr(0)2798 2167 y Fs(2)24 b Fv(R)2951 2131 y Ft(d)3002 2167 y FD(.)30 b(The)24 b(desired)294 2287 y(result)k(no)n(w)f(follo)n (ws)g(from)g(the)h(strong)g(continuity)e(of)31 b Fw(x)k Fs(7!)27 b Fw(k)2523 2302 y Ft(t)2554 2287 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)27 b FD(in)h(Lemma)f(1.7\(iii\).)p 414 2415 41 90 v 394 2756 a(W)-8 b(e)23 b(will)g(e)n(v)o(entually)e (pro)o(v)o(e)h(Theorem)h(1.10)g(by)g(sho)n(wing)e(the)i(operator)g (equality)29 b Fw(K)3392 2771 y Ft(t)3446 2756 y Fs(D)294 2876 y FD(e)338 2840 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)670 2876 y FD(.)71 b(As)38 b(an)h(initial)e(step)h(we)g(recall)h(De\002nition)e(2.1)h(and)h(emplo) o(y)e(Lemma)h(2.2)294 2995 y(in)28 b(order)h(to)f(establish)f(strong)g (con)l(v)o(er)n(gence)i(of)f(the)h(re)o(gularized)f(operator)g(e)o (xponentials)294 3115 y(e)338 3079 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)629 3089 y Fl(R)668 3079 y Fu(/)724 3115 y FD(to)31 b Fw(K)905 3130 y Ft(t)961 3115 y FD(on)24 b(L)1146 3078 y Fz(2)1146 3142 y(G)1201 3115 y Fx(.)p Fv(R)1305 3079 y Ft(d)1356 3115 y Fx(/)h FD(as)33 b Fw(R)c Fs(!)c(1)p FD(.)396 3366 y(L)t FG(E)t(M)t(M)t(A)33 b FD(3)t(.)t(2)t(.)125 b Fw(Let)27 b(t)35 b Fx(>)25 b FD(0)p Fw(,)j Fx( )35 b Fs(2)26 b FD(L)1720 3330 y Fz(2)1720 3393 y(G)1775 3366 y Fx(.)p Fv(R)1878 3330 y Ft(d)1929 3366 y Fx(/)h Fw(and)g(suppose)f(the)h(assumptions)d(of)j(Theo-)294 3486 y(r)l(em)e FD(1.10)p Fw(.)31 b(Then)1282 3754 y FD(lim)1255 3817 y Ft(R)s Fr(!1)1463 3754 y Fp(k)p FD(e)1557 3712 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)1848 3722 y Fl(R)1886 3712 y Fu(/)1917 3754 y Fx( )29 b Fs(\000)d Fw(K)2192 3769 y Ft(t)2222 3754 y Fx( )9 b Fp(k)2351 3769 y Fz(2)2416 3754 y Fs(D)25 b FD(0)764 b(\(3.7\))294 4053 y Fw(holds.)399 4304 y(Pr)l(oof)23 b(.)120 b FD(W)-8 b(e)23 b(recall)g(from)f(Thm.)g(6.1)g(in)g([10])h(for)g(the)f(case)i Fw(d)30 b Fs(\025)23 b FD(2,)g(respecti)n(v)o(ely)e(from)294 4424 y(Eq.)32 b(\(6.6\))g(in)g([40])g(or)h(from)e(Eqs.)h(\(1.3.3\),)i (\(1.3.4\))e(and)g(Ex)o(ercise)g(1.4.2)f(in)h([46])g(for)h(the)294 4543 y(case)i Fw(d)i Fs(D)30 b FD(1,)36 b(the)e(Fe)o(ynman-Kac-It\364)g (formula)f(for)i(the)e(bounded)h(semigroup)e(with)i(the)294 4663 y(re)o(gularized)25 b(potential)1145 4978 y(e)1189 4937 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)1480 4947 y Fl(R)1519 4937 y Fu(/)1550 4978 y Fx( )34 b Fs(D)1757 4842 y Fo(Z)1809 5067 y Fn(R)1862 5047 y Fl(d)1899 4978 y FD(d)16 b Fw(y)30 b(k)2089 4937 y Fu(.)6 b Ft(R)s Fu(/)2085 5005 y Ft(t)2198 4978 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)14 b(;)660 b FD(\(3.8\))294 5309 y(v)n(alid)31 b(for)h(all)f Fx( )38 b Fs(2)29 b FD(L)1063 5273 y Fz(2)1102 5309 y Fx(.)p Fv(R)1205 5273 y Ft(d)1257 5309 y Fx(/)p FD(.)51 b(No)n(w)-6 b(,)32 b(gi)n(v)o(en)f(an)o(y)g Fx( )38 b Fs(2)28 b FD(L)2297 5273 y Fz(2)2297 5336 y(G)2352 5309 y Fx(.)p Fv(R)2455 5273 y Ft(d)2506 5309 y Fx(/)k FD(there)g(e)o(xists)52 b Fs(Q)-54 b Fx(\032)35 b(>)28 b FD(0)k(such)294 5442 y(that)f Fp(k)p FD(e)584 5406 y Fr(Q)-38 b Fu(\032)t Fi(j)p Fr(\001)p Fi(j)678 5380 y Fk(2)713 5442 y Fx( )9 b Fp(k)842 5457 y Fz(1)910 5442 y Fx(<)29 b Fs(1)i FD(by)g(Remark)h (1.6\(i\).)50 b(Lemma)31 b(2.2)g(then)g(yields)f(the)i(e)o(xistence)e (of)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 294 90 a FD(24)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y Fx(\032)31 b(>)25 b FD(0)g(such)f(that)h(the)g(right-hand)f(side)g(of)h(the)g(estimate) 418 699 y Fp(k)p FD(e)512 658 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)803 668 y Fl(R)842 658 y Fu(/)873 699 y Fx( )29 b Fs(\000)c Fw(K)1147 714 y Ft(t)1178 699 y Fx( )9 b Fp(k)1307 658 y 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(1.11\(iii\))o(.)394 2820 y(The)c(ne)o(xt)g(lemma)f(concerns)i(a)f (certain)h(stability)d(of)j(strong-resolv)o(ent)d(con)l(v)o(er)n (gence.)31 b(It)294 2939 y(will)24 b(be)h(the)f(basis)g(for)h(an)g(ar)n (gument)f(similar)g(to)g(the)h(one)f(pro)o(vided)g(by)g(Thm.)g(3.1)g (in)h([43)o(].)396 3191 y(L)t FG(E)t(M)t(M)t(A)33 b FD(3)t(.)t(4)t(.) 125 b Fw(F)-10 b(or)28 b(n)k Fs(2)c Fv(N)45 b Fw(let)39 b(A)1687 3206 y Ft(n)1760 3191 y Fw(and)g(A)32 b(be)e(self-adjoint)e (oper)o(ator)o(s)g(acting)h(on)h(a)294 3310 y(comple)n(x)20 b(Hilbert)f(space)h(and)g(let)h(G)27 b FD(:)20 b Fv(R)31 b Fs(!)20 b Fv(R)31 b Fw(be)20 b(a)g(continuous)e(function.)28 b(De\002ne)22 b(G)6 b Fx(.)k Fw(A)3443 3325 y Ft(n)3486 3310 y Fx(/)294 3430 y Fw(for)26 b(n)k Fs(2)25 b Fv(N)41 b Fw(and)28 b(G)6 b Fx(.)k Fw(A)r Fx(/)27 b Fw(via)f(the)g(spectr)o(al) f(theor)l(em)h(and)g(the)g(functional)f(calculus)g(as)h(self-)294 3550 y(adjoint)38 b(oper)o(ator)o(s.)74 b(Then)40 b(str)l(ong-r)l (esolvent)e(con)l(ver)l(g)o(ence)j(of)49 b(A)2751 3565 y Ft(n)2833 3550 y Fw(to)h(A)41 b(as)f(n)d Fs(!)c(1)294 3669 y Fw(implies)24 b(str)l(ong-r)l(esolvent)f(con)l(ver)l(g)o(ence)j (of)g(G)6 b Fx(.)k Fw(A)2089 3684 y Ft(n)2133 3669 y Fx(/)25 b Fw(to)h(G)6 b Fx(.)k Fw(A)r Fx(/)p Fw(.)399 3921 y(Pr)l(oof)23 b(.)120 b FD(F)o(or)29 b Fw(z)h Fs(2)c Fv(C)53 b FD(with)26 b(Im)15 b Fw(z)31 b Fs(6D)26 b FD(0)g(we)i (de\002ne)f(the)g(bounded)f(continuous)f(function)302 4040 y Fw(R)364 4055 y Ft(z)428 4040 y FD(:)h Fv(R)38 b Fs(!)27 b Fv(C)20 b FD(,)35 b Fx(\025)27 b Fs(7!)34 b Fw(R)1132 4055 y Ft(z)1169 4040 y Fx(.\025/)27 b FD(:)p Fs(D)f Fx(.\025)21 b Fs(\000)h Fw(z)5 b Fx(/)1751 4004 y Fr(\000)p Fz(1)1845 4040 y FD(.)42 b(Hence,)30 b(the)e(composition)35 b Fw(R)2963 4055 y Ft(z)3020 4040 y Fs(B)23 b Fw(G)35 b FD(is)28 b(also)g(a)294 4160 y(bounded)i(and)h(continuous)e(function) g(on)i Fv(R)5 b FD(.)54 b(Therefore,)32 b Fx(.)8 b Fw(R)2505 4175 y Ft(z)2564 4160 y Fs(B)24 b Fw(G)6 b Fx(/.)k Fw(A)2858 4175 y Ft(n)2901 4160 y Fx(/)28 b Fs(D)36 b Fw(R)3142 4175 y Ft(z)3179 4160 y Fx(.)r Fw(G)6 b Fx(.)k Fw(A)3406 4175 y Ft(n)3449 4160 y Fx(//)294 4279 y FD(con)l(v)o(er)n(ges)23 b(strongly)f(to)h Fx(.)8 b Fw(R)1272 4294 y Ft(z)1326 4279 y Fs(B)20 b Fw(G)6 b Fx(/.)k Fw(A)r Fx(/)24 b Fs(D)31 b Fw(R)1848 4294 y Ft(z)1885 4279 y Fx(.)r Fw(G)6 b Fx(.)k Fw(A)r Fx(//)24 b FD(as)f Fw(n)28 b Fs(!)22 b(1)h FD(by)g(Thm.)f (VIII.20\(b\))i(in)294 4399 y([38])h(or)g(Thm.)f(9.17)h(in)f([49].)p 1444 4408 V 394 4578 a(Ha)n(ving)h(these)h(auxiliary)f(results)h(at)g (our)f(disposal,)g(we)h(can)h(proceed)f(to)g(pro)o(v)o(e)f(\226)g(as)h (an)294 4698 y(intermediate)19 b(step)g(\226)h(Theorem)f(1.10\(ii\))o (,)i(which)e(is)g(analogous)g(to)g(the)h(claim)f(of)g(Remark)i(1)294 4818 y(after)26 b(Thm.)e(1.2)g(in)h([43].)396 5069 y(L)t FG(E)t(M)t(M)t(A)33 b FD(3)t(.)t(5)t(.)125 b Fw(Let)36 b(t)j Fx(>)31 b FD(0)p Fw(.)63 b(Under)35 b(the)h(assumptions)e(of)h (Theor)l(em)h FD(1.10)f Fw(one)h(has)294 5189 y FD(L)355 5152 y Fz(2)355 5215 y(G)411 5189 y Fx(.)p Fv(R)514 5152 y Ft(d)565 5189 y Fx(/)25 b Fs(\022)g FD(dom)907 5108 y Fq(\000)952 5189 y FD(e)996 5152 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1328 5108 y Fq(\001)1399 5189 y Fw(and)24 b(the)h(F)-7 b(e)m(ynman-Kac-It\364)25 b(formula)1524 5442 y FD(e)1568 5401 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1899 5442 y Fx( )34 b Fs(D)d Fw(K)2184 5457 y Ft(t)2215 5442 y Fx( )998 b FD(\(3.10\))p eop end %%Page: 25 25 TeXDict begin 25 24 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(25)294 384 y Fw(holds)20 b(for)g(all)g Fx( )30 b Fs(2)21 b FD(L)1039 347 y Fz(2)1039 410 y(G)1094 384 y Fx(.)p Fv(R)1197 348 y Ft(d)1248 384 y Fx(/)p Fw(.)30 b(In)20 b(particular)-11 b(,)20 b FD(e)1927 348 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2280 384 y Fw(and)20 b(thus)26 b(K)2715 399 y Ft(t)2766 384 y Fw(ar)l(e)21 b(both)f(symmetric)294 503 y(on)25 b FD(L)480 467 y Fz(2)480 530 y(G)535 503 y Fx(.)p Fv(R)638 467 y Ft(d)689 503 y Fx(/)p Fw(.)399 755 y(Pr)l(oof)48 b FD(\(of)25 b(Lemma)g(3.5\))p Fw(.)119 b FD(The)26 b(Schr\366dinger)f(operators)33 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)26 b FD(and)33 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)3411 770 y Ft(R)3461 755 y Fx(/)p FD(,)302 874 y Fw(R)25 b Fx(>)20 b FD(0,)g(are)h(all)e(essentially)g(self-adjoint)f(on)i Fj(C)1959 838 y Fr(1)1942 901 y Fz(0)2035 874 y Fx(.)p Fv(R)2138 838 y Ft(d)2189 874 y Fx(/)g FD(according)g(to)f(Proposition) f(1.3)i(and)294 994 y(De\002nition)26 b(1.4.)33 b(Moreo)o(v)o(er)l(,)g Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)1662 1009 y Ft(R)1713 994 y Fx(/)26 b FD(con)l(v)o(er)n(ges)g(strongly)f(to)33 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)26 b FD(on)g Fj(C)3256 957 y Fr(1)3239 1021 y Fz(0)3332 994 y Fx(.)p Fv(R)3435 958 y Ft(d)3486 994 y Fx(/)294 1114 y FD(as)33 b Fw(R)d Fs(!)24 b(1)p FD(.)31 b(This)24 b(can)h(be)g(inferred)h(from)e(\(1.4\))h(and)g(the)g(estimate)411 1383 y Fp(k)8 b Fw(H)i Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)778 1398 y Ft(R)828 1383 y Fx(/')25 b Fs(\000)i Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/')5 b Fp(k)1520 1342 y Fz(2)1520 1410 y(2)1619 1383 y Fs(D)1757 1248 y Fo(Z)1809 1473 y Fn(R)1862 1453 y Fl(d)1898 1383 y FD(d)13 b Fw(x)2039 1299 y Fq(\014)2039 1358 y(\014)2075 1383 y Fw(V)2151 1335 y Fu(.)6 b Ft(R)s Fu(/)2133 1410 y Fz(2)2260 1383 y Fx(.)s Fw(x)j Fx(/)20 b Fs(\000)i Fw(V)2568 1398 y Fz(2)2608 1383 y Fx(.)s Fw(x)9 b Fx(/)2738 1299 y Fq(\014)2738 1358 y(\014)2771 1326 y Fz(2)2825 1383 y Fp(j)p Fx(')c(.)s Fw(x)k Fx(/)p Fp(j)3072 1342 y Fz(2)1621 1648 y Fs(\024)1757 1513 y Fo(Z)1809 1737 y Fn(R)1862 1717 y Fl(d)1898 1648 y FD(d)k Fw(x)34 b Fx(2)s(.)p Fp(j)s Fw(x)9 b Fp(j)19 b Fs(\000)27 b Fw(R)t Fx(/)2511 1567 y Fq(\000)2558 1648 y Fx(")s Fp(j)s Fw(x)9 b Fp(j)2718 1607 y Fz(2)2776 1648 y Fs(C)20 b Fx(v)2923 1663 y Fu(")2961 1567 y Fq(\001)3006 1590 y Fz(2)3060 1648 y Fp(j)p Fx(')5 b(.)s Fw(x)k Fx(/)p Fp(j)3307 1607 y Fz(2)3360 1648 y Fx(;)3283 1845 y FD(\(3.11\))294 2071 y(which)31 b(is)f(v)n(alid)g(for)h(all)g Fx(")g(>)d FD(0)j(and)g(all)f Fx(')j Fs(2)28 b Fj(C)2004 2034 y Fr(1)1986 2097 y Fz(0)2080 2071 y Fx(.)p Fv(R)2183 2035 y Ft(d)2234 2071 y Fx(/)p FD(.)49 b(The)31 b(right-hand)f(side)h(of)g(\(3.11\))294 2190 y(v)n(anishes,)j(if)41 b Fw(R)d FD(is)32 b(lar)n(ge)i(enough.)54 b(Therefore,)36 b(Thm.)c(VIII.25\(a\))i(in)e([38])h(implies)f(that)302 2310 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)611 2325 y Ft(R)662 2310 y Fx(/)26 b FD(con)l(v)o(er)n(ges)h(to)34 b Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)27 b FD(in)f(strong-resolv)o(ent)f(sense)h(as)35 b Fw(R)30 b Fs(!)25 b(1)p FD(,)i(and)f(thus,)294 2430 y(thanks)35 b(to)g(Lemma)g(3.4,)i(e)1274 2393 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)1565 2403 y Fl(R)1605 2393 y Fu(/)1671 2430 y FD(con)l(v)o(er)n(ges)35 b(to)g(e)2260 2393 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2627 2430 y FD(as)44 b Fw(R)36 b Fs(!)30 b(1)35 b FD(in)g(strong-)294 2549 y(resolv)o(ent)20 b(sense)i(for)f(all)g Fw(t)30 b Fx(>)21 b FD(0.)30 b(Since)22 b(the)f(operators)g(e)2259 2513 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2550 2523 y Fl(R)2589 2513 y Fu(/)2641 2549 y FD(and)21 b(e)2850 2513 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)3203 2549 y FD(are)22 b(self-)294 2669 y(adjoint,)j (strong-resolv)o(ent)e(con)l(v)o(er)n(gence)j(is)f(equi)n(v)n(alent)f (to)h(e)2490 2633 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2847 2669 y FD(being)25 b(the)g(strong-)294 2788 y(graph)g(limit)e(of)h(e)910 2752 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)1201 2762 y Fl(R)1240 2752 y Fu(/)1296 2788 y FD(as)32 b Fw(R)e Fs(!)24 b(1)g FD(by)g(Thm.)g(VIII.26)h(in)f([38)o(].)31 b(Thus,)24 b(by)g(de\002nition)294 2908 y(of)h(this)f(limit,)f(the)i(graph)389 3134 y Fj(G)450 3149 y Ft(t)505 3134 y FD(:)p Fs(D)636 3053 y Fq(\010)694 3134 y Fx(. )s(;)14 b(\036)5 b(/)25 b Fs(2)g FD(L)1125 3098 y Fz(2)1164 3134 y Fx(.)p Fv(R)1268 3098 y Ft(d)1319 3134 y Fx(/)19 b Fs(\002)h FD(L)1534 3098 y Fz(2)1574 3134 y Fx(.)p Fv(R)1677 3098 y Ft(d)1728 3134 y Fx(/)25 b FD(:)g Fx( )34 b Fs(2)25 b FD(dom)2212 3053 y Fq(\000)2257 3134 y FD(e)2301 3098 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2633 3053 y Fq(\001)2679 3134 y Fx(;)i(\036)29 b Fs(D)c FD(e)2960 3098 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)3292 3134 y Fx( )3371 3053 y Fq(\011)3283 3283 y FD(\(3.12\))294 3509 y(of)23 b(e)444 3473 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)799 3509 y FD(consists)22 b(of)g(all)h(pairs)g Fx(. )s(;)14 b(\036)5 b(/)23 b Fs(2)g FD(L)2010 3473 y Fz(2)2050 3509 y Fx(.)p Fv(R)2153 3473 y Ft(d)2204 3509 y Fx(/)18 b Fs(\002)g FD(L)2416 3473 y Fz(2)2456 3509 y Fx(.)p Fv(R)2559 3473 y Ft(d)2610 3509 y Fx(/)23 b FD(for)g(which)g(there)g(e)o(xists) 294 3629 y(a)i(sequence)h Fs(f)p Fx( )867 3644 y Ft(R)917 3629 y Fs(g)960 3644 y Ft(R)1036 3629 y FD(with)e Fx( )1314 3644 y Ft(R)1390 3629 y Fs(2)g FD(dom)1655 3548 y Fq(\000)1700 3629 y FD(e)1744 3593 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2035 3603 y Fl(R)2074 3593 y Fu(/)2105 3548 y Fq(\001)2176 3629 y Fs(D)25 b FD(L)2340 3593 y Fz(2)2379 3629 y Fx(.)p Fv(R)2482 3593 y Ft(d)2534 3629 y Fx(/)g FD(such)f(that)972 3869 y(lim)945 3933 y Ft(R)s Fr(!1)1139 3789 y Fq(\000)1184 3869 y Fp(k)p Fx( )1310 3884 y Ft(R)1380 3869 y Fs(\000)c Fx( )9 b Fp(k)1607 3884 y Fz(2)1666 3869 y Fs(C)20 b Fp(k)p FD(e)1858 3828 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2149 3838 y Fl(R)2188 3828 y Fu(/)2219 3869 y Fx( )2295 3884 y Ft(R)2365 3869 y Fs(\000)19 b Fx(\036)5 b Fp(k)2575 3884 y Fz(2)2615 3789 y Fq(\001)2685 3869 y Fs(D)25 b FD(0)14 b Fx(:)404 b FD(\(3.13\))294 4143 y(According)26 b(to)g(Lemma)g(3.2)g(the)g(con)l(v)o(er)n(gence)h(in)f(\(3.13\))g (holds)f(for)i(e)n(v)o(ery)f Fx( )34 b Fs(2)26 b FD(L)3252 4106 y Fz(2)3252 4169 y(G)3307 4143 y Fx(.)p Fv(R)3410 4107 y Ft(d)3461 4143 y Fx(/)p FD(,)294 4262 y(if)f(we)g(set)g Fx( )733 4277 y Ft(R)809 4262 y Fs(D)f Fx( )34 b FD(and)25 b Fx(\036)30 b Fs(D)g Fw(K)1452 4277 y Ft(t)1483 4262 y Fx( )9 b FD(,)25 b(that)f(is,)684 4488 y Fj(G)745 4503 y Ft(t)800 4488 y Fs(\023)903 4408 y Fq(\010)961 4488 y Fx(. )s(;)14 b(\036)5 b(/)25 b Fs(2)g FD(L)1392 4447 y Fz(2)1432 4488 y Fx(.)p Fv(R)1535 4447 y Ft(d)1586 4488 y Fx(/)20 b Fs(\002)f FD(L)1801 4447 y Fz(2)1841 4488 y Fx(.)p Fv(R)1944 4447 y Ft(d)1995 4488 y Fx(/)26 b FD(:)e Fx( )34 b Fs(2)25 b FD(L)2363 4447 y Fz(2)2363 4515 y(G)2418 4488 y Fx(.)p Fv(R)2521 4447 y Ft(d)2572 4488 y Fx(/;)14 b(\036)30 b Fs(D)g Fw(K)2924 4503 y Ft(t)2955 4488 y Fx( )3034 4408 y Fq(\011)3106 4488 y Fx(:)150 b FD(\(3.14\))294 4724 y(This)28 b(implies)g(L)884 4688 y Fz(2)884 4751 y(G)939 4724 y Fx(.)p Fv(R)1042 4688 y Ft(d)1093 4724 y Fx(/)f Fs(\022)g FD(dom)1439 4644 y Fq(\000)1485 4724 y FD(e)1529 4688 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1861 4644 y Fq(\001)1935 4724 y FD(and)29 b(\(3.10\).)42 b(Moreo)o(v)o(er)l(,)29 b(the)g(restriction)e(of)294 4844 y(the)e(self-adjoint)f(operator)h(e) 1321 4808 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1678 4844 y FD(to)24 b(L)1841 4807 y Fz(2)1841 4870 y(G)1896 4844 y Fx(.)p Fv(R)1999 4808 y Ft(d)2050 4844 y Fx(/)i FD(yields)e(a)h(symmetric)e(operator)-5 b(.)p 3362 4853 41 90 v 394 5023 a(Ha)n(ving)29 b(settled)f(Lemma)h (3.5,)h(we)g(are)g(in)f(a)g(position)f(to)h(establish)f(Theorem)h(1.12) g(on)294 5143 y(the)c(semigroup)f(properties)g(of)h(the)g(f)o(amily)f Fs(f)p FD(e)1930 5107 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2262 5143 y Fs(g)2299 5158 y Ft(t)6 b Fr(\025)p Fz(0)2417 5143 y FD(.)399 5322 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Theorem)g(1.12\))p Fw(.)374 b FD(\(i\))100 b(The)60 b(v)n(alidity)e(of)i(the)g(semigroup)f(la)o(w) 294 5442 y(\(1.18\))36 b(on)g(L)768 5405 y Fz(2)768 5468 y(G)824 5442 y Fx(.)p Fv(R)927 5406 y Ft(d)978 5442 y Fx(/)g FD(relies)h(on)f(the)g(functional)f(calculus)h(for)h(unbounded)e (functions)g(of)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 294 90 a FD(26)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(unbounded)29 b(self-adjoint)g(operators,)h(see)g(e.g.)g(Chap.)g(5)g(in)f([7],)i(on)f (Lemma)f(3.5)g(and)h(on)294 503 y(the)d(inclusion)k Fw(K)913 518 y Ft(t)944 503 y FD(L)1005 467 y Fz(2)1005 530 y(G)1060 503 y Fx(.)p Fv(R)1164 467 y Ft(d)1215 503 y Fx(/)26 b Fs(\022)g FD(L)1443 467 y Fr(1)1443 530 y Fz(G)1519 503 y Fx(.)p Fv(R)1622 467 y Ft(d)1673 503 y Fx(/)p FD(,)i(which)e(w)o (as)h(pro)o(v)o(en)f(in)g(Lemma)g(3.1.)37 b(The)27 b(latter)294 623 y(tw)o(o)e(ensure)g(that)f(both)g(sides)h(of)f(\(1.18\))h(are)h (well)e(de\002ned)i(on)e(L)2574 586 y Fz(2)2574 649 y(G)2629 623 y Fx(.)p Fv(R)2733 587 y Ft(d)2784 623 y Fx(/)p FD(.)521 826 y(\(ii\))100 b(Strong)36 b(continuity)f(of)h(the)h(orbit)e(mapping) h Fw(u)2429 841 y Fu( )2526 826 y FD(for)g Fx( )41 b Fs(2)31 b FD(L)2944 789 y Fz(2)2944 853 y(G)2999 826 y Fx(.)p Fv(R)3102 790 y Ft(d)3153 826 y Fx(/)37 b FD(follo)n(ws)294 946 y(from)25 b(the)g(functional)f(calculus,)g(too,)g(in)h(that)690 1264 y Fp(k)p Fw(u)795 1279 y Fu( )856 1264 y Fx(.)p Fw(t)j Fs(C)19 b Fw(h)6 b Fx(/)20 b Fs(\000)f Fw(u)1311 1279 y Fu( )1372 1264 y Fx(.)p Fw(t)9 b Fx(/)p Fp(k)1533 1223 y Fz(2)1533 1291 y(2)1597 1264 y Fs(D)1700 1128 y Fo(Z)1753 1353 y Fn(R)1794 1264 y Fp(h)p Fx( )s(;)23 b Fw(P)8 b Fx(.)p FD(d)17 b Fw(E)9 b Fx(/ )g Fp(i)25 b Fx(.)p FD(e)2466 1223 y Fr(\000)p Fu(.)p Ft(t)6 b Fr(C)p Ft(h)t Fu(/)f Ft(E)2770 1264 y Fs(\000)19 b FD(e)2911 1223 y Fr(\000)p Ft(t)11 b(E)3050 1264 y Fx(/)3087 1223 y Fz(2)3283 1264 y FD(\(3.15\))294 1589 y(for)22 b(all)e Fw(t)30 b Fs(2)21 b FD([0)p Fx(;)14 b Fs(1)p FD([)21 b(and)g(all)g Fw(h)27 b Fs(2)21 b FD([)p Fs(\000)p Fw(t)8 b Fx(;)14 b Fs(1)p FD([.)29 b(Here)h Fw(P)f FD(denotes)21 b(the)g(projection-v)n(alued)e(spec-)294 1709 y(tral)25 b(measure)g(of)h(the)e(Schr\366dinger)i(operator)33 b Fw(H)i FD(:)p Fs(D)e Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)p FD(,)26 b(that)e(is,)34 b Fw(P)8 b Fx(.)f Fw(I)12 b Fx(/)26 b FD(:)p Fs(D)3248 1691 y Fx(\037)3321 1724 y Ft(I)3359 1709 y Fx(.)8 b Fw(H)i Fx(/)294 1828 y FD(for)24 b(Borel)g(sets)30 b Fw(I)37 b Fs(\022)24 b Fv(R)5 b FD(.)36 b(Indeed,)24 b(the)f(inte)o(gral)g(in)g(\(3.15\))g (v)n(anishes)g(in)g(the)g(limit)f Fw(h)29 b Fs(!)23 b FD(0)h(by)294 1948 y(the)29 b(dominated-con)l(v)o(er)n(gence)g (theorem,)h(because)g(we)g(may)f(assume)g Fw(h)k Fs(2)27 b FD([)p Fs(\000)p Fw(t)8 b Fx(;)14 b Fw(h)3245 1963 y Fz(0)3284 1948 y FD(])29 b(with)294 2068 y(some)e Fw(h)588 2083 y Fz(0)653 2068 y Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([)27 b(so)f(that)h(the)g(function)f Fv(R)37 b Fs(3)c Fw(E)i Fs(7!)25 b Fx(.)p FD(1)c Fs(C)f FD(2)14 b(e)2547 2031 y Fr(\000)p Fu(.)p Ft(t)6 b Fr(C)p Ft(h)2745 2041 y Fk(0)2776 2031 y Fu(/)f Ft(E)2861 2068 y Fx(/)2898 2031 y Fz(2)2965 2068 y FD(dominates)26 b(the)294 2187 y(inte)o(grand)h(of)g(\(3.15\))g(and)h(is)f Fp(h)p Fx( )s(;)c Fw(P)8 b Fx(.)p Fs(\001)p Fx(/ )h Fp(i)p FD(-inte)o(grable)27 b(due)g(to)g Fx( )35 b Fs(2)27 b FD(L)2788 2150 y Fz(2)2788 2214 y(G)2843 2187 y Fx(.)p Fv(R)2946 2151 y Ft(d)2997 2187 y Fx(/)p FD(.)39 b(In)27 b(the)g(spe-)294 2307 y(cial)e(case)h Fw(t)33 b Fs(D)25 b FD(0,)f(this)g(procedure)i(gi)n(v)o(es)d(the)i (only)f(meaningful)g(right-sided)g(limit)f Fw(h)30 b Fs(#)c FD(0.)494 2510 y(\(iii\))99 b(First)73 b(we)g(claim)f Fj(C)1565 2473 y Fr(1)1548 2536 y Fz(0)1642 2510 y Fx(.)p Fv(R)1745 2474 y Ft(d)1796 2510 y Fx(/)52 b Fs(\032)f FD(dom)o Fx(.)8 b Fw(H)i FD(e)2362 2474 y Fr(\000)p Ft(t)h(H)2512 2510 y Fx(/)p FD(.)175 b(Since)73 b Fj(C)3126 2473 y Fr(1)3109 2536 y Fz(0)3203 2510 y Fx(.)p Fv(R)3306 2474 y Ft(d)3357 2510 y Fx(/)52 b Fs(\032)294 2629 y FD(dom)o Fx(.)p FD(e)552 2593 y Fr(\000)p Ft(t)11 b(H)701 2629 y Fx(/)p FD(,)25 b(this)f(follo)n(ws)g(from)g(Thm.)g(5.2.9\(c\))h(in)g ([7],)g(if)1280 2816 y Fo(Z)1333 3041 y Fn(R)1374 2951 y Fp(h)p Fx(')5 b(;)23 b Fw(P)8 b Fx(.)p FD(d)17 b Fw(E)9 b Fx(/')c Fp(i)24 b Fx(.)7 b Fw(E)i FD(e)2092 2910 y Fr(\000)p Ft(t)i(E)2231 2951 y Fx(/)2268 2910 y Fz(2)2333 2951 y Fx(<)25 b Fs(1)745 b FD(\(3.16\))294 3292 y(for)30 b(all)g Fx(')j Fs(2)27 b Fj(C)833 3256 y Fr(1)815 3319 y Fz(0)909 3292 y Fx(.)p Fv(R)1012 3256 y Ft(d)1063 3292 y Fx(/)p FD(.)46 b(The)30 b(latter)g(holds)f(true,)i(because)g Fx(.)7 b Fw(E)i FD(e)2552 3256 y Fr(\000)p Ft(t)i(E)2691 3292 y Fx(/)2728 3256 y Fz(2)2795 3292 y Fs(\024)35 b Fw(E)2975 3256 y Fz(2)3036 3292 y Fs(C)22 b FD(e)3180 3256 y Fr(\000)p Fz(2)p Ft(t)3288 3266 y Fk(0)3324 3256 y Ft(E)3407 3292 y FD(for)294 3412 y(all)30 b Fw(E)i Fs(2)23 b Fv(R)34 b FD(with)22 b(some)h Fw(t)1160 3427 y Fz(0)1223 3412 y Fx(>)g Fw(t)31 b FD(and)23 b(because)h Fj(C)1972 3375 y Fr(1)1955 3438 y Fz(0)2048 3412 y Fx(.)p Fv(R)2151 3376 y Ft(d)2203 3412 y Fx(/)f Fs(\032)g FD(dom)o Fx(.)8 b Fw(H)i Fx(/)19 b Fs(\\)e FD(dom)o Fx(.)p FD(e)3066 3376 y Fr(\000)p Ft(t)3139 3386 y Fk(0)3176 3376 y Ft(H)3239 3412 y Fx(/)p FD(.)30 b(Ne)o(xt)294 3531 y(we)22 b(compute)g(the)f (strong)g(deri)n(v)n(ati)n(v)o(e)f(of)i Fw(u)1784 3546 y Fu(')1854 3531 y FD(for)g Fx(')27 b Fs(2)22 b Fj(C)2243 3495 y Fr(1)2226 3558 y Fz(0)2319 3531 y Fx(.)p Fv(R)2423 3495 y Ft(d)2474 3531 y Fx(/)p FD(.)30 b(T)-8 b(o)21 b(this)g(end,)i(we)f(consider)294 3651 y(the)j(squared)g(norm)474 3861 y Fq(\015)474 3921 y(\015)529 3946 y Fw(h)585 3905 y Fr(\000)p Fz(1)679 3865 y Fq(\000)725 3946 y FD(e)769 3905 y Fr(\000)p Fu(.)p Ft(t)6 b Fr(C)p Ft(h)t Fu(/)f Ft(H)1063 3946 y Fx(')24 b Fs(\000)19 b FD(e)1284 3905 y Fr(\000)p Ft(t)11 b(H)1433 3946 y Fx(')1494 3865 y Fq(\001)1559 3946 y Fs(C)28 b Fw(H)10 b FD(e)1791 3905 y Fr(\000)p Ft(t)h(H)1940 3946 y Fx(')2001 3861 y Fq(\015)2001 3921 y(\015)2056 3888 y Fz(2)2056 3984 y(2)1019 4160 y Fs(D)1121 4024 y Fo(Z)1174 4249 y Fn(R)1216 4160 y Fp(h)p Fx(')5 b(;)23 b Fw(P)8 b Fx(.)p FD(d)17 b Fw(E)9 b Fx(/')c Fp(i)1776 4079 y Fq(\002)1817 4160 y Fw(h)1873 4119 y Fr(\000)p Fz(1)1967 4079 y Fq(\000)2013 4160 y FD(e)2057 4119 y Fr(\000)p Fu(.)p Ft(t)h Fr(C)p Ft(h)t Fu(/)f Ft(E)2360 4160 y Fs(\000)20 b FD(e)2502 4119 y Fr(\000)p Ft(t)11 b(E)2641 4079 y Fq(\001)2706 4160 y Fs(C)27 b Fw(E)9 b FD(e)2925 4119 y Fr(\000)p Ft(t)i(E)3063 4079 y Fq(\003)3105 4102 y Fz(2)3283 4160 y FD(\(3.17\))294 4485 y(for)32 b Fw(h)i Fs(2)p FD(])22 b Fs(\000)g Fw(t)8 b Fx(;)14 b FD(1])22 b Fs(n)g(f)p FD(0)p Fs(g)31 b FD(and)h(claim)f (that)g(it)g(v)n(anishes)f(in)h(the)g(limit)f Fw(h)35 b Fs(!)28 b FD(0.)50 b(\(In)32 b(the)f(spe-)294 4605 y(cial)f(case)g Fw(t)36 b Fs(D)28 b FD(0,)i(the)g(limit)e(gi)n(v)o(es)g (the)h(only)g(meaningful)g(right-sided)f(deri)n(v)n(ati)n(v)o(e.\))43 b(This)294 4724 y(follo)n(ws)33 b(from)g(the)h(dominated-con)l(v)o(er)n (gence)f(theorem)h(and)g(the)g Fw(h)6 b FD(-independent)33 b(upper)294 4844 y(bound)i(2)7 b Fw(E)706 4808 y Fz(2)745 4763 y Fq(\000)791 4844 y FD(2)23 b Fs(C)h FD(e)1010 4808 y Fr(\000)p Fz(2)p Ft(t)11 b(E)1207 4844 y Fs(C)23 b FD(2e)1402 4808 y Fr(\000)p Fz(2)p Fu(.)p Ft(t)6 b Fr(C)p Fz(1)p Fu(/)f Ft(E)1718 4763 y Fq(\001)1799 4844 y FD(for)35 b(the)h(inte)o(grand)e(in)h(\(3.17\).)63 b(This)34 b(bound)h(is)294 4963 y Fp(h)p Fx(')5 b(;)23 b Fw(P)8 b Fx(.)p Fs(\001)p Fx(/')d Fp(i)p FD(-inte)o(grable)23 b(as)i(a)f(function)g(of)31 b Fw(E)i FD(because)25 b(of)f Fx(')30 b Fs(2)24 b Fj(C)2619 4927 y Fr(1)2602 4990 y Fz(0)2695 4963 y Fx(.)p Fv(R)2798 4927 y Ft(d)2850 4963 y Fx(/)g Fs(\032)g FD(dom)o Fx(.)8 b Fw(H)i Fx(/)26 b FD(and)294 5083 y(\(3.16\).)294 5203 y(It)40 b(remains)f(to)h(sho)n(w)f (that)g Fw(u)1360 5218 y Fu(')1448 5203 y FD(is)g(the)h Fw(unique)f FD(solution)f(of)i(the)f(initial-v)n(alue)f(problem)294 5322 y(\(1.19\).)31 b(T)-8 b(o)24 b(this)f(end,)h(let)g Fx(8)g FD(be)g(an)h(arbitrary)f(solution)e(of)j(\(1.19\))e(and)i(\002x) f Fw(t)33 b Fx(>)24 b FD(0)g(arbitrary)-6 b(.)294 5442 y(By)32 b(the)g(abo)o(v)o(e)e(reasoning)h(one)h(has)1649 5402 y Fz(d)p 1633 5419 77 4 v 1633 5477 a(d)10 b Ft(s)1735 5442 y FD(e)1779 5406 y Fr(\000)p Fu(.)p Ft(t)c Fr(\000)p Ft(s)t Fu(/)f Ft(H)2067 5442 y Fw(g)33 b Fs(D)j Fw(H)10 b FD(e)2390 5406 y Fr(\000)p Fu(.)p Ft(t)c Fr(\000)p Ft(s)t Fu(/)f Ft(H)2679 5442 y Fw(g)35 b FD(in)d(the)f(strong)g(sense)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(27)294 384 y(for)25 b(arbitrary)g Fw(s)31 b Fs(2)p FD(]0)p Fx(;)14 b Fw(t)9 b FD([)24 b(and)h(arbitrary)i Fw(g)i Fs(2)c Fj(C)1949 347 y Fr(1)1932 410 y Fz(0)2025 384 y Fx(.)p Fv(R)2128 348 y Ft(d)2179 384 y Fx(/)p FD(.)31 b(As)25 b(a)g(consequence,)g(one)g(\002nds)404 612 y(d)p 381 659 105 4 v 381 753 a(d)10 b Fw(s)512 682 y Fp(h)p FD(e)595 641 y Fr(\000)p Fu(.)p Ft(t)c Fr(\000)p Ft(s)t Fu(/)f Ft(H)883 682 y Fw(g)t Fx(;)14 b(8)s(.)p Fw(s)6 b Fx(/)p Fp(i)24 b Fs(D)h Fp(h)8 b Fw(H)i FD(e)1524 641 y Fr(\000)p Fu(.)p Ft(t)c Fr(\000)p Ft(s)t Fu(/)f Ft(H)1812 682 y Fw(g)t Fx(;)14 b(8)s(.)p Fw(s)6 b Fx(/)p Fp(i)19 b Fs(\000)h Fp(h)p FD(e)2353 641 y Fr(\000)p Fu(.)p Ft(t)6 b Fr(\000)p Ft(s)t Fu(/)f Ft(H)2641 682 y Fw(g)t Fx(;)22 b Fw(H)10 b Fx(8)s(.)p Fw(s)c Fx(/)p Fp(i)25 b Fs(D)f FD(0)3283 861 y(\(3.18\))294 1104 y(by)k(the)f(assumptions)f(on)i Fx(8)g FD(and)f(the)h(self-adjointness)f(of)35 b Fw(H)10 b FD(.)41 b(Hence,)29 b(the)e(fundamental)294 1224 y(theorem)e(of)g (calculus)f(implies)631 1527 y(0)60 b Fs(D)878 1391 y Fo(Z)976 1418 y Ft(t)931 1616 y Fz(0)1007 1527 y FD(d)10 b Fw(s)1171 1457 y FD(d)p 1148 1504 V 1148 1597 a(d)g Fw(s)1279 1527 y Fp(h)p FD(e)1362 1485 y Fr(\000)p Fu(.)p Ft(t)c Fr(\000)p Ft(s)t Fu(/)f Ft(H)1650 1527 y Fw(g)t Fx(;)14 b(8)s(.)p Fw(s)6 b Fx(/)p Fp(i)24 b Fs(D)h Fp(h)r Fw(g)t Fx(;)14 b(8)s(.)p Fw(t)9 b Fx(/)p Fp(i)19 b Fs(\000)g Fp(h)p FD(e)2691 1485 y Fr(\000)p Ft(t)11 b(H)2841 1527 y Fw(g)t Fx(;)j(8)s(.)p FD(0)p Fx(/)p Fp(i)741 1744 y Fs(D)59 b Fp(h)r Fw(g)t Fx(;)14 b(8)s(.)p Fw(t)9 b Fx(/)p Fp(i)19 b Fs(\000)g Fp(h)r Fw(g)t Fx(;)14 b FD(e)1554 1703 y Fr(\000)p Ft(t)d(H)1702 1744 y Fx(')5 b Fp(i)25 b Fs(D)f Fp(h)r Fw(g)t Fx(;)14 b(8)s(.)p Fw(t)9 b Fx(/)19 b Fs(\000)h Fw(u)2436 1759 y Fu(')2484 1744 y Fx(.)p Fw(t)9 b Fx(/)p Fp(i)14 b Fx(:)608 b FD(\(3.19\))294 1987 y(The)25 b(denseness)g(of)g Fj(C)1089 1951 y Fr(1)1072 2014 y Fz(0)1165 1987 y Fx(.)p Fv(R)1268 1951 y Ft(d)1319 1987 y Fx(/)g FD(in)g(L)1545 1951 y Fz(2)1585 1987 y Fx(.)p Fv(R)1688 1951 y Ft(d)1739 1987 y Fx(/)g FD(completes)f(the)h (proof)g(of)f(uniqueness.)p 3313 1996 41 90 v 394 2189 a(An)h(immediate)e(consequence)i(of)g(the)g(just-pro)o(v)o(en)e (Theorem)i(1.12)f(is)396 2441 y(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)35 b FD(3)t(.)t(6)t(.)125 b Fw(Assume)22 b(the)f(situation)f(of)h(Theor)l (em)h FD(1.10)p Fw(.)29 b(Then)22 b FD(L)3067 2404 y Fz(2)3067 2467 y(G)3123 2441 y Fx(.)p Fv(R)3226 2405 y Ft(d)3277 2441 y Fx(/)g Fw(is)f(an)294 2560 y(oper)o(ator)j(cor)l(e)h (for)f FD(e)1050 2524 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1407 2560 y Fw(for)24 b(all)g(t)34 b Fx(>)24 b FD(0)p Fw(.)399 2812 y(Pr)l(oof)f(.)120 b FD(By)40 b(Theorem)f(1.12)g(and)h(the)f(symmetry)f(of)i(e)2510 2776 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2881 2812 y FD(on)39 b(L)3081 2775 y Fz(2)3081 2838 y(G)3137 2812 y Fx(.)p Fv(R)3240 2776 y Ft(d)3291 2812 y Fx(/)p FD(,)k(see)294 2931 y(Lemma)38 b(3.5,)i(all)e(three)g (assumptions)e(of)i(Thm.)f(1)h(in)g([35)o(])h(are)f(ful\002lled)g(by)g (choosing)294 3051 y(there)29 b Fx(\013)i Fs(D)c Fw(t)35 b Fs(2)14 b FD(]0)p Fx(;)g Fs(1)p FD([,)34 b Fw(S)1227 3066 y Ft(t)1285 3051 y Fs(D)27 b FD(e)1434 3015 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1794 3051 y FD(with)28 b(dom)o Fx(.)5 b Fw(S)2267 3066 y Ft(t)2298 3051 y Fx(/)27 b Fs(D)g FD(L)2528 3014 y Fz(2)2528 3078 y(G)2583 3051 y Fx(.)p Fv(R)2686 3015 y Ft(d)2737 3051 y Fx(/)i FD(and)37 b Fw(D)30 b Fs(D)d FD(L)3252 3014 y Fz(2)3252 3078 y(G)3307 3051 y Fx(.)p Fv(R)3410 3015 y Ft(d)3461 3051 y Fx(/)p FD(.)294 3171 y(In)d(this)e(conte)o(xt,)h(we) g(recall)h(from)f(Lemma)g(3.5)g(that)g(e)2203 3135 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2559 3171 y FD(is)22 b(symmetric)h(on)g(L)3277 3134 y Fz(2)3277 3197 y(G)3332 3171 y Fx(.)p Fv(R)3435 3135 y Ft(d)3486 3171 y Fx(/)294 3290 y FD(and)f(from)g(Theorem)f(1.12) h(that)f(the)h(mapping)e([0)p Fx(;)14 b Fs(1)p FD([)p Fs(3)22 b Fw(t)31 b Fs(7!)22 b Fp(h)p Fx( )s(;)14 b Fw(u)2694 3305 y Fu( )2754 3290 y Fx(.)p Fw(t)9 b Fx(/)p Fp(i)22 b FD(is)f(continuous)g(\226)294 3410 y(and)k(hence)g(Borel)h (measurable)e(\226)h(for)g(e)n(v)o(ery)f Fx( )34 b Fs(2)25 b FD(L)2156 3373 y Fz(2)2156 3436 y(G)2211 3410 y Fx(.)p Fv(R)2314 3374 y Ft(d)2365 3410 y Fx(/)g FD(due)g(to)g(the)f(strong)g (continuity)294 3529 y(of)h(the)g(orbit)f(mapping)g Fw(u)1191 3544 y Fu( )1251 3529 y FD(.)31 b(Therefore)26 b(the)f(claim)f(follo)n (ws)f(from)i(Thm.)f(1)h(in)f([35].)p 3384 3538 V 394 3732 a(The)h(remaining)f(part)h(of)g(the)f(proof)h(of)g(Theorem)g(1.10) f(is)h(pro)o(vided)e(by)396 3983 y(L)t FG(E)t(M)t(M)t(A)33 b FD(3)t(.)t(7)t(.)125 b Fw(Assume)24 b(the)h(situation)e(of)i(Theor)l (em)g FD(1.10)g Fw(and)f(let)31 b(K)2947 3998 y Ft(t)3003 3983 y Fw(be)25 b(de\002ned)g(as)294 4103 y(in)g(Lemma)g FD(3.1)p Fw(.)30 b(Then)c(one)f(has)f(the)h(equality)1588 4346 y(K)1660 4361 y Ft(t)1716 4346 y Fs(D)g FD(e)1863 4304 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2208 4346 y Fx(:)1048 b FD(\(3.20\))399 4597 y Fw(Pr)l(oof)23 b(.)120 b FD(W)-8 b(e)27 b(follo)n(w)f([3])h(or)g([45)o(])g(and)g (introduce)f(the)h(restriction)32 b Fw(K)2918 4561 y Fz(0)2910 4624 y Ft(t)2984 4597 y FD(:)p Fs(D)g Fw(K)3194 4612 y Ft(t)3224 4597 y Fp(j)3252 4620 y Fz(dom)p Fu(.)t Ft(K)3463 4594 y Fk(0)3457 4638 y Fl(t)3492 4620 y Fu(/)294 4717 y FD(of)25 b(the)g(maximal)f(Carleman)h(operator)31 b Fw(K)1770 4732 y Ft(t)1826 4717 y FD(to)24 b(the)h(subspace)964 4960 y(dom)o Fx(.)6 b Fw(K)1264 4919 y Fz(0)1256 4986 y Ft(t)1305 4960 y Fx(/)25 b FD(:)p Fs(D)1497 4879 y Fq(\010)1555 4960 y Fx( )34 b Fs(2)25 b FD(dom)o Fx(.)6 b Fw(K)2039 4975 y Ft(t)2070 4960 y Fx(/)25 b FD(:)g Fx(\024)2234 4975 y Ft(t)2265 4960 y Fx( )34 b Fs(2)24 b FD(L)2517 4919 y Fz(1)2557 4960 y Fx(.)p Fv(R)2660 4919 y Ft(d)2711 4960 y Fx(/)2748 4879 y Fq(\011)2821 4960 y Fx(;)429 b FD(\(3.21\))294 5203 y(where)24 b(the)e(function)h Fv(R)1127 5166 y Ft(d)1201 5203 y Fs(3)j Fw(x)31 b Fs(7!)23 b Fx(\024)1547 5218 y Ft(t)1578 5203 y Fx(.)s Fw(x)9 b Fx(/)23 b FD(:)p Fs(D)g Fp(k)p Fw(k)1956 5218 y Ft(t)1986 5203 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)p Fp(k)2241 5218 y Fz(2)2304 5203 y Fs(D)23 b FD([)p Fw(k)2484 5218 y Fz(2)p Ft(t)2550 5203 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)p FD(])2816 5166 y Fz(1)p Fu(=)p Fz(2)2946 5203 y FD(is)23 b(well)f(de\002ned)294 5322 y(and)g(continuous)e(because)i (of)f(Lemma)h(1.7\(iii\))o(.)29 b(The)22 b(estimate)f(\(1.13\))g(in)g (Lemma)g(1.7)h(and)294 5442 y(Remark)33 b(1.6\(i\))f(imply)f(L)1219 5405 y Fz(2)1219 5468 y(G)1274 5442 y Fx(.)p Fv(R)1377 5406 y Ft(d)1428 5442 y Fx(/)e Fs(\022)g FD(dom)o Fx(.)6 b Fw(K)1901 5406 y Fz(0)1893 5469 y Ft(t)1942 5442 y Fx(/)p FD(.)52 b(Thus,)34 b(the)e(Fe)o(ynman-Kac-It\364)g(formula)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 294 90 a FD(28)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(from)25 b(Lemma)f(3.5)h(leads) g(to)950 626 y(e)994 585 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1326 626 y Fp(j)1354 649 y Fz(L)1397 623 y Fk(2)1397 673 y(G)1437 649 y Fu(.)p Fn(R)1516 628 y Fl(d)1547 649 y Fu(/)1603 626 y Fs(D)31 b Fw(K)1784 641 y Ft(t)1815 626 y Fp(j)1843 649 y Fz(L)1886 623 y Fk(2)1886 673 y(G)1926 649 y Fu(.)p Fn(R)2005 628 y Fl(d)2036 649 y Fu(/)2092 626 y Fs(D)f Fw(K)2280 585 y Fz(0)2272 653 y Ft(t)2321 626 y Fp(j)2349 649 y Fz(L)2392 623 y Fk(2)2392 673 y(G)2432 649 y Fu(.)p Fn(R)2511 628 y Fl(d)2542 649 y Fu(/)2598 626 y Fs(\022)g Fw(K)2786 585 y Fz(0)2778 653 y Ft(t)2840 626 y Fx(:)416 b FD(\(3.22\))294 868 y(Here,)31 b(as)e(usual,)g(the)g(notation)37 b Fw(A)29 b Fs(\022)35 b Fw(B)g FD(means)29 b(that)f(the)h(operator)37 b Fw(B)e FD(is)28 b(an)h(e)o(xtension)e(of)294 987 y(the)h(operator)39 b Fw(A)r FD(.)i(By)29 b(Thm.)e(10.1)h(in)g([45])h(the)f(operator)34 b Fw(K)2430 951 y Fz(0)2422 1014 y Ft(t)2499 987 y FD(is)28 b(symmetric,)g(hence)h(clos-)294 1107 y(able.)43 b(T)-8 b(aking)28 b(the)g(closure)h(of)g(\(3.22\))f(with)g(respect)h(to)g(the) f(graph)h(norm)f(and)h(e)o(xploiting)294 1227 y(Corollary)j(3.6,)h(we)e (get)h(e)1236 1190 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1596 1227 y Fs(\022)p 1702 1138 127 4 v 34 w Fw(K)1788 1198 y Fz(0)1780 1253 y Ft(t)1828 1227 y FD(.)51 b(Since)38 b Fw(K)2243 1190 y Fz(0)2235 1253 y Ft(t)2315 1227 y FD(is)31 b(symmetric,)h(so)f(is)g(its)g (closure)p 294 1272 V 300 1360 a Fw(K)380 1332 y Fz(0)372 1387 y Ft(t)420 1360 y FD(.)g(Therefore)26 b(we)f(conclude)1571 1617 y(e)1615 1575 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1971 1617 y Fs(D)p 2074 1529 V 31 w Fw(K)2160 1588 y Fz(0)2152 1643 y Ft(t)2214 1617 y Fx(;)1036 b FD(\(3.23\))294 1859 y(because)37 b(self-adjoint)e (operators)i(are)g(maximally)e(symmetric.)64 b(Furthermore,)39 b(we)e(ob-)294 1978 y(serv)o(e)23 b(the)f(equalities)p 1075 1890 V 28 w Fw(K)1161 1949 y Fz(0)1153 2005 y Ft(t)1224 1978 y Fs(D)1325 1898 y Fq(\000)p 1393 1890 V 1399 1978 a Fw(K)1479 1949 y Fz(0)1471 2005 y Ft(t)1542 1898 y Fq(\001)1588 1920 y Fr(\003)1652 1978 y Fs(D)h Fx(.)6 b Fw(K)1876 1942 y Fz(0)1868 2005 y Ft(t)1916 1978 y Fx(/)1953 1942 y Fr(\003)2018 1978 y Fs(D)28 b Fw(K)2196 1993 y Ft(t)2227 1978 y FD(,)23 b(which)g(hold)f(according)g(to)h (\(3.23\),)294 2098 y(Thm.)h(VIII.1\(c\))i(in)f([38)o(])h(and)e(Thm.)h (10.1)f(in)g([45].)31 b(This)24 b(completes)g(the)h(proof.)p 3276 2107 41 90 v 394 2294 a(Finally)-6 b(,)24 b(we)h(gather)f(our)h (pre)n(vious)f(results)g(to)g(complete)h(the)399 2490 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Theorem)g(1.10\))p Fw(.)119 b FD(Corollary)39 b(3.6)g(has)g(established)f(that)h(L)3038 2453 y Fz(2)3038 2517 y(G)3093 2490 y Fx(.)p Fv(R)3196 2454 y Ft(d)3247 2490 y Fx(/)h FD(is)e(an)294 2610 y(operator)e(core)g (for)f(e)1064 2574 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)1396 2610 y FD(.)63 b(The)35 b(remaining)g (assertions)f(of)i(Theorem)f(1.10)g(follo)n(w)294 2729 y(from)25 b(Lemma)f(3.7,)h(Lemma)f(3.1)h(and)f(Lemma)h(1.7\(iii\))o(.)p 2412 2738 V 371 3048 a(4)t(.)105 b(P)t FG(R)q(O)t(O)t(F)t(S)35 b(O)t(F)c FD(T)t FG(H)t(E)t(O)t(R)t(E)t(M)h FD(1.14)r(,)d(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)k FD(1.16)27 b FG(A)t(N)t(D)j FD(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)j FD(1.18)394 3268 y(The)27 b(follo)n(wing)e(lemma)h(is)g(in)h(the)f(spirit)g(of)h (Thm.)f(B.7.8)h(in)f([42],)h(b)n(ut,)g(among)f(others,)294 3387 y(we)f(do)g(not)f(assume)h(that)f(the)h(operator)32 b Fw(M)h FD(is)24 b(bounded.)396 3639 y(L)t FG(E)t(M)t(M)t(A)33 b FD(4)t(.)t(1)t(.)125 b Fw(Let)30 b(M)h(be)22 b(the)h(maximal)f (self-adjoint)f(Carleman)h(oper)o(ator)f(induced)294 3758 y(by)27 b(the)f(Bor)l(el-measur)o(able)f(and)h(Hermitian)g(inte)l (gr)o(al)e(k)o(ernel)j(m)32 b FD(:)25 b Fv(R)2751 3722 y Ft(d)2822 3758 y Fs(\002)c Fv(R)2987 3722 y Ft(d)3064 3758 y Fs(!)k Fv(C)53 b Fw(in)26 b(the)294 3878 y(sense)f(that)402 4147 y Fj(C)485 4106 y Fr(1)468 4174 y Fz(0)561 4147 y Fx(.)p Fv(R)664 4106 y Ft(d)715 4147 y Fx(/)60 b Fs(\032)g FD(dom)o Fx(.)7 b Fw(M)h Fx(/)25 b FD(:)p Fs(D)1455 4036 y Fq(n)1521 4147 y Fx( )34 b Fs(2)25 b FD(L)1774 4106 y Fz(2)1814 4147 y Fx(.)p Fv(R)1917 4106 y Ft(d)1968 4147 y Fx(/)g FD(:)2083 4011 y Fo(Z)2135 4236 y Fn(R)2188 4216 y Fl(d)2238 4147 y FD(d)16 b Fw(y)31 b(m)6 b Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)24 b Fs(2)h FD(L)3058 4106 y Fz(2)3098 4147 y Fx(.)p Fv(R)3201 4106 y Ft(d)3252 4147 y Fx(/)3289 4036 y Fq(o)3369 4147 y Fx(;)582 4412 y Fw(M)8 b Fx( )69 b Fs(D)950 4276 y Fo(Z)1002 4501 y Fn(R)1055 4481 y Fl(d)1091 4412 y FD(d)16 b Fw(y)31 b(m)6 b Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)1595 b FD(\(4.1\))294 4702 y Fw(for)27 b(all)g Fx( )36 b Fs(2)26 b FD(dom)o Fx(.)7 b Fw(M)h Fx(/)p Fw(,)29 b(m)6 b Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)26 b Fs(D)g Fw(m)1688 4666 y Fr(\003)1729 4702 y Fx(.)6 b Fw(y)g Fx(;)17 b Fw(x)9 b Fx(/)28 b Fw(for)f(Lebesgue-almost)g(all)g(pair)o(s)g Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)26 b Fs(2)294 4822 y Fv(R)360 4786 y Ft(d)436 4822 y Fs(\002)e Fv(R)604 4786 y Ft(d)693 4822 y Fw(and)37 b(m)43 b(has)37 b(the)g(Carleman)g(pr) l(operty)g FD(\(1.15\))p Fw(.)68 b(Assume)37 b(further)g(that)i(x)i Fs(7!)294 4941 y Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)27 b Fw(de\002nes)g(a)g(str)l(ongly)f(continuous)g (mapping)g(fr)l(om)g Fv(R)2453 4905 y Ft(d)2531 4941 y Fw(to)h FD(L)2697 4905 y Fz(2)2737 4941 y Fx(.)p Fv(R)2840 4905 y Ft(d)2891 4941 y Fx(/)p Fw(.)38 b(F)l(inally)-5 b(,)26 b(let)35 b(B)294 5061 y(be)27 b(a)f(bounded)g(oper)o(ator)e(on)i FD(L)1421 5025 y Fz(2)1461 5061 y Fx(.)p Fv(R)1564 5025 y Ft(d)1615 5061 y Fx(/)h Fw(suc)o(h)f(that)32 b(M)r(B)h(and)f(M)r(B) 2589 5025 y Fr(\003)2656 5061 y Fw(ar)l(e)27 b(also)e(bounded)h(and)294 5181 y(that)31 b(M)r(B)14 b(M)32 b(admits)24 b(a)h(bounded)f(closed)h (e)n(xtension)p 2169 5102 258 4 v 31 w(M)r(B)14 b(M)32 b(to)25 b(all)f(of)g FD(L)2847 5145 y Fz(2)2887 5181 y Fx(.)p Fv(R)2990 5145 y Ft(d)3041 5181 y Fx(/)p Fw(.)31 b(Then)549 5322 y FD(\(i\))p 743 5244 V 107 w Fw(M)r(B)14 b(M)46 b(is)38 b(a)h(bounded)f(Carleman)g(oper)o(ator)f(induced)i(by)f (the)h(continuous)294 5442 y(inte)l(gr)o(al)23 b(k)o(ernel)h Fx(\014)32 b FD(:)24 b Fv(R)1107 5406 y Ft(d)1177 5442 y Fs(\002)19 b Fv(R)1340 5406 y Ft(d)1415 5442 y Fs(!)24 b Fv(C)c Fw(,)30 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(7!)g Fx(\014)7 b(.)s Fw(x)i Fx(;)20 b Fw(y)6 b Fx(/)25 b FD(:)p Fs(D)e Fp(h)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)22 b Fw(B)7 b(m)f Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)p Fp(i)22 b Fw(in)i(the)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(29)294 384 y Fw(sense)25 b(that)p 1290 556 258 4 v 1297 635 a(M)r(B)14 b(M)8 b Fx( )34 b Fs(D)1754 499 y Fo(Z)1807 724 y Fn(R)1860 704 y Fl(d)1896 635 y FD(d)16 b Fw(y)30 b Fx(\014)7 b(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)806 b FD(\(4.2\))294 921 y Fw(for)24 b(all)h Fx( )34 b Fs(2)24 b FD(L)818 885 y Fz(2)858 921 y Fx(.)p Fv(R)961 885 y Ft(d)1012 921 y Fx(/)h Fw(and)g Fx(\014)32 b Fw(has)24 b(the)h(Carleman)g(pr)l(operty)e FD(\(1.15\))p Fw(.)521 1085 y FD(\(ii\))100 b Fw(the)28 b(left-hand)g(side)g(of)g FD(\(4.2\))h Fw(has)f(a)h(continuous)e(r)l (epr)l(esentative)h(in)g FD(L)3267 1049 y Fz(2)3307 1085 y Fx(.)p Fv(R)3410 1049 y Ft(d)3461 1085 y Fx(/)p Fw(,)294 1204 y(whic)o(h)d(is)f(given)h(by)g(the)g(right-hand)e(side)h(of)h FD(\(4.2\))p Fw(.)494 1368 y FD(\(iii\))99 b Fw(for)38 b(any)h Fx(w)d Fs(2)d FD(L)1347 1332 y Fr(1)1423 1368 y Fx(.)p Fv(R)1526 1332 y Ft(d)1577 1368 y Fx(/)39 b Fw(with)1864 1288 y Fo(R)1910 1402 y Fn(R)1963 1382 y Fl(d)1994 1402 y Fr(\002)p Fn(R)2102 1382 y Fl(d)2151 1368 y FD(d)13 b Fw(x)c FD(d)16 b Fw(y)j Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2658 1332 y Fz(2)2712 1368 y Fp(j)p Fw(m)d Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)3079 1332 y Fz(2)3150 1368 y Fx(<)33 b Fs(1)39 b Fw(the)294 1504 y(pr)l(oduct)p 625 1426 V 31 w(M)r(B)14 b(M)33 b Fs(O)-59 b Fx(w)28 b Fw(is)d(a)f(Hilbert-Sc)o(hmidt)f(oper)o(ator)g (with)i(squar)l(ed)f(norm)h(given)f(by)713 1774 y FD(T)m(race)936 1694 y Fq(\002)1003 1774 y Fs(O)-58 b Fx(w)1054 1733 y Fr(\003)1095 1774 y Fp(j)p 1123 1696 V 7 w Fw(M)r(B)14 b(M)7 b Fp(j)1408 1733 y Fz(2)1473 1774 y Fs(O)-58 b Fx(w)1524 1694 y Fq(\003)1590 1774 y Fs(D)1693 1639 y Fo(Z)1746 1864 y Fn(R)1799 1844 y Fl(d)1835 1774 y FD(d)13 b Fw(x)33 b Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2237 1733 y Fz(2)2291 1639 y Fo(Z)2343 1864 y Fn(R)2396 1844 y Fl(d)2432 1774 y FD(d)16 b Fw(y)31 b Fp(j)p Fx(\014)7 b(.)s Fw(x)i Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)2925 1733 y Fz(2)2978 1774 y Fx(:)328 b FD(\(4.3\))294 2052 y Fw(Her)l(e)51 b Fs(O)-59 b Fx(w)28 b Fw(is)c(the)h(bounded)e(multiplication)f(oper)o (ator)h(uniquely)h(corr)l(esponding)f(to)h Fx(w)s Fw(,)h(and)320 2172 y Fs(O)-59 b Fx(w)370 2136 y Fr(\003)437 2172 y Fw(denotes)24 b(its)g(Hilbert)g(adjoint.)399 2423 y(Pr)l(oof)f(.)120 b FD(The)33 b(strong)g(continuity)e(of)j(the)f(mapping)f Fv(R)2419 2387 y Ft(d)2500 2423 y Fs(!)d FD(L)2700 2387 y Fz(2)2740 2423 y Fx(.)p Fv(R)2843 2387 y Ft(d)2894 2423 y Fx(/)p FD(,)38 b Fw(x)h Fs(7!)29 b Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)p FD(,)294 2543 y(the)26 b(triangle)f(and)h(the)g(Cauchy-Schw)o(arz)h(inequality)e(imply)g(the)g (continuity)f(of)i(the)g(func-)294 2663 y(tion)48 b Fj(M)53 b FD(:)38 b Fv(R)791 2626 y Ft(d)880 2663 y Fs(!)f Fv(R)5 b FD(,)64 b Fw(x)46 b Fs(7!)38 b Fj(M)16 b Fx(.)s Fw(x)9 b Fx(/)38 b FD(:)p Fs(D)f Fp(k)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)p Fp(k)2238 2678 y Fz(2)2325 2663 y FD(because)49 b Fp(j)p Fj(M)16 b Fx(.)s Fw(x)9 b Fx(/)28 b Fs(\000)g Fj(M)16 b Fx(.)s Fw(x)3322 2626 y Fr(0)3346 2663 y Fx(/)p Fp(j)37 b Fs(\024)294 2782 y Fp(k)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)23 b Fs(\000)f Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)996 2746 y Fr(0)1019 2782 y Fx(/)p Fp(k)1106 2797 y Fz(2)1146 2782 y FD(.)58 b(No)n(w)-6 b(,)35 b(for)f(e)n(v)o(ery)f Fx(')i Fs(2)30 b Fj(C)2143 2746 y Fr(1)2126 2809 y Fz(0)2219 2782 y Fx(.)p Fv(R)2323 2746 y Ft(d)2374 2782 y Fx(/)k FD(and)g(e)n(v)o(ery)f Fx( )39 b Fs(2)30 b FD(L)3137 2746 y Fz(2)3176 2782 y Fx(.)p Fv(R)3280 2746 y Ft(d)3331 2782 y Fx(/)k FD(the)294 2902 y(Cauchy-Schw)o(arz)27 b(inequality)c(pro)o(vides)h(the)g(estimate)472 3037 y Fo(Z)524 3262 y Fn(R)577 3242 y Fl(d)608 3262 y Fr(\002)p Fn(R)716 3242 y Fl(d)763 3173 y FD(d)13 b Fw(x)c FD(d)16 b Fw(y)30 b Fp(j)p Fx( )9 b(.)d Fw(y)g Fx(/)p Fp(j)14 b(j)p Fw(m)6 b Fx(.)g Fw(y)g Fx(;)17 b Fw(x)9 b Fx(/)p Fp(j)14 b(j)p Fx(')5 b(.)s Fw(x)k Fx(/)p Fp(j)23 b Fs(\024)i Fp(k)p Fx( )9 b Fp(k)2228 3188 y Fz(2)2282 3173 y Fp(k)p Fx(')c Fp(k)2443 3188 y Fz(2)2496 3173 y Fp(k)p Fj(M)2670 3155 y Fx(\037)2738 3188 y Fz(supp)10 b Fu(')2927 3173 y Fp(k)2977 3188 y Fz(2)3042 3173 y Fx(<)25 b Fs(1)86 b FD(\(4.4\))294 3452 y(due)25 b(to)g(the)f(continuity)f(of)i Fj(M)16 b FD(.)31 b(Therefore,)25 b(\(4.1\))g(and)g(Fubini')-5 b(s)24 b(theorem)g(yield)1133 3723 y Fp(h)7 b Fw(M)h Fx(')d(;)14 b( )9 b Fp(i)25 b Fs(D)1623 3587 y Fo(Z)1676 3812 y Fn(R)1729 3792 y Fl(d)1765 3723 y FD(d)13 b Fw(x)33 b Fx(')1966 3682 y Fr(\003)2008 3723 y Fx(.)s Fw(x)9 b Fx(/)14 b Fp(h)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )9 b Fp(i)14 b Fx(;)647 b FD(\(4.5\))294 3994 y(where)34 b(the)f(scalar)h(product)e(in)h(the)g(inte)o(grand)f (is)h(well)g(de\002ned,)i(because,)h(by)d(hypothe-)294 4113 y(sis,)h Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)29 b Fs(2)g FD(L)924 4077 y Fz(2)964 4113 y Fx(.)p Fv(R)1067 4077 y Ft(d)1118 4113 y Fx(/)k FD(for)h(all)h Fw(x)j Fs(2)29 b Fv(R)1713 4077 y Ft(d)1765 4113 y FD(.)54 b(Ne)o(xt,)34 b(we)g(consider)e(a)h(sequence)h Fx(. )3199 4128 y Ft(n)3242 4113 y Fx(/)3279 4128 y Ft(n)s Fr(2)p Fn(N)3446 4113 y Fs(\032)294 4233 y Fj(C)377 4196 y Fr(1)360 4260 y Fz(0)454 4233 y Fx(.)p Fv(R)557 4197 y Ft(d)608 4233 y Fx(/)e FD(with)g(lim)1020 4248 y Ft(n)s Fr(!1)1224 4233 y Fp(k)p Fx( )1344 4248 y Ft(n)1410 4233 y Fs(\000)22 b Fx( )9 b Fp(k)1639 4248 y Fz(2)1707 4233 y Fs(D)29 b FD(0)j(and)g(sup)2211 4258 y Ft(n)s Fr(2)p Fn(N)2348 4233 y Fs(f)p Fp(k)p Fx( )2505 4248 y Ft(n)2548 4233 y Fp(k)2598 4248 y Fz(2)2638 4233 y Fs(g)c(\024)h FD(2)p Fp(k)p Fx( )9 b Fp(k)3036 4248 y Fz(2)3076 4233 y FD(.)53 b(From)32 b(the)294 4353 y(boundedness)k(of)p 967 4274 V 45 w Fw(M)r(B)14 b(M)7 b FD(,)40 b(the)d(continuity)f(of)h(the)g (scalar)h(product)f Fp(h)p Fs(\001)p Fx(;)14 b Fs(\001)p Fp(i)36 b FD(and)h(\(4.5\))g(we)294 4472 y(conclude)804 4696 y Fp(h)p Fx(')5 b(;)p 951 4617 V 21 w Fw(M)r(B)14 b(M)7 b Fx( )i Fp(i)60 b Fs(D)86 b FD(lim)1523 4756 y Ft(n)s Fr(!1)1709 4696 y Fp(h)p Fx(')5 b(;)21 b Fw(M)r(B)14 b(M)7 b Fx( )2183 4711 y Ft(n)2226 4696 y Fp(i)1386 4881 y Fs(D)86 b FD(lim)1523 4941 y Ft(n)s Fr(!1)1709 4881 y Fp(h)7 b Fw(M)h Fx(')d(;)22 b Fw(B)14 b(M)8 b Fx( )2198 4896 y Ft(n)2240 4881 y Fp(i)1386 5128 y Fs(D)86 b FD(lim)1523 5187 y Ft(n)s Fr(!1)1723 4992 y Fo(Z)1776 5217 y Fn(R)1829 5197 y Fl(d)1865 5128 y FD(d)13 b Fw(x)33 b Fx(')2066 5086 y Fr(\003)2108 5128 y Fx(.)s Fw(x)9 b Fx(/)14 b Fp(h)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)22 b Fw(B)14 b(M)8 b Fx( )2865 5143 y Ft(n)2906 5128 y Fp(i)1386 5392 y Fs(D)86 b FD(lim)1523 5452 y Ft(n)s Fr(!1)1723 5257 y Fo(Z)1776 5482 y Fn(R)1829 5462 y Fl(d)1865 5392 y FD(d)13 b Fw(x)33 b Fx(')2066 5351 y Fr(\003)2108 5392 y Fx(.)s Fw(x)9 b Fx(/)14 b Fp(h)7 b Fw(M)r(B)2451 5351 y Fr(\003)2491 5392 y Fw(m)f Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )2891 5407 y Ft(n)2933 5392 y Fp(i)g Fx(:)320 b FD(\(4.6\))p eop end %%Page: 30 30 TeXDict begin 30 29 bop 294 90 a FD(30)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(Since)952 624 y(sup)955 710 y Ft(n)s Fr(2)p Fn(N)1090 539 y Fq(\014)1090 599 y(\014)1123 624 y Fp(h)7 b Fw(M)r(B)1322 582 y Fr(\003)1363 624 y Fw(m)f Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )1763 639 y Ft(n)1805 624 y Fp(i)1844 539 y Fq(\014)1844 599 y(\014)1902 624 y Fs(\024)25 b FD(2)14 b Fp(k)7 b Fw(M)r(B)2276 582 y Fr(\003)2316 624 y Fp(k)14 b(k)p Fx( )9 b Fp(k)2559 639 y Fz(2)2612 624 y Fj(M)16 b Fx(.)s Fw(x)9 b Fx(/)467 b FD(\(4.7\))294 935 y(for)21 b(all)j Fw(x)30 b Fs(2)21 b Fv(R)779 898 y Ft(d)830 935 y FD(,)29 b Fw(M)r(B)1037 898 y Fr(\003)1098 935 y FD(is)20 b(bounded)h(and)f Fj(M)37 b FD(is)20 b(continuous,)g(the)h(dominated-con)l(v)o(er)n (gence)294 1054 y(theorem)k(and)g(the)f(continuity)f(of)i(the)g(scalar) g(product)g(yield)977 1342 y Fp(h)p Fx(')5 b(;)p 1124 1263 258 4 v 21 w Fw(M)r(B)14 b(M)7 b Fx( )i Fp(i)24 b Fs(D)1626 1206 y Fo(Z)1679 1431 y Fn(R)1732 1411 y Fl(d)1768 1342 y FD(d)13 b Fw(x)33 b Fx(')1969 1301 y Fr(\003)2011 1342 y Fx(.)s Fw(x)9 b Fx(/)14 b Fp(h)7 b Fw(M)r(B)2354 1301 y Fr(\003)2394 1342 y Fw(m)f Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )9 b Fp(i)491 b FD(\(4.8\))294 1645 y(for)40 b(all)f Fx(')f Fs(2)33 b Fj(C)862 1608 y Fr(1)845 1671 y Fz(0)938 1645 y Fx(.)p Fv(R)1042 1609 y Ft(d)1093 1645 y Fx(/)39 b FD(and)h(all)f Fx( )j Fs(2)33 b FD(L)1761 1609 y Fz(2)1801 1645 y Fx(.)p Fv(R)1904 1609 y Ft(d)1955 1645 y Fx(/)p FD(.)75 b(Moreo)o(v)o(er)l(,) 42 b(the)d(function)g Fv(R)3145 1609 y Ft(d)3229 1645 y Fs(3)d Fw(x)42 b Fs(7!)294 1764 y Fp(h)7 b Fw(M)r(B)493 1728 y Fr(\003)533 1764 y Fw(m)f Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )9 b Fp(i)22 b FD(belongs)f(to)g(L)1495 1728 y Fr(1)1495 1791 y Fz(loc)1585 1764 y Fx(.)p Fv(R)1689 1728 y Ft(d)1740 1764 y Fx(/)p FD(,)h(confer)h(\(4.7\),)f(so)g(that)g (the)f(lemma)h(of)g(Du)g(Bois-)294 1884 y(Re)o(ymond)29 b(\226)h(also)g(kno)n(wn)e(as)i(the)g(fundamental)f(lemma)h(of)g(the)f (calculus)h(of)g(v)n(ariations,)294 2004 y(see)25 b(e.g.)g(Lemma)g (3.26)f(in)g([5])i(\226)e(implies)601 2163 y Fq(\000)p 646 2165 V 653 2243 a Fw(M)r(B)14 b(M)8 b Fx( )983 2163 y Fq(\001)1028 2243 y Fx(.)s Fw(x)h Fx(/)60 b Fs(D)g Fp(h)7 b Fw(M)r(B)1555 2202 y Fr(\003)1595 2243 y Fw(m)f Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b( )9 b Fp(i)1218 2468 y Fs(D)1356 2333 y Fo(Z)1409 2558 y Fn(R)1462 2538 y Fl(d)1498 2468 y FD(d)16 b Fw(y)1627 2328 y Fq(\024)1680 2333 y Fo(Z)1732 2558 y Fn(R)1785 2538 y Fl(d)1821 2468 y FD(d)c Fw(z)29 b(m)6 b Fx(.)g Fw(y)g Fx(;)16 b Fw(z)5 b Fx(/)2266 2388 y Fq(\000)2319 2468 y Fw(B)2387 2427 y Fr(\003)2427 2468 y Fw(m)h Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)2710 2388 y Fq(\001)2755 2468 y Fx(.)r Fw(z)c Fx(/)2875 2328 y Fq(\025)2928 2351 y Fr(\003)3008 2468 y Fx( )k(.)d Fw(y)g Fx(/)1218 2739 y Fs(D)1356 2603 y Fo(Z)1409 2828 y Fn(R)1462 2808 y Fl(d)1498 2739 y FD(d)16 b Fw(y)30 b Fp(h)p Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)22 b Fw(B)7 b(m)f Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)p Fp(i)14 b Fx( )9 b(.)d Fw(y)g Fx(/)705 b FD(\(4.9\))294 3042 y(for)28 b(Lebesgue-almost)f(all)k Fw(x)k Fs(2)26 b Fv(R)1512 3005 y Ft(d)1591 3042 y FD(and)i(all)g Fx( )35 b Fs(2)27 b FD(L)2147 3005 y Fz(2)2187 3042 y Fx(.)p Fv(R)2290 3005 y Ft(d)2341 3042 y Fx(/)p FD(.)40 b(T)-8 b(o)27 b(get)h(the)g(last)f(equality)-6 b(,)27 b(we)294 3161 y(ha)n(v)o(e)e(also)f(used)g(the)g(Hermiticity)-6 b(,)23 b Fw(m)6 b Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(D)h Fw(m)2061 3125 y Fr(\003)2102 3161 y Fx(.)6 b Fw(y)g Fx(;)17 b Fw(x)9 b Fx(/)24 b FD(for)h(Lebesgue-almost)e(all)i (pairs)294 3281 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b Fs(2)g Fv(R)706 3245 y Ft(d)776 3281 y Fs(\002)20 b Fv(R)940 3245 y Ft(d)991 3281 y FD(.)31 b(This)24 b(pro)o(v)o(es)g (\(4.2\).)394 3400 y(The)34 b(Carleman)h(property)f(\(1.15\))h(for)f Fx(\014)42 b FD(follo)n(ws)33 b(from)h(part)g(\(iii\))g(of)h(the)f (lemma)g(\(to)294 3520 y(be)28 b(pro)o(v)o(en)e(belo)n(w\).)37 b(Indeed,)28 b(since)f Fw(m)34 b FD(is)27 b(Hermitian)f(and)h(since)g Fj(M)43 b FD(is)27 b(continuous,)g(one)294 3640 y(may)d(choose)f Fx(w)k Fs(D)991 3622 y Fx(\037)1060 3655 y Fu(3)1144 3640 y FD(in)c(\(4.3\))h(for)g(an)g(arbitrary)f(bounded)g(Borel)h (subset)f Fx(3)h Fs(\032)g Fv(R)3240 3603 y Ft(d)3291 3640 y FD(.)30 b(This)294 3759 y(completes)24 b(the)h(proof)g(of)g (part)g(\(i\).)394 3879 y(The)j(proof)h(of)f(assertion)g(\(ii\))g (follo)n(ws)f(from)h(the)g(\002rst)g(equality)g(in)g(\(4.9\),)h(the)f (f)o(act)h(that)294 3998 y(the)21 b(mapping)f Fv(R)873 3962 y Ft(d)945 3998 y Fs(!)h FD(L)1137 3962 y Fz(2)1177 3998 y Fx(.)p Fv(R)1280 3962 y Ft(d)1331 3998 y Fx(/)p FD(,)k Fw(x)30 b Fs(7!)21 b Fw(m)6 b Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/)p FD(,)21 b(is)g(strongly)e(continuous,)27 b Fw(M)r(B)3031 3962 y Fr(\003)3093 3998 y FD(is)20 b(bounded)294 4118 y(and)25 b Fp(h)p Fs(\001)p Fx(;)14 b Fs(\001)p Fp(i)24 b FD(is)g(continuous.)394 4238 y(F)o(or)g(the)g(proof)g(of)g (assertion)f(\(iii\))h(we)g(e)o(xploit)e(our)i(assumption)e(on)i Fx(w)s FD(,)h(the)e(maximality)294 4357 y(of)29 b(the)f(Carleman)h (operator)35 b Fw(M)8 b FD(,)30 b(\(4.1\))e(and)h(Thm.)e(VI.23)h(in)g ([38])h(to)f(conclude)g(that)35 b Fw(M)f Fs(O)-59 b Fx(w)294 4477 y FD(is)34 b(Hilbert-Schmidt.)59 b(Therefore,)45 b Fw(M)r(B)14 b(M)33 b Fs(O)-58 b Fx(w)33 b Fs(D)p 2055 4398 V 37 w Fw(M)r(B)14 b(M)33 b Fs(O)-59 b Fx(w)38 b FD(is)d(Hilbert-Schmidt,)g(too,)i(by)294 4596 y(the)d(boundedness)e(of) 41 b Fw(M)r(B)e FD(and)34 b(the)f(H\366lder)g(inequality)f(for)i (Schatten)g(norms,)h(see)f(e.g.)294 4716 y(Thm.)f(2.8)h(in)g([41].)58 b(Thanks)33 b(to)h Fx(w)f Fs(2)d FD(L)1760 4680 y Fr(1)1836 4716 y Fx(.)p Fv(R)1939 4680 y Ft(d)1990 4716 y Fx(/)k FD(and)g(Eq.)g(\(4.2\))g(we)g(ha)n(v)o(e)p 3003 4637 V 41 w Fw(M)r(B)14 b(M)33 b Fs(O)-59 b Fx(w)t( )39 b Fs(D)294 4755 y Fo(R)340 4870 y Fn(R)393 4849 y Fl(d)443 4836 y FD(d)16 b Fw(y)j Fx(\014)7 b(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)14 b(w)s(.)6 b Fw(y)g Fx(/)14 b( )9 b(.)d Fw(y)g Fx(/)29 b FD(for)g(all)g Fx( )36 b Fs(2)27 b FD(L)1843 4799 y Fz(2)1883 4836 y Fx(.)p Fv(R)1986 4799 y Ft(d)2037 4836 y Fx(/)p FD(.)43 b(Hence)29 b(\(4.3\))g(follo)n(ws)e(from)i(an)g (ane)n(w)294 4955 y(application)24 b(of)h(Thm.)f(VI.23)h(in)f([38].)p 1776 4964 41 90 v 394 5139 a(After)h(these)g(preparations)f(it)h(is)f (easy)h(to)g(deduce)g(Theorem)g(1.14)f(as)h(a)g(special)g(case.)399 5322 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Theorem)g(1.14\))p Fw(.)119 b FD(W)-8 b(e)55 b(apply)e(Lemma)h(4.1)f(with)h(the)g(choices) 60 b Fw(M)50 b Fs(D)294 5442 y FD(e)338 5406 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)695 5442 y FD(and)33 b Fw(B)d Fs(D)25 b FD(e)1110 5406 y Fz(2)p Ft(t)12 b(H)d Fu(.)e Ft(A)r Fu(;)r Ft(V)j Fu(/)1429 5442 y Fw(F)1499 5361 y Fq(\000)1553 5442 y Fw(H)g Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1911 5361 y Fq(\001)1956 5442 y FD(,)25 b(where)h Fw(t)33 b Fs(2)p FD(]0)p Fx(;)14 b(\034)9 b(=)p FD(2[.)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(31)394 384 y(This)37 b(is)f(allo)n(wed,)k(because)e(Theorem)f(1.10)g(ensures)g (that)g(e)2620 348 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2989 384 y FD(is)37 b(a)h(maximal)294 503 y(Carleman)22 b(operator)g(with)f(the)g(required)h(properties,)g (recall)g(Remark)g(1.6\(ii\),)g(Lemma)f(1.7)294 623 y(and)k(Remark)h (1.8\(i)n(v\))o(.)394 743 y(Furthermore,)20 b(we)g(observ)o(e)e(from)h (\(1.20\))g(and)g(the)f(functional)h(calculus)f(for)i(unbounded)294 862 y(functions)35 b(of)h(unbounded)f(self-adjoint)g(operators,)j(see)e (e.g.)g(Chap.)g(5)g(in)g([7],)i(that)e(the)294 982 y(operator)25 b(product)32 b Fw(B)f Fs(D)25 b FD(e)1227 946 y Fz(2)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1546 982 y Fw(F)1616 901 y Fq(\000)1669 982 y Fw(H)f Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)2027 901 y Fq(\001)2098 982 y FD(is)24 b(bounded.)30 b(The)25 b(functional)f(calculus)294 1101 y(also)36 b(guarantees)h(that)e(the)h(tw)o(o)g(operator)g (products)43 b Fw(M)r(B)f FD(and)h Fw(M)r(B)2765 1065 y Fr(\003)2842 1101 y FD(are)37 b(bounded)f(and)294 1221 y(that)24 b(the)h(equality)31 b Fw(M)r(B)14 b(M)32 b Fs(D)f Fw(F)1422 1140 y Fq(\000)1476 1221 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)1833 1140 y Fq(\001)1904 1221 y FD(holds)24 b(on)g(dom)o Fx(.)7 b Fw(M)h Fx(/)p FD(.)31 b(The)25 b(latter)g(implies)e(the)294 1352 y(boundedness)h(of)p 942 1274 258 4 v 32 w Fw(M)r(B)14 b(M)32 b Fs(D)g Fw(F)1404 1272 y Fq(\000)1457 1352 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)1815 1272 y Fq(\001)1861 1352 y FD(,)25 b(because)32 b Fw(F)i Fs(2)25 b FD(L)2502 1316 y Fr(1)2578 1352 y Fx(.)p Fv(R)5 b Fx(/)p FD(.)394 1472 y(Finally)-6 b(,)21 b(the)h(\002niteness)g(of)g(the)f(inte)o(gral)1833 1392 y Fo(R)1879 1506 y Fn(R)1932 1486 y Fl(d)1963 1506 y Fr(\002)p Fn(R)2071 1486 y Fl(d)2121 1472 y FD(d)13 b Fw(x)c FD(d)16 b Fw(y)27 b Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2636 1436 y Fz(2)2676 1472 y Fp(j)p Fw(k)2750 1487 y Ft(t)2780 1472 y Fx(.)s Fw(x)g Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)3041 1436 y Fz(2)3103 1472 y FD(for)22 b(all)f Fx(w)26 b Fs(2)294 1605 y FD(L)355 1568 y Fr(1)355 1631 y Fz(G)431 1605 y Fx(.)p Fv(R)535 1569 y Ft(d)586 1605 y Fx(/)d FD(follo)n(ws)f(from)h(the)f(estimate)h(\(1.13\))g(with)f (suf)n(\002ciently)g(small)g Fx(\016)28 b(>)23 b FD(0,)g(inequality)294 1724 y(\(2.14\))30 b(and)g(Remark)g(1.6\(i\).)45 b(Thus,)31 b(all)e(assumptions)f(of)i(Lemma)f(4.1)h(are)g(ful\002lled)g(and)294 1844 y(Theorem)25 b(1.14)f(holds)g(with)46 b Fw(f)g Fs(D)24 b Fx(\014)32 b FD(and)25 b(for)g(all)g Fx(w)j Fs(2)d FD(L)2292 1807 y Fr(1)2292 1870 y Fz(G)2368 1844 y Fx(.)p Fv(R)2471 1808 y Ft(d)2522 1844 y Fx(/)p FD(.)p 2704 1853 41 90 v 394 2068 a(Ne)o(xt)f(we)h(sho)n(w)f(ho)n(w)g(to)h(deduce)g (Corollary)g(1.16)f(from)h(Theorem)f(1.14.)399 2292 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Corollary)g(1.16\))p Fw(.)119 b FD(Clearly)-6 b(,)30 b(choosing)35 b Fw(F)i Fs(D)2512 2274 y Fx(\037)2585 2307 y Ft(I)2651 2292 y FD(in)29 b(Theorem)g(1.14)f(is)h(in)294 2411 y(accordance)e(with)d(\(1.20\))h (because)h(of)f(sup)20 b Fw(I)39 b Fx(<)24 b Fs(1)p FD(.)32 b(Therefore,)26 b(part)f(\(i\))g(of)h(this)e(theorem)294 2531 y(yields)37 b(the)h(e)o(xistence)f(and)h(continuity)e(of)i(the)f (inte)o(gral)g(k)o(ernel)49 b Fw(p)2734 2546 y Ft(I)2810 2531 y FD(of)2930 2513 y Fx(\037)3004 2546 y Ft(I)3041 2450 y Fq(\000)3095 2531 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)3453 2450 y Fq(\001)3499 2531 y FD(.)294 2661 y(T)-8 b(o)49 b(deri)n(v)o(e)e(\(1.24\))h(we)h(note)f (that)g(the)h(operator)74 b Fs(O)-59 b Fx(w)2238 2624 y Fr(\003)2280 2643 y Fx(\037)2353 2676 y Ft(I)2391 2580 y Fq(\000)2444 2661 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)2802 2580 y Fq(\001)2873 2661 y Fs(O)-58 b Fx(w)52 b FD(is)c(trace)h(class)294 2780 y(by)f(Theorem)g (1.14\(iii\))e(and)1413 2762 y Fx(\037)1482 2744 y Fz(2)1487 2807 y Ft(I)1562 2780 y Fs(D)1677 2762 y Fx(\037)1751 2795 y Ft(I)1788 2780 y FD(.)100 b(Moreo)o(v)o(er)l(,)52 b(thanks)c(to)f Fx(w)41 b Fs(2)c FD(L)3086 2743 y Fr(1)3086 2807 y Fz(G)3162 2780 y Fx(.)p Fv(R)3266 2744 y Ft(d)3317 2780 y Fx(/)48 b FD(the)294 2900 y(L)355 2864 y Fz(2)395 2900 y Fx(.)p Fv(R)498 2864 y Ft(d)577 2900 y Fs(\002)28 b Fv(R)749 2864 y Ft(d)800 2900 y Fx(/)p FD(-function)47 b Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)38 b Fs(7!)f Fx(w)1744 2864 y Fr(\003)1785 2900 y Fx(.)s Fw(x)9 b Fx(/)i Fw(p)1981 2915 y Ft(I)2019 2900 y Fx(.)s Fw(x)e Fx(;)20 b Fw(y)6 b Fx(/w)s(.)g Fw(y)g Fx(/)48 b FD(is)f(an)g(inte)o (gral)g(k)o(ernel)g(for)320 3019 y Fs(O)-59 b Fx(w)370 2983 y Fr(\003)412 3001 y Fx(\037)485 3034 y Ft(I)523 2939 y Fq(\000)576 3019 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)934 2939 y Fq(\001)1005 3019 y Fs(O)-58 b Fx(w)s FD(.)30 b(Recalling)24 b(that)e Fx(3)1768 3034 y Fu(`)1806 3019 y Fx(.)s Fw(x)9 b Fx(/)23 b FD(is)g(the)g(open)g(cube) g(in)g Fv(R)2789 2983 y Ft(d)2863 3019 y FD(with)f(edge)h(length)294 3139 y Fx(`)29 b(>)g FD(0)j(and)g(centre)j Fw(x)j Fs(2)29 b Fv(R)1253 3103 y Ft(d)1304 3139 y FD(,)34 b(an)e(application)f(of)h (Thm.)g(3.1)g(in)f([8],)k(see)d(also)g([9])g(or)g([6],)294 3259 y(gi)n(v)o(es)24 b(the)g(equality)555 3505 y(T)m(race)778 3424 y Fq(\002)845 3505 y Fs(O)-58 b Fx(w)896 3464 y Fr(\003)937 3487 y Fx(\037)1011 3520 y Ft(I)1048 3424 y Fq(\000)1102 3505 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)1459 3424 y Fq(\001)1531 3505 y Fs(O)-59 b Fx(w)1582 3424 y Fq(\003)674 3716 y Fs(D)777 3580 y Fo(Z)830 3805 y Fn(R)883 3785 y Fl(d)933 3716 y FD(d)13 b Fw(x)47 b FD(lim)1098 3779 y Fu(`)p Fr(#)p Fz(0)1248 3716 y Fx(`)1295 3674 y Fr(\000)p Fz(2)p Ft(d)1442 3580 y Fo(Z)1495 3805 y Fu(3)1551 3815 y Fm(`)1579 3805 y Fu(.)r Ft(x)6 b Fu(/)p Fr(\002)p Fu(3)1781 3815 y Fm(`)1809 3805 y Fu(.)r Ft(x)g Fu(/)1905 3716 y FD(d)13 b Fw(x)2021 3674 y Fr(0)2044 3716 y FD(d)j Fw(y)2160 3674 y Fr(0)2208 3716 y Fx(w)2284 3674 y Fr(\003)2326 3716 y Fx(.)s Fw(x)2419 3674 y Fr(0)2443 3716 y Fx(/)25 b Fw(p)2560 3731 y Ft(I)2598 3716 y Fx(.)s Fw(x)2691 3674 y Fr(0)2714 3716 y Fx(;)20 b Fw(y)2817 3674 y Fr(0)2841 3716 y Fx(/)14 b(w)s(.)6 b Fw(y)3061 3674 y Fr(0)3085 3716 y Fx(/)14 b(:)120 b FD(\(4.10\))294 4022 y(The)64 b(continuity)e(of)74 b Fw(p)1193 4037 y Ft(I)1294 4022 y FD(and)64 b(the)f(Lebesgue)h(dif)n (ferentiation)e(theorem,)73 b(see)64 b(e.g.)294 4141 y(Sects.)25 b(I.1.3)g(and)g(I.1.8)g(in)f([44],)h(no)n(w)f(complete)g (the)h(proof)g(because)813 4435 y(lim)825 4499 y Fu(`)p Fr(#)p Fz(0)974 4435 y Fx(`)1021 4394 y Fr(\000)p Fz(2)p Ft(d)1169 4300 y Fo(Z)1221 4525 y Fu(3)1277 4535 y Fm(`)1305 4525 y Fu(.)r Ft(x)6 b Fu(/)p Fr(\002)p Fu(3)1507 4535 y Fm(`)1535 4525 y Fu(.)r Ft(x)g Fu(/)1631 4435 y FD(d)13 b Fw(x)1747 4394 y Fr(0)1770 4435 y FD(d)j Fw(y)1886 4394 y Fr(0)1935 4435 y Fx(w)2011 4394 y Fr(\003)2052 4435 y Fx(.)s Fw(x)2145 4394 y Fr(0)2169 4435 y Fx(/)25 b Fw(p)2286 4450 y Ft(I)2324 4435 y Fx(.)s Fw(x)2417 4394 y Fr(0)2441 4435 y Fx(;)20 b Fw(y)2544 4394 y Fr(0)2567 4435 y Fx(/)14 b(w)s(.)6 b Fw(y)2787 4394 y Fr(0)2811 4435 y Fx(/)1500 4742 y Fs(D)36 b Fw(p)1669 4757 y Ft(I)1706 4742 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)39 b FD(lim)1989 4805 y Fu(`)p Fr(#)p Fz(0)2111 4597 y Fq(\014)2111 4657 y(\014)2111 4717 y(\014)2111 4776 y(\014)2144 4742 y Fx(`)2191 4700 y Fr(\000)p Ft(d)2304 4606 y Fo(Z)2357 4831 y Fu(3)2413 4841 y Fm(`)2441 4831 y Fu(.)r Ft(x)6 b Fu(/)2537 4742 y FD(d)13 b Fw(x)2653 4700 y Fr(0)2701 4742 y Fx(w)s(.)s Fw(x)2870 4700 y Fr(0)2894 4742 y Fx(/)2931 4597 y Fq(\014)2931 4657 y(\014)2931 4717 y(\014)2931 4776 y(\014)2965 4624 y Fz(2)1500 4971 y Fs(D)36 b Fw(p)1669 4986 y Ft(I)1706 4971 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)14 b Fp(j)p Fx(w)s(.)s Fw(x)9 b Fx(/)p Fp(j)2215 4930 y Fz(2)3283 4971 y FD(\(4.11\))294 5218 y(for)25 b(Lebesgue-almost)f(all)k Fw(x)34 b Fs(2)24 b Fv(R)1500 5182 y Ft(d)1551 5218 y FD(.)p 1696 5227 V 394 5442 a(No)n(w)g(we)h (are)h(concerned)f(with)f(the)h(second)g(corollary)g(to)f(Theorem)h (1.14.)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 294 90 a FD(32)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)399 384 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Corollary)g(1.18\))p Fw(.)119 b FD(W)-8 b(e)33 b(\002x)i Fw(x)9 b Fx(;)20 b Fw(y)34 b Fs(2)28 b Fv(R)2215 348 y Ft(d)2266 384 y FD(.)51 b(In)32 b(the)g(\002rst)f (case)i(we)f(apply)f(the)294 503 y(functional)24 b(calculus)h(to)f(the) h(right-hand)f(side)h(of)g(\(1.21\).)30 b(This)24 b(gi)n(v)o(es)1192 782 y Fw(f)d Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(D)1601 646 y Fo(Z)1654 871 y Fn(R)1709 782 y FD(d)10 b Fx(#)1828 797 y Ft(t)1858 782 y Fx(.)d Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(e)2286 741 y Fz(2)p Ft(t)11 b(E)2426 782 y Fw(F)e Fx(.)e Fw(E)i Fx(/)636 b FD(\(4.12\))294 1060 y(for)85 b(an)o(y)e Fw(t)67 b Fs(2)p FD(]0)p Fx(;)14 b(\034)9 b(=)p FD(2[)83 b(with)g(the)h(comple) o(x)f(spectral)h(\223distrib)n(ution\224)e(function)294 1180 y Fx(#)353 1195 y Ft(t)384 1180 y Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(:)p Fs(D)900 1099 y Fq(\012)947 1180 y Fw(k)993 1195 y Ft(t)1024 1180 y Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)1276 1162 y(\037)1344 1195 y Fz(])p Fr(\0001)p Fu(;)c Ft(E)g Fz([)1598 1099 y Fq(\000)1651 1180 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)2009 1099 y Fq(\001)2055 1180 y Fw(k)2101 1195 y Ft(t)2132 1180 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)2337 1099 y Fq(\013)2384 1180 y FD(.)35 b(Here,)27 b Fx(\034)38 b(>)26 b FD(0)g(is)g(the)g(constant) 294 1299 y(required)d(to)f(e)o(xist)f(for)30 b Fw(F)i FD(in)22 b(\(1.20\).)30 b(In)22 b(particular)l(,)h(for)30 b Fw(F)i Fs(D)2463 1281 y Fx(\037)2532 1314 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)2753 1324 y Fk(0)2784 1314 y Fz([)2834 1299 y FD(with)29 b Fw(E)3106 1314 y Fz(0)3169 1299 y Fs(2)22 b Fv(R)d FD(,)29 b(Eq.)294 1419 y(\(4.12\))c(tak)o(es)g (the)f(form)1166 1714 y Fw(p)s Fx(.)7 b Fw(E)1328 1729 y Fz(0)1368 1714 y Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(D)1741 1578 y Fo(Z)1844 1605 y Ft(E)1889 1615 y Fk(0)1794 1803 y Fr(\0001)1925 1714 y FD(d)10 b Fx(#)2044 1729 y Ft(t)2074 1714 y Fx(.)d Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(e)2502 1672 y Fz(2)p Ft(t)11 b(E)2635 1714 y Fx(:)621 b FD(\(4.13\))294 1998 y(This)26 b(equation)f(holds)h(for)g(arbitrary)h Fw(t)34 b Fx(>)26 b FD(0,)g(because)h Fx(\034)39 b FD(can)26 b(be)h(chosen)f(arbitrarily)g(lar)n(ge)294 2117 y(in)f(this)f (particular)g(case.)32 b(T)-8 b(ak)o(en)25 b(together)l(,)f(\(4.12\))h (and)g(\(4.13\))g(yield)f(the)h(claim)f(\(1.25\).)394 2237 y(In)h(the)g(second)f(case)i(we)f(may)f(write)401 2515 y Fw(k)447 2530 y Ft(t)478 2515 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)25 b Fs(D)839 2435 y Fq(\012)886 2515 y Fw(k)932 2530 y Ft(t)6 b Fu(=)p Fz(2)1030 2515 y Fx(.)p Fs(\001)p Fx(;)17 b Fw(x)9 b Fx(/;)14 b Fw(k)1328 2530 y Ft(t)6 b Fu(=)p Fz(2)1426 2515 y Fx(.)p Fs(\001)p Fx(;)20 b Fw(y)6 b Fx(/)1631 2435 y Fq(\013)1703 2515 y Fs(D)1806 2380 y Fo(Z)1858 2605 y Fn(R)1914 2515 y FD(d)k Fx(#)2033 2530 y Ft(t)c Fu(=)p Fz(2)2131 2515 y Fx(.)h Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b Fs(D)2617 2380 y Fo(Z)2670 2605 y Fn(R)2725 2515 y FD(d)d Fw(p)s Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)24 b FD(e)3277 2474 y Fr(\000)p Ft(t)11 b(E)3283 2712 y FD(\(4.14\))294 2943 y(for)28 b(all)f Fw(t)36 b Fx(>)26 b FD(0.)38 b(Here,)29 b(the)f(\002rst)f (equality)g(is)g(due)h(to)f(the)h(Hermiticity)e(and)h(the)h(semigroup) 294 3063 y(property)e(of)h(the)f(k)o(ernel)h Fw(k)1237 3078 y Ft(t)1267 3063 y FD(,)g(the)f(second)h(equality)e(is)h(just)g (the)g(de\002nition)f(of)i Fx(#)3107 3078 y Ft(t)6 b Fu(=)p Fz(2)3231 3063 y FD(and)27 b(the)294 3182 y(last)e(equality)f (follo)n(ws)f(from)i(\(4.13\).)p 1729 3191 41 90 v 394 3362 a(F)o(or)h(con)l(v)o(enience,)f(we)h(formulate)g(and)g(pro)o(v)o (e)e(simple)h(estimates)g(on)h(the)f(inte)o(gral)g(k)o(er)n(-)294 3481 y(nel)33 b(of)f(a)h(spectral)g(projection)e(in)h(the)h(remainder)g (of)f(this)g(section.)53 b(W)-8 b(e)33 b(will)f(only)f(need)294 3601 y(these)25 b(estimates)f(for)h(the)g(applications)e(to)i(random)f (Schr\366dinger)h(operators.)396 3852 y(L)t FG(E)t(M)t(M)t(A)33 b FD(4)t(.)t(2)t(.)125 b Fw(Assume)29 b(the)h(situation)f(of)g(Cor)l (ollary)g FD(1.16)p Fw(.)47 b(Then)30 b(the)g(dia)o(gonal)f(of)294 3972 y(the)g(continuous)f(inte)l(gr)o(al)f(k)o(ernel)40 b(p)1594 3987 y Ft(I)1661 3972 y Fw(of)29 b(the)g(spectr)o(al)f(pr)l (ojection)2702 3954 y Fx(\037)2776 3987 y Ft(I)2813 3891 y Fq(\000)2867 3972 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)e Fx(/)3225 3891 y Fq(\001)3300 3972 y Fw(obe)m(ys)294 4091 y(the)25 b(estimates)1315 4322 y FD(0)f Fs(\024)36 b Fw(p)1555 4337 y Ft(I)1593 4322 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)25 b Fs(\024)g FD(e)1995 4281 y Ft(t)17 b Fz(sup)d Ft(I)2193 4322 y Fw(k)2239 4337 y Ft(t)2270 4322 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)780 b FD(\(4.15\))294 4553 y Fw(for)24 b(all)k(x)33 b Fs(2)25 b Fv(R)800 4517 y Ft(d)876 4553 y Fw(with)g(any)f(t)34 b Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fw(.)399 4805 y(Pr)l(oof)23 b(.)120 b FD(Fix)26 b Fw(x)33 b Fs(2)23 b Fv(R)1179 4769 y Ft(d)1254 4805 y FD(arbitrary)-6 b(,)23 b(pick)g Fx(')29 b Fs(2)23 b Fj(C)2088 4768 y Fr(1)2071 4831 y Fz(0)2165 4805 y Fx(.)p Fv(R)2268 4769 y Ft(d)2319 4805 y Fx(/)h FD(and)f(de\002ne)h Fx(')2875 4769 y Fu(.")r(/)2872 4832 y Ft(x)2989 4805 y FD(by)g Fx(')3174 4769 y Fu(.")r(/)3171 4832 y Ft(x)3264 4805 y Fx(.)6 b Fw(y)g Fx(/)24 b FD(:)p Fs(D)294 4924 y Fx(")342 4888 y Fr(\000)p Ft(d)442 4924 y Fx(')503 4844 y Fq(\000)548 4924 y Fx(.)6 b Fw(y)32 b Fs(\000)d Fw(x)9 b Fx(/=")959 4844 y Fq(\001)1046 4924 y FD(for)42 b(e)n(v)o(ery)47 b Fw(y)40 b Fs(2)34 b Fv(R)1716 4888 y Ft(d)1809 4924 y FD(and)42 b(e)n(v)o(ery)f Fx(")c Fs(2)p FD(]0)p Fx(;)14 b FD(1].)81 b(Then)41 b Fs(f)p Fx(')3062 4888 y Fu(.")r(/)3059 4951 y Ft(x)3153 4924 y Fs(g)3190 4939 y Fu(")r Fr(2)p Fz(]0)p Fu(;)p Fz(1])3446 4924 y Fs(\032)294 5044 y FD(L)355 5008 y Fz(2)395 5044 y Fx(.)p Fv(R)498 5008 y Ft(d)549 5044 y Fx(/)26 b FD(is)f(a)h(f)o (amily)e(of)i(approximating)e(delta)h(functions)f(at)29 b Fw(x)34 b Fs(2)25 b Fv(R)2722 5008 y Ft(d)2773 5044 y FD(.)32 b(By)26 b(the)f(continuity)294 5164 y(of)36 b Fw(p)468 5179 y Ft(I)531 5164 y FD(and)25 b(the)f(dominated-con)l(v)o (er)n(gence)h(theorem)f(one)h(gets)f(the)h(representation)1103 5395 y Fw(p)1158 5410 y Ft(I)1196 5395 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)24 b Fs(D)h FD(lim)1567 5459 y Fu(")r Fr(#)p Fz(0)1689 5395 y Fp(h)p Fx(')1789 5353 y Fu(.")r(/)1786 5421 y Ft(x)1879 5395 y Fx(;)1926 5377 y(\037)2000 5410 y Ft(I)2037 5314 y Fq(\000)2091 5395 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)d Fx(/)2448 5314 y Fq(\001)2494 5395 y Fx(')2555 5353 y Fu(.")r(/)2552 5421 y Ft(x)2646 5395 y Fp(i)g Fx(:)557 b FD(\(4.16\))p eop end %%Page: 33 33 TeXDict begin 33 32 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(33)294 384 y(The)25 b(same)g(ar)n(guments)f(yield)1222 684 y Fw(k)1268 699 y Ft(t)1299 684 y Fx(.)s Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)25 b Fs(D)f FD(lim)1670 748 y Fu(")r Fr(#)p Fz(0)1792 684 y Fp(h)p Fx(')1892 643 y Fu(.")r(/)1889 711 y Ft(x)1983 684 y Fx(;)14 b FD(e)2074 643 y Fr(\000)p Ft(t)d(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)2405 684 y Fx(')2466 643 y Fu(.")r(/)2463 711 y Ft(x)2557 684 y Fp(i)687 b FD(\(4.17\))294 1030 y(for)24 b(an)o(y)e Fw(t)32 b Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([.)30 b(The)23 b(claim)g(\(4.15\))g(no)n(w)f(follo)n(ws)g(from)h(the)f(functional)h (calculus)g(and)294 1150 y(the)i(elementary)g(inequalities)1438 1450 y(0)f Fs(\024)1612 1432 y Fx(\037)1686 1465 y Ft(I)1723 1450 y Fx(.)7 b Fw(E)i Fx(/)25 b Fs(\024)g FD(e)2043 1409 y Ft(t)8 b Fu(.)p Fz(sup)16 b Ft(I)8 b Fr(\000)d Ft(E)g Fu(/)3283 1450 y FD(\(4.18\))294 1751 y(for)25 b(all)32 b Fw(E)i Fs(2)24 b Fv(R)19 b FD(.)p 980 1760 41 90 v 420 2070 a(5)t(.)105 b(P)t FG(R)q(O)t(O)t(F)t(S)35 b(O)t(F)c FD(L)t FG(E)t(M)t(M)t(A)f FD(1.22)r(,)g(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)j FD(1.27)26 b FG(A)t(N)t(D)31 b FD(C)t FG(O)t(R)q(O)t(L)t(L)t(A)t(R)o(Y)i FD(1.29)399 2289 y Fw(Pr)l(oof)48 b FD(\(of)25 b(Lemma)g(1.22\))p Fw(.)119 b FD(W)-8 b(e)25 b(mimic)e(the)i(proof)f(of)h([30)o(],)g(see)g (also)f(Prop.)h(V)-13 b(.3.2)24 b(in)294 2409 y([12].)33 b(By)26 b(the)f(de\002nition)g(of)36 b Fw(p)s Fx(.)p Fw(d)7 b Fx(/)27 b FD(in)e(property)g(\()p Fh(S)p FD(\))h(and)g(since)f Fx(.)p Fw(d)7 b Fx(=)p FD(2)p Fx(/)k Fw(p)2852 2424 y Fz(1)2893 2409 y Fx(=)p FD([)g Fw(p)3034 2424 y Fz(1)3094 2409 y Fs(\000)30 b Fw(p)s Fx(.)p Fw(d)7 b Fx(/)p FD(])27 b Fx(<)305 2528 y Fw(p)355 2543 y Fz(2)396 2528 y FD(,)d(we)h(can)h (\002nd)f Fx(\027)31 b Fs(2)p FD(]0)p Fx(;)14 b FD(2[)24 b(and)f Fw(r)35 b Fs(2)p FD(])11 b Fw(p)s Fx(.)p Fw(d)c Fx(/;)25 b Fw(p)1948 2543 y Fz(1)1989 2528 y FD([)g(such)f(that)1594 2806 y Fw(d)p 1594 2853 58 4 v 1596 2947 a Fx(\027)1782 2806 y Fw(p)1832 2821 y Fz(1)p 1689 2853 265 4 v 1700 2947 a Fw(p)1750 2962 y Fz(1)1810 2947 y Fs(\000)17 b Fw(r)1991 2876 y Fx(<)35 b Fw(p)2154 2891 y Fz(2)2208 2876 y Fx(:)1098 b FD(\(5.1\))294 3226 y(Ne)o(xt,)24 b(we)h(pick)g(a)g(constant)f Fw(c)j Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([)25 b(and)g(de\002ne)900 3526 y Fw(V)976 3477 y Fu(.!)q(/)958 3553 y Fz(2)1080 3526 y Fx(.)s Fw(x)9 b Fx(/)60 b FD(:)p Fs(D)i Fw(V)1514 3485 y Fu(.!)q(/)1618 3526 y Fx(.)s Fw(x)9 b Fx(/)14 b(2)1842 3445 y Fq(\000)1888 3526 y Fw(c)r Fx(.)p FD(1)20 b Fs(C)f Fp(j)s Fw(x)9 b Fp(j)2250 3485 y Fu(\027)2292 3526 y Fx(/)19 b Fs(\000)h Fp(j)s Fw(V)2552 3485 y Fu(.!)q(/)2656 3526 y Fx(.)s Fw(x)9 b Fx(/)p Fp(j)2814 3445 y Fq(\001)2874 3526 y Fx(;)381 b FD(\(5.2a\))900 3698 y Fw(V)976 3649 y Fu(.!)q(/)958 3724 y Fz(1)1080 3698 y Fx(.)s Fw(x)9 b Fx(/)60 b FD(:)p Fs(D)i Fw(V)1514 3657 y Fu(.!)q(/)1618 3698 y Fx(.)s 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b FD(1)p Fx(/)26 b Fs(\024)g Fp(j)p Fx(\030)11 b Fp(j)2236 4718 y Fu(\024)2307 4754 y FD(with)26 b Fx(\024)34 b Fs(D)j Fw(p)2759 4769 y Fz(1)2819 4754 y Fs(\000)18 b Fw(r)36 b Fx(>)26 b FD(0)g(to)g(obtain)294 4874 y(for)f(all)g Fx(!)i Fs(2)e Fx(\177)f FD(the)h(estimate)341 5231 y Fp(k)s Fw(V)470 5182 y Fu(.!)q(/)452 5258 y Fz(1)574 5213 y Fx(\037)643 5246 y Fu(3.)t Ft(y)t Fu(/)794 5231 y Fp(k)843 5190 y Ft(r)843 5258 y(r)907 5231 y Fs(D)1010 5095 y Fo(Z)1062 5320 y Fu(3.)t Ft(y)t Fu(/)1228 5231 y FD(d)13 b Fw(x)22 b Fp(j)s Fw(V)1464 5190 y Fu(.!)q(/)1568 5231 y Fx(.)s Fw(x)9 b Fx(/)p Fp(j)1725 5190 y Ft(r)1777 5231 y Fx(2)1871 5090 y Fq(\022)1992 5161 y Fp(j)s Fw(V)2099 5125 y Fu(.!)q(/)2203 5161 y Fx(.)s Fw(x)g Fx(/)p Fp(j)p 1956 5208 441 4 v 1956 5302 a Fw(c)r Fx(.)p FD(1)20 b Fs(C)f Fp(j)s Fw(x)9 b Fp(j)2318 5273 y Fu(\027)2360 5302 y Fx(/)2428 5231 y Fs(\000)20 b FD(1)2576 5090 y Fq(\023)2674 5231 y Fs(\024)2797 5151 y(Q)-44 b Fw(c)2831 5115 y Ft(r)2884 5151 y Fp(k)s Fw(V)3012 5115 y Fu(.!)q(/)3116 5133 y Fx(\037)3185 5166 y Fu(3.)t Ft(y)t Fu(/)3337 5151 y Fp(k)3395 5115 y Ft(p)3430 5125 y Fk(1)3395 5178 y Ft(p)3430 5188 y Fk(1)p 2786 5208 679 4 v 2845 5302 a Fx(.)p FD(1)20 b Fs(C)f Fp(j)6 b Fw(y)g Fp(j)3161 5273 y Fu(\027)3203 5302 y Fx(/)3248 5273 y Ft(p)3283 5283 y Fk(1)3313 5273 y Fr(\000)o Ft(r)3333 5442 y FD(\(5.3\))p eop end %%Page: 34 34 TeXDict begin 34 33 bop 294 90 a FD(34)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(for)35 b(all)40 b Fw(y)c Fs(2)29 b Fv(Z)826 348 y Ft(d)903 384 y FD(with)34 b(some)f(constant)45 b Fs(Q)-44 b Fw(c)32 b Fs(2)p FD(]0)p Fx(;)14 b Fs(1)p FD([,)36 b(which)e(is)g(independent)f(of)40 b Fw(y)c Fs(2)30 b Fv(Z)3456 348 y Ft(d)3499 384 y FD(.)294 503 y(This)24 b(implies)588 735 y Fq(X)582 955 y Ft(y)t Fr(2)p Fn(Z)713 935 y Fl(d)755 830 y Fv(P)814 749 y Fq(\002)857 830 y Fp(k)s Fw(V)968 845 y Fz(1)1008 812 y Fx(\037)1076 845 y Fu(3.)t Ft(y)t Fu(/)1228 830 y Fp(k)1277 845 y Ft(r)1341 830 y Fx(>)g FD(1)1493 749 y Fq(\003)1594 830 y Fs(\024)1739 735 y Fq(X)1733 955 y Ft(y)t Fr(2)p Fn(Z)1865 935 y Fl(d)1906 830 y Fv(E)1981 660 y Fq(")2045 830 y Fx(2)2139 660 y Fq( )2273 750 y Fs(Q)-45 b Fw(c)27 b Fp(k)s Fw(V)2461 732 y Fx(\037)2530 765 y Fu(3.)t Ft(y)t Fu(/)2681 750 y Fp(k)2739 714 y Ft(p)2774 724 y Fk(1)2804 714 y Fu(=)s Ft(r)2739 777 y(p)2774 787 y Fk(1)p 2229 807 681 4 v 2229 901 a Fx(.)p FD(1)20 b Fs(C)f Fp(j)6 b Fw(y)g Fp(j)2545 872 y Fu(\027)2587 901 y Fx(/)2624 872 y Fu(.)i Ft(p)2693 882 y Fk(1)2723 872 y Fr(\000)o Ft(r)e Fu(/=)s Ft(r)2942 830 y Fs(\000)19 b FD(1)3089 660 y Fq(!#)1594 1183 y Fs(\024)72 b(Q)-45 b Fw(c)1775 1142 y Ft(q)1844 1089 y Fq(X)1838 1308 y Ft(y)t Fr(2)p Fn(Z)1970 1288 y Fl(d)2023 1104 y Fv(E)2098 1023 y Fq(\002)2145 1104 y Fp(k)s Fw(V)2274 1086 y Fx(\037)2342 1119 y Fu(3.)t Ft(y)t Fu(/)2494 1104 y Fp(k)2552 1067 y Ft(p)2587 1077 y Fk(1)2617 1067 y Ft(q)5 b Fu(=)s Ft(r)2552 1130 y(p)2587 1140 y Fk(1)2731 1023 y Fq(\003)p 2023 1161 750 4 v 2037 1254 a Fx(.)p FD(1)20 b Fs(C)f Fp(j)6 b Fw(y)g Fp(j)2353 1225 y Fu(\027)2395 1254 y Fx(/)2432 1225 y Fu(.)i Ft(p)2501 1235 y Fk(1)2531 1225 y Fr(\000)o Ft(r)e Fu(/)p Ft(q)f Fu(=)s Ft(r)2798 1183 y Fx(:)508 b FD(\(5.4\))294 1531 y(In)29 b(order)h(to)e(get)h(the)g(second)g(inequality)f(in)g(\(5.4\),) i(we)g(used)e(the)h(\223Chebyshe)n(v-Mark)o(o)o(v\224)294 1650 y(inequality)24 b(with)g Fx(\024)33 b Fs(D)25 b Fw(q)7 b FD(,)24 b(where)i Fw(q)31 b FD(is)25 b(chosen)f(such)h(that) 1439 1869 y Fw(d)p 1439 1916 58 4 v 1441 2010 a Fx(\027)1627 1869 y Fw(p)1677 1884 y Fz(1)p 1534 1916 265 4 v 1545 2010 a Fw(p)1595 2025 y Fz(1)1655 2010 y Fs(\000)17 b Fw(r)1836 1939 y Fx(<)1962 1869 y Fw(p)2012 1884 y Fz(1)2052 1869 y Fw(q)p 1951 1916 159 4 v 2004 2010 a(r)2146 1939 y Fx(<)35 b Fw(p)2309 1954 y Fz(2)2363 1939 y Fx(:)943 b FD(\(5.5\))294 2244 y(The)27 b(numerator)g(in)f(the)h(second)f(line)g (of)h(\(5.4\))g(is)f(uniformly)g(bounded)g(in)32 b Fw(y)g Fs(2)26 b Fv(Z)3206 2208 y Ft(d)3275 2244 y FD(due)h(to)294 2363 y(the)20 b(right)f(inequality)f(in)i(\(5.5\),)h(Jensen')-5 b(s)19 b(inequality)f(and)i(property)f(\()p Fh(S)p FD(\))q(.)29 b(The)20 b(left)g(inequal-)294 2483 y(ity)33 b(in)f(\(5.5\))h(then)g (assures)g(that)g(the)g(series)g(in)f(the)h(second)g(line)g(of)g (\(5.4\))g(is)g(summable,)294 2603 y(which)25 b(implies)e(by)i(the)f (\002rst)h(Borel-Cantelli)h(lemma)847 2843 y Fv(P)906 2762 y Fq(\002)949 2843 y Fp(k)s Fw(V)1060 2858 y Fz(1)1100 2825 y Fx(\037)1169 2858 y Fu(3.)t Ft(y)t Fu(/)1320 2843 y Fp(k)1369 2858 y Ft(r)1433 2843 y Fx(>)e FD(1)50 b(for)25 b(in\002nitely)f(man)o(y)55 b Fw(y)30 b Fs(2)25 b Fv(Z)2668 2802 y Ft(d)2711 2762 y Fq(\003)2777 2843 y Fs(D)g FD(0)14 b Fx(:)362 b FD(\(5.6\))294 3083 y(This)24 b(deli)n(v)o(ers)471 3389 y Fv(P)544 3219 y Fq(")616 3389 y FD(sup)608 3481 y Ft(y)t Fr(2)p Fn(Z)740 3461 y Fl(d)781 3389 y Fp(k)s Fw(V)892 3404 y Fz(1)932 3372 y Fx(\037)1000 3404 y Fu(3.)t Ft(y)t Fu(/)1152 3389 y Fp(k)1201 3404 y Ft(r)1264 3389 y Fs(D)h(1)1469 3219 y Fq(#)1587 3389 y Fs(D)59 b Fv(P)1783 3309 y Fq(\002)1827 3389 y Fp(k)s Fw(V)1938 3404 y Fz(1)1977 3372 y Fx(\037)2046 3404 y Fu(3.)t Ft(y)2163 3414 y Fk(0)2192 3404 y Fu(/)2223 3389 y Fp(k)2272 3404 y Ft(r)2336 3389 y Fs(D)24 b(1)50 b FD(for)25 b(some)55 b Fw(y)3041 3404 y Fz(0)3106 3389 y Fs(2)24 b Fv(Z)3262 3348 y Ft(d)3305 3309 y Fq(\003)1588 3659 y Fs(\024)1734 3564 y Fq(X)1728 3784 y Ft(y)t Fr(2)p Fn(Z)1860 3763 y Fl(d)1901 3659 y Fv(P)1960 3578 y Fq(\002)2004 3659 y Fp(k)s Fw(V)2115 3674 y Fz(1)2154 3641 y Fx(\037)2223 3674 y Fu(3.)t Ft(y)t Fu(/)2374 3659 y Fp(k)2423 3674 y Ft(r)2487 3659 y Fs(D)h(1)2692 3578 y Fq(\003)1588 3952 y Fs(\024)1734 3857 y Fq(X)1728 4077 y Ft(y)t Fr(2)p Fn(Z)1860 4057 y Fl(d)1901 3952 y Fv(P)1960 3871 y Fq(\002)2004 3952 y Fp(k)s Fw(V)2132 3934 y Fx(\037)2201 3967 y Fu(3.)t Ft(y)t Fu(/)2352 3952 y Fp(k)2410 3967 y Ft(p)2445 3977 y Fk(1)2505 3952 y Fs(D)g(1)2710 3871 y Fq(\003)1587 4209 y Fs(D)59 b FD(0)14 b Fx(;)1512 b FD(\(5.7\))294 4449 y(where)31 b(we)g(ha)n(v)o(e)f(used)g (the)g(countable)g(subadditi)n(vity)d(of)k Fv(P)h FD(for)f(the)f (\002rst)h(inequality)e(and)294 4569 y Fp(j)s Fw(V)383 4584 y Fz(1)423 4569 y Fp(j)d Fs(\024)h Fp(j)s Fw(V)14 b Fp(j)28 b FD(as)h(well)e(as)f Fw(r)37 b Fx(<)g Fw(p)1402 4584 y Fz(1)1471 4569 y FD(for)28 b(the)g(second)g(inequality)-6 b(.)39 b(The)28 b(last)g(equality)f(in)h(\(5.7\))294 4688 y(follo)n(ws)c(from)g(property)h(\()p Fh(S)p FD(\))q(.)31 b(Thus,)24 b(we)h(ha)n(v)o(e)g(sho)n(wn)1262 4928 y Fv(P)1321 4848 y Fq(\002)1367 4928 y Fw(V)1425 4943 y Fz(1)1490 4928 y Fs(2)g FD(L)1638 4887 y Ft(r)1639 4955 y Fz(unif)p Fu(;)p Fz(loc)1864 4928 y Fx(.)p Fv(R)1968 4887 y Ft(d)2019 4928 y Fx(/)2056 4848 y Fq(\003)2122 4928 y Fs(D)g FD(1)14 b Fx(:)p 2515 4937 41 90 v 1017 w FD(\(5.8\))394 5169 y(F)o(or)28 b(the)g(proof)f(of)h(Corollary)g(1.27)g(and)g(Corollary)g (1.29)f(we)h(need)g(suitable)f(measura-)294 5288 y(bility)d(properties) g(of)h(the)g(in)l(v)n(olv)o(ed)e(inte)o(gral)h(k)o(ernels,)h(which)f (we)h(establish)f(in)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(35)396 384 y(L)t FG(E)t(M)t(M)t(A)33 b FD(5)t(.)t(1)t(.)125 b Fw(Let)50 b(A)41 b(be)f(a)f(vector)h(potential)e(with)h(pr)l(operty)f FD(\()p Fh(A)p FD(\))j Fw(and)e(let)j(V)54 b(be)294 503 y(a)39 b(r)o(andom)f(scalar)g(potential)g(with)h(pr)l(operty)e FD(\()p Fh(S)p FD(\))q Fw(.)74 b(Then)39 b(ther)l(e)h(e)n(xists)e Fx(\177)3042 518 y Fz(0)3114 503 y Fs(2)33 b Fj(A)62 b Fw(with)294 623 y Fv(P)p Fx(.\177)468 638 y Fz(0)510 623 y Fx(/)25 b Fs(D)f FD(1)h Fw(suc)o(h)f(that)g(for)h(e)o(very)g Fx(!)i Fs(2)e Fx(\177)1774 638 y Fz(0)549 765 y FD(\(i\))100 b Fw(the)26 b(oper)o(ator)f(e)n(xponential)h FD(e)1799 728 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2100 703 y Fm(.!)q(/)2175 728 y Fu(/)2233 765 y Fw(has)26 b(a)h(continuous)e(inte)l(gr)o(al)f(k)o(ernel)294 884 y(k)344 835 y Fu(.!)q(/)340 902 y Ft(t)474 884 y Fw(for)g(any)h(t)33 b Fx(>)25 b FD(0)g Fw(and)f(the)h(mapping)1156 1125 y Fx(\177)1234 1140 y Fz(0)1273 1125 y Fs(\002)p FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fs(\002)p Fv(R)1760 1089 y Ft(d)1831 1125 y Fs(\002)20 b Fv(R)1995 1089 y Ft(d)2106 1125 y Fs(!)216 b Fv(C)1384 1258 y Fx(.!)5 b(;)14 b Fw(t)8 b Fx(;)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)289 b Fs(7!)59 b Fw(k)2325 1209 y Fu(.!)q(/)2321 1276 y Ft(t)2429 1258 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)3333 1193 y FD(\(5.9\))294 1514 y Fw(is)25 b Fj(A)466 1529 y Fz(0)525 1514 y Fs(\012)19 b Fj(B)t Fx(.)p FD(]0)p Fx(;)14 b Fs(1)p FD([)p Fx(/)21 b Fs(\012)e Fj(B)t Fx(.)p Fv(R)1375 1478 y Ft(d)1426 1514 y Fx(/)h Fs(\012)f Fj(B)t Fx(.)p Fv(R)1780 1478 y Ft(d)1831 1514 y Fx(/)p Fw(-measur)o(able)o(.)521 1685 y FD(\(ii\))100 b Fw(the)34 b(spectr)o(al)f(pr)l(ojection)1693 1668 y Fx(\037)1762 1700 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)2016 1605 y Fq(\000)2069 1685 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)2390 1649 y Fu(.!)q(/)2494 1685 y Fx(/)2531 1605 y Fq(\001)2611 1685 y Fw(has)34 b(a)h(continuous)e(inte-)294 1805 y(gr)o(al)24 b(k)o(ernel)36 b(p)821 1769 y Fu(.!)q(/)925 1805 y Fx(.)7 b Fw(E)i Fs(I)14 b(\001)p Fx(;)g Fs(\001)p Fx(/)24 b Fw(for)g(any)32 b(E)i Fs(2)24 b Fv(R)36 b Fw(and)25 b(the)f(mapping)1143 2046 y Fx(\177)1221 2061 y Fz(0)1280 2046 y Fs(\002)c Fv(R)30 b Fs(\002)20 b Fv(R)1633 2010 y Ft(d)1704 2046 y Fs(\002)f Fv(R)1867 2010 y Ft(d)1978 2046 y Fs(!)287 b Fv(C)1294 2166 y Fx(.!)5 b(;)21 b Fw(E)9 b Fx(;)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)210 b Fs(7!)70 b Fw(p)2211 2130 y Fu(.!)q(/)2316 2166 y Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)3283 2107 y FD(\(5.10\))294 2422 y Fw(is)25 b Fj(A)466 2437 y Fz(0)525 2422 y Fs(\012)19 b Fj(B)t Fx(.)p Fv(R)6 b Fx(/)26 b Fs(\012)19 b Fj(B)t Fx(.)p Fv(R)1182 2386 y Ft(d)1233 2422 y Fx(/)h Fs(\012)f Fj(B)t Fx(.)p Fv(R)1587 2386 y Ft(d)1638 2422 y Fx(/)p Fw(-measur)o(able)o(.)294 2564 y(Her)l(e)o(,)38 b Fj(A)631 2579 y Fz(0)705 2564 y Fw(is)c(the)h(r)l(estriction)e(of)h(the)h(sigma-alg)o(ebr)o(a)d Fj(A)58 b Fw(of)34 b Fx(\177)g Fw(to)g Fx(\177)2843 2579 y Fz(0)2883 2564 y Fw(,)j(and)d(given)h(any)294 2684 y(Bor)l(el)27 b(set)35 b(B)d Fs(\022)26 b Fv(R)948 2647 y Ft(d)1026 2684 y Fw(we)i(denote)f(by)g Fj(B)t Fx(.)8 b Fw(B)f Fx(/)27 b Fw(the)g(sub-sigma-alg)o(ebr)o(a)d(of)j(Bor)l(el)g (sets)g(in)f Fv(R)3472 2647 y Ft(d)294 2803 y Fw(whic)o(h)f(ar)l(e)g (contained)f(in)32 b(B)7 b(.)399 3055 y(Pr)l(oof)23 b(.)120 b FD(The)26 b(e)o(xistence)g(and)g(continuity)e(of)i(the)g(inte)o(gral) g(k)o(ernels)g(is)f(guaranteed)i(by)294 3174 y(Corollary)34 b(1.24,)i(Lemma)d(1.7,)j(Theorem)e(1.10)f(and)h(Corollary)g(1.16)f (\(see)i(also)e(Corol-)294 3294 y(lary)h(1.18\).)59 b(The)34 b(measurability)f(claimed)g(in)h(\(i\))g(follo)n(ws)f(from)h(the)g(Bro) n(wnian-bridge)294 3413 y(representation)c(\(1.11\))g(for)h Fw(k)1357 3365 y Fu(.!)q(/)1353 3432 y Ft(t)1461 3413 y FD(.)47 b(The)30 b(claim)g(of)g(\(ii\))g(follo)n(ws)f(from)h(\(i\),)i (Corollary)e(1.18)294 3533 y(and)25 b(the)g(in)l(v)o(ertibility)d(of)j (the)g(Laplace)g(transformation.)p 2426 3542 41 90 v 399 3759 a Fw(Pr)l(oof)48 b FD(\(of)25 b(Corollary)g(1.27\))p Fw(.)119 b FD(W)-8 b(e)35 b(\002x)41 b Fw(E)e Fs(2)29 b Fv(R)45 b FD(arbitrary)-6 b(.)58 b(Lemma)34 b(5.1\(ii\))f(guaran-)294 3878 y(tees)k(the)g(e)o(xistence,)i(continuity)c(and)i(suitable)f (measurability)f(properties)h(of)h(the)g(inte-)294 3998 y(gral)h(k)o(ernel)48 b Fw(p)836 3962 y Fu(.!)q(/)941 3998 y Fx(.)7 b Fw(E)i Fs(I)14 b(\001)p Fx(;)g Fs(\001)p Fx(/)36 b FD(of)h(the)g(spectral)h(projection)2349 3980 y Fx(\037)2418 4013 y Fz(])p Fr(\0001)p Fu(;)5 b Ft(E)g Fz([)2672 3917 y Fq(\000)2725 3998 y Fw(H)10 b Fx(.)g Fw(A)5 b Fx(;)17 b Fw(V)3046 3962 y Fu(.!)q(/)3150 3998 y Fx(/)3187 3917 y Fq(\001)3270 3998 y FD(for)38 b(all)294 4117 y Fx(!)27 b Fs(2)e Fx(\177)552 4132 y Fz(0)616 4117 y Fs(2)g Fj(A)48 b FD(with)24 b Fv(P)p Fx(.\177)1208 4132 y Fz(0)1249 4117 y Fx(/)h Fs(D)g FD(1.)31 b(Eq.)24 b(\(1.24\))h(and)g(Proposition)e(1.25)i(imply)e(that)1291 4421 y Fw(N)12 b Fx(.)7 b Fw(E)i Fx(/)25 b Fs(D)g Fv(E)1708 4280 y Fq(\024)1767 4285 y Fo(Z)1819 4510 y Fu(0)1901 4351 y FD(d)13 b Fw(x)p 1898 4398 124 4 v 1898 4492 a Fp(j)p Fx(0)t Fp(j)2069 4421 y Fw(p)s Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)2482 4280 y Fq(\025)3283 4421 y FD(\(5.11\))294 4738 y(is)21 b(\002nite.)29 b(No)n(w)20 b(the)h(claim)g(follo)n(ws)e(from)i(Fubini')-5 b(s)20 b(theorem,)h(because)32 b Fw(p)2892 4702 y Fu(.!)q(/)2997 4738 y Fx(.)7 b Fw(E)i Fs(I)17 b Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)20 b Fs(\025)i FD(0)294 4857 y(for)39 b(all)f Fx(!)d Fs(2)d Fx(\177)860 4872 y Fz(0)938 4857 y FD(and)38 b(all)k Fw(x)f Fs(2)32 b Fv(R)1508 4821 y Ft(d)1559 4857 y FD(,)42 b(see)d(Lemma)f(4.2,)j(and)d(because)h Fv(E)13 b FD([)e Fw(p)s Fx(.)d Fw(E)h Fs(I)17 b Fw(x)9 b Fx(;)17 b Fw(x)9 b Fx(/)p FD(])43 b(is)294 4977 y(independent)24 b(of)k Fw(x)34 b Fs(2)25 b Fv(R)1149 4941 y Ft(d)1225 4977 y FD(due)f(to)h(the)g Fv(R)1709 4941 y Ft(d)1760 4977 y FD(-er)n(godicity)f(of)k Fw(V)15 b FD(.)p 2551 4986 41 90 v 399 5203 a Fw(Pr)l(oof)48 b FD(\(of)25 b(Corollary)g (1.29\))p Fw(.)119 b FD(W)-8 b(e)43 b(\002x)g Fw(t)g Fx(>)34 b FD(0)42 b(arbitrary)-6 b(.)83 b(Lemma)41 b(5.1\(i\))h (guaran-)294 5322 y(tees)37 b(the)g(e)o(xistence,)i(continuity)c(and)i (suitable)f(measurability)f(properties)h(of)h(the)g(inte-)294 5442 y(gral)28 b(k)o(ernel)g Fw(k)803 5393 y Fu(.!)q(/)799 5460 y Ft(t)936 5442 y FD(of)f(the)h(operator)g(e)o(xponential)f(e)2092 5406 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)2393 5380 y Fm(.!)q(/)2468 5406 y Fu(/)2527 5442 y FD(for)28 b(all)g Fx(!)h Fs(2)d Fx(\177)3060 5457 y Fz(0)3126 5442 y Fs(2)h Fj(A)50 b FD(with)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 294 90 a FD(36)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y Fv(P)p Fx(.\177)468 399 y Fz(0)510 384 y Fx(/)j Fs(D)f FD(1.)29 b(Jensen')-5 b(s)21 b(inequality)-6 b(,)20 b(Fubini')-5 b(s)20 b(theorem)h(and)h (property)f(\()p Fh(L)p FD(\))h(imply)e(for)i Fx(\026)3384 348 y Fz(0)p Fu(;)p Ft(t)3386 411 y(x)6 b Fu(;)t Ft(y)3490 384 y FD(-)294 503 y(almost)24 b(e)n(v)o(ery)g(path)h Fw(b)i FD(of)d(the)h(Bro)n(wnian)g(bridge)f(the)h(estimate)340 780 y Fv(E)415 640 y Fq(\024)473 780 y FD(e)o(xp)615 640 y Fq(\032)690 780 y Fs(\000)782 644 y Fo(Z)880 671 y Ft(t)834 869 y Fz(0)911 780 y FD(d)10 b Fw(s)33 b(V)15 b Fx(.)p Fw(b)r Fx(.)p Fw(s)6 b Fx(//)1364 640 y Fq(\033\025)1516 780 y Fs(\024)1617 644 y Fo(Z)1715 671 y Ft(t)1669 869 y Fz(0)1771 710 y FD(d)k Fw(s)p 1771 757 105 4 v 1805 851 a(t)1913 780 y Fv(E)1973 699 y Fq(\002)2021 780 y FD(e)o(xp)2163 699 y Fq(\010)2221 780 y Fs(\000)p Fw(t)i(V)j Fx(.)p Fw(b)r Fx(.)p Fw(s)6 b Fx(//)2660 699 y Fq(\011\003)2784 780 y Fs(\024)25 b Fj(L)2971 795 y Ft(t)3027 780 y Fx(<)g Fs(1)14 b Fx(;)3283 982 y FD(\(5.12\))294 1200 y(which)25 b(sho)n(ws)e(that)i(the)f(inte)o(gral)g(k)o(ernel)p 1756 1121 77 4 v 25 w Fw(k)1802 1215 y Ft(t)1858 1200 y FD(is)h(well)f (de\002ned)h(and)g(obe)o(ys)f(the)h(inequality)1031 1494 y Fp(j)p 1059 1415 V Fw(k)1105 1509 y Ft(t)1136 1494 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)24 b Fs(\024)h Fv(E)1582 1413 y Fq(\002)1629 1494 y Fp(j)p Fw(k)1703 1509 y Ft(t)1734 1494 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)1995 1413 y Fq(\003)2061 1494 y Fs(\024)25 b Fj(L)2248 1509 y Ft(t)2316 1424 y FD(e)2360 1388 y Fr(\000)p Fi(j)r Ft(x)6 b Fr(\000)t Ft(y)t Fi(j)2594 1363 y Fk(2)2624 1388 y Fu(=.)p Fz(2)p Ft(t)g Fu(/)p 2316 1471 459 4 v 2373 1565 a Fx(.)p FD(2)p Fx(\031)k Fw(t)f Fx(/)2604 1536 y Ft(d)c Fu(=)p Fz(2)3283 1494 y FD(\(5.13\))294 1776 y(for)28 b(all)h Fw(x)9 b Fx(;)20 b Fw(y)32 b Fs(2)26 b Fv(R)904 1740 y Ft(d)955 1776 y FD(,)h(thereby)g(pro)o(ving)f(\(1.33\).)37 b(The)27 b(Hermiticity)f(of) p 2751 1698 77 4 v 27 w Fw(k)2797 1791 y Ft(t)2855 1776 y FD(is)g(inherited)h(from)294 1896 y(that)k(of)g Fw(k)635 1911 y Ft(t)666 1896 y FD(,)i(see)e(Lemma)g(1.7\(i\).)49 b(The)31 b(estimate)g(\(5.13\))g(also)g(yields)p 2794 1817 V 30 w Fw(k)2840 1911 y Ft(t)2871 1896 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)28 b Fs(2)g FD(L)3256 1859 y Fr(1)3256 1923 y Fz(G)3332 1896 y Fx(.)p Fv(R)3435 1860 y Ft(d)3486 1896 y Fx(/)294 2016 y FD(for)d(all)i Fw(x)33 b Fs(2)25 b Fv(R)793 1980 y Ft(d)844 2016 y FD(,)f(and)h(hence) f(the)h(Carleman)g(property)f(\(1.15\))g(for)p 2645 1937 V 25 w Fw(k)2691 2031 y Ft(t)2722 2016 y FD(.)30 b(W)-8 b(e)25 b(defer)g(the)g(proof)294 2135 y(of)f(the)g(continuity)e(of)p 1082 2057 V 24 w Fw(k)1128 2150 y Ft(t)1183 2135 y FD(to)h(the)h(end,)g (b)n(ut)f(e)o(xploit)f(its)h(consequences)h(right)f(no)n(w)-6 b(.)29 b(Jensen')-5 b(s)294 2255 y(inequality)f(,)37 b(Fubini')-5 b(s)35 b(theorem)g(and)h(the)g(almost-surely)e(applicable) i(Mark)o(o)o(v)f(property)294 2374 y(\(1.12\))25 b(yield)f(the)h (estimate)489 2639 y Fp(k)p 539 2561 V Fw(k)585 2654 y Ft(t)615 2639 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)20 b Fs(\000)p 937 2561 V 19 w Fw(k)983 2654 y Ft(t)1014 2639 y Fx(.)r Fw(z)5 b Fx(;)14 b Fs(\001)p Fx(/)p Fp(k)1259 2598 y Fz(2)1259 2666 y(2)1359 2639 y Fs(\024)1495 2504 y Fo(Z)1548 2729 y Fn(R)1601 2709 y Fl(d)1637 2639 y FD(d)i Fw(y)30 b Fv(E)1838 2559 y Fq(\002)1885 2639 y Fp(j)p Fw(k)1959 2654 y Ft(t)1990 2639 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)19 b Fs(\000)h Fw(k)2386 2654 y Ft(t)2417 2639 y Fx(.)r Fw(z)5 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)2668 2598 y Fz(2)2707 2559 y Fq(\003)1358 2855 y Fs(D)p 1495 2777 112 4 v 59 w Fw(k)1541 2870 y Fz(2)p Ft(t)1607 2855 y Fx(.)s Fw(x)j Fx(;)17 b Fw(x)9 b Fx(/)19 b Fs(\000)p 1957 2777 V 20 w Fw(k)2003 2870 y Fz(2)p Ft(t)2069 2855 y Fx(.)r Fw(z)5 b Fx(;)17 b Fw(x)9 b Fx(/)19 b Fs(\000)p 2408 2777 V 19 w Fw(k)2454 2870 y Fz(2)p Ft(t)2520 2855 y Fx(.)s Fw(x)9 b Fx(;)16 b Fw(z)5 b Fx(/)19 b Fs(C)p 2859 2777 V 19 w Fw(k)2905 2870 y Fz(2)p Ft(t)2971 2855 y Fx(.)r Fw(z)5 b Fx(;)16 b Fw(z)5 b Fx(/)14 b(;)52 b FD(\(5.14\))294 3073 y(sho)n(wing)32 b(that)h(the)g(continuity)f(of)p 1555 2994 V 33 w Fw(k)1601 3088 y Fz(2)p Ft(t)1701 3073 y FD(implies)f(the)j(strong)e(continuity)g (of)h(the)h(mapping)294 3192 y Fv(R)360 3156 y Ft(d)436 3192 y Fs(!)25 b FD(L)632 3156 y Fz(2)671 3192 y Fx(.)p Fv(R)775 3156 y Ft(d)826 3192 y Fx(/)p FD(,)j Fw(x)33 b Fs(7!)p 1128 3113 77 4 v 25 w Fw(k)1174 3207 y Ft(t)1205 3192 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)p FD(.)394 3312 y(The)25 b(estimate)f(\(5.13\))h(deli)n(v)o(ers)1435 3529 y Fp(j)r Fw(T)1517 3544 y Ft(t)1547 3529 y Fx( )9 b Fp(j)25 b Fs(\024)g Fj(L)1866 3544 y Ft(t)1911 3529 y FD(e)1955 3488 y Fr(\000)p Ft(t)11 b(H)e Fu(.)p Fz(0)p Fu(;)p Fz(0)p Fu(/)2248 3529 y Fp(j)p Fx( )g Fp(j)900 b FD(\(5.15\))294 3746 y(for)21 b(all)f Fx( )29 b Fs(2)21 b FD(L)795 3710 y Fz(2)834 3746 y Fx(.)p Fv(R)937 3710 y Ft(d)989 3746 y Fx(/)p FD(,)g(where)i Fw(T)1390 3761 y Ft(t)1440 3746 y FD(is)d(de\002ned)h(as)f(in)g(\(1.34\).)29 b(Consequently)-6 b(,)21 b Fw(T)2978 3761 y Ft(t)3029 3746 y FD(is)f(a)g(bounded)294 3866 y(Carleman)33 b(operator)f(on)g(L) 1271 3830 y Fz(2)1311 3866 y Fx(.)p Fv(R)1414 3830 y Ft(d)1465 3866 y Fx(/)p FD(.)53 b(Moreo)o(v)o(er)l(,)35 b Fw(T)2079 3881 y Ft(t)2141 3866 y FD(is)d(self-adjoint)f(because)i (of)f(the)g(Her)n(-)294 3986 y(miticity)k(of)p 763 3907 V 38 w Fw(k)809 4001 y Ft(t)878 3986 y FD(and)i(an)g(interchange)g(of)g (inte)o(grations)e(thanks)h(to)g(\(5.13\))h(and)g(Fubini')-5 b(s)294 4105 y(theorem.)38 b(The)27 b(continuity)e(of)j(an)o(y)e(image) j Fw(T)1902 4120 y Ft(t)1932 4105 y Fx( )37 b FD(follo)n(ws)26 b(from)h(the)g(strong)f(continuity)f(of)p 294 4146 V 294 4225 a Fw(k)340 4240 y Ft(t)371 4225 y Fx(.)s Fw(x)9 b Fx(;)14 b Fs(\001)p Fx(/)25 b FD(by)f(proceeding)h(along)f(the)h (lines)f(of)h(Eq.)g(\(3.6\))g(in)f(the)h(proof)g(of)g(Lemma)f(3.1.)394 4344 y(No)n(w)k(let)h Fx( )37 b Fs(2)27 b FD(L)1001 4308 y Fz(2)1001 4371 y(G)1056 4344 y Fx(.)p Fv(R)1159 4308 y Ft(d)1210 4344 y Fx(/)j FD(so)e(that)h(the)g(equality)h Fw(T)2128 4359 y Ft(t)2159 4344 y Fx( )36 b Fs(D)27 b Fv(E)2431 4264 y Fq(\002)2478 4344 y FD(e)2522 4308 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)12 b Fu(/)2854 4344 y Fx( )2933 4264 y Fq(\003)3004 4344 y FD(follo)n(ws)28 b(from)294 4464 y(\(1.17\))k(and)g(an)g(interchange) g(of)g(inte)o(grations.)50 b(This)31 b(interchange)h(is)f(again)h(allo) n(wed)f(by)294 4584 y(Fubini')-5 b(s)25 b(theorem)g(and)h(\(5.13\).)34 b(The)25 b(inequalities)g(\(5.13\))g(and)h(\(2.14\))g(imply)e(that)k Fw(T)3326 4599 y Ft(t)3356 4584 y Fx( )35 b Fs(2)294 4703 y FD(L)355 4667 y Fr(1)355 4730 y Fz(G)431 4703 y Fx(.)p Fv(R)535 4667 y Ft(d)586 4703 y Fx(/)25 b FD(for)g(all)f Fx( )34 b Fs(2)25 b FD(L)1166 4667 y Fz(2)1166 4730 y(G)1221 4703 y Fx(.)p Fv(R)1324 4667 y Ft(d)1376 4703 y Fx(/)p FD(.)31 b(Remark)25 b(1.11\(iii\))f(applies)g(accordingly)-6 b(.)394 4823 y(Ne)o(xt)33 b(we)i(establish)e(the)h(positi)n(vity)d(of) 36 b Fw(T)1889 4838 y Ft(t)1920 4823 y FD(.)58 b(Gi)n(v)o(en)33 b(an)o(y)g Fx( )40 b Fs(2)29 b FD(L)2716 4786 y Fz(2)2716 4849 y(G)2772 4823 y Fx(.)p Fv(R)2875 4787 y Ft(d)2926 4823 y Fx(/)p FD(,)36 b(one)e(deduces)294 4942 y(from)k(the)g(just-pro) o(v)o(en)e(equality)h(\(1.35\),)k(the)d(estimate)f(\(5.13\))h(and)g (Fubini')-5 b(s)37 b(theorem)294 5062 y(that)19 b Fp(h)p Fx( )s(;)d Fw(T)676 5077 y Ft(t)707 5062 y Fx( )9 b Fp(i)20 b Fs(D)g Fv(E)1003 4981 y Fq(\002)1051 5062 y Fp(h)p Fx( )s(;)14 b FD(e)1254 5026 y Fr(\000)p Ft(t)d(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)k Fu(/)1585 5062 y Fx( )e Fp(i)1703 4981 y Fq(\003)1765 5062 y Fs(\025)20 b FD(0,)g(where)g(the)g(lo)n(wer) f(bound)f(follo)n(ws)g(from)i(the)294 5203 y(positi)n(vity)27 b(of)i(e)854 5166 y Fr(\000)p Ft(t)11 b(H)e Fu(.)e Ft(A)r Fu(;)r Ft(V)1155 5141 y Fm(.!)q(/)1230 5166 y Fu(/)1291 5203 y FD(for)29 b Fv(P)p FD(-almost)h(all)f Fx(!)h Fs(2)d Fx(\177)p FD(.)43 b(No)n(w)-6 b(,)29 b(the)g(denseness)g(of)g(L)3277 5166 y Fz(2)3277 5229 y(G)3332 5203 y Fx(.)p Fv(R)3435 5166 y Ft(d)3486 5203 y Fx(/)294 5322 y FD(in)j(L)465 5286 y Fz(2)505 5322 y Fx(.)p Fv(R)608 5286 y Ft(d)659 5322 y Fx(/)p FD(,)i(the)e(boundedness)f(of)j Fw(T)1625 5337 y Ft(t)1688 5322 y FD(and)e(the)g(continuity)e(of)i(the)g(scalar)h (product)e(yield)294 5442 y Fp(h)p Fx( )s(;)16 b Fw(T)507 5457 y Ft(t)538 5442 y Fx( )9 b Fp(i)25 b Fs(\025)g FD(0)g(for)g(all)f Fx( )34 b Fs(2)25 b FD(L)1374 5406 y Fz(2)1414 5442 y Fx(.)p Fv(R)1517 5406 y Ft(d)1568 5442 y Fx(/)p FD(.)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(37)394 384 y(Finally)-6 b(,)26 b(we)h(turn)f(to)h(the)f(postponed)g(proof)h (of)f(the)h(continuity)e(of)i(the)f(mapping)g Fv(R)3374 348 y Ft(d)3445 384 y Fs(\002)294 503 y Fv(R)360 467 y Ft(d)438 503 y Fs(!)g Fv(C)34 b Fx(;)14 b(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)32 b Fs(7!)p 1102 425 77 4 v 26 w Fw(k)1148 518 y Ft(t)1178 503 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p FD(.)39 b(This)27 b(continuity)f(will)g (follo)n(w)h(from)g(Lemma)g(5.1\(i\))h(and)294 623 y(the)d (dominated-con)l(v)o(er)n(gence)f(theorem,)h(pro)o(vided)e(we)j(sho)n (w)1399 918 y Fv(E)1460 777 y Fq(\024)1569 918 y FD(sup)1521 1005 y Ft(x)6 b Fu(;)t Ft(y)t Fr(2)p Ff(K)1771 918 y Fp(j)p Fw(k)1845 933 y Ft(t)1876 918 y Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)2137 777 y Fq(\025)2214 918 y Fx(<)25 b Fs(1)864 b FD(\(5.16\))294 1240 y(for)37 b(an)o(y)e(bounded)h(set)g Fj(K)62 b Fs(\032)31 b Fv(R)1480 1204 y Ft(d)1555 1240 y Fs(\002)24 b Fv(R)1723 1204 y Ft(d)1774 1240 y FD(.)65 b(In)36 b(order)g(to)g(do)g(so,)i(let)e(us)g (\002x)h Fx(!)c Fs(2)e Fx(\177)3304 1255 y Fz(0)3380 1240 y FD(and)297 1360 y Fw(x)9 b Fx(;)20 b Fw(y)34 b Fs(2)28 b Fj(K)61 b FD(arbitrary)-6 b(.)48 b(By)31 b(using)f(\(1.11\),) h(the)g(triangle)f(inequality)-6 b(,)30 b(Jensen')-5 b(s)30 b(inequality)294 1479 y(and)25 b(Fubini')-5 b(s)24 b(theorem,)g(we)h(get)515 1777 y Fp(j)p Fw(k)593 1736 y Fu(.!)q(/)589 1804 y Ft(t)697 1777 y Fx(.)s Fw(x)9 b Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)61 b Fs(\024)g Fx(.)p FD(2)p Fx(\031)10 b Fw(t)f Fx(/)1386 1736 y Fr(\000)p Ft(d)c Fu(=)p Fz(2)1567 1641 y Fo(Z)1665 1668 y Ft(t)1620 1866 y Fz(0)1722 1707 y FD(d)10 b Fw(s)p 1722 1754 105 4 v 1756 1848 a(t)1852 1641 y Fo(Z)1964 1777 y Fx(\026)2032 1736 y Fz(0)p Fu(;)p Ft(t)2034 1804 y(x)c Fu(;)t Ft(y)2139 1777 y Fx(.)p FD(d)k Fw(b)r Fx(/)25 b FD(e)2394 1736 y Fr(\000)p Ft(t)8 b(V)2529 1711 y Fm(.!)q(/)2605 1736 y Fu(.)p Ft(b)q Fu(.)p Ft(s)t Fu(//)1018 2063 y Fs(D)59 b Fx(.)p FD(2)p Fx(\031)10 b Fw(t)f Fx(/)1386 2022 y Fr(\000)p Ft(d)c Fu(=)p Fz(2)1567 1927 y Fo(Z)1665 1954 y Fz(1)1620 2152 y(0)1705 2063 y FD(d)10 b Fx(\033)1845 1927 y Fo(Z)1897 2152 y Fn(R)1950 2132 y Fl(d)1987 2063 y FD(d)i Fw(z)31 b(g)2169 2078 y Fu(\033)2220 1982 y Fq(\000)2267 2063 y Fw(z)24 b Fs(\000)c Fw(m)2508 2078 y Ft(x)6 b Fu(;)t Ft(y)2612 2063 y Fx(.\033)13 b(/)2752 1982 y Fq(\001)2812 2063 y FD(e)2856 2022 y Fr(\000)p Ft(t)8 b(V)2991 1997 y Fm(.!)q(/)3067 2022 y Fu(.)q Ft(z)t Fu(/)3170 2063 y Fx(;)80 b FD(\(5.17\))294 2350 y(where)38 b(the)f(equality)f(follo)n(ws)g(from)h(an)g(e)o(xplicit)f(computation)f (with)h Fw(m)2952 2365 y Ft(x)6 b Fu(;)t Ft(y)3056 2350 y Fx(.\033)13 b(/)33 b FD(:)p Fs(D)h Fw(x)f Fs(C)294 2470 y Fx(.)6 b Fw(y)26 b Fs(\000)c Fw(x)9 b Fx(/)14 b(\033)38 b FD(and)1185 2772 y Fw(g)1235 2787 y Fu(\033)1287 2772 y Fx(.)r Fw(z)5 b Fx(/)24 b FD(:)p Fs(D)1573 2702 y FD(e)o(xp)p Fs(f\000)p Fp(j)r Fw(z)5 b Fp(j)1933 2666 y Fz(2)1971 2702 y Fx(=)p FD([2)p Fx(.)p FD(1)20 b Fs(\000)f Fx(\033)13 b(/\033)g Fw(t)c FD(])p Fs(g)p 1573 2749 1008 4 v 1724 2843 a FD([2)p Fx(\031)e(.)p FD(1)20 b Fs(\000)f Fx(\033)13 b(/\033)g Fw(t)c FD(])2317 2814 y Ft(d)c Fu(=)p Fz(2)2607 2772 y Fx(:)649 b FD(\(5.18\))294 3056 y(Ne)o(xt)25 b(we)h(apply)f(H\366lder')-5 b(s)26 b(inequality)e(with)h(the)g (conjugated)g(e)o(xponents)36 b Fw(p)28 b Fs(2)p FD(]1)p Fx(;)14 b Fs(1)p FD([)26 b(and)305 3176 y Fw(p)358 3139 y Fr(0)414 3176 y FD(:)p Fs(D)k Fx(.)p FD(1)24 b Fs(\000)35 b Fw(p)827 3139 y Fr(\000)p Fz(1)922 3176 y Fx(/)959 3139 y Fr(\000)p Fz(1)1089 3176 y FD(to)h(the)g(inte)o(gral)g(with)f (respect)i(to)h Fw(z)i FD(in)c(\(5.17\),)j(which)d(yields)g(the)294 3295 y(upper)25 b(bound)467 3474 y Fq(\022)540 3479 y Fo(Z)593 3704 y Fn(R)646 3684 y Fl(d)682 3615 y FD(d)12 b Fw(z)29 b FD(e)856 3574 y Fr(\000)8 b Ft(p)r(t)g(V)1036 3548 y Fm(.!)q(/)1112 3574 y Fu(.)q Ft(z)t Fu(/)1215 3615 y FD(e)1259 3574 y Fr(\000)g Ft(p)r Fi(j)q Ft(z)t Fi(j)1443 3474 y Fq(\023)1516 3497 y Fz(1)p Fu(=)g Ft(p)1647 3474 y Fq(\022)1721 3479 y Fo(Z)1773 3704 y Fn(R)1826 3684 y Fl(d)1862 3615 y FD(d)k Fw(z)30 b FD(e)2045 3574 y Ft(p)2082 3548 y Fg(0)2100 3574 y Fi(j)q Ft(z)t Fi(j)2184 3530 y Fq(\014)2184 3590 y(\014)2220 3615 y Fw(g)2270 3630 y Fu(\033)2321 3534 y Fq(\000)2368 3615 y Fw(z)24 b Fs(\000)c Fw(m)2609 3630 y Ft(x)6 b Fu(;)t Ft(y)2713 3615 y Fx(.\033)13 b(/)2853 3534 y Fq(\001)2899 3530 y(\014)2899 3590 y(\014)2940 3557 y Ft(p)2977 3532 y Fg(0)3001 3474 y Fq(\023)3074 3497 y Fz(1)p Fu(=)8 b Ft(p)3187 3472 y Fg(0)3224 3615 y Fx(:)32 b FD(\(5.19\))294 3906 y(The)25 b(second)g(inte)o(gral)f(in)g(\(5.19\))h(is)f(bounded)g (from)h(abo)o(v)o(e)f(by)448 4192 y(e)500 4150 y Ft(p)537 4125 y Fg(0)565 4150 y Fz(max)q Fr(f)p Fi(j)r Ft(x)6 b Fi(j)p Fu(;)p Fi(j)t Ft(y)t Fi(j)p Fr(g)952 4056 y Fo(Z)1005 4281 y Fn(R)1058 4261 y Fl(d)1094 4192 y FD(d)12 b Fw(z)29 b FD(e)1276 4150 y Ft(p)1313 4125 y Fg(0)1331 4150 y Fi(j)q Ft(z)t Fi(j)1429 4192 y Fp(j)r Fw(g)1509 4207 y Fu(\033)1560 4192 y Fx(.)r Fw(z)5 b Fx(/)p Fp(j)1716 4150 y Ft(p)1753 4125 y Fg(0)1801 4192 y Fs(\024)25 b FD(e)1953 4150 y Ft(p)1990 4125 y Fg(0)2018 4150 y Fz(max)q Fr(f)p Fi(j)r Ft(x)6 b Fi(j)p Fu(;)p Fi(j)t Ft(y)t Fi(j)p Fr(g)2405 4192 y FD([)p Fx(.)p FD(1)20 b Fs(\000)f Fx(\033)13 b(/\033)g Fw(t)c FD(])2881 4150 y Fu(.)p Fz(1)p Fr(\000)f Ft(p)3041 4125 y Fg(0)3060 4150 y Fu(/)p Ft(d)d Fu(=)p Fz(2)3220 4192 y Fw(I)3261 4207 y Ft(p)3298 4186 y Fg(0)3337 4192 y Fx(;)3283 4388 y FD(\(5.20\))294 4627 y(where)32 b Fw(I)610 4642 y Ft(p)647 4621 y Fg(0)696 4627 y FD(:)p Fs(D)23 b Fx(.)p FD(2)p Fx(\031)10 b(/)1019 4590 y Fr(\000)p Ft(d)5 b Fu(=)p Fz(2)1200 4546 y Fo(R)1246 4661 y Fn(R)1299 4640 y Fl(d)1335 4627 y FD(d)10 b Fx(\020)26 b FD(e)1508 4590 y Fr(\000)8 b Ft(p)1607 4565 y Fg(0)1626 4590 y Fu(.)p Fi(j)p Fu(\020)h Fi(j)1739 4565 y Fk(2)1769 4590 y Fr(\000)p Fi(j)p Fu(\020)g Fi(j)1910 4536 y Fr(p)p 1968 4536 26 4 v 1968 4590 a Ft(t)e Fu(/=)p Fz(2)2117 4627 y Fx(<)24 b Fs(1)f FD(for)i(an)o(y)34 b Fw(p)2715 4590 y Fr(0)2763 4627 y Fx(>)24 b FD(1.)30 b(This)23 b(gi)n(v)o(es)g(the)294 4746 y(estimate)378 5032 y Fv(E)438 4891 y Fq(\024)547 5032 y FD(sup)499 5119 y Ft(x)6 b Fu(;)t Ft(y)t Fr(2)p Ff(K)749 5032 y Fp(j)p Fw(k)823 5047 y Ft(t)854 5032 y Fx(.)s Fw(x)j Fx(;)20 b Fw(y)6 b Fx(/)p Fp(j)1115 4891 y Fq(\025)1227 5032 y Fs(\024)60 b Fx(.)p FD(2)p Fx(\031)10 b Fw(t)f Fx(/)1593 4991 y Fr(\000)p Ft(d)c Fu(=)p Fz(2)1767 5032 y Fw(I)1812 4981 y Fz(1)p Fu(=)j Ft(p)1925 4956 y Fg(0)1808 5058 y Ft(p)1845 5038 y Fg(0)1963 4891 y Fq(\022)2052 5032 y FD(sup)2037 5119 y Ft(z)t Fr(2)p Ff(K)2220 5032 y FD(e)2264 4991 y Fi(j)q Ft(z)t Fi(j)2348 4891 y Fq(\023)2435 4896 y Fo(Z)2533 4923 y Fz(1)2488 5121 y(0)2573 5032 y FD(d)i Fx(\033)j FD([)p Fx(.)p FD(1)20 b Fs(\000)f Fx(\033)13 b(/\033)g Fw(t)c FD(])3175 4991 y Fr(\000)p Ft(d)c Fu(=.)p Fz(2)j Ft(p)r Fu(/)1645 5357 y Fs(\002)14 b Fv(E)1812 5187 y Fq(")1876 5217 y(\022)1949 5222 y Fo(Z)2002 5446 y Fn(R)2055 5426 y Fl(d)2091 5357 y FD(d)e Fw(z)29 b FD(e)2265 5316 y Fr(\000)8 b Ft(p)r(t)g(V)k Fu(.)q Ft(z)t Fu(/)2548 5357 y FD(e)2592 5316 y Fr(\000)c Ft(p)r Fi(j)q Ft(z)t Fi(j)2776 5217 y Fq(\023)2849 5240 y Fz(1)p Fu(=)g Ft(p)2967 5187 y Fq(#)3053 5357 y Fx(:)203 b FD(\(5.21\))p eop end %%Page: 38 38 TeXDict begin 38 37 bop 294 90 a FD(38)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)294 384 y FD(The)29 b(e)o(xpectation)f(v)n (alue)h(on)f(the)h(right-hand)f(side)h(of)g(\(5.21\))g(is)f(\002nite)h (for)g(an)o(y)40 b Fw(p)30 b Fx(>)d FD(1)i(by)294 503 y(Jensen')-5 b(s)28 b(inequality)-6 b(,)28 b(property)g(\()p Fh(L)p FD(\))i(and)e(Fubini')-5 b(s)28 b(theorem.)42 b(Therefore)29 b(\(5.16\))g(follo)n(ws)294 623 y(from)c(the)g (boundedness)e(of)i Fj(K)57 b FD(and)24 b(by)h(choosing)35 b Fw(p)28 b Fx(>)d FD(max)o Fs(f)p FD(1)p Fx(;)14 b Fw(d)7 b Fx(=)p FD(2)p Fs(g)p FD(.)p 2965 632 41 90 v 1474 1005 a(A)p FG(C)t(K)t(N)t(O)q(W)t(L)t(E)e(D)t(G)g(M)g(E)t(N)g(T)394 1200 y(It)17 b(is)g(a)g(pleasure)h(to)f(thank)g(V)-9 b(adim)17 b(K)m(ostrykin)g(and)g(Simone)h(W)-6 b(arzel)17 b(for)f(helpful)h(discussions)h(and)f(comments)294 1294 y(on)j(the)g(manuscript.)1629 1613 y FD(R)t FG(E)t(F)t(E)t(R)t(E)t(N)5 b(C)g(E)g(S)314 1808 y(1.)40 b(R.)20 b(J.)g(Adler)m(,)f(\223The)h (geometry)g(of)g(random)f(\002elds,)-6 b(\224)22 b(W)m(ile)o(y)-5 b(,)20 b(Chichester)m(,)h(1981.)314 1944 y(2.)40 b(M.)19 b(Aizenman)h(and)g(B.)f(Simon,)h(Bro)n(wnian)f(motion)g(and)h(Harnack)f (inequality)h(for)f(Schr\366dinger)g(operators,)414 2038 y FB(Commun.)g(Pur)m(e)h(Appl.)f(Math.)h Fy(35)g FG(\(1982\),)e (209\226273.)314 2174 y(3.)40 b(N.)22 b(I.)h(Akhiezer)g(and)g(I.)g(M.)g (Glazman,)h(\223Theory)e(of)h(linear)h(operators)e(in)h(Hilbert)g (space,)-6 b(\224)25 b(v)n(ol.)f(II,)e(Pitman,)414 2269 y(Boston,)e(1981.)314 2404 y(4.)40 b(J.)18 b(E.)g(A)-6 b(vron)17 b(and)h(I.)g(W)-7 b(.)18 b(Herbst,)f(Spectral)j(and)e (scattering)h(theory)f(of)g(Schr\366dinger)g(operators)g(related)h(to)f (the)414 2499 y(Stark)i(ef)n(fect,)g FB(Commun.)f(Math.)h(Phys.)g Fy(52)g FG(\(1977\),)e(239\226254.)314 2635 y(5.)40 b(R.)20 b(A.)f(Adams,)h(\223Sobole)n(v)h(spaces,)-6 b(\224)22 b(Academic,)f(Ne)n(w)e(Y)-9 b(ork,)19 b(1975.)314 2771 y(6.)40 b(M.)23 b(Sh.)h(Birman,)g(A)f(proof)g(of)g(the)h(Fredholm)g (trace)g(formula)f(as)i(an)f(application)h(of)e(a)h(simple)g(embedding) 414 2865 y(for)i(k)o(ernels)h(of)f(inte)o(gral)h(operators)g(of)f (trace)i(class)g(in)e(L)2070 2837 y Fe(2)2105 2865 y Fd(.)p Fc(R)2189 2837 y Fa(m)2244 2865 y Fd(/)p FG(,)g(Report)h (Lith-Mat-R-89-30,)e(Link\366ping)414 2960 y(Uni)n(v)o(ersity)-5 b(,)19 b(Sweden,)h(1989)g(\(unpublished\).)314 3096 y(7.)40 b(J.)31 b(Blank,)h(P)-9 b(.)33 b(Exner)e(and)g(M.)h(Ha)n(vl\355)1496 3095 y(\020)1492 3096 y(cek,)h(\223Hilbert)f(space)g(operators)g(in)f (quantum)h(physics,)-6 b(\224)32 b(American)414 3191 y(Institute)20 b(of)g(Physics,)g(Ne)n(w)f(Y)-9 b(ork,)19 b(1994.)314 3326 y(8.)40 b(C.)20 b(Brisla)o(wn,)f(K)n(ernels)h(of)g (trace)h(class)g(operators,)e FB(Pr)l(oc.)h(Amer)-9 b(.)20 b(Math.)g(Soc.)g Fy(104)g FG(\(1988\),)e(1181\2261190.)314 3462 y(9.)40 b(C.)20 b(Brisla)o(wn,)f(T)m(race)i(class)g(inte)o(gral)f (k)o(ernels,)h FB(Pr)l(oc.)f(Symp.)g(Pur)m(e)g(Math.)g Fy(51)p FG(,)f(Pt.)i(2)e(\(1990\),)g(61\22664.)274 3598 y(10.)40 b(K.)23 b(Broderix,)h(D.)f(Hundertmark)h(and)g(H.)f(Leschk)o (e,)i(Continuity)g(properties)f(of)f(Schr\366dinger)h(semigroups)414 3693 y(with)c(magnetic)h(\002elds,)f FB(Re)o(v)-6 b(.)21 b(Math.)f(Phys.)g Fy(12)f FG(\(2000\),)g(181\226225.)274 3828 y(11.)40 b(R.)26 b(Carmona,)h(Re)o(gularity)g(properties)g(of)e (Schr\366dinger)i(and)f(Dirichlet)h(semigroups,)f FB(J)n(.)h(Funct.)g (Anal.)f Fy(33)414 3923 y FG(\(1979\),)18 b(259\226296.)274 4059 y(12.)40 b(R.)c(Carmona)g(and)h(J.)f(Lacroix,)g(\223Spectral)i (theory)d(of)h(random)g(Schr\366dinger)f(operators,)-6 b(\224)37 b(Birkh\344user)m(,)414 4154 y(Boston,)20 b(1990.)274 4289 y(13.)40 b(Kai)17 b(Lai)f(Chung)h(and)f(Zhongxin)g(Zhao,)h (\223From)f(Bro)n(wnian)g(motion)h(to)g(Schr\366dinger')l(s)e (equation,)-6 b(\224)18 b(Springer)m(,)414 4384 y(Berlin,)i(1995.)274 4520 y(14.)40 b(H.)29 b(L.)h(Cycon,)g(R.)g(G.)f(Froese,)i(W)-7 b(.)29 b(Kirsch)h(and)g(B.)g(Simon,)g(Schr\366dinger)f(operators,)h (Springer)m(,)f(Berlin,)414 4615 y(1987.)274 4750 y(15.)40 b(E.)19 b(B.)h(Da)n(vies,)h(\223One-parameter)g(semigroups,)-6 b(\224)20 b(Academic,)h(London,)e(1980.)274 4886 y(16.)40 b(M.)24 b(D.)f(Donsk)o(er)g(and)h(S.)g(R.)g(S.)h(V)-9 b(aradhan,)23 b(Asymptotics)h(for)f(the)i(polaron,)e FB(Commun.)g(Pur)m(e)h(Appl.)f(Math.)414 4981 y Fy(36)d FG(\(1983\),)e(505\226528.)274 5117 y(17.)40 b(K.)34 b(Efeto)o(v)-5 b(,)34 b(\223Supersymmetry)h(in)f(disorder)g(and)h (chaos,)-6 b(\224)36 b(Cambridge)e(Uni)n(v)o(ersity)g(Press,)h (Cambridge,)414 5211 y(1997.)274 5347 y(18.)40 b(J.)23 b(Fr\366hlich,)g(Unbounded,)f(symmetric)i(semigroups)f(on)g(a)g (separable)h(Hilbert)f(space)i(are)e(essentially)i(self-)414 5442 y(adjoint,)20 b FB(Adv)-6 b(.)20 b(Appl.)g(Math.)g Fy(1)g FG(\(1980\),)e(237\226256.)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 603 90 a FG(INTEGRAL)18 b(KERNELS)h(FOR)h (UNBOUNDED)d(SCHR\326DINGER)g(SEMIGR)m(OUPS)209 b FD(39)274 384 y FG(19.)40 b(A.)19 b(Galindo)h(and)g(P)-9 b(.)21 b(P)o(ascual,)g(\223Quantum)f(mechanics)h(I,)-6 b(\224)21 b(Springer)m(,)e(Berlin,)h(1990.)274 521 y(20.)40 b(A.)19 b(Galindo)h(and)g(P)-9 b(.)21 b(P)o(ascual,)g(\223Quantum)f(mechanics)h (II,)-6 b(\224)20 b(Springer)m(,)f(Berlin,)i(1991.)274 659 y(21.)40 b(J.)17 b(G\344rtner)g(and)h(W)-7 b(.)17 b(K\366nig,)g(Moment)h(asymptotics)g(for)e(the)i(continuous)g (parabolic)g(Anderson)e(model,)i FB(Ann.)414 754 y(Appl.)h(Pr)l(obab)m (.)h Fy(10)f FG(\(2000\),)g(192\226217.)274 891 y(22.)40 b(J.)21 b(G\344rtner)m(,)f(W)-7 b(.)20 b(K\366nig)h(and)g(S.)g(A.)f (Molchano)o(v)-5 b(,)21 b(Almost)f(sure)h(asymptotics)h(for)e(the)h (continuous)g(parabolic)414 986 y(Anderson)e(model,)h FB(Pr)l(obab)m(.)f(Theory)i(Relat.)f(F)l(ields)h Fy(118)f FG(\(2000\),)e(547\226573.)274 1123 y(23.)40 b(A.)22 b(Gulisashvili)h(and)f(M.)g(A.)g(K)m(on,)g(Exact)h(smoothing)f (properties)g(of)g(Schr\366dinger)g(semigroups,)g FB(Amer)-9 b(.)22 b(J)n(.)414 1218 y(Math.)e Fy(118)g FG(\(1996\),)e (1215\2261248.)274 1356 y(24.)40 b(A.)g(M.)h(Hinz)g(and)g(G.)f(Stolz,)i (Polynomial)g(boundedness)f(of)f(eigensolutions)h(and)h(the)f(spectrum) g(of)414 1450 y(Schr\366dinger)20 b(operators,)f FB(Math.)h(Ann.)g Fy(294)f FG(\(1992\),)g(195\226211.)274 1588 y(25.)40 b(R.)17 b(J.)g(Hughes,)f(Semigroups)h(of)f(unbounded)g(linear)i (operators)e(in)h(Banach)h(space,)g FB(T)l(r)o(ans.)f(Amer)-9 b(.)16 b(Math.)h(Soc.)414 1683 y Fy(230)i FG(\(1977\),)g(113\226145.) 274 1820 y(26.)40 b(T)-6 b(.)17 b(Hupfer)m(,)e(H.)i(Leschk)o(e,)g(P)-9 b(.)18 b(M\374ller)f(and)g(S.)g(W)-6 b(arzel,)18 b(Existence)g(and)f (uniqueness)h(of)e(the)h(inte)o(grated)h(density)414 1915 y(of)25 b(states)h(for)f(Schr\366dinger)g(operators)g(with)g (magnetic)h(\002elds)h(and)e(unbounded)g(random)g(potentials,)h FB(Re)o(v)-6 b(.)414 2010 y(Math.)20 b(Phys.)g Fy(13)g FG(\(2001\),)e(1547\2261581.)274 2147 y(27.)40 b(K.)17 b(It\364)h(and)g(H.)g(P)-9 b(.)18 b(McK)n(ean,)h(Jr)l(.,)e(\223Dif)n (fusion)g(processes)i(and)f(their)g(sample)h(paths,)-6 b(\224)20 b(2nd)d(corrected)i(printing,)414 2242 y(Springer)m(,)g (Berlin,)h(1974.)274 2379 y(28.)40 b(T)-6 b(.)20 b(Kato,)f (Schr\366dinger)h(operators)g(with)f(singular)h(potentials,)h FB(Isr)o(ael)f(J)n(.)g(Math.)g Fy(13)g FG(\(1972\),)f(135\226148.)274 2517 y(29.)40 b(W)-7 b(.)24 b(Kirsch,)f(Random)h(Schr\366dinger)g (operators:)f(a)h(course,)g(in)g(\223Schr\366dinger)g(operators,)-6 b(\224)24 b(H.)g(Holden)f(and)414 2612 y(A.)c(Jensen,)i(eds.,)f (Lecture)g(Notes)g(in)g(Physics,)g(V)-10 b(ol.)19 b(345,)h(Springer)m (,)f(Berlin,)h(1989,)g(pp.)f(264\226370.)274 2749 y(30.)40 b(W)-7 b(.)22 b(Kirsch)g(and)g(F)-6 b(.)22 b(Martinelli,)i(On)d(the)i (essential)h(self)o(adjointness)f(of)f(stochastic)i(Schr\366dinger)e (operators,)414 2844 y FB(Duk)o(e)e(Math.)g(J)n(.)h Fy(50)e FG(\(1983\),)g(1255\2261260.)274 2982 y(31.)40 b(A.)22 b(Klein)i(and)f(L.)g(J.)g(Landau,)g(Construction)g(of)f(a)i(unique)f (self-adjoint)g(generator)g(for)f(a)i(symmetric)f(local)414 3076 y(semigroup,)c FB(J)n(.)i(Funct.)f(Anal.)f Fy(44)h FG(\(1981\),)e(121\226137.)274 3214 y(32.)40 b(H.)15 b(Leschk)o(e,)h(P)-9 b(.)16 b(M\374ller)g(and)g(S.)g(W)-6 b(arzel,)16 b(A)f(surv)o(e)o(y)g(of)g(rigorous)f(results)i(on)f(random) g(Schr\366dinger)h(operators)414 3309 y(for)23 b(amorphous)h(solids,)g FB(Mark)o(o)o(v)h(Pr)l(ocess.)g(Relat.)g(F)l(ields)g Fy(9)f FG(\(2003\),)f(729\226760.)g(F)o(or)h(a)g(slightly)h(e)o (xtended)414 3403 y(and)20 b(updated)g(v)o(ersion)g(see)h(the)f (e-print)f(mp_arc)h(03-536.)274 3541 y(33.)40 b(H.)18 b(Leschk)o(e)i(and)e(S.)h(W)-6 b(onneber)o(ger)m(,)18 b(Long-time)g(asymptotics)i(of)e(dif)n(fusion)f(in)i(random)f(media)i (and)f(related)414 3635 y(problems,)g FB(J)n(.)i(Phys.)f(A)f Fy(22)h FG(\(1989\),)e(L1009\226L1014.)g(Corrigendum:)i FB(ibid.)g Fy(23)f FG(\(1990\),)g(1475.)274 3773 y(34.)40 b(M.)20 b(A.)f(Lifshits,)h(\223Gaussian)g(random)g(functions,)-6 b(\224)20 b(Kluwer)m(,)f(Dordrecht,)g(1995.)274 3911 y(35.)40 b(A.)15 b(E.)g(Nussbaum,)h(Spectral)g(representation)h(of)e (certain)h(one-parametric)h(f)o(amilies)f(of)g(symmetric)g(operators) 414 4005 y(in)k(Hilbert)g(space,)h FB(T)l(r)o(ans.)e(Amer)-9 b(.)20 b(Math.)g(Soc.)g Fy(152)f FG(\(1970\),)g(419\226429.)274 4143 y(36.)40 b(L.)28 b(P)o(astur)h(and)f(A.)g(Figotin,)h(\223Spectra)h (of)e(random)h(and)f(almost-periodic)h(operators,)-6 b(\224)29 b(Springer)m(,)f(Berlin,)414 4238 y(1992.)274 4375 y(37.)40 b(P)-9 b(.)39 b(Protter)m(,)g(\223Stochastic)j(inte)o (gration)d(and)g(dif)n(ferential)h(equations:)g(a)f(ne)n(w)g(approach,) -6 b(\224)41 b(3rd)d(printing,)414 4470 y(Springer)m(,)19 b(Berlin,)h(1995.)274 4607 y(38.)40 b(M.)25 b(Reed)i(and)f(B.)g(Simon,) g(\223Methods)g(of)f(modern)g(mathematical)k(physics)c(I:)h(Functional) h(analysis,)-6 b(\224)27 b(re)n(v)-5 b(.)414 4702 y(and)20 b(enl.)g(ed.,)g(Academic,)h(San)g(Die)o(go,)e(1980.)274 4840 y(39.)40 b(B.)34 b(I.)g(Shklo)o(vskii)g(and)h(A.)e(L.)h(Efros,)f (\223Electronic)j(properties)e(of)f(doped)i(semiconductors,)-6 b(\224)35 b(Springer)m(,)414 4934 y(Berlin,)20 b(1984.)274 5072 y(40.)40 b(B.)20 b(Simon,)g(\223Functional)h(inte)o(gration)g(and) f(quantum)g(physics,)-6 b(\224)20 b(Academic,)h(Ne)n(w)f(Y)-9 b(ork,)19 b(1979.)274 5210 y(41.)40 b(B.)20 b(Simon,)g(\223T)m(race)h (ideals)g(and)f(their)g(applications,)-6 b(\224)22 b(Cambridge)e(Uni)n (v)o(ersity)g(Press,)g(Cambridge,)g(1979.)274 5347 y(42.)40 b(B.)27 b(Simon,)h(Schr\366dinger)f(semigroups,)h FB(Bull.)f(Amer)-9 b(.)27 b(Math.)h(Soc.)g(\(N.S.\))e Fy(7)i FG(\(1982\),)e(447\226526.)g (Erratum:)414 5442 y FB(ibid.)20 b Fy(11)f FG(\(1984\),)g(426.)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 294 90 a FD(40)843 b FG(BR)m(ODERIX,)18 b(LESCHKE)h(AND)g(M\334LLER)274 384 y(43.)40 b(B.)19 b(Simon,)g(A)g(Fe)o(ynman-Kac)h(formula)e(for)h(unbounded)f (semigroups,)h FB(Canadian)g(Math.)g(Soc.)g(Conf)o(.)g(Pr)l(oc.)414 478 y Fy(28)h FG(\(2000\),)e(317\226321.)274 613 y(44.)40 b(E.)18 b(M.)g(Stein,)h(\223Singular)h(inte)o(grals)e(and)h(dif)n (ferentiability)g(properties)f(of)g(functions,)-6 b(\224)19 b(Princeton)g(Uni)n(v)o(ersity)414 708 y(Press,)h(Princeton,)h(Ne)n(w)e (Jerse)o(y)-5 b(,)20 b(1970.)274 842 y(45.)40 b(M.)20 b(H.)g(Stone,)h(\223Linear)f(transformations)g(in)g(Hilbert)g(space,)-6 b(\224)23 b(7th)d(printing,)f(American)i(Mathematical)i(So-)414 937 y(ciety)-5 b(,)21 b(Pro)o(vidence,)f(Rhode)g(Island,)g(1970.)274 1071 y(46.)40 b(A.-S.)19 b(Sznitman,)i(\223Bro)n(wnian)f(motion,)g (obstacles)h(and)f(random)g(media,)-6 b(\224)21 b(Springer)m(,)f (Berlin,)g(1998.)274 1206 y(47.)40 b(W)-7 b(.)20 b(Thirring,)e (\223Quantum)i(mathematical)j(physics,)-6 b(\224)20 b(Springer)m(,)g (Berlin,)g(2002.)274 1341 y(48.)40 b(J.)18 b(V)-10 b(oigt,)18 b(Absorption)f(semigroups,)h(their)h(generators,)f(and)h (Schr\366dinger)f(semigroups,)g FB(J)n(.)g(Funct.)h(Anal.)f Fy(67)414 1435 y FG(\(1986\),)g(167\226205.)274 1570 y(49.)40 b(J.)20 b(W)-6 b(eidmann,)20 b(\223Linear)h(operators)e(in)h (Hilbert)g(space,)-6 b(\224)22 b(Springer)m(,)d(Berlin,)h(1980.)1552 1889 y FD(C)t FG(I)t(T)m(A)-5 b(T)t(I)t(O)t(N)34 b FD(I)t FG(N)t(D)t(E)t(X)321 2194 y([1])544 b(13)321 2288 y([2])584 b(4)321 2383 y([3])464 b(9,)20 b(27)321 2478 y([4])584 b(3)321 2572 y([5])544 b(30)321 2667 y([6])g(31)321 2762 y([7])265 b(2,)19 b(9,)h(26,)g(31)321 2856 y([8])544 b(31)321 2951 y([9])g(31)281 3046 y([10])125 b(2,)20 b(7,)g(10,)f(15,)h(22,)959 3140 y(23)281 3235 y([11])584 b(2)281 3330 y([12])225 b(3,)20 b(15,)f(17,)h(33)281 3424 y([13])584 b(2)281 3519 y([14])504 b(2,)20 b(3)281 3614 y([15])584 b(3)281 3709 y([16])544 b(17)281 3803 y([17])464 b(3,)20 b(15)1524 2194 y([18])463 b(3,)20 b(10)1524 2288 y([19])583 b(2)1524 2383 y([20])g(2)1524 2478 y([21])g(3)1524 2572 y([22])g(3)1524 2667 y([23])384 b(2,)19 b(4,)h(10)1524 2762 y([24])583 b(6)1524 2856 y([25])463 b(3,)20 b(10)1524 2951 y([26])424 b(14\22616)1524 3046 y([27])583 b(6)1524 3140 y([28])g(4)1524 3235 y([29])224 b(3,)20 b(13,)g(15,)f(17)1524 3330 y([30])543 b(33)1524 3424 y([31])463 b(3,)20 b(10)1524 3519 y([32])583 b(3)1524 3614 y([33])344 b(3,)20 b(15,)f(17)1524 3709 y([34])543 b(13)1524 3803 y([35])344 b(3,)20 b(10,)f(27)2766 2194 y([36])344 b(3,)20 b(15,)g(17)2766 2288 y([37])584 b(7)2766 2383 y([38])185 b(24,)20 b(25,)f(28,)h(30)2766 2478 y([39])464 b(3,)20 b(15)2766 2572 y([40])384 b(2,)20 b(7,)g(23)2766 2667 y([41])544 b(30)2766 2762 y([42])125 b(2,)20 b(3,)g(10,)f(11,)h (15,)3324 2856 y(28,)g(33)2766 2951 y([43])105 b(3,)20 b(6,)g(8,)f(9,)h(18,)g(24)2766 3046 y([44])544 b(31)2766 3140 y([45])344 b(9,)20 b(27,)g(28)2766 3235 y([46])165 b(2,)20 b(3,)f(7,)h(18,)g(22,)3444 3330 y(23)2766 3424 y([47])584 b(2)2766 3519 y([48])g(4)2766 3614 y([49])344 b(9,)20 b(22,)g(24)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0403080948370--