This is a multi-part message in MIME format. ---------------0303111137980 Content-Type: text/plain; name="03-106.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-106.comments" 3 figures ---------------0303111137980 Content-Type: text/plain; name="03-106.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-106.keywords" Schroedinger Operators, Magnetic Stark Resonances, Spectral Analysis ---------------0303111137980 Content-Type: application/postscript; name="mstark.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mstark.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: life-new.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -N0 -f life-new.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.03.11:1826 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fg(!)2742 5132 y Fr(:)1828 5755 y Fs(17)p eop %%Page: 18 18 18 17 bop 21 219 a Fh(Conclusion)21 439 y Fs(Theorem)32 b(5.2)f(giv)m(es)h(a)f(partial)f(generalisation)f(of)i(the)h(result)f (obtained)g(in)g([GM].)43 b(As)32 b(exp)s(ected,)21 596 y(the)39 b(fact)e(that)h(the)h(lo)m(w)m(er)f(b)s(ound)g(on)g(the)g (resonance)h(life-times)c(is)j(Gaussian)f(in)h Fr(F)3264 560 y Fk(\000)p Fx(1)3395 596 y Fs(and)h(not)21 752 y(exp)s(onen)m (tial)d(is)f(due)i(to)f(the)g(presence)j(of)c(the)i(magnetic)e (\014eld.)54 b(Ho)m(w)m(ev)m(er,)39 b(further)e(comparison)21 909 y(with)j(the)g(purely)g(electric)f(Stark)h(e\013ect)h(sho)m(ws)g(m) m(uc)m(h)f(larger)f(restriction)g(on)h(the)g(class)g(of)f(ad-)21 1065 y(missible)d(p)s(oten)m(tials,)j(in)f(particular)e(the)j (condition)e(on)h(the)h(Gaussian)e(deca)m(y)j(of)e Fr(V)21 b Fs(\()p Fr(x;)c(y)t Fs(\).)60 b(Let)21 1222 y(us)39 b(no)m(w)g(brie\015y)g(discuss)h(the)f(issue)g(of)f(Gaussian)g(v)m (ersus)i(exp)s(onen)m(tial)e(b)s(eha)m(viour.)62 b(As)39 b(follo)m(ws)21 1378 y(from)k(the)i(analysis)f(of)f(the)i(Stark)g (resonances,)k([Op])44 b([HaSi])g([Sig)o(],)k(the)c(exp)s(onen)m(tial)g (la)m(w)g(for)21 1535 y(the)33 b(resonan)m(t)h(states)f(is)f(in)g(that) h(case)g(directly)f(connected)i(with)f(the)g(exp)s(onen)m(tial)f(deca)m (y)i(of)e(the)21 1691 y(eigenfunctions)27 b(of)g(a)f(\\free")h (Hamiltonian,)e(i.e.)41 b(without)27 b(electric)f(\014eld.)42 b(If)27 b(w)m(e)h(supp)s(ose)g(that)f(the)21 1848 y(same)32 b(connection)g(exists)h(also)e(in)g(the)h(magnetic)f(case,)i(then)f (our)g(result)g(should)g(hold)f(whenev)m(er)21 2004 y(the)41 b(eigenfunctions)f(of)g Fr(H)8 b Fs(\(0\))41 b(=)g Fr(H)1414 2019 y Fp(L)1493 2004 y Fs(+)28 b Fr(V)21 b Fs(,)43 b(asso)s(ciated)d (with)g(the)h(discrete)h(sp)s(ectrum,)h(fall)38 b(o\013)21 2161 y(as)f(a)f(Gaussian.)53 b(Su\016cien)m(t)38 b(condition)c(for)i (the)h(latter)e(is)h(the)g(Gaussian)g(deca)m(y)i(of)d Fr(V)22 b Fs(\()p Fr(x;)17 b(y)t Fs(\),)36 b(see)21 2317 y([CN)q(],)g(whic)m(h)g(is)g(compatible)d(with)i(our)h(assumption)f(\() p Fr(b)p Fs(\).)53 b(Up)36 b(to)f(no)m(w,)i(the)f(optimal)d(condition) 21 2474 y(is)e(kno)m(wn)h(only)f(for)g(the)g(ground)h(state,)g(in)e (whic)m(h)i(case)g(a)f(sort)g(of)g(exp)s(onen)m(tial)f(deca)m(y)j(of)e Fr(V)21 b Fs(\()p Fr(x;)c(y)t Fs(\))21 2630 y(is)44 b(sho)m(wn)h(to)f (b)s(e)h(su\016cien)m(t)g(and)f(necessary)j(for)c(Gaussian)h(b)s(eha)m (viour)g(of)g(the)g(corresp)s(onding)21 2787 y(eigenfunctions)32 b(at)h(in\014nit)m(y)-8 b(,)32 b([Er].)167 2943 y(Suc)m(h)h(a)e (restriction)f(is)h(in)g(con)m(trast)h(with)f(the)g(non)h(magnetic)e (Sc)m(hr\177)-49 b(odinger)31 b(op)s(erator,)g(whose)21 3100 y(eigenfunctions)k(decrease)j(exp)s(onen)m(tially)c(in)h(the)g (classically)f(forbidden)h(region)g(indep)s(enden)m(tly)21 3256 y(on)i(the)g(rate)g(at)g(whic)m(h)g Fr(V)22 b Fs(\()p Fr(x;)17 b(y)t Fs(\))36 b(tends)i(to)f(zero)g(at)f(in\014nit)m(y)-8 b(.)56 b(This)37 b(migh)m(t)f(indicate)g(a)h(principal)21 3413 y(di\013erence)30 b(b)s(et)m(w)m(een)i(the)e(b)s(eha)m(viour)f(of) g(resonan)m(t)h(states)g(in)f(the)h(presence)h(resp)s(ectiv)m(ely)g (absence)21 3569 y(of)h(magnetic)g(\014eld.)21 3957 y Ft(A)161 b(Estimate)54 b(of)g Fc(k)p Fb(K)10 b Fa(\()p Fb(z)c Fa(;)24 b Fb(ib)p Fa(\))p Fc(k)21 4212 y Fs(Here)37 b(w)m(e)f(estimate)f(the)h(norm)f(of)g(eac)m(h)i(term)e(in)g(the)h (de\014nition)f(of)g Fr(K)7 b Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))37 b(separately)-8 b(.)53 b(Since)21 4369 y(the)33 b(calculations)e(are)h(often)h(analogous,)e(w)m(e)j(skip) f(the)g(details)e(in)h(man)m(y)h(places.)21 4664 y Fh(Norm)j(of)i Fr(M)10 b Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))21 4885 y(T)-8 b(erms)33 b Fq(k)p Fr(A)440 4900 y Fx(1)479 4885 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)g Fs(and)g Fq(k)p Fr(A)1025 4900 y Fx(5)1064 4885 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)p Fs(:)849 5141 y Fq(k)p Fr(A)972 5156 y Fx(1)1011 5141 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)84 b(\024)f(k)p Fr(V)22 b Fs(\()p Fr(ib)p Fs(\))p Fr(J)1788 5156 y Fx(1)1828 5141 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)2028 5156 y Fk(1)2103 5141 y Fq(k)p Fr(R)2227 5156 y Fx(1)2266 5141 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(kk)2633 5116 y Fs(~)2609 5141 y Fr(J)2663 5156 y Fx(1)2703 5141 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)1295 5323 y(\024)83 b(C)6 b(k)p Fr(V)22 b Fs(\()p Fr(ib)p Fs(\))p Fr(J)1846 5338 y Fx(1)1886 5323 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)2086 5338 y Fk(1)2161 5323 y Fq(k)p Fr(R)2285 5338 y Fx(1)2324 5323 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)890 b Fs(\(A.1\))1828 5755 y(18)p eop %%Page: 19 19 19 18 bop 21 219 a Fs(and)33 b(for)f Fr(F)46 b Fs(su\016cien)m(tly)33 b(small)687 475 y Fq(k)p Fr(V)21 b Fs(\()p Fr(ib)p Fs(\))p Fr(J)1019 490 y Fx(1)1059 475 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)1259 490 y Fk(1)1418 475 y Fs(=)86 b(sup)1578 560 y Fx(\()p Fp(x;y)r Fx(\))1746 475 y Fq(j)p Fr(V)21 b Fs(\()p Fr(x)h Fs(+)g Fr(ib;)17 b(y)t Fs(\))p Fq(jj)p Fr(J)2383 490 y Fk(\000)2441 475 y Fs(\()p Fr(x)23 b Fs(+)f Fr(ib)p Fs(\))p Fq(jj)p Fr(J)2877 490 y Fp(c)2911 475 y Fs(\()p Fr(y)t Fs(\))p Fq(j)1417 774 y(\024)84 b Fs(sup)1631 853 y Fp(x)1741 774 y Fq(j)p Fr(V)21 b Fs(\()p Fr(x)i Fs(+)f Fr(ib;)i Fs(^)-56 b Fr(y)s Fs(\))p Fq(j)2483 706 y Fr(e)2528 670 y Fx(2)p Fp(\015)t Fx(\()p Fp(x)p Fk(\000)p Fp(x)2765 679 y Fl(2)2800 670 y Fx(\))p 2306 751 704 4 v 2306 865 a Fs(\()p Fr(e)2389 836 y Fx(4)p Fp(\015)t Fx(\()p Fp(x)p Fk(\000)p Fp(x)2626 845 y Fl(2)2661 836 y Fx(\))2715 865 y Fs(+)22 b(1\))2899 812 y Fx(1)p Fp(=)p Fx(2)21 1072 y Fs(W)-8 b(e)33 b(estimate)f(this)g(term)g(as)h (max)o Fq(f)p Fr(a;)17 b(b;)g(c)p Fq(g)33 b Fs(where)g Fr(a)p Fs(,)g Fr(b)p Fs(,)g Fr(c)g Fs(are)261 1328 y Fr(a)83 b Fs(=)112 b(sup)554 1414 y Fk(j)p Fp(x)p Fk(j)p Fp(a)726 1984 y Fl(0)760 1975 y Fx(+)p Fp(\016)865 1890 y Fr(V)922 1905 y Fx(0)961 1890 y Fr(e)1006 1848 y Fk(\000)p Fp(\027)t(x)1140 1825 y Fl(2)1207 1890 y Fq(\024)28 b Fr(V)1369 1905 y Fx(0)1408 1890 y Fr(e)1453 1836 y Fk(\000)1588 1800 y Fj(\016)1616 1815 y Fl(0)1651 1779 y(2)p 1518 1821 237 3 v 1518 1879 a Fj(F)1565 1861 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))21 2196 y Fs(and)33 b Fr(\016)e Fs(=)d Fr(\016)432 2211 y Fx(0)472 2196 y Fr(F)549 2160 y Fk(\000)p Fx(\(1)p Fk(\000)p Fp(")p Fx(\))813 2196 y Fr(<)f(x)971 2211 y Fx(2)1011 2196 y Fs(.)44 b(This)32 b(leads)h(to)1190 2452 y Fq(k)p Fr(A)1313 2467 y Fx(1)1352 2452 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)c(\024)f(C)6 b Fr(e)1789 2398 y Fk(\000)1955 2371 y Fi(C)p 1854 2383 V 1854 2441 a Fj(F)1901 2423 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))2105 2452 y Fq(k)p Fr(R)2229 2467 y Fx(1)2269 2452 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)21 2708 y Fs(In)33 b(the)g(same)f(w)m(a)m(y)i(w) m(e)g(pro)m(v)m(e)g(the)f(estimate)e(for)i Fq(k)p Fr(A)1991 2723 y Fx(5)2030 2708 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)p Fs(.)21 3021 y(T)-8 b(erm)33 b Fq(k)p Fr(A)402 3036 y Fx(2)441 3021 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)p Fs(:)418 3277 y Fq(k)p Fr(A)541 3292 y Fx(2)581 3277 y Fs(\()p Fr(ib)p Fs(\))p Fq(k)83 b(\024)h Fr(F)14 b Fq(k)p Fs(\()p Fr(x)22 b Fs(+)g Fr(ib)p 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Fk(\000)2984 5380 y Fi(C)p 2884 5392 237 3 v 2884 5451 a Fj(F)2931 5433 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))1828 5755 y Fs(20)p eop %%Page: 21 21 21 20 bop 21 219 a Fs(for)32 b Fr(F)42 b Fq(!)27 b Fs(0)32 b(due)h(to)g(\(3.2\))f(and)g(similarly)d(in)j(other)h(cases.)45 b(Therefore)1109 348 y Fg(\015)1109 407 y(\015)1109 467 y(\015)1109 527 y(\015)1109 587 y(\015)1219 397 y Fx(5)1165 427 y Fg(X)1179 637 y Fp(i)p Fx(=1)1325 522 y Fr(J)1379 537 y Fp(i)1407 522 y Fs(\()p Fr(ib)p Fs(\))1581 497 y(~)1557 522 y Fr(J)1611 537 y Fp(i)1640 522 y Fs(\()p Fr(ib)p Fs(\))23 b Fq(\000)f Fs(1)1961 348 y Fg(\015)1961 407 y(\015)1961 467 y(\015)1961 527 y(\015)1961 587 y(\015)2016 651 y Fk(1)2119 522 y Fq(\024)28 b(C)6 b Fr(e)2327 468 y Fk(\000)2493 441 y Fi(C)p 2392 453 237 3 v 2392 511 a Fj(F)2439 493 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))21 822 y Fs(Finally)-8 b(,)728 978 y Fq(k)p Fr(M)10 b Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)29 b(\024)f(C)6 b Fr(e)1412 925 y Fk(\000)1578 897 y Fi(C)p 1477 909 V 1477 968 a Fj(F)1524 950 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))1745 978 y Fs(\()p Fq(k)p Fr(R)1907 993 y Fx(1)1946 978 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)24 b Fs(+)e Fq(k)p Fr(R)2485 993 y Fx(2)2524 978 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)23 b Fs(+)f(1\))21 1273 y Fh(Norm)36 b(of)i Fr(K)552 1288 y Fx(3)591 1273 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))39 b Fh(and)f Fr(K)1172 1288 y Fx(4)1212 1273 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))21 1494 y(T)-8 b(o)38 b(con)m(trol)f(the)h(op)s(erator) f(norm)g(w)m(e)i(will)d(use)i(alternativ)m(ely)f(the)h(Hilb)s(ert-Sc)m (hmidt)d(norm)i(and)21 1651 y(the)30 b(follo)m(wing)d(inequalit)m(y)h (for)h(the)h(norm)f(of)g(an)h(in)m(tegral)e(op)s(erator)h(whic)m(h)h (can)f(b)s(e)h(found)g(in)f([Ka)o(,)21 1807 y(p.)44 b(144])729 1985 y Fq(k)p Fr(A)p Fq(k)27 b(\024)h Fs(max)1232 1844 y Fg(\032)1307 1985 y Fs(sup)1359 2064 y Fd(x)1470 1849 y Fg(Z)1587 1985 y Fq(j)p Fr(A)p Fs(\()p Fh(x)p Fr(;)17 b Fh(x)1888 1944 y Fk(0)1911 1985 y 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2680 y Fk(0)1081 2716 y Fq(j)27 b(\025)h Fs(1)i Fy(and)g(let)h Fr(F)44 b Fy(b)-5 b(e)30 b(smal)5 b(l)29 b(enough.)43 b(Then)30 b(ther)-5 b(e)30 b(exist)g(some)g(strictly)21 2873 y(p)-5 b(ositive)34 b(c)-5 b(onstants)35 b Fr(G)878 2888 y Fx(0)917 2873 y Fr(;)e(!)t Fs(\()p Fr(z)t Fs(\))i Fy(and)g Fr(\033)t Fs(\()p Fr(z)t Fs(\))28 b Fq(\025)g Fs(1)35 b Fy(such)f(that)727 3129 y Fq(j)p Fr(@)811 3088 y Fp(n)806 3153 y(x;y)907 3129 y Fr(G)984 3144 y Fx(1)1023 3129 y Fs(\()p Fh(x)p Fr(;)17 b Fh(x)1223 3088 y Fk(0)1246 3129 y Fs(;)g Fr(z)t Fs(\))p Fq(j)28 b(\024)g Fr(G)1615 3144 y Fx(0)1671 3129 y Fr(\014)6 b Fs(\()p Fr(z)t Fs(\))1857 3088 y Fk(\000)p Fp(\033)r Fx(\()p Fp(z)s Fx(\))2066 3129 y Fr(e)2111 3088 y Fk(\000)p Fp(\014)s Fx(\()p Fp(z)s Fx(\))p Fk(j)p Fp(y)2356 3064 y Fi(0)2379 3088 y Fk(\000)p Fp(y)r Fk(j)2512 3129 y Fr(e)2557 3088 y Fk(\000)p Fp(!)r Fx(\()p Fp(z)s Fx(\)\()p Fp(x)2815 3064 y Fi(0)2838 3088 y Fk(\000)p Fp(x)p Fx(\))2960 3064 y Fl(2)2999 3129 y Fr(;)21 3385 y Fy(wher)-5 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1649 y(\034)3458 1678 y Fs(\(A.61\))167 1934 y(T)-8 b(aking)42 b(in)m(to)g(accoun)m(t)g(all)e(the)j(estimates)f (\(A.32\))o(,)j(\(A.58\),)f(\(A.61\))e(made)g(ab)s(o)m(v)m(e,)j(w)m(e)e (can)21 2091 y(claim)30 b(that)j(for)f Fr(F)46 b Fs(small)30 b(enough)328 2347 y Fq(k)p Fr(K)461 2362 y Fx(2)501 2347 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)28 b(\024)g(C)23 b Fr(F)1079 2306 y Fk(\000C)1195 2347 y Fr(\014)6 b Fs(\()p Fr(z)t Fs(\))1381 2306 y Fk(\000)p Fp(\033)r Fx(\()p Fp(z)s Fx(\))1591 2236 y Fg(\020)1650 2347 y Fr(e)1695 2306 y Fk(\000)1760 2272 y Fj(\014)s Fl(\()p Fj(z)r Fl(\))p 1760 2291 V 1776 2332 a Fj(F)1823 2318 y(\034)1913 2347 y Fs(+)22 b Fr(e)2056 2293 y Fk(\000)2222 2266 y Fi(C)p 2121 2278 237 3 v 2121 2336 a Fj(F)2168 2318 y Fl(2\(1)p Fi(\000)p Fj(")p Fl(\))2372 2236 y Fg(\021)2448 2347 y Fs(\(1)g(+)g Fq(k)p Fr(R)2779 2362 y Fx(2)2819 2347 y Fs(\()p Fr(z)t Fs(;)17 b Fr(ib)p Fs(\))p Fq(k)p Fs(\))308 b(\(A.62\))167 2603 y(Inequalit)m(y)44 b(\(A.62\))e(pla)m(ys)g(an)h 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b(hosts)g(at)f(Institute)h(for)f (Theoretical)g(Ph)m(ysics,)49 b(EPF)44 b(Lausanne)g(for)f(a)h(w)m(arm)f (hospitalit)m(y)21 4185 y(extended)c(to)e(him.)56 b(C.F.)37 b(thanks)i(the)e(Math.)58 b(departmen)m(t)37 b(at)g(Stuttgart)g(Univ)m (ersit)m(y)h(for)e(hos-)21 4341 y(pitalit)m(y)-8 b(,)32 b(where)j(part)f(of)f(the)h(presen)m(t)i(w)m(ork)f(w)m(as)f(done.)48 b(The)35 b(w)m(ork)g(of)e(C.F.)h(w)m(as)h(supp)s(orted)f(b)m(y)21 4498 y(the)f(F)-8 b(onds)33 b(National)d(Suisse)j(de)h(la)d(Rec)m(herc) m(he)k(Scien)m(ti\014que)e(No.)44 b(20-55694.98.)21 4885 y Ft(References)21 5140 y Fs([A)m(C])50 b(J.)k(Aguilar,)j(J.)d(M.)g (Com)m(b)s(es:)87 b(A)54 b(Class)g(of)f(Analytic)g(P)m(erturbations)h (for)f(One-b)s(o)s(dy)222 5297 y(Sc)m(hr\177)-49 b(odinger)32 b(Hamiltonians,)e Fy(Commun.)k(Math.)h(Phys.)d Fh(22)p Fs(,)h(\(1971\),)f(269-279.)1828 5755 y(34)p eop %%Page: 35 35 35 34 bop 21 219 a Fs([AH])49 b(J.)41 b(E.)g(Avron,)i(I.)e(W.)g (Herbst:)60 b(Sp)s(ectral)40 b(and)h(Scattering)f(Theory)i(of)e(Sc)m (hr\177)-49 b(odinger)41 b(Op-)222 375 y(erators)h(Related)g(to)g(the)g (Stark)h(E\013ect,)j Fy(Commun.)c(Math.)i(Phys.)f Fh(52)p Fs(,)h(\(1977\),)g(247-274.)21 795 y([BC])50 b(E.)41 b(Balslev,)h(J.)e(M.)h(Com)m(b)s(es:)60 b(Sp)s(ectral)40 b(Prop)s(erties)h(of)f(Man)m(y-)h(b)s(o)s(dy)g(Sc)m(hr\177)-49 b(odinger)41 b(Op-)222 951 y(erators)i(with)h(Dilatation-analytic)37 b(In)m(teractions,)47 b Fy(Commun.)d(Math.)h(Phys.)f Fh(22)p Fs(,)i(\(1971\),)222 1108 y(280-294.)21 1371 y([BCD])j(P)-8 b(.)31 b(Briet,)f(J.)g(M.)h(Com)m(b)s(es)g(and)f(P)-8 b(.)31 b(Duclos:)42 b(Sp)s(ectral)30 b(stabilit)m(y)e(under)j (tunneling,)f Fy(Com-)222 1527 y(mun.)k(Math.)h(Phys.)e Fh(126)p Fs(,)g(\(1989\),)e(133-156.)21 1790 y([BG])49 b(F.)26 b(Ben)m(tosela)g(and)g(V.)h(Grecc)m(hi:)40 b(Stark)27 b(W)-8 b(annier)25 b(Ladders,)k Fy(Commun.)f(Math.)h(Phys.)d Fh(142)p Fs(,)222 1947 y(\(1991\),)31 b(169-192.)21 2210 y([CN])50 b(H.)25 b(D.)f(Cornean,)j(G.)e(Nenciu:)40 b(On)25 b(Eigenfunction)f(Deca)m(y)i(of)e(Tw)m(o)i(Dimensional)c(Magnetic)222 2367 y(Sc)m(hr\177)-49 b(odinger)32 b(Op)s(erators,)h Fy(Commun.)h(Math.)h(Phys.)d Fh(192)p Fs(,)h(\(1998\),)f(671-685.)21 2630 y([Er])49 b(L.)35 b(Erd\177)-49 b(os:)47 b(Gaussian)34 b(deca)m(y)i(of)e(the)g(Magnetic)g(Eigenfunctions,)h Fy(Ge)-5 b(om.)36 b(F)-7 b(unct.)36 b(A)n(nal.)d Fh(6,)222 2786 y(No.)k(2)32 b Fs(,)h(\(1996\),)f(231-248.)21 3049 y([FK])49 b(C.)42 b(F)-8 b(errari)41 b(and)h(H.)g(Ko)m(v)-5 b(a)g(\024)-43 b(r)-11 b(\023)-38 b(\020k:)61 b(Exp)s(onen)m(tial)41 b(Deca)m(y)i(for)e(Magnetic)h(Stark)h(Resonances.)222 3206 y Fy(mp-ar)-5 b(c/03-31)p Fs(.)21 3469 y([FM])49 b(C.)c(F)-8 b(errari)42 b(and)i(N.)g(Macris:)66 b(In)m(termixture)44 b(of)g(Extended)i(Edge)e(and)g(Lo)s(calized)f(Bulk)222 3625 y(Energy)h(Lev)m(els)g(in)f(Macroscopic)h(Hall)d(Systems,)p Fy(J.)46 b(Phys.)f(A:)g(Math.)g(Gen.)e Fh(35)p Fs(,)j(\(2002\),)222 3782 y(6339-6358.)21 4045 y([GR])i(I.)25 b(S.)g(Gradsh)m(tein,)h(I.)f (M.)g(Ryzhik:)40 b Fy(T)-7 b(ables)25 b(of)j(Inte)-5 b(gr)g(als,)28 b(Series,)g(and)f(Pr)-5 b(o)g(ducts)p Fs(,)26 b(Academic)222 4202 y(Press,)34 b(New)f(Y)-8 b(ork)33 b(1980.)21 4465 y([GM])49 b(S.)38 b(Gyger)f(and)h(Ph.)g(A.)f (Martin:)52 b(Lifetimes)36 b(of)h(Impurit)m(y)g(States)h(in)f(Crossed)i (Magnetic)222 4621 y(and)32 b(Electric)g(Fields,)g Fy(J.)j(Math.)g (Phys.)d Fh(40)p Fs(,)h(\(1999\),)f(3275-3282.)21 4884 y([HaSi])48 b(E.)54 b(Harrell)d(and)i(B.)g(Simon:)82 b(The)54 b(Mathematical)d(Theory)j(of)e(Resonances)i(Whose)222 5041 y(Widths)32 b(Are)h(Exp)s(onen)m(tially)f(Small,)e Fy(Duke)k(Math.)h(J.)e Fh(47)p Fs(,)g(\(1980\),)e(845-902.)21 5304 y([HS])49 b(P)-8 b(.)41 b(D.)f(Hislop,)h(I.)g(M.)g(Sigal)d(:)59 b Fy(Intr)-5 b(o)g(duction)41 b(to)i(Sp)-5 b(e)g(ctr)g(al)41 b(The)-5 b(ory,)42 b Fs(Springer,)g(New)f(Y)-8 b(ork)222 5461 y(1996.)1828 5755 y(35)p eop %%Page: 36 36 36 35 bop 21 219 a Fs([Ka])49 b(T.)33 b(Kato:)43 b Fy(Perturb)-5 b(ation)35 b(The)-5 b(ory)34 b(for)h(Line)-5 b(ar)34 b(Op)-5 b(er)g(ators,)32 b Fs(Springer,)h(Heidelb)s(erg)e(1966.)21 482 y([MR])49 b(M.)36 b(Melgaard)e(and)h(G.)g(Rozen)m(blum:)47 b(Eigen)m(v)-5 b(alue)35 b(Asymptotics)g(for)f(Ev)m(en-Dimensional)222 638 y(P)m(erturb)s(ed)48 b(Dirac)d(and)i(Sc)m(h\177)-49 b(odinger)47 b(Op)s(erators)g(with)g(Constan)m(t)h(Magnetic)e(Field,)j Fy(mp-)222 795 y(ar)-5 b(c/02-140)p Fs(.)21 1058 y([Op])49 b(R.)26 b(Opp)s(enheimer:)40 b(Three)27 b(Notes)g(on)f(the)g(Quan)m (tum)g(Theory)h(of)e(Ap)s(erio)s(dic)g(E\013ects,)j Fy(Phys.)222 1214 y(R)-5 b(ev.)32 b Fh(31)p Fs(,)h(\(1928\),)e(66-81.)21 1477 y([Ra])49 b(G.)34 b(D.)g(Raik)m(o)m(v:)47 b(Eigen)m(v)-5 b(alue)34 b(Asymptotics)g(for)g(the)h(Sc)m(hr\177)-49 b(odinger)35 b(Op)s(erator)f(with)g(Homo-)222 1634 y(geneous)i (Magnetic)f(P)m(oten)m(tial)f(and)i(Decreasing)f(Electric)f(P)m(oten)m (tial,)i Fy(Comm.)g(Part.)h(Di\013.)222 1790 y(Eq.)p Fs(,)32 b Fh(1990)p Fs(,)h(15)f(no.)h(3,)f(407-434.)21 2054 y([RS])49 b(M.)36 b(Reed)h(and)e(B.)h(Simon:)49 b Fy(Metho)-5 b(ds)37 b(of)h(Mo)-5 b(dern)37 b(Mathematic)-5 b(al)38 b(Physics,)g(I.)f(F)-7 b(unctional)222 2210 y(A)n(nalysis,)43 b(II.)f(F)-7 b(ourier)41 b(A)n(nalysis,)j(Self-A)-5 b(djointness,)42 b(IV.)h(A)n(nalysis)e(of)i(Op)-5 b(er)g(ators,)42 b Fs(Aca-)222 2367 y(demic)31 b(Press,)k(New)e(Y)-8 b(ork,)33 b(1972,)f(1975,)f (1978.)21 2630 y([Sig])48 b(I.)66 b(M.)f(Sigal:)107 b(Geometric)64 b(Theory)j(of)e(Stark)g(Resonances)i(in)e(Multielectron)f(Sys-)222 2786 y(tems,)p Fy(Commun.)34 b(Math.)h(Phys.)d Fh(119)p Fs(,)h(\(1988\),)f(287-314.)21 3049 y([Ti])49 b(E.)30 b(C.)h(Titc)m(hmarsh:)42 b Fy(Eigenfunction)31 b(Exp)-5 b(ansions)31 b(Asso)-5 b(ciate)g(d)33 b(with)f(Se)-5 b(c)g(ond)31 b(Or)-5 b(der)32 b(Di\013er-)222 3206 y(ential)i (Equations,)e Fs(Oxford,)h(Oxford)g(Univ)m(ersit)m(y)g(Press)h(1958.) 1828 5755 y(36)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303111137980--