Content-Type: multipart/mixed; boundary="-------------0309061749134" This is a multi-part message in MIME format. ---------------0309061749134 Content-Type: text/plain; name="03-405.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-405.keywords" Dirac operator, long-range potential, absolutely continuous spectrum ---------------0309061749134 Content-Type: application/postscript; name="dirac2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dirac2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: dirac2.dvi %%CreationDate: Sat Sep 06 15:40:07 2003 %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\PROGRA~1\WINEDT\MIKTEX\BIN\DVIPS.EXE dirac2 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.06:1539 %%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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2551 y Fl(\023)1264 2668 y Fo(;)42 b(\033)1376 2680 y Fk(1)1436 2668 y Fq(=)1524 2551 y Fl(\022)1627 2617 y Fq(0)82 b(1)1627 2717 y(1)g(0)1834 2551 y Fl(\023)1909 2668 y Fo(;)42 b(\033)2021 2680 y Fk(2)2081 2668 y Fq(=)2169 2551 y Fl(\022)2272 2617 y Fq(0)82 b Fn(\000)p Fo(i)2278 2717 y(i)115 b Fq(0)2531 2551 y Fl(\023)2606 2668 y Fo(;)41 b(\033)2717 2680 y Fk(3)2778 2668 y Fq(=)2866 2551 y Fl(\022)2968 2617 y Fq(1)115 b(0)2968 2717 y(0)83 b Fn(\000)p Fq(1)3241 2551 y Fl(\023)456 2864 y Fq(Matrices)29 b Fo(\033)845 2876 y Fm(j)933 2864 y Fq(are)g(called)h(the)g(P)n(auli)f (matrices.)44 b(Denote)31 b(the)f(exterior)f(of)h(the)g(unit)h(ball)f (in)456 2964 y Fj(R)510 2934 y Fk(3)577 2964 y Fq(b)n(y)24 b(\012.)36 b(W)-7 b(e)25 b(use)f(a)g(notation)g(\006)h(for)e(the)i (unit)g(sphere)f(in)h Fj(R)2407 2934 y Fk(3)2450 2964 y Fq(.)36 b(Consider)23 b Fo(H)31 b Fq(in)25 b(the)g(Hilb)r(ert)456 3063 y(space)31 b([)p Fo(L)762 3033 y Fk(2)799 3063 y Fq(\(\012\)])946 3033 y Fk(4)984 3063 y Fq(.)52 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3812 y(0)82 b(0)h Fn(\000)p Fq(1)114 b(0)1759 3911 y(0)82 b(0)115 b(0)g Fn(\000)p Fq(1)2345 3546 y Fl(3)2345 3692 y(7)2345 3742 y(7)2345 3795 y(5)456 4059 y Fq(The)27 b(main)h(result)f(of)h(the) g(pap)r(er)f(is)g(the)h(follo)n(wing)f(statemen)n(t.)555 4158 y Fv(Theorem)37 b(3.1)p Fq(.)51 b Fh(Assume)34 b Fo(V)19 b Fq(\()p Fo(x)p Fq(\))33 b(=)e Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)t Fh(,)38 b(wher)l(e)d Fo(v)s Fq(\()p Fo(x)p Fq(\))h Fh(is)e(a)h(r)l(e)l(al-value)l(d,)i(uniformly) 456 4258 y(b)l(ounde)l(d,)30 b(sc)l(alar)h(function)f(satisfying)h(the) f(fol)t(lowing)i(c)l(ondition)1658 4345 y Fl(Z)1616 4583 y Fm(x)p Fg(2)p Fk(\012)1829 4401 y Fo(v)1872 4371 y Fk(2)1909 4401 y Fq(\()p Fo(x)p Fq(\))p 1788 4438 274 4 v 1788 4515 a Fn(j)p Fo(x)p Fn(j)1881 4491 y Fk(2)1937 4515 y Fq(+)18 b(1)2095 4458 y Fo(<)23 b Fn(1)1072 b Fq(\(2\))456 4717 y Fh(Then,)30 b Fo(\033)744 4729 y Fm(ac)815 4717 y Fq(\()p Fi(H)q Fq(\))22 b(=)h Fj(R)p Fh(.)555 4817 y Fq(This)k(result)g(can)f(b)r(e)i(regarded)d(as)h(a)g (PDE)h(\(P)n(artial)e(Di\013eren)n(tial)i(Equations\))f(analog)g(of)456 4917 y(the)k(celebrated)g(results)g(b)n(y)g(Szeg\177)-42 b(o)30 b(on)g(p)r(olynomials)g(orthogonal)e(on)i(the)h(unit)g(circle)f (with)456 5016 y Fo(`)491 4986 y Fk(2)557 5016 y Fq(V)-7 b(erblunsky)30 b(parameters)e([17)o(,)j(19)o(].)44 b(If)30 b(one)g(considers)e(the)j(p)r(o)n(w)n(er-deca)n(ying)c(p)r(oten)n (tials)456 5116 y Fo(v)s Fq(\()p Fo(x)p Fq(\))32 b(:)f Fn(j)p Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fn(j)i(\024)d Fo(C)6 b Fq(\()p Fn(j)p Fo(x)p Fn(j)23 b Fq(+)e(1\))1397 5086 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)p Fg(\000)p Fm(")1622 5116 y Fq(,)34 b(then)f(condition)f(\(2\))h(is)f(satis\014ed)g(for)g (an)n(y)g Fo(")f(>)f Fq(0.)51 b(In)456 5216 y(this)27 b(case,)g(w)n(e)h(also)e(obtain)h(an)h(asymptotics)e(for)i(the)g (Green's)f(function.)1933 5315 y Fk(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fk(2)1117 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)555 450 y Fq(F)-7 b(or)30 b(the)g(massless)f(Dirac)g(op)r(erator,)g (theorem)h(3.1)f(solv)n(es)f(Simon's)i(conjecture)g([18)o(])g(for)456 550 y(Sc)n(hr\177)-42 b(odinger)27 b(op)r(erators.)42 b(Under)29 b(more)g(conditions)h(on)f Fo(V)19 b Fq(,)30 b(the)g(sp)r(ectrum)g(of)g Fi(H)f Fq(is)h(purely)456 649 y(a.c.)k(on)22 b Fj(R)27 b Fq([22].)35 b(One)21 b(can)h(easily)f (construct)g(an)g(example)h(when)g(conditions)f(of)h(the)g(theorem)456 749 y(3.1)37 b(are)g(satis\014ed)g(and)h(the)g(ric)n(h)g(singular)e(sp) r(ectrum)i(o)r(ccurs.)67 b(In)38 b(the)h(one-dimensional)456 849 y(case,)24 b(the)h(\014rst)f(result)g(on)g(the)h(presence)e(of)i (a.c.)35 b(sp)r(ectrum)25 b(for)e(slo)n(wly)h(deca)n(ying)f(p)r(oten)n (tials)456 948 y(is)41 b(due)g(to)g(M.)g(Krein)g([9)o(].)78 b(See)41 b(also)f([4)o(,)i(12)o(,)f(6)o(])h(for)e(the)i(mo)r(dern)f (dev)n(elopmen)n(t.)76 b(The)456 1048 y(existence)38 b(of)g(w)n(a)n(v)n(e)f(op)r(erators)f(for)i(the)h(one-dimensional)e (Dirac)h(op)r(erator)f(with)i(square)456 1147 y(summable)32 b(p)r(oten)n(tial)g(w)n(as)f(pro)n(v)n(ed)f(in)j([7)o(].)51 b(An)32 b(in)n(teresting)g(pap)r(er)f([10)o(])i(discusses)e(Szeg\177) -42 b(o-)456 1247 y(t)n(yp)r(e)37 b(inequalities)f(for)g(Sc)n(hr\177) -42 b(odinger)35 b(op)r(erators)g(with)i(short-range)d(p)r(oten)n (tials.)64 b(W)-7 b(e)37 b(will)456 1347 y(use)26 b(some)g(ideas)g (from)g([10)o(].)37 b(F)-7 b(or)26 b(discrete)g(m)n(ultidimensional)g (Sc)n(hr\177)-42 b(odinger)25 b(op)r(erator)g(with)456 1446 y(random)h(slo)n(w)h(deca)n(y)-7 b(,)27 b(the)h(existence)f(of)g (w)n(a)n(v)n(e)f(op)r(erators)g(w)n(as)h(pro)n(v)n(ed)f(b)n(y)h (Bourgain)f([3)o(].)555 1546 y(The)h(structure)f(of)h(the)g(pap)r(er)f (is)h(as)f(follo)n(ws.)35 b(In)27 b(the)h(second)e(section,)g(w)n(e)h (consider)e(one-)456 1646 y(dimensional)i(Dirac)g(systems)g(with)h (matrix-v)-5 b(alued)27 b(p)r(oten)n(tials.)36 b(Then,)28 b(in)g(the)g(third)g(part,)456 1745 y(w)n(e)f(deal)g(with)h(a)f(m)n (ultidimensional)h(Dirac)f(op)r(erator.)555 1847 y(The)g(follo)n(wing)f (notations)g(will)h(b)r(e)g(used.)37 b Fn(j)p Fo(M)9 b Fn(j)23 b Fq(=)2175 1776 y Fn(p)p 2244 1776 218 4 v 71 x Fo(M)2334 1823 y Fg(\003)2372 1847 y Fo(M)36 b Fq(denotes)26 b(the)h(absolute)g(v)-5 b(alue)456 1946 y(of)19 b(matrix)g Fo(M)9 b Fq(,)21 b(sym)n(b)r(ol)e(1)g(will)h(often)g(stand)f(for)g(the) h(iden)n(tit)n(y)g(matrix)f(or)g(op)r(erator.)32 b(As)20 b(usual,)456 2046 y Fo(C)521 2016 y Fg(1)515 2066 y Fk(0)591 2046 y Fq(\()p Fj(R)677 2016 y Fk(+)739 2046 y Fq(\))j(denotes)f(the)h (space)f(of)h(in\014nitely)g(smo)r(oth)g(functions)g(\(or)f(v)n (ector-functions\))g(with)456 2146 y(the)35 b(compact)f(supp)r(ort)g (inside)g(\(0)p Fo(;)14 b Fn(1)p Fq(\).)58 b Fo(\037)1860 2158 y Fm(K)1924 2146 y Fq(\()p Fo(x)p Fq(\))36 b(denotes)e(the)h(c)n (haracteristic)d(function)k(of)456 2245 y(the)28 b(set)f Fo(K)6 b Fq(.)37 b Fn(h)p Fo(x;)14 b(y)s Fn(i)28 b Fq(stands)f(for)g (the)h(inner)g(pro)r(duct)f(of)h(t)n(w)n(o)e(v)n(ectors)h Fo(x)h Fq(and)f Fo(y)j Fq(in)e Fj(R)3130 2215 y Fk(3)3174 2245 y Fq(.)550 2454 y(2.)41 b Fp(One-dimensional)31 b(Dira)n(c)h(opera)-6 b(tor)32 b(with)g(ma)-6 b(trix-v)f(alued)31 b(potential)555 2603 y Fq(In)k(this)f(section,)h(w)n(e)f(study)g(the)h (one-dimensional)e(case.)55 b(Let)34 b(us)g(consider)f(the)i(Dirac)456 2703 y(op)r(erator)25 b(in)j(the)g(follo)n(wing)f(form)1071 2945 y Fo(D)e Fq(=)1253 2828 y Fl(\024)1401 2894 y Fq(0)146 b Fn(\000)p Fo(d=dr)1338 2994 y(d=dr)181 b Fq(0)1863 2828 y Fl(\025)1925 2945 y Fq(+)18 b Fo(V)5 b(;)97 b(V)42 b Fq(=)2358 2828 y Fl(\024)2448 2894 y Fn(\000)p Fo(b)86 b Fn(\000)p Fo(a)2444 2994 y Fn(\000)p Fo(a)119 b(b)2785 2828 y Fl(\025)3338 2945 y Fq(\(3\))456 3125 y(where)39 b Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(b)p Fq(\()p Fo(r)r Fq(\))41 b(are)e Fo(m)26 b Fn(\002)g Fo(m)40 b Fq(self-adjoin)n(t)f(matrices)g(with)h(lo)r(cally)f(in)n(tegrable)f(en) n(tries.)456 3225 y(This)31 b(form)h(of)f(a)h(general)e(Dirac)h(op)r (erator)f(is)i(called)f(a)g(canonical)g(form)g(\([11)o(],)i (pp.48{50\).)456 3329 y(The)d(elemen)n(ts)h(of)g(the)g(Hilb)r(ert)g (space)f(are)g Fo(f)36 b Fq(=)28 b(\()p Fo(f)2115 3341 y Fk(1)2152 3329 y Fo(;)14 b(f)2230 3341 y Fk(2)2267 3329 y Fq(\))2299 3299 y Fm(t)2360 3329 y Fq(with)31 b Fo(f)2593 3344 y Fk(1\(2\))2743 3329 y Fn(2)2826 3262 y Fl(\002)2861 3329 y Fo(L)2918 3299 y Fk(2)2955 3329 y Fq(\()p Fj(R)3041 3299 y Fk(+)3102 3329 y Fq(\))3134 3262 y Fl(\003)3169 3279 y Fm(m)3232 3329 y Fq(.)46 b(The)456 3429 y(b)r(oundary)31 b(condition)i Fo(f)1245 3441 y Fk(2)1282 3429 y Fq(\(0\))e(=)g(0)h(de\014nes)h(the)g(self-adjoin)n(t)f (op)r(erator)f Fi(D)p Fq(.)52 b(W)-7 b(e)33 b(start)f(with)456 3529 y(an)27 b(elemen)n(tary)f(sp)r(ectral)i(theory)e(of)i Fi(D)p Fq(.)37 b(Consider)26 b(solutions)h(of)h(the)g(follo)n(wing)e (equation)1013 3722 y Fo(D)1098 3605 y Fl(\024)1185 3672 y Fq(\010)1183 3771 y(\011)1289 3605 y Fl(\025)1356 3722 y Fq(=)d Fo(\025)1506 3605 y Fl(\024)1593 3672 y Fq(\010)1591 3771 y(\011)1697 3605 y Fl(\025)1755 3722 y Fo(;)14 b Fq(\010\(0)p Fo(;)g(\025)p Fq(\))23 b(=)g(1)p Fo(;)14 b Fq(\011\(0)p Fo(;)g(\025)p Fq(\))22 b(=)h(0)p Fo(;)14 b(\025)23 b Fn(2)g Fj(C)456 3916 y Fq(F)-7 b(or)27 b(an)n(y)f Fo(f)9 b Fq(,)28 b(consider)e(the)i(generalized)e(F)-7 b(ourier)27 b(transform)1107 4152 y Fo(F)12 b Fq(\()p Fo(\025)p Fq(\))24 b(=)1423 4022 y Fg(1)1396 4039 y Fl(Z)1403 4274 y Fk(0)1493 4152 y Fq(\010)1553 4117 y Fg(\003)1591 4152 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(f)1816 4164 y Fk(1)1854 4152 y Fq(\()p Fo(r)r Fq(\))p Fo(dr)23 b Fq(+)2170 4022 y Fg(1)2143 4039 y Fl(Z)2149 4274 y Fk(0)2240 4152 y Fq(\011)2305 4117 y Fg(\003)2342 4152 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(f)2567 4164 y Fk(2)2605 4152 y Fq(\()p Fo(r)r Fq(\))p Fo(dr)456 4402 y Fq(There)28 b(exists)h(non-decreasing)e Fo(m)20 b Fn(\002)f Fo(m)29 b Fq(matrix-function)g Fo(\033)s Fq(\()p Fo(\025)p Fq(\))p Fo(;)g(\025)e Fn(2)f Fj(R)35 b Fq(\(sp)r(ectral)29 b(matrix-)456 4502 y(v)-5 b(alued)27 b(measure\))g(suc)n(h)g(that)h (\([14],)f(p.106\))1104 4613 y Fg(1)1077 4629 y Fl(Z)1084 4865 y Fk(0)1174 4675 y Fl(\000)1212 4742 y Fn(j)p Fo(f)1276 4754 y Fk(1)1313 4742 y Fq(\()p Fo(r)r Fq(\))p Fn(j)1439 4708 y Fk(2)1496 4742 y Fq(+)18 b Fn(j)p Fo(f)1643 4754 y Fk(2)1680 4742 y Fq(\()p Fo(r)r Fq(\))p Fn(j)1806 4708 y Fk(2)1845 4675 y Fl(\001)1897 4742 y Fo(dr)26 b Fq(=)2135 4613 y Fg(1)2108 4629 y Fl(Z)2073 4863 y Fg(\0001)2222 4742 y Fo(F)2287 4708 y Fg(\003)2325 4742 y Fq(\()p Fo(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\))p Fo(F)12 b Fq(\()p Fo(\025)p Fq(\))519 b(\(4\))1707 4983 y Fg(1)1680 4999 y Fl(Z)1644 5234 y Fg(\0001)1816 5056 y Fo(d\033)s Fq(\()p Fo(\025)p Fq(\))p 1805 5093 229 4 v 1805 5169 a(1)18 b(+)g Fo(\025)1996 5145 y Fk(2)2066 5112 y Fo(<)23 b Fn(1)1101 b Fq(\(5\))p eop %%Page: 3 3 3 2 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)254 b(3)456 450 y Fq(The)27 b(resolv)n(en)n(t)f(k)n(ernel)h Fo(R)1283 462 y Fm(z)1321 450 y Fq(\()p Fo(r)n(;)14 b(s)p Fq(\))28 b(of)g(op)r(erator)e Fi(D)h Fq(has)g(the)h(follo)n(wing)f (form)705 687 y Fo(R)768 699 y Fm(z)806 687 y Fq(\()p Fo(r)n(;)14 b(s)p Fq(\))24 b(=)1137 558 y Fg(1)1110 574 y Fl(Z)1075 809 y Fg(\0001)1310 631 y Fq(1)p 1234 668 193 4 v 1234 744 a Fo(\025)19 b Fn(\000)f Fo(z)1450 570 y Fl(\024)1538 637 y Fq(\010\()p Fo(r)n(;)c(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\)\010)2047 606 y Fg(\003)2088 637 y Fq(\()p Fo(s;)g(\025)p Fq(\))88 b(\010\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\)\011)2878 606 y Fg(\003)2919 637 y Fq(\()p Fo(s;)g(\025)p Fq(\))1536 736 y(\011\()p Fo(r)n(;)g(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\)\010)2050 706 y Fg(\003)2090 736 y Fq(\()p Fo(s;)g(\025)p Fq(\))84 b(\011\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\)\011)2881 706 y Fg(\003)2921 736 y Fq(\()p Fo(s;)g(\025)p Fq(\))3151 570 y Fl(\025)3338 687 y Fq(\(6\))456 935 y(where)27 b(the)h(in)n(tegral)e(is)h(understo)r(o)r(d)h(in)g(the)g (distributional)f(sense.)36 b(Notice)28 b(that)1144 1273 y(Im)14 b Fo(R)1320 1285 y Fm(z)1359 1273 y Fq(\(0)p Fo(;)g Fq(0\))22 b(=)1654 1057 y Fl(2)1654 1203 y(6)1654 1253 y(6)1654 1306 y(4)1751 1216 y Fq(Im)1908 1086 y Fg(1)1881 1103 y Fl(Z)1845 1337 y Fg(\0001)1982 1216 y Fq(\()p Fo(\025)d Fn(\000)f Fo(z)t Fq(\))2239 1182 y Fg(\000)p Fk(1)2328 1216 y Fo(d\033)s Fq(\()p Fo(\025)p Fq(\))84 b(0)2122 1422 y(0)453 b(0)2700 1057 y Fl(3)2700 1203 y(7)2700 1253 y(7)2700 1306 y(5)3338 1273 y Fq(\(7\))456 1550 y(where)28 b(the)h(in)n(tegral)f(con)n(v)n(erges)f(due)i(to)g (\(5\).)41 b(Assume)29 b Fo(a)p Fq(\()p Fo(r)r Fq(\))e(=)d Fo(b)p Fq(\()p Fo(r)r Fq(\))j(=)e(0,)k(if)g Fo(r)f(>)d(R)q Fq(.)41 b(Then,)456 1650 y(w)n(e)27 b(can)g(alw)n(a)n(ys)f(\014nd)i (the)g(Jost)f(solution)g Fo(F)12 b Fq(\()p Fo(r)n(;)i(\025)p Fq(\):)722 1845 y Fo(D)r(F)35 b Fq(=)22 b Fo(\025F)r(;)29 b(F)12 b Fq(\()p Fo(r)n(;)i(\025)p Fq(\))24 b(=)1484 1728 y Fl(\024)1569 1794 y Fo(F)1622 1806 y Fk(1)1660 1794 y Fq(\()p 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Fl(\025)2771 4382 y Fo(;)41 b Fq(if)29 b Fo(r)c(>)e(s)456 4577 y Fq(In)35 b(the)g(free)g(case)g(\()p Fo(a)g Fq(=)h Fo(b)f Fq(=)g(0\),)i(\010\()p Fo(r)n(;)14 b(\025)p Fq(\))37 b(=)e(cos)o(\()p Fo(\025r)r Fq(\),)k(\011\()p Fo(r)n(;)14 b(\025)p Fq(\))36 b(=)f Fn(\000)14 b Fq(sin\()p Fo(\025r)r Fq(\),)39 b Fo(F)3122 4589 y Fk(1)3159 4577 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))37 b(=)456 4676 y Fn(\000)p Fo(i)14 b Fq(exp)n(\()p Fo(i\025r)r Fq(\),)30 b Fo(F)975 4688 y Fk(2)1012 4676 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))24 b(=)f(exp\()p Fo(i\025r)r Fq(\),)29 b Fo(\033)s Fq(\()p Fo(\025)p Fq(\))24 b(=)f Fo(d\025=\031)s Fq(,)28 b(and)698 5020 y Fo(R)762 4986 y Fk(0)761 5041 y Fm(\025)805 5020 y Fq(\()p Fo(r)n(;)14 b(s)p Fq(\))23 b(=)1091 4751 y Fl(8)1091 4825 y(>)1091 4850 y(>)1091 4875 y(>)1091 4900 y(>)1091 4925 y(<)1091 5074 y(>)1091 5099 y(>)1091 5124 y(>)1091 5149 y(>)1091 5174 y(:)1206 4754 y(\024)1319 4820 y Fo(i)14 b Fq(cos)o(\()p Fo(\025r)r Fq(\))g(exp)q(\()p Fo(i\025s)p Fq(\))112 b Fn(\000)14 b Fq(cos)n(\()p Fo(\025r)r Fq(\))g(exp)r(\()p Fo(i\025s)p Fq(\))1292 4920 y Fn(\000)p Fo(i)g Fq(sin)o(\()p Fo(\025r)r Fq(\))g(exp)q(\()p Fo(i\025s)p Fq(\))127 b(sin\()p Fo(\025r)r Fq(\))14 b(exp)r(\()p Fo(i\025s)p Fq(\))2763 4754 y Fl(\025)2820 4871 y Fo(;)42 b Fq(if)28 b Fo(r)e(<)c(s)1206 5053 y Fl(\024)1310 5119 y Fo(i)14 b Fq(exp)o(\()p Fo(i\025r)r Fq(\))g(cos\()p Fo(\025s)p Fq(\))102 b Fn(\000)p Fo(i)14 b Fq(exp)o(\()p Fo(i\025r)r Fq(\))g(sin)q(\()p Fo(\025s)p Fq(\))1292 5219 y Fn(\000)g Fq(exp)o(\()p Fo(i\025r)r Fq(\))g(cos\()p Fo(\025s)p Fq(\))138 b(exp\()p Fo(i\025r)r Fq(\))14 b(sin)q(\()p Fo(\025s)p Fq(\))2763 5053 y Fl(\025)2820 5170 y Fo(;)42 b Fq(if)28 b Fo(r)e(>)c(s)p eop %%Page: 4 4 4 3 bop 456 251 a Fk(4)1117 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)456 491 y Fq(Notice)27 b(that)h Fo(Z)6 b Fq(\(0)p Fo(;)14 b(\025)p Fq(\))23 b(=)1261 374 y Fl(\024)1346 440 y Fq(1)83 b Fn(\000)p Fo(F)1589 452 y Fk(1)1626 440 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(F)1882 405 y Fg(\000)p Fk(1)1870 462 y(2)1971 440 y Fq(\(0)p Fo(;)g(\025)p Fq(\))1346 543 y(0)256 b Fo(F)1709 507 y Fg(\000)p Fk(1)1697 565 y(2)1798 543 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))2204 374 y Fl(\025)2276 491 y Fq(and)1232 735 y(Im)g Fo(R)1408 747 y Fm(\025)1452 735 y Fq(\(0)p Fo(;)g Fq(0\))22 b(=)1771 679 y(1)p 1757 716 71 4 v 1757 792 a(2)p Fo(i)1851 735 y Fq([)p Fo(R)1937 747 y Fm(\025)1980 735 y Fq(\(0)p Fo(;)14 b Fq(0\))k Fn(\000)g Fo(R)2330 701 y Fg(\003)2329 756 y Fm(\025)2373 735 y Fq(\(0)p Fo(;)c Fq(0\)])22 b(=)840 976 y(=)952 920 y(1)p 938 957 V 938 1033 a(2)p Fo(i)1032 859 y Fl(\024)1117 927 y Fo(F)1182 892 y Fg(\003\000)p Fk(1)1170 949 y(2)1305 927 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(F)1561 897 y Fg(\003)1549 948 y Fk(1)1600 927 y Fq(\(0)p Fo(;)g(\025)p Fq(\))19 b Fn(\000)f Fo(F)1946 939 y Fk(1)1983 927 y Fq(\(0)p Fo(;)c(\025)p Fq(\))p Fo(F)2239 892 y Fg(\000)p Fk(1)2227 949 y(2)2329 927 y Fq(\(0)p Fo(;)g(\025)p Fq(\))83 b(0)1798 1027 y(0)763 b(0)2686 859 y Fl(\025)2744 976 y Fo(;)28 b(\025)23 b Fn(2)h Fj(C)2998 942 y Fk(+)3338 976 y Fq(\(8\))456 1154 y(F)-7 b(or)27 b Fo(\025)c Fn(2)g Fj(C)15 b Fq(,)34 b(let)28 b(us)g(in)n(tro)r(duce)f(the)h(follo)n(wing)e(functions:)38 b Fo(f)2386 1166 y Fk(1)2422 1154 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))25 b(=)d(exp\()p Fn(\000)p Fo(i\025r)r Fq(\))p Fo(F)3143 1166 y Fk(1)3182 1154 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\),)456 1254 y Fo(f)497 1266 y Fk(2)533 1254 y Fq(\()p Fo(r)n(;)g(\025)p Fq(\))25 b(=)d(exp\()p Fn(\000)p Fo(i\025r)r Fq(\))p Fo(F)1254 1266 y Fk(2)1293 1254 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))q(.)37 b(F)-7 b(unctions)27 b Fo(f)1958 1266 y Fk(1)2023 1254 y Fq(and)g Fo(f)2225 1266 y Fk(2)2290 1254 y Fq(satisfy)g(the)h(follo)n(wing)f(equations) 1335 1330 y Fl(\032)1439 1397 y Fo(f)1489 1367 y Fg(0)1480 1417 y Fk(1)1540 1397 y Fq(=)c Fn(\000)p Fo(i\025f)1811 1409 y Fk(1)1866 1397 y Fq(+)18 b Fo(af)2034 1409 y Fk(1)2089 1397 y Fq(+)g Fo(\025f)2261 1409 y Fk(2)2317 1397 y Fn(\000)g Fo(bf)2477 1409 y Fk(2)1439 1496 y Fo(f)1489 1466 y Fg(0)1480 1517 y Fk(2)1540 1496 y Fq(=)23 b Fn(\000)p Fo(\025f)1782 1508 y Fk(1)1837 1496 y Fn(\000)18 b Fo(bf)1997 1508 y Fk(1)2052 1496 y Fn(\000)g Fo(i\025f)2253 1508 y Fk(2)2308 1496 y Fn(\000)g Fo(af)2476 1508 y Fk(2)456 1636 y Fq(Therefore,)26 b(a)h(simple)h(algebra)e(yields)1246 1769 y Fo(d)p 1227 1806 83 4 v 1227 1883 a(dr)1333 1709 y Fl(\022)1404 1769 y Fo(f)1454 1739 y Fg(\003)1445 1790 y Fk(1)1492 1769 y Fo(f)1533 1781 y Fk(2)1589 1769 y Fn(\000)18 b Fo(f)1722 1739 y Fg(\003)1713 1790 y Fk(2)1759 1769 y Fo(f)1800 1781 y Fk(1)p 1404 1806 433 4 v 1607 1883 a Fo(i)1847 1709 y Fl(\023)1931 1826 y Fq(=)23 b Fn(\000)p Fq(2)14 b(Im)f Fo(\025)h Fn(j)p Fo(f)2378 1838 y Fk(1)2434 1826 y Fq(+)k Fo(if)2587 1838 y Fk(2)2623 1826 y Fn(j)2646 1791 y Fk(2)3338 1826 y Fq(\(9\))456 2015 y(Using)27 b(the)h(asymptotics)f(at)g(in\014nit)n(y)-7 b(,)28 b(w)n(e)g(obtain)579 2190 y Fo(f)629 2160 y Fg(\003)620 2211 y Fk(1)666 2190 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(f)891 2202 y Fk(2)929 2190 y Fq(\()p Fo(r)n(;)g(\025)p Fq(\))20 b Fn(\000)e Fo(f)1266 2160 y Fg(\003)1257 2211 y Fk(2)1304 2190 y Fq(\()p Fo(r)n(;)c(\025)p Fq(\))p Fo(f)1529 2202 y Fk(1)1567 2190 y Fq(\()p Fo(r)n(;)g(\025)p Fq(\))p 579 2227 1173 4 v 1151 2303 a Fo(i)1785 2246 y Fq(=)22 b(2)c(+)g(2)c(Im)g Fo(\025)2273 2117 y Fg(1)2246 2133 y Fl(Z)2253 2368 y Fm(r)2343 2246 y Fn(j)p Fo(f)2407 2258 y Fk(1)2444 2246 y Fq(\()p Fo(\032;)g(\025)p Fq(\))19 b(+)f Fo(if)2808 2258 y Fk(2)2844 2246 y Fq(\()p Fo(\032;)c(\025)p Fq(\))p Fn(j)3059 2212 y Fk(2)3098 2246 y Fo(d\032)113 b Fq(\(10\))456 2497 y Fv(Lemma)29 b(2.1.)40 b Fh(The)31 b(fol)t(lowing)h(r)l(elations)f(hold)1136 2628 y Fo(F)1201 2598 y Fg(\003)1189 2649 y Fk(1)1239 2628 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))p Fo(F)1476 2640 y Fk(2)1515 2628 y Fq(\()p Fo(r)n(;)g(\025)p Fq(\))19 b Fn(\000)f Fo(F)1866 2598 y Fg(\003)1854 2649 y Fk(2)1904 2628 y Fq(\()p Fo(r)n(;)c(\025)p Fq(\))p Fo(F)2141 2640 y Fk(1)2180 2628 y Fq(\()p Fo(r)n(;)g(\025)p Fq(\))p 1136 2665 1230 4 v 1736 2741 a Fo(i)2398 2684 y Fq(=)23 b(2)p Fo(;)14 b(\025)23 b Fn(2)g Fj(R)529 b Fq(\(11\))1096 2866 y Fo(F)1161 2835 y Fg(\003)1149 2886 y Fk(1)1199 2866 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(F)1443 2878 y Fk(2)1481 2866 y Fq(\(0)p Fo(;)g(\025)p Fq(\))19 b Fn(\000)f Fo(F)1839 2835 y Fg(\003)1827 2886 y Fk(2)1877 2866 y Fq(\(0)p Fo(;)c(\025)p Fq(\))p Fo(F)2121 2878 y Fk(1)2159 2866 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p 1096 2903 1255 4 v 1709 2979 a Fo(i)2383 2922 y Fn(\025)23 b Fq(2)p Fo(;)14 b(\025)23 b Fn(2)g Fj(C)2753 2887 y Fk(+)3297 2922 y Fq(\(12\))555 3203 y(Pro)r(of.)36 b(It)28 b(su\016ces)f(to)h(use)f(\(10\))g(and)h(de\014nition)g(of)f Fo(f)2273 3215 y Fk(1)2338 3203 y Fq(and)g Fo(f)2540 3215 y Fk(2)2577 3203 y Fq(.)37 b Ff(\004)555 3303 y Fq(Notice)f(that)h(the)f(lemma)g(is)g(wrong)f(if)h(the)h(Dirac)f(op)r (erator)e(is)i(not)g(in)g(the)h(canonical)456 3402 y(form.)f(\(11\))27 b(implies)h(that)g Fo(F)1374 3414 y Fk(2)1412 3402 y Fq(\(0)p Fo(;)14 b(\025)p 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b(0)1893 5186 y Fn(\000)p Fo(F)2023 5150 y Fg(\000)p Fk(1)2011 5208 y(2)2112 5186 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))83 b(0)2469 5016 y Fl(\025)2527 5133 y Fo(;)42 b(\025)23 b Fn(2)p 2741 5064 V 23 w Fj(C)2795 5109 y Fk(+)p eop %%Page: 5 5 5 4 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)254 b(5)456 450 y Fv(Lemma)31 b(2.2.)40 b Fh(Consider)33 b Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(b)p Fq(\()p Fo(r)r Fq(\))27 b Fn(2)e Fo(C)1836 420 y Fg(1)1830 471 y Fk(0)1907 450 y Fq(\()p Fj(R)1993 420 y Fk(+)2054 450 y Fq(\))p Fh(.)43 b(Then,)32 b(for)g(the)f(function)g Fo(F)3051 462 y Fk(2)3088 450 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fh(,)32 b(the)456 550 y(fol)t(lowing)g(ine)l(qualities)f(hold)g (true)1047 781 y Fn(k)p Fo(F)1142 793 y Fk(2)1180 781 y Fq(\(0)p Fo(;)14 b(iy)s Fq(\))p Fn(k)21 b(\024)i Fq(exp)1688 614 y Fl(2)1688 764 y(4)1743 781 y Fo(C)6 b(y)1852 747 y Fg(\000)p Fk(1)1982 652 y Fg(1)1955 668 y Fl(Z)1961 904 y Fk(0)2038 781 y Fq([)p Fn(k)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2292 747 y Fk(2)2348 781 y Fq(+)18 b Fn(k)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2654 747 y Fk(2)2691 781 y Fq(])p Fo(dr)2796 614 y Fl(3)2796 764 y(5)3297 781 y Fq(\(15\))555 1034 y Fh(The)31 b(asymptotics)g(of)f Fo(F)1327 1046 y Fk(2)1365 1034 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))30 b Fh(as)g Fo(\025)24 b Fn(2)p 1842 964 116 4 v 23 w Fj(C)1896 1010 y Fk(+)1957 1034 y Fo(;)14 b Fn(j)p Fo(\025)p Fn(j)24 b(!)f(1)30 b Fh(is)1074 1307 y Fo(F)1127 1319 y Fk(2)1164 1307 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))24 b(=)e(1)c Fn(\000)1658 1251 y Fq(1)p 1619 1288 119 4 v 1619 1364 a(2)p Fo(i\025)1789 1178 y Fg(1)1762 1194 y Fl(Z)1768 1430 y Fk(0)1859 1307 y Fn(j)p Fo(b)p Fq(\()p Fo(r)r Fq(\))h(+)f Fo(ia)p Fq(\()p Fo(r)r Fq(\))p Fn(j)2322 1273 y Fk(2)2360 1307 y Fo(dr)k Fq(+)c Fo(O)r Fq(\()p Fn(j)p Fo(\025)p Fn(j)2736 1273 y Fg(\000)p Fk(2)3297 1307 y Fq(\(16\))555 1617 y(Pro)r(of.)36 b(Consider)27 b(a)g(matrix)1667 1847 y Fo(J)k Fq(=)1832 1730 y Fl(\024)1917 1797 y Fq(0)83 b Fn(\000)p Fq(1)1917 1896 y(1)115 b(0)2189 1730 y Fl(\025)456 2012 y Fq(Up)r(on)28 b(m)n(ultiplying)f(b)n(y)h Fo(J)8 b Fq(,)27 b(the)h(equation)f Fo(D)r(Y)42 b Fq(=)23 b Fo(\025Y)47 b Fq(can)27 b(b)r(e)h(rewritten)f(as)1424 2213 y Fo(Y)1490 2179 y Fg(0)1532 2213 y Fq(+)18 b Fo(QY)41 b Fq(=)23 b Fn(\000)p Fo(\025J)8 b(Y)d(;)14 b(Q)22 b Fq(=)h Fn(\000)p Fo(J)8 b(V)839 b Fq(\(17\))456 2332 y(T)-7 b(ak)n(e)1468 2564 y Fo(Y)1516 2576 y Fk(0)1576 2564 y Fq(=)1664 2446 y Fl(\024)1749 2513 y Fn(\000)p Fo(ie)1882 2483 y Fm(i\025r)2110 2513 y Fo(e)2149 2483 y Fg(\000)p Fm(i\025r)1796 2613 y Fo(e)1835 2583 y Fm(i\025r)2064 2613 y Fn(\000)p Fo(ie)2197 2583 y Fg(\000)p Fm(i\025r)2388 2446 y Fl(\025)456 2733 y Fq(and)27 b(\014nd)h(the)g(solution)f(to)h (\(17\))f(in)h(the)g(follo)n(wing)e(form)i Fo(Y)41 b Fq(=)23 b Fo(Y)2529 2745 y Fk(0)2566 2733 y Fo(S)5 b Fq(.)555 2833 y(Th)n(us)889 2999 y Fo(S)945 2965 y Fg(0)991 2999 y Fq(=)22 b Fn(\000)p Fo(Y)1209 2964 y Fg(\000)p Fk(1)1191 3021 y(0)1299 2999 y Fo(QY)1413 3011 y Fk(0)1450 2999 y Fo(S)27 b Fq(=)1616 2882 y Fl(\024)1936 2949 y Fq(0)317 b(\()p Fn(\000)p Fo(b)17 b Fq(+)i Fo(ia)p Fq(\))p Fo(e)2673 2919 y Fg(\000)p Fk(2)p Fm(i\025r)1701 3049 y Fq(\()p Fn(\000)p Fo(b)f Fn(\000)g Fo(ia)p Fq(\))p Fo(e)2079 3019 y Fk(2)p Fm(i\025r)2555 3049 y Fq(0)2898 2882 y Fl(\025)2956 2999 y Fo(S)290 b Fq(\(18\))456 3182 y(Imp)r(osing)27 b(condition)1644 3329 y Fo(S)5 b Fq(\()p Fn(1)p Fo(;)14 b(\025)p Fq(\))24 b(=)2044 3212 y Fl(\024)2129 3278 y Fq(1)2129 3378 y(0)2212 3212 y Fl(\025)456 3493 y Fq(on)j(the)h(2)p Fo(m)18 b Fn(\002)g Fo(m)27 b Fq(matrix)1737 3724 y Fo(S)h Fq(=)1904 3607 y Fl(\024)1989 3673 y Fo(S)2040 3685 y Fk(1)1989 3773 y Fo(S)2040 3785 y Fk(2)2119 3607 y Fl(\025)456 3893 y Fq(w)n(e)f(get)g Fo(F)35 b Fq(=)23 b Fo(Y)940 3905 y Fk(0)977 3893 y Fo(S)5 b Fq(.)37 b(System)28 b(\(18\))f(can)g(b)r(e)h(rewritten)f(in)h(the)g(follo)n(wing)f(form) 1107 4166 y Fo(S)1158 4178 y Fk(1)1195 4166 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))24 b(=)f(1)18 b(+)1661 4036 y Fg(1)1634 4053 y Fl(Z)1641 4287 y Fm(r)1731 4166 y Fo(e)1770 4131 y Fg(\000)p Fk(2)p Fm(i\025s)1953 4166 y Fq(\()p Fo(b)p Fq(\()p Fo(s)p Fq(\))h Fn(\000)f Fo(ia)p Fq(\()p Fo(s)p Fq(\)\))p Fo(S)2485 4178 y Fk(2)2522 4166 y Fq(\()p Fo(s;)c(\025)p Fq(\))p Fo(ds)505 b Fq(\(19\))1204 4497 y Fo(S)1255 4509 y Fk(2)1293 4497 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))24 b(=)1615 4367 y Fg(1)1588 4384 y Fl(Z)1595 4618 y Fm(r)1685 4497 y Fo(e)1724 4462 y Fk(2)p Fm(i\025s)1855 4497 y Fq(\()p Fo(b)p Fq(\()p Fo(s)p Fq(\))19 b(+)f Fo(ia)p Fq(\()p Fo(s)p Fq(\)\))p Fo(S)2387 4509 y Fk(1)2425 4497 y Fq(\()p Fo(s;)c(\025)p Fq(\))p Fo(ds)602 b Fq(\(20\))456 4712 y(These)26 b(t)n(w)n(o)h(equations)f(are)g(easy)g(to)h(iterate.)36 b(W)-7 b(e)27 b(obtain)g(the)g(follo)n(wing)f(in)n(tegral)g(equation) 456 4811 y(for)h Fo(S)634 4823 y Fk(1)632 5119 y Fo(S)683 5131 y Fk(1)720 5119 y Fq(\()p Fo(r)n(;)14 b(\025)p Fq(\))24 b(=)f(1)18 b(+)1186 4990 y Fg(1)1159 5006 y Fl(Z)1166 5240 y Fm(r)1256 5119 y Fo(e)1295 5085 y Fg(\000)p Fk(2)p Fm(i\025s)1478 5119 y Fq(\()p Fo(b)p Fq(\()p Fo(s)p Fq(\))h Fn(\000)f Fo(ia)p Fq(\()p Fo(s)p Fq(\)\))2000 4990 y Fg(1)1973 5006 y Fl(Z)1980 5240 y Fm(s)2070 5119 y Fo(e)2109 5085 y Fk(2)p Fm(i\025t)2234 5119 y Fq(\()p Fo(b)p Fq(\()p Fo(t)p Fq(\))h(+)f Fo(ia)p Fq(\()p Fo(t)p Fq(\)\))p Fo(S)2748 5131 y Fk(1)2785 5119 y Fq(\()p Fo(t;)c(\025)p Fq(\))p Fo(dtds)178 b Fq(\(21\))p eop %%Page: 6 6 6 5 bop 456 251 a Fk(6)1117 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)456 450 y Fq(One)k(needs)g(to)g(use)h(Gron)n(w)n(all's)d(inequalit)n(y)i (to)h(obtain)f(the)h(\014rst)f(statemen)n(t)g(of)h(the)f(lemma.)456 550 y(Iterating)g(\(21\))i(and)f(in)n(tegrating)f(b)n(y)i(parts,)f(one) g(pro)n(v)n(es)f(\(16\).)36 b Ff(\004)555 649 y Fv(Remark.)54 b Fq(Notice)34 b(that)g(the)g(usual)g(scattering)e(co)r(e\016cien)n(ts) i Fo(A)p Fq(\()p Fo(\025)p Fq(\))h(and)f Fo(B)t Fq(\()p Fo(\025)p Fq(\))h(can)e(b)r(e)456 749 y(calculated)27 b(b)n(y)g(the)h(form)n(ulas)1190 975 y Fo(A)p Fq(\()p Fo(\025)p Fq(\))d(=)1486 919 y(1)p 1486 956 42 4 v 1486 1032 a(2)1538 975 y(\()p Fo(F)1623 987 y Fk(2)1661 975 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))k(+)g Fo(iF)2035 987 y Fk(1)2073 975 y Fq(\(0)p Fo(;)c(\025)p Fq(\)\))23 b(=)g Fo(S)2458 987 y Fk(1)2495 975 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(;)1029 1142 y(B)t Fq(\()p Fo(\025)p Fq(\))25 b(=)1330 1086 y(1)p 1330 1123 V 1330 1199 a(2)1382 1142 y(\()p Fo(F)1467 1154 y Fk(2)1505 1142 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))k Fn(\000)g Fo(iF)1879 1154 y Fk(1)1917 1142 y Fq(\(0)p Fo(;)c(\025)p Fq(\)\))23 b(=)g Fn(\000)p Fo(iS)2396 1154 y Fk(2)2432 1142 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(;)g(\025)24 b Fn(2)g Fj(C)3297 1053 y Fq(\(22\))456 1300 y(If)k Fo(a;)14 b(b)22 b Fn(2)i Fo(C)822 1269 y Fg(1)816 1320 y Fk(0)892 1300 y Fq(\()p Fj(R)978 1269 y Fk(+)1039 1300 y Fq(\),)29 b(then)f(the)g(asymptotics)e(of)i Fo(A)p Fq(\()p Fo(\025)p Fq(\))h(is)811 1535 y Fo(A)p Fq(\()p Fo(\025)p Fq(\))c(=)d(1)c Fn(\000)1289 1479 y Fq(1)p 1250 1516 119 4 v 1250 1592 a(2)p Fo(i\025)1419 1406 y Fg(1)1393 1422 y Fl(Z)1399 1658 y Fk(0)1489 1535 y Fn(j)p Fo(b)p Fq(\()p Fo(r)r Fq(\))i(+)e Fo(ia)p Fq(\()p Fo(r)r Fq(\))p Fn(j)1953 1501 y Fk(2)1991 1535 y Fo(dr)j Fq(+)d Fo(O)r Fq(\()p Fn(j)p Fo(\025)p Fn(j)2366 1501 y Fg(\000)p Fk(2)2457 1535 y Fq(\))p Fo(;)c(\025)24 b Fn(!)f(1)p Fo(;)14 b(\025)23 b Fn(2)p 2974 1465 116 4 v 24 w Fj(C)3028 1511 y Fk(+)3297 1535 y Fq(\(23\))456 1780 y(Using)k(\(11\),)g(w)n(e)h(get)f(the)h(conserv)-5 b(ation)26 b(la)n(w)1425 1925 y Fn(j)p Fo(A)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)1645 1891 y Fk(2)1707 1925 y Fq(=)c(1)c(+)g Fn(j)p Fo(B)t Fq(\()p Fo(\025)p Fq(\))p Fn(j)2162 1891 y Fk(2)2201 1925 y Fo(;)41 b(\025)24 b Fn(2)f Fj(R)828 b Fq(\(24\))456 2068 y(F)-7 b(rom)27 b(\(12\),)1397 2178 y Fn(j)p Fo(A)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)1617 2143 y Fk(2)1679 2178 y Fn(\025)c Fq(1)18 b(+)g Fn(j)p Fo(B)t Fq(\()p Fo(\025)p Fq(\))p Fn(j)2135 2143 y Fk(2)2173 2178 y Fo(;)42 b(\025)23 b Fn(2)h Fj(C)2442 2143 y Fk(+)3297 2178 y Fq(\(25\))555 2306 y(Notice)k(that)h(if)f Fo(\030)h Fn(2)24 b Fj(C)1271 2276 y Fm(m)1340 2306 y Fo(;)42 b(\030)28 b Fn(6)p Fq(=)c(0)p Fo(;)j Fq(then)i(the)g(function)f(ln)14 b Fn(jj)p Fo(F)2503 2270 y Fg(\000)p Fk(1)2491 2328 y(2)2593 2306 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(\030)t Fn(jj)28 b Fq(is)g(subharmonic)456 2406 y(in)f Fj(C)606 2376 y Fk(+)668 2406 y Fq(.)37 b(Th)n(us,)27 b(the)h(follo)n(wing)e(inequalit) n(y)i(holds)599 2637 y Fo(\031)17 b Fq(ln)d Fn(jj)p Fo(F)857 2601 y Fg(\000)p Fk(1)845 2659 y(2)946 2637 y Fq(\(0)p Fo(;)g(iy)s Fq(\))p Fo(\030)t Fn(jj)23 b(\024)f Fo(y)1460 2507 y Fg(1)1433 2524 y Fl(Z)1397 2758 y Fg(\0001)1558 2580 y Fq(ln)14 b Fn(jj)p Fo(F)1752 2545 y Fg(\000)p Fk(1)1740 2603 y(2)1841 2580 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(\030)t Fn(jj)p 1558 2618 561 4 v 1704 2694 a Fo(\025)1752 2670 y Fk(2)1808 2694 y Fq(+)k Fo(y)1935 2670 y Fk(2)2129 2637 y Fo(d\025)23 b Fq(=)2341 2580 y Fo(y)p 2341 2618 44 4 v 2342 2694 a Fq(2)2453 2507 y Fg(1)2426 2524 y Fl(Z)2390 2758 y Fg(\0001)2550 2580 y Fq(ln)14 b(\()q Fo(\031)s(\033)2766 2550 y Fg(0)2790 2580 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p 2550 2618 502 4 v 2667 2694 a Fo(\025)2715 2670 y Fk(2)2771 2694 y Fq(+)k Fo(y)2898 2670 y Fk(2)3062 2637 y Fo(d\025)144 b Fq(\(26\))456 2902 y(Here)29 b(w)n(e)h(used)g (asymptotics)g(of)g Fo(F)1596 2866 y Fg(\000)p Fk(1)1584 2924 y(2)1685 2902 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))31 b(as)e Fn(j)p Fo(\025)p Fn(j)f(!)f(1)k Fq(and)f(its)g(con)n(tin)n(uit)n (y)g(in)p 3133 2832 116 4 v 30 w Fj(C)3187 2878 y Fk(+)3248 2902 y Fo(:)g Fq(The)456 3001 y(follo)n(wing)c(theorem)h(is)h(straigh)n (tforw)n(ard.)456 3156 y Fv(Theorem)36 b(2.1.)44 b Fh(Assume)34 b(that)h Fn(jj)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(jj)p Fo(;)14 b Fn(jj)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(jj)34 b(2)f Fo(L)2237 3126 y Fk(2)2274 3156 y Fq(\()p Fj(R)2360 3126 y Fk(+)2421 3156 y Fq(\))p Fh(,)k Fo(\030)f Fn(2)d Fj(C)2729 3126 y Fm(m)2798 3156 y Fh(,)k Fn(k)p Fo(\030)t Fn(k)31 b Fq(=)h(1)p Fh(.)53 b(Then,)456 3255 y(the)30 b(fol)t(lowing)i(ine)l(quality)e(holds)963 3486 y Fo(y)1007 3452 y Fk(2)1102 3357 y Fg(1)1075 3373 y Fl(Z)1040 3608 y Fg(\0001)1200 3430 y Fq(ln\()p Fo(\031)s(\033)1401 3400 y Fg(0)1425 3430 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)14 b(\030)t Fq(\))p 1200 3467 488 4 v 1310 3543 a Fo(y)1354 3519 y Fk(2)1409 3543 y Fq(+)k Fo(\025)1540 3519 y Fk(2)1698 3486 y Fo(d\025)23 b(>)g Fn(\000)p Fo(C)2071 3357 y Fg(1)2044 3373 y Fl(Z)2050 3609 y Fk(0)2141 3419 y Fl(\002)2175 3486 y Fn(k)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2406 3452 y Fk(2)2462 3486 y Fq(+)18 b Fn(k)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2768 3452 y Fk(2)2805 3419 y Fl(\003)2854 3486 y Fo(dr)363 b Fq(\(27\))555 3752 y(Pro)r(of.)41 b(Assume)29 b Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(b)p Fq(\()p Fo(r)r Fq(\))28 b Fn(2)e Fj(C)1615 3722 y Fg(1)1615 3773 y Fk(0)1691 3752 y Fq(\()p Fj(R)1778 3722 y Fk(+)1839 3752 y Fq(\))j(\014rst.)42 b(Then,)30 b(the)g(estimate)f(\(15\))g(from) g(lemma)456 3852 y(2.2)d(implies)959 4090 y Fn(j)p Fo(F)1047 4055 y Fg(\000)p Fk(1)1035 4112 y(2)1136 4090 y Fq(\(0)p Fo(;)14 b(iy)s Fq(\))p Fn(j)23 b(\025)f Fq(exp)1626 3923 y Fl(0)1626 4073 y(@)1699 4090 y Fn(\000)p Fo(C)6 b(y)1873 4056 y Fg(\000)p Fk(1)2002 3961 y Fg(1)1975 3977 y Fl(Z)1982 4213 y Fk(0)2072 4023 y Fl(\002)2107 4090 y Fn(k)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2338 4056 y Fk(2)2394 4090 y Fq(+)18 b Fn(k)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2700 4056 y Fk(2)2737 4023 y Fl(\003)2785 4090 y Fo(dr)2867 3923 y Fl(1)2867 4073 y(A)3297 4090 y Fq(\(28\))456 4341 y(So,)34 b(\(26\))f(giv)n(es)g(the)h(needed)f(estimate.)55 b(No)n(w,)34 b(consider)f(an)n(y)f Fn(jj)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(jj)p Fo(;)14 b Fn(jj)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(jj)36 b(2)d Fo(L)3205 4311 y Fk(2)3242 4341 y Fq(\()p Fj(R)3328 4311 y Fk(+)3389 4341 y Fq(\).)456 4440 y(W)-7 b(e)28 b(can)f(\014nd)h(the)g(sequence)f Fo(a)1448 4452 y Fm(n)1493 4440 y Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(b)1669 4452 y Fm(n)1715 4440 y Fq(\()p Fo(r)r Fq(\))24 b Fn(2)g Fj(C)1975 4410 y Fg(1)1975 4461 y Fk(0)2051 4440 y Fq(\()p Fj(R)2137 4410 y Fk(+)2199 4440 y Fq(\))j(suc)n(h)h (that)1379 4546 y Fg(1)1352 4563 y Fl(Z)1359 4799 y Fk(0)1435 4676 y Fq(\()p Fn(k)p Fo(a)1553 4688 y Fm(n)1617 4676 y Fn(\000)18 b Fo(a)p Fn(k)1786 4642 y Fk(2)1841 4676 y Fq(+)g Fn(k)p Fo(b)2002 4688 y Fm(n)2064 4676 y Fn(\000)g Fo(b)p Fn(k)2225 4642 y Fk(2)2262 4676 y Fq(\))p Fo(dr)26 b Fn(!)d Fq(0)456 4917 y(as)38 b Fo(n)43 b Fn(!)g(1)p Fq(.)73 b(Let)39 b Fo(\033)1174 4929 y Fm(n)1260 4917 y Fq(denote)g(the)h(sp)r(ectral)f(measure)f(for)h Fo(a)2536 4929 y Fm(n)2581 4917 y Fo(;)14 b(b)2654 4929 y Fm(n)2699 4917 y Fq(.)73 b(Then,)42 b(the)e(second)456 5016 y(resolv)n(en)n(t)26 b(iden)n(tit)n(y)i(yields)g(the)g(w)n(eak)f(con)n(v)n(ergence)e(of)j (measures)f Fo(\033)2655 5028 y Fm(n)2729 5016 y Fq(to)g Fo(\033)s Fq(.)39 b(F)-7 b(or)27 b(eac)n(h)g(indi-)456 5116 y(vidual)c Fo(\033)750 5128 y Fm(n)795 5116 y Fq(,)h(w)n(e)f(ha)n (v)n(e)f(the)i(estimate)f(\(27\).)35 b(Then,)24 b(using)f(the)h (standard)e(argumen)n(t)g(in)n(v)n(olving)456 5216 y(semicon)n(tin)n (uit)n(y)27 b(of)g(the)h(en)n(trop)n(y)e([8],)i(one)f(gets)g(\(28\).)37 b Ff(\004)p eop %%Page: 7 7 7 6 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)254 b(7)555 450 y Fv(Remark.)65 b Fq(Constan)n(ts)36 b(in)h(the)h(inequalit)n(y)f(\(27\))f(do)h(not)h(dep)r(end)f(on)g Fo(m)p Fq(,)j(the)e(size)f(of)456 550 y(the)k(matrix.)76 b(Therefore,)43 b(one)e(can)f(easily)g(pro)n(v)n(e)g(an)g(analogous)f (result)h(for)h(the)g(Dirac)456 649 y(op)r(erator)30 b(with)j(square)d(summable)i(op)r(erator-v)-5 b(alued)30 b(p)r(oten)n(tial.)51 b(As)32 b(a)g(simple)g(corollary)-7 b(,)456 749 y Fo(\033)503 761 y Fm(ac)573 749 y Fq(\()p Fi(D)p Fq(\))23 b(=)g Fj(R)p Fq(.)555 849 y Fv(Remark.)42 b Fq(The)30 b(similar)f(results)g(can)h(b)r(e)g(pro)n(v)n(ed)e(for)h (the)h(Krein)f(systems)h([9)o(,)g(15)o(])g(with)456 948 y(matrix-v)-5 b(alued)31 b(or)h(op)r(erator-v)-5 b(alued)31 b(co)r(e\016cien)n(ts.)52 b(Notice)33 b(that)g(the)g(Krein)f(system)h (with)456 1048 y(co)r(e\016cien)n(t)e Fn(\000)p Fo(A)p Fq(\()p Fo(r)r Fq(\))h(in)g(a)f(standard)f(w)n(a)n(y)g(generates)g(the) h(Dirac)g(op)r(erator)f Fi(D)2957 1060 y Fm(g)3024 1048 y Fq(=)f Fn(\000)p Fi(D)i Fq(with)456 1147 y Fi(D)c Fq(in)n(tro)r (duced)h(b)n(y)f(\(3\).)555 1247 y(Assume)c(that)f Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(b)p Fq(\()p Fo(r)r Fq(\))25 b(are)c(in)n(tegrable)g(with)i(\014nite)g(supp)r(ort.)35 b(Consider)21 b(a)h(\014xed)h(v)n(ector)456 1347 y Fo(\030)46 b Fn(2)c Fj(C)689 1317 y Fm(m)758 1347 y Fq(.)71 b(Let)39 b(us)g(\014nd)h(an)e(elemen)n(t)h(of)g(the)h(Hilb)r(ert)f(space)f(with) i(generalized)d(F)-7 b(ourier)456 1446 y(transform)26 b(\()p Fo(\025)19 b Fn(\000)f Fo(iy)s Fq(\))1124 1416 y Fg(\000)p Fk(1)1213 1446 y Fo(\030)t Fq(.)37 b(This)28 b(function)g(is)f(giv)n(en)g(b)n(y)g(the)h(form)n(ula)1358 1589 y Fl(2)1358 1735 y(6)1358 1785 y(6)1358 1835 y(6)1358 1885 y(6)1358 1935 y(6)1358 1984 y(6)1358 2038 y(4)1502 1619 y Fg(1)1475 1635 y Fl(Z)1439 1870 y Fg(\0001)1575 1748 y Fq(\()p Fo(\025)19 b Fn(\000)f Fo(iy)s Fq(\))1862 1714 y Fg(\000)p Fk(1)1951 1748 y Fq(\010\()p Fo(r)n(;)c(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\))p Fo(\030)1499 1918 y Fg(1)1472 1935 y Fl(Z)1436 2169 y Fg(\0001)1573 2048 y Fq(\()p Fo(\025)19 b Fn(\000)f Fo(iy)s Fq(\))1860 2014 y Fg(\000)p Fk(1)1949 2048 y Fq(\011\()p Fo(r)n(;)c(\025)p Fq(\))p Fo(d\033)s Fq(\()p Fo(\025)p Fq(\))p Fo(\030)2487 1589 y Fl(3)2487 1735 y(7)2487 1785 y(7)2487 1835 y(7)2487 1885 y(7)2487 1935 y(7)2487 1984 y(7)2487 2038 y(5)456 2307 y Fq(and)34 b(is)h(equal)f(to)h Fo(R)1113 2319 y Fm(iy)1176 2307 y Fq(\()p Fo(r)n(;)14 b Fq(0\)[)p Fo(\030)t(;)g Fq(0])1519 2277 y Fm(t)1583 2307 y Fq(due)35 b(to)g(\(6\).)59 b(A)n(t)35 b(the)g(same)f(time,)j(\(14\))e(sa)n(ys)e(that)i(this)456 2407 y(v)n(ector)26 b(is)h(equal)g(to)1471 2586 y Fn(\000)1550 2469 y Fl(\024)1635 2535 y Fo(F)1688 2547 y Fk(1)1725 2535 y Fq(\()p Fo(r)n(;)14 b(iy)s Fq(\))p Fo(F)1999 2500 y Fg(\000)p Fk(1)1987 2557 y(2)2088 2535 y Fq(\(0)p Fo(;)g(iy)s Fq(\))p Fo(\030)1635 2638 y(F)1688 2650 y Fk(2)1725 2638 y Fq(\()p Fo(r)n(;)g(iy)s Fq(\))p Fo(F)1999 2603 y Fg(\000)p Fk(1)1987 2660 y(2)2088 2638 y Fq(\(0)p Fo(;)g(iy)s Fq(\))p Fo(\030)2385 2469 y Fl(\025)456 2782 y Fq(Therefore,)26 b(the)i(follo)n(wing)f(argumen)n(ts)f(are)g(v)-5 b(alid.)65 b(If)28 b(the)g(v)n(ector-function)1435 2948 y Fo(h)p Fq(\()p Fo(r)r Fq(\))c(=)f(\()p Fo(h)1778 2960 y Fk(1)1815 2948 y Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(h)2003 2960 y Fk(2)2042 2948 y Fq(\()p Fo(r)r Fq(\)\))2177 2914 y Fm(t)2230 2948 y Fn(2)24 b Fj(C)2363 2914 y Fk(2)p Fm(m)3297 2948 y Fq(\(29\))456 3114 y(satis\014es)h(equation)h Fo(D)r(h)d Fq(=)g Fo(iy)s(h)i Fq(and)i(deca)n(ys)e(at)h(in\014nit)n(y)-7 b(,)28 b(then)f Fo(h)2535 3126 y Fk(1)2572 3114 y Fq(\()p Fo(r)r Fq(\))d(=)f Fo(F)2840 3126 y Fk(1)2877 3114 y Fq(\()p Fo(r)n(;)14 b(iy)s Fq(\))p Fo(\021)s(;)g(h)3215 3126 y Fk(2)3252 3114 y Fq(\()p Fo(r)r Fq(\))25 b(=)456 3213 y Fo(F)509 3225 y Fk(2)546 3213 y Fq(\()p Fo(r)n(;)14 b(iy)s Fq(\))p Fo(\021)s Fq(,)35 b(where)e Fo(\021)j Fn(2)d Fj(C)1322 3183 y Fm(m)1391 3213 y Fq(.)h(T)-7 b(ak)n(e)32 b Fo(\030)37 b Fq(=)32 b Fo(F)1877 3225 y Fk(2)1915 3213 y Fq(\(0)p Fo(;)14 b(iy)s Fq(\))p Fo(\021)35 b Fq(=)e Fo(h)2353 3225 y Fk(2)2390 3213 y Fq(\(0\).)54 b(The)34 b(generalized)d(F)-7 b(ourier)456 3313 y(transform)26 b(for)h(the)h(function)g Fo(h)p Fq(\()p Fo(r)r Fq(\))h(is)f Fn(\000)p Fq(\()p Fo(\025)18 b Fn(\000)g Fo(iy)s Fq(\))2047 3283 y Fg(\000)p Fk(1)2136 3313 y Fo(h)2184 3325 y Fk(2)2221 3313 y Fq(\(0\).)37 b(The)28 b(sp)r(ectral)f(measure)f(of)i Fo(h)f Fq(is)1349 3586 y Fo(\033)s Fq(\()p Fo(\025;)14 b(h)p Fq(\))25 b(=)1766 3457 y Fm(\025)1726 3473 y Fl(Z)1691 3707 y Fg(\0001)1850 3530 y Fo(d)p Fq(\()p Fo(\033)s Fq(\()p Fo(\025)p Fq(\))p Fo(h)2135 3542 y Fk(2)2174 3530 y Fq(\(0\))p Fo(;)14 b(h)2365 3542 y Fk(2)2402 3530 y Fq(\(0\)\))p 1850 3567 691 4 v 2061 3643 a Fo(\025)2109 3619 y Fk(2)2165 3643 y Fq(+)k Fo(y)2292 3619 y Fk(2)456 3859 y Fq(Using)27 b(the)h(estimate)g(\(26\),)f(w)n(e)g(ha)n(v)n(e)670 4117 y Fo(y)772 3988 y Fg(1)745 4004 y Fl(Z)709 4238 y Fg(\0001)869 4056 y Fq(ln)952 3988 y Fl(\002)987 4056 y Fo(\031)s Fq(\()p Fo(\033)1119 4026 y Fg(0)1143 4056 y Fq(\()p Fo(\025;)14 b(h)p Fq(\)\()p Fo(\025)1420 4026 y Fk(2)1477 4056 y Fq(+)k Fo(y)1604 4026 y Fk(2)1641 4056 y Fq(\)\))1705 3988 y Fl(\003)p 869 4098 872 4 v 1171 4174 a Fo(\025)1219 4150 y Fk(2)1275 4174 y Fq(+)g Fo(y)1402 4150 y Fk(2)1750 4117 y Fo(d\025)24 b Fn(\025)f Fq(2)p Fo(\031)16 b Fq(ln)e Fn(jj)p Fo(F)2252 4082 y Fg(\000)p Fk(1)2240 4139 y(2)2342 4117 y Fq(\(0)p Fo(;)g(iy)s Fq(\))p Fo(h)2606 4129 y Fk(2)2642 4117 y Fq(\(0\))p Fn(jj)23 b Fq(=)g(2)p Fo(\031)17 b Fq(ln)d Fn(jj)p Fo(\021)s Fn(jj)456 4364 y Fq(or)1282 4566 y Fo(y)1384 4436 y Fg(1)1357 4453 y Fl(Z)1321 4687 y Fg(\0001)1481 4510 y Fq(ln)g([\()p Fo(\033)1669 4480 y Fg(0)1693 4510 y Fq(\()p Fo(\025;)g(h)p Fq(\)])p 1481 4547 433 4 v 1564 4623 a Fo(\025)1612 4599 y Fk(2)1668 4623 y Fq(+)k Fo(y)1795 4599 y Fk(2)1924 4566 y Fo(d\025)24 b Fn(\025)e Fo(C)j Fq(+)18 b(2)p Fo(\031)f Fq(ln)d Fn(jj)p Fo(\021)s Fn(jj)679 b Fq(\(30\))456 4817 y(where)24 b(the)h(constan)n(t)g Fo(C)31 b Fq(dep)r(ends)25 b(on)g Fo(y)s Fq(.)36 b(This)25 b(inequalit)n(y)f(will)h(pla)n(y)g(the) g(crucial)f(role)g(later.)456 4917 y(Roughly)j(sp)r(eaking,)f(it)i (means)g(the)g(follo)n(wing.)555 5016 y Fh(If)i(we)g(found)g(at)g(le)l (ast)f(one)h(solution)g(having)h(the)f(\\fr)l(e)l(e")f(asymptotics)i (at)e(some)h(p)l(oint)g(in)456 5116 y Fj(C)510 5086 y Fk(+)571 5116 y Fh(,)g(then)g(the)g(entr)l(opy)h(of)g(the)f(sp)l(e)l (ctr)l(al)g(me)l(asur)l(e)g(c)l(an)g(b)l(e)h(c)l(ontr)l(ol)t(le)l(d)g (by)f(the)g(\\amplitude",)456 5216 y(i.e.)39 b Fn(jj)p Fo(\021)s Fn(jj)p Fh(.)p eop %%Page: 8 8 8 7 bop 456 251 a Fk(8)1117 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)555 450 y Fq(The)28 b(de\014nition)g(of)f(the)h(sp)r(ectral)f(measure)g (giv)n(es)f(an)i(equalit)n(y)1603 543 y Fg(1)1576 559 y Fl(Z)1540 794 y Fg(\0001)1690 672 y Fo(\033)1740 638 y Fg(0)1764 672 y Fq(\()p Fo(\025;)14 b(h)p Fq(\))p Fo(d\025)25 b Fq(=)d Fn(jj)p Fo(h)p Fn(jj)2304 638 y Fk(2)456 914 y Fh(Ther)l(efor)l(e,)32 b(the)e(lower)g(b)l(ound)g(for)h Fn(jj)p Fo(\021)s Fn(jj)f Fh(yields)h(the)f(lower)h(b)l(ound)e(for)1667 1012 y Fg(1)1641 1028 y Fl(Z)1605 1262 y Fg(\0001)1765 1085 y Fq(ln)1834 1048 y Fg(\000)1904 1085 y Fo(\033)1954 1055 y Fg(0)1978 1085 y Fq(\()p Fo(\025;)14 b(h)p Fq(\))p 1765 1122 411 4 v 1836 1198 a Fo(\025)1884 1174 y Fk(2)1940 1198 y Fq(+)k Fo(y)2067 1174 y Fk(2)2185 1141 y Fo(d\025)1021 b Fq(\(31\))474 1388 y(Consider)17 b Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(;)d(b)p Fq(\()p Fo(r)r Fq(\))26 b Fn(2)d Fj(C)1292 1358 y Fg(1)1292 1409 y Fk(0)1368 1388 y Fq(\()p Fj(R)1455 1358 y Fk(+)1516 1388 y Fq(\))p Fo(;)14 b Fn(jj)p Fo(\030)t Fn(jj)23 b Fq(=)g(1.)33 b(The)19 b(functions)f(ln)c Fn(j)p Fq(\()p Fo(F)2639 1358 y Fg(\000)p Fk(1)2729 1388 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p Fn(j)p Fq(,)21 b(ln)14 b Fn(j)p Fq(\()p Fo(A)3336 1358 y Fg(\000)p Fk(1)3426 1388 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p Fn(j)p Fq(,)456 1494 y(ln)g Fn(k)p Fo(A)p Fq(\()p Fo(\025)p Fq(\))p Fn(k)33 b Fq(are)f(subharmonic)g(in)i Fj(C)1626 1464 y Fk(+)1720 1494 y Fq(and)f(tend)h(to)f(0)f(as)h Fn(j)p Fo(\025)p Fn(j)g(!)f(1)p Fo(;)14 b(\025)32 b Fn(2)p 2901 1424 116 4 v 33 w Fj(C)2955 1470 y Fk(+)3016 1494 y Fq(.)54 b(W)-7 b(rite)33 b(the)456 1594 y(corresp)r(onding)19 b(inequalit)n(y)h(at)h Fo(\025)i Fq(=)g Fo(iy)s Fq(.)34 b(T)-7 b(aking)20 b Fo(y)26 b Fn(!)d(1)p Fq(,)f(one)f(gets)g(the)g (follo)n(wing)f(estimates)456 1693 y(b)n(y)27 b(comparing)f(the)i(co)r (e\016cien)n(ts)f(against)g Fo(y)1871 1663 y Fg(\000)p Fk(1)1959 1693 y Fq(.)1244 1869 y Fg(1)1217 1886 y Fl(Z)1181 2120 y Fg(\0001)1331 1999 y Fq(ln\()p Fo(\031)s(\033)1532 1965 y Fg(0)1557 1999 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)14 b(\030)t Fq(\))p Fo(d\025)25 b Fq(=)d(2)2121 1869 y Fg(1)2094 1886 y Fl(Z)2058 2120 y Fg(\0001)2209 1999 y Fq(ln)14 b Fn(k)p Fo(F)2399 1964 y Fg(\000)p Fk(1)2387 2021 y(2)2487 1999 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(\030)t Fn(k)p Fo(d\025)24 b Fn(\025)771 2332 y(\025)f Fq(2)959 2202 y Fg(1)932 2219 y Fl(Z)896 2453 y Fg(\0001)1046 2332 y Fq(ln)14 b Fn(j)p Fq(\()p Fo(F)1249 2296 y Fg(\000)p Fk(1)1237 2354 y(2)1339 2332 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p Fn(j)p Fo(d\025)24 b(>)f Fn(\000)p Fo(C)2076 2202 y Fg(1)2049 2219 y Fl(Z)2055 2455 y Fk(0)2146 2264 y Fl(\002)2180 2332 y Fn(jj)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)h Fq(+)18 b Fo(ia)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)t Fn(jj)2770 2297 y Fk(2)2808 2264 y Fl(\003)2857 2332 y Fo(dr)r Fq(;)1040 2535 y Fg(1)1013 2551 y Fl(Z)977 2786 y Fg(\0001)1127 2664 y Fq(ln)c Fn(j)p Fq(\()p Fo(A)1327 2630 y Fg(\000)p Fk(1)1417 2664 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p Fn(j)p Fo(d\025)25 b(>)e Fn(\000)p Fo(C)2076 2535 y Fg(1)2049 2551 y Fl(Z)2055 2787 y Fk(0)2146 2597 y Fl(\002)2180 2664 y Fn(jj)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)h Fq(+)18 b Fo(ia)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)t Fn(jj)2770 2630 y Fk(2)2808 2597 y Fl(\003)2857 2664 y Fo(dr)r Fq(;)1247 2867 y Fg(1)1220 2884 y Fl(Z)1185 3118 y Fg(\0001)1335 2997 y Fq(ln)c Fn(jj)p Fo(A)1526 2963 y Fg(\000)p Fk(1)1615 2997 y Fq(\()p Fo(\025)p Fq(\))p Fn(jj)p Fo(d\025)25 b(>)e Fn(\000)p Fo(C)2147 2867 y Fg(1)2120 2884 y Fl(Z)2127 3120 y Fk(0)2203 2997 y Fq([)p Fn(k)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2457 2963 y Fk(2)2513 2997 y Fq(+)18 b Fn(k)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2819 2963 y Fk(2)2857 2997 y Fq(])p Fo(dr)456 3244 y Fq(Notice)27 b(that)h(estimates)f Fn(jj)p Fo(A)1369 3214 y Fg(\000)p Fk(1)1459 3244 y Fn(jj)c(\024)g Fq(1)p Fo(;)k Fn(jj)p Fo(A)1816 3214 y Fg(\003\000)p Fk(1)1940 3244 y Fn(jj)c(\024)f Fq(1)28 b(imply)912 3358 y Fl(Z)908 3596 y Fk(\001)1009 3471 y Fq(ln)14 b Fn(j)p Fq(\()p Fo(A)1209 3437 y Fg(\000)p Fk(1)1299 3471 y Fq(\()p Fo(\025)p Fq(\))p Fo(\030)t(;)g(\030)t Fq(\))p Fn(j)p Fo(d\025)25 b(>)e Fn(\000)p Fo(C)1957 3342 y Fg(1)1931 3358 y Fl(Z)1937 3594 y Fk(0)2027 3404 y Fl(\002)2062 3471 y Fn(jj)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)h Fq(+)18 b Fo(ia)p Fq(\()p Fo(r)r Fq(\))p Fo(\030)t Fn(jj)2652 3437 y Fk(2)2690 3404 y Fl(\003)2739 3471 y Fo(dr)1005 3654 y Fl(Z)1000 3893 y Fk(\001)1101 3767 y Fq(ln)c Fn(jj)p Fo(A)1292 3733 y Fg(\000)p Fk(1)1382 3767 y Fq(\()p Fo(\025)p Fq(\))p Fn(jj)p Fo(d\025)25 b(>)d Fn(\000)p Fo(C)1914 3638 y Fg(1)1887 3654 y Fl(Z)1894 3890 y Fk(0)1970 3767 y Fq([)p Fn(k)p Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2224 3733 y Fk(2)2280 3767 y Fq(+)c Fn(k)p Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fn(k)2586 3733 y Fk(2)2623 3767 y Fq(])p Fo(dr)3297 3625 y Fq(\(32\))456 4006 y(for)27 b(an)n(y)h(\001)c Fn(\032)f Fj(R)p Fq(.)45 b(Consider)27 b(Dirac)h(op)r(erator)e(\(3\).)39 b(Fix)28 b(some)g Fo(\030)g Fn(2)c Fj(C)2675 3976 y Fm(m)2744 4006 y Fo(;)14 b Fn(k)p Fo(\030)t Fn(k)23 b Fq(=)h(1.)38 b(Estimate)456 4106 y(\(32\))27 b(suggests)f(that)i(condition)f(\()p Fo(b)p Fq(\()p Fo(r)r Fq(\))20 b(+)e Fo(ia)p Fq(\()p Fo(r)r Fq(\)\))p Fo(\030)28 b Fn(2)c Fo(L)2181 4076 y Fk(2)2218 4106 y Fq(\()p Fj(R)2304 4076 y Fk(+)2365 4106 y Fq(\))k(guaran)n(tees)d Fo(\033)2885 4118 y Fm(ac)2956 4106 y Fq(\()p Fi(D)p Fq(\))e(=)g Fj(R)p Fq(.)456 4239 y Fv(Theorem)h(2.2.)36 b Fh(L)l(et)25 b Fo(a)p Fq(\()p Fo(r)r Fq(\))h Fh(and)g Fo(b)p Fq(\()p Fo(r)r Fq(\))g Fh(have)h(lo)l(c)l(al)t(ly)g(summable)e(entries.)38 b(Assume)24 b(that)h(ther)l(e)456 4339 y(is)30 b(a)g(\014xe)l(d)f(ve)l(ctor)h Fo(\030)d Fn(2)d Fj(C)1249 4309 y Fm(m)1348 4339 y Fh(such)29 b(that)h Fq(\()p Fo(b)18 b Fq(+)g Fo(ia)p Fq(\))p Fo(\030)27 b Fn(2)d Fo(L)2179 4309 y Fk(2)2216 4339 y Fq(\()p Fj(R)2302 4309 y Fk(+)2363 4339 y Fq(\))p Fh(.)39 b(Then,)31 b Fo(\033)2748 4351 y Fm(ac)2818 4339 y Fq(\()p Fi(D)p Fq(\))24 b(=)e Fj(R)p Fo(:)555 4472 y Fq(Pro)r(of.)50 b(Without)33 b(loss)e(of)i(generalit)n(y)-7 b(,)32 b(assume)f Fo(\030)k Fq(=)c Fo(e)2332 4484 y Fk(1)2399 4472 y Fq(=)g(\(1)p Fo(;)14 b Fq(0)p Fo(;)g(:)g(:)g(:)f(;)h Fq(0\))2906 4442 y Fm(t)2934 4472 y Fq(.)51 b(Let)33 b(us)f(start)456 4572 y(with)23 b(the)h(compactly)e(supp)r(orted)h Fo(a)g Fq(and)g Fo(b)p Fq(.)35 b(In)23 b(this)h(case,)f(the)g(Jost)g(solution) f Fo(F)12 b Fq(\()p Fo(r)n(;)i(\025)p Fq(\))25 b(mak)n(es)456 4674 y(sense.)33 b(Consider)17 b(t)n(w)n(o)h(functions)h Fo(f)1573 4686 y Fk(1)1609 4674 y Fq(\()p Fo(\025)p Fq(\))25 b(=)d Fo(A)p Fq(\()p Fo(\025)p Fq(\))p Fo(F)2072 4638 y Fg(\000)p Fk(1)2060 4696 y(2)2162 4674 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)2392 4686 y Fk(1)2448 4674 y Fq(and)k Fo(f)2641 4686 y Fk(2)2678 4674 y Fq(\()p Fo(\025)p Fq(\))24 b(=)f Fo(B)t Fq(\()p Fo(\025)p Fq(\))p Fo(F)3146 4638 y Fg(\000)p Fk(1)3134 4696 y(2)3236 4674 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)3466 4686 y Fk(1)3504 4674 y Fq(.)456 4777 y(F)-7 b(rom)24 b(\(24\),)i Fn(k)p Fo(f)951 4789 y Fk(1)987 4777 y Fq(\()p Fo(\025)p Fq(\))p Fn(k)1141 4747 y Fk(2)1202 4777 y Fq(=)d Fn(k)p Fo(F)1397 4742 y Fg(\000)p Fk(1)1385 4800 y(2)1485 4777 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)1715 4789 y Fk(1)1752 4777 y Fn(k)1794 4747 y Fk(2)1845 4777 y Fq(+)g Fn(k)p Fo(f)2007 4789 y Fk(2)2042 4777 y Fq(\()p Fo(\025)p Fq(\))p Fn(k)2196 4747 y Fk(2)2259 4777 y Fq(for)25 b(real)f Fo(\025)p Fq(.)37 b(Since)25 b Fo(f)2906 4789 y Fk(1)2957 4777 y Fq(+)14 b Fo(f)3077 4789 y Fk(2)3136 4777 y Fq(=)22 b Fo(e)3262 4789 y Fk(1)3324 4777 y Fq(\(b)n(y)456 4881 y(\(22\)\),)e(w)n(e)e(ha)n(v)n(e)f Fn(j)p Fq(\()p Fo(f)1070 4893 y Fk(1)1107 4881 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)d(e)1295 4893 y Fk(1)1333 4881 y Fq(\))p Fn(j)1388 4851 y Fk(2)1449 4881 y Fq(=)22 b Fn(k)p Fo(F)1643 4845 y Fg(\000)p Fk(1)1631 4903 y(2)1732 4881 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)1962 4893 y Fk(1)1999 4881 y Fn(k)2041 4851 y Fk(2)2078 4881 y Fq(+)p Fn(j)p Fq(\()p Fo(f)2239 4893 y Fk(2)2276 4881 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)g(e)2464 4893 y Fk(1)2502 4881 y Fq(\))p Fn(j)2557 4851 y Fk(2)2617 4881 y Fn(\025)23 b(j)p Fq(\()p Fo(F)2825 4845 y Fg(\000)p Fk(1)2813 4903 y(2)2914 4881 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)3144 4893 y Fk(1)3182 4881 y Fo(;)g(e)3258 4893 y Fk(1)3294 4881 y Fq(\))p Fn(j)3349 4851 y Fk(2)3387 4881 y Fq(+)456 4980 y Fn(j)p Fq(\()p Fo(f)552 4992 y Fk(2)589 4980 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)g(e)777 4992 y Fk(1)815 4980 y Fq(\))p Fn(j)870 4950 y Fk(2)907 4980 y Fq(.)37 b(Consider)1037 5163 y Fo(\013)p Fq(\()p Fo(\025)p Fq(\))24 b(=)1439 5107 y(\()p Fo(f)1512 5119 y Fk(1)1549 5107 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)1737 5119 y Fk(1)1775 5107 y Fq(\))p 1324 5144 599 4 v 1324 5227 a(\()p Fo(F)1421 5191 y Fg(\000)p Fk(1)1409 5249 y(2)1510 5227 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(e)1740 5239 y Fk(1)1778 5227 y Fo(;)g(e)1854 5239 y Fk(1)1890 5227 y Fq(\))1933 5163 y Fo(;)g(\014)t Fq(\()p Fo(\025)p Fq(\))24 b(=)2370 5107 y(\()p Fo(f)2443 5119 y Fk(2)2480 5107 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)2668 5119 y Fk(1)2706 5107 y Fq(\))p 2254 5144 V 2254 5227 a(\()p Fo(F)2351 5191 y Fg(\000)p Fk(1)2339 5249 y(2)2441 5227 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(e)2671 5239 y Fk(1)2708 5227 y Fo(;)g(e)2784 5239 y Fk(1)2821 5227 y Fq(\))p eop %%Page: 9 9 9 8 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)254 b(9)456 450 y Fq(Then,)1369 587 y Fo(\013)p Fq(\()p Fo(\025)p Fq(\))20 b(+)e Fo(\014)t Fq(\()p Fo(\025)p Fq(\))24 b(=)2201 531 y(1)p 1922 568 599 4 v 1922 651 a(\()p Fo(F)2019 615 y Fg(\000)p Fk(1)2007 673 y(2)2108 651 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)2338 663 y Fk(1)2376 651 y Fo(;)g(e)2452 663 y Fk(1)2489 651 y Fq(\))456 759 y(and)1451 861 y Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)1662 826 y Fk(2)1724 861 y Fn(\025)23 b Fq(1)17 b(+)h Fn(j)p Fo(\014)t Fq(\()p Fo(\025)p Fq(\))p Fn(j)2163 826 y Fk(2)2202 861 y Fo(;)c(\025)24 b Fn(2)f Fj(R)456 980 y Fq(Therefore,)1312 1117 y Fn(j)p Fq(\()p Fo(F)1432 1081 y Fg(\000)p Fk(1)1420 1139 y(2)1521 1117 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)1751 1129 y Fk(1)1789 1117 y Fo(;)g(e)1865 1129 y Fk(1)1901 1117 y Fq(\))p Fn(j)24 b(\025)2184 1061 y Fq(1)p 2077 1098 255 4 v 2077 1174 a(2)p Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)2341 1117 y Fo(;)14 b(\025)24 b Fn(2)f Fj(R)456 1299 y Fq(F)-7 b(unction)32 b(\()p Fo(F)904 1264 y Fg(\000)p Fk(1)892 1321 y(2)993 1299 y Fq(\(0)p Fo(;)14 b(\025)p Fq(\))p Fo(e)1223 1311 y Fk(1)1260 1299 y Fo(;)g(e)1336 1311 y Fk(1)1373 1299 y Fq(\))32 b(is)f(analytic)g(in)p 1945 1229 116 4 v 32 w Fj(C)1999 1275 y Fk(+)2092 1299 y Fq(and)g(tends)h(to)g(1)f(as)g Fn(j)p Fo(\025)p Fn(j)f(!)f(1)p Fq(,)k Fo(\025)d Fn(2)p 3306 1229 V 30 w Fj(C)3360 1275 y Fk(+)3421 1299 y Fq(.)456 1406 y(The)h(n)n(umerator)e(of)i Fo(\013)p Fq(\()p Fo(\025)p Fq(\),)i(function)e(\()p Fo(f)1757 1418 y Fk(1)1795 1406 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)1983 1418 y Fk(1)2020 1406 y Fq(\),)32 b(do)r(es)f(not)g (ha)n(v)n(e)e(zero)r(es)h(in)p 2994 1336 V 31 w Fj(C)3048 1382 y Fk(+)3109 1406 y Fq(.)47 b(Indeed,)456 1512 y(in)p 555 1442 V 30 w Fj(C)609 1488 y Fk(+)670 1512 y Fq(,)31 b(w)n(e)f(ha)n(v)n(e)g Fn(k)p Fo(f)1127 1524 y Fk(1)1163 1512 y Fq(\()p Fo(\025)p Fq(\))p Fn(k)1317 1482 y Fk(2)1382 1512 y Fn(\025)e Fq(1)19 b(+)h Fn(k)p Fo(f)1704 1524 y Fk(2)1741 1512 y Fq(\()p Fo(\025)p Fq(\))p Fn(k)1895 1482 y Fk(2)1963 1512 y Fq(\(b)n(y)30 b(\(25\)\))h(and)f Fo(f)2529 1524 y Fk(1)2566 1512 y Fq(\()p Fo(\025)p Fq(\))21 b(+)f Fo(f)2825 1524 y Fk(2)2862 1512 y Fq(\()p Fo(\025)p Fq(\))28 b(=)g Fo(e)3134 1524 y Fk(1)3170 1512 y Fq(.)46 b(Th)n(us,)456 1612 y Fn(j)p Fq(\()p Fo(f)552 1624 y Fk(1)589 1612 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)777 1624 y Fk(1)815 1612 y Fq(\))p Fn(j)870 1581 y Fk(2)938 1612 y Fn(\025)30 b Fq(1)21 b(+)g Fn(j)p Fq(\()p Fo(f)1278 1624 y Fk(2)1315 1612 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)1503 1624 y Fk(1)1541 1612 y Fq(\))p Fn(j)1596 1581 y Fk(2)1664 1612 y Fn(\025)30 b Fq(1.)50 b(So,)33 b(the)f(function)h (ln)14 b Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)34 b Fq(is)e(sup)r(erharmonic)456 1711 y(and)27 b(nonnegativ)n(e)f(on)i Fj(R)p Fq(.)42 b(F)-7 b(or)27 b(an)n(y)g(\001)d Fn(\032)e Fj(R)p Fq(,)34 b(w)n(e)27 b(ha)n(v)n(e)g(the)g(estimates)541 1773 y Fl(Z)537 2011 y Fk(\001)638 1886 y Fq(ln[)p Fo(\031)s Fq(\()p Fo(\033)862 1852 y Fg(0)886 1886 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1037 1898 y Fk(1)1075 1886 y Fo(;)14 b(e)1151 1898 y Fk(1)1188 1886 y Fq(\)])p Fo(d\025)24 b Fn(\025)f Fq(2)1502 1773 y Fl(Z)1498 2011 y Fk(\001)1598 1886 y Fq(ln)14 b Fn(k)p Fo(F)1788 1852 y Fg(\000)p Fk(1)1876 1886 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(e)2106 1898 y Fk(1)2144 1886 y Fn(k)p Fo(d\025)23 b Fn(\025)g Fq(2)2444 1773 y Fl(Z)2440 2011 y Fk(\001)2540 1886 y Fq(ln)14 b Fn(j)p Fq(\()p Fo(F)2743 1852 y Fg(\000)p Fk(1)2832 1886 y Fq(\(0)p Fo(;)g(\025)p Fq(\))p Fo(e)3062 1898 y Fk(1)3099 1886 y Fo(;)g(e)3175 1898 y Fk(1)3212 1886 y Fq(\))p Fn(j)p Fo(d\025)3297 2090 y Fq(\(33\))1392 2234 y Fn(\025)23 b(\000)p Fq(2)1601 2121 y Fl(Z)1597 2359 y Fk(\001)1696 2234 y Fq(ln)14 b Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)p Fo(d\025)21 b Fn(\000)d Fq(2)c Fn(j)p Fq(\001)p Fn(j)27 b Fq(ln)14 b(2)456 2458 y(Sup)r(erharmonicit)n(y)26 b(of)i(ln)14 b Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)29 b Fq(implies)1405 2688 y Fo(\031)17 b Fq(ln)d Fn(j)p Fo(\013)p Fq(\()p Fo(iy)s Fq(\))p Fn(j)23 b(\025)g Fo(y)2001 2558 y Fg(1)1974 2575 y Fl(Z)1938 2809 y Fg(\0001)2098 2631 y Fq(ln)14 b Fn(j)p Fo(\013)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)p 2098 2668 296 4 v 2112 2744 a Fo(\025)2160 2721 y Fk(2)2216 2744 y Fq(+)k Fo(y)2343 2721 y Fk(2)2404 2688 y Fo(d\025)802 b Fq(\(34\))456 2937 y(Consider)22 b Fo(a)842 2949 y Fm(")878 2937 y Fo(;)14 b(b)951 2949 y Fm(")1009 2937 y Fn(2)23 b Fo(C)1152 2907 y Fg(1)1146 2958 y Fk(0)1223 2937 y Fq(\()p Fj(R)1309 2907 y Fk(+)1370 2937 y Fq(\))h(that)g(appro)n (ximate)e Fo(a;)14 b(b)23 b Fq(in)h Fo(L)2368 2907 y Fk(2)2405 2937 y Fq(\()p Fj(R)2491 2907 y Fk(+)2552 2937 y Fq(\))g(norm)f(as)g Fo(")g Fn(!)g Fq(0.)35 b(Denote)456 3037 y(the)28 b(corresp)r(onding)e(sp)r(ectral)h(measure)g(b)n(y)h Fo(\033)1932 3049 y Fm(")1968 3037 y Fq(.)38 b(Apply)28 b(\(33\))g(and)g(\(34\).)38 b(T)-7 b(ak)n(e)27 b Fo(y)f Fn(!)e Fq(+)p Fn(1)j Fq(in)456 3137 y(\(34\))g(and)g(use)h(lemma)f(2.2) g(and)g(\(23\).)37 b(W)-7 b(e)28 b(obtain)1155 3236 y Fg(1)1128 3253 y Fl(Z)1092 3487 y Fg(\0001)1242 3366 y Fq(ln)14 b Fn(j)p Fo(\013)1401 3378 y Fm(")1437 3366 y Fq(\()p Fo(\025)p Fq(\))p Fn(j)p Fo(d\025)25 b Fn(\024)1786 3310 y Fo(\031)p 1786 3347 51 4 v 1790 3423 a Fq(2)1886 3236 y Fg(1)1860 3253 y Fl(Z)1866 3489 y Fk(0)1956 3366 y Fn(k)p Fq(\()p Fo(b)2066 3378 y Fm(")2101 3366 y Fq(\()p Fo(r)r Fq(\))20 b(+)e Fo(ia)2380 3378 y Fm(")2415 3366 y Fq(\()p Fo(r)r Fq(\)\))p Fo(e)2589 3378 y Fk(1)2628 3366 y Fn(k)2670 3332 y Fk(2)2707 3366 y Fo(dr)510 b Fq(\(35\))456 3610 y(Estimates)27 b(\(33\))g(and)g(\(35\))g(yield)922 3727 y Fl(Z)918 3965 y Fk(\001)1019 3840 y Fq(ln\()p Fo(\033)1170 3805 y Fg(0)1167 3860 y Fm(")1203 3840 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1354 3852 y Fk(1)1392 3840 y Fo(;)14 b(e)1468 3852 y Fk(1)1505 3840 y Fq(\))p Fo(d\025)24 b Fn(\025)f Fo(C)1799 3852 y Fk(1)1855 3840 y Fq(+)18 b Fo(C)1997 3852 y Fk(2)2075 3710 y Fg(1)2048 3727 y Fl(Z)2055 3962 y Fk(0)2145 3840 y Fn(k)p Fq(\()p Fo(b)2255 3852 y Fm(")2290 3840 y Fq(\()p Fo(r)r Fq(\))i(+)e Fo(ia)2569 3852 y Fm(")2604 3840 y Fq(\()p Fo(r)r Fq(\)\))p Fo(e)2778 3852 y Fk(1)2816 3840 y Fn(k)2858 3805 y Fk(2)2895 3840 y Fo(dr)322 b Fq(\(36\))456 4076 y(Since)27 b Fo(d\033)762 4088 y Fm(")826 4076 y Fq(con)n(v)n(erges)e(w)n(eakly)i(to)g Fo(d\033)s Fq(,)h(w)n(e)g(ha)n(v)n(e)962 4188 y Fl(Z)958 4426 y Fk(\001)1059 4301 y Fq(ln\()p Fo(\033)1210 4267 y Fg(0)1234 4301 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1385 4313 y Fk(1)1423 4301 y Fo(;)14 b(e)1499 4313 y Fk(1)1536 4301 y Fq(\))p Fo(d\025)24 b Fn(\025)f Fo(C)1830 4313 y Fk(1)1886 4301 y Fq(+)18 b Fo(C)2028 4313 y Fk(2)2106 4171 y Fg(1)2079 4188 y Fl(Z)2086 4424 y Fk(0)2176 4301 y Fn(k)p Fq(\()p Fo(b)p Fq(\()p Fo(r)r Fq(\))h(+)f Fo(ia)p Fq(\()p Fo(r)r Fq(\)\))p Fo(e)2738 4313 y Fk(1)2776 4301 y Fn(k)2818 4267 y Fk(2)2855 4301 y Fo(dr)362 b Fq(\(37\))456 4542 y(No)n(w,)41 b(tak)n(e)d(arbitrary)f Fo(a)p Fq(\()p Fo(r)r Fq(\))j(and)f Fo(b)p Fq(\()p Fo(r)r Fq(\).)71 b(Consider)38 b(truncations)h Fo(a)2684 4554 y Fm(n)2729 4542 y Fq(\()p Fo(r)r Fq(\))k(=)e Fo(a)p Fq(\()p Fo(r)r Fq(\))p Fo(\037)3180 4557 y Fk([0)p Fm(;n)p Fk(])3317 4542 y Fq(\()p Fo(r)r Fq(\),)456 4642 y Fo(b)492 4654 y Fm(n)536 4642 y Fq(\()p Fo(r)r Fq(\))k(=)e Fo(b)p Fq(\()p Fo(r)r Fq(\))p Fo(\037)983 4657 y Fk([0)p Fm(;n)p Fk(])1119 4642 y Fq(\()p Fo(r)r Fq(\).)75 b(Since)40 b(the)g(functions)g Fo(a)2118 4654 y Fm(n)2203 4642 y Fq(and)f Fo(b)2412 4654 y Fm(n)2497 4642 y Fq(are)g(compactly)g(supp)r(orted,)456 4742 y(estimate)30 b(\(37\))f(holds)h(for)g(the)g(corresp)r(onding)e (measure)h Fo(\033)2372 4754 y Fm(n)2418 4742 y Fq(\()p Fo(\025)p Fq(\).)46 b(It)30 b(is)g(the)h(general)d(fact)j(of)456 4842 y(the)j(sp)r(ectral)g(theory)-7 b(,)35 b(that)f Fo(\033)1442 4854 y Fm(n)1522 4842 y Fq(con)n(v)n(erges)d(to)j Fo(\033)k Fq(w)n(eakly)33 b(as)g Fo(n)h Fn(!)g(1)p Fq(.)57 b(T)-7 b(ak)n(e)33 b Fo(n)g Fn(!)h(1)h Fq(to)456 4942 y(obtain)962 5002 y Fl(Z)958 5240 y Fk(\001)1059 5115 y Fq(ln\()p Fo(\033)1210 5081 y Fg(0)1234 5115 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1385 5127 y Fk(1)1423 5115 y Fo(;)14 b(e)1499 5127 y Fk(1)1536 5115 y Fq(\))p Fo(d\025)24 b Fn(\025)f Fo(C)1830 5127 y Fk(1)1886 5115 y Fq(+)18 b Fo(C)2028 5127 y Fk(2)2106 4986 y Fg(1)2079 5002 y Fl(Z)2086 5238 y Fk(0)2176 5115 y Fn(k)p Fq(\()p Fo(b)p Fq(\()p Fo(r)r Fq(\))h(+)f Fo(ia)p Fq(\()p Fo(r)r Fq(\)\))p Fo(e)2738 5127 y Fk(1)2776 5115 y Fn(k)2818 5081 y Fk(2)2855 5115 y Fo(dr)362 b Fq(\(38\))p eop %%Page: 10 10 10 9 bop 456 251 a Fk(10)1084 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)456 450 y Fq(That)27 b(completes)g(the)h(pro)r(of.)37 b Ff(\004)555 550 y Fv(Remark.)e Fq(The)27 b(absence)e(of)i(zero)r(es)e(for)h(the)h (function)g(\()p Fo(f)2427 562 y Fk(1)2464 550 y Fq(\()p Fo(\025)p Fq(\))p Fo(;)14 b(e)2652 562 y Fk(1)2690 550 y Fq(\))26 b(is)h(a)f(v)n(ery)f(imp)r(ortan)n(t)456 649 y(observ)-5 b(ation.)34 b(It)25 b(has)f(a)h(coun)n(terpart)e(for)h(the) h(Sc)n(hr\177)-42 b(odinger)23 b(op)r(erators)g(with)i(matrix-v)-5 b(alued)456 749 y(p)r(oten)n(tials)35 b(and)h(it)h(mak)n(es)e(p)r (ossible)h(to)g(obtain)g(the)g(new)g(trace-form)n(ulas)e(for)i(that)g (case.)456 849 y(This)27 b(analysis)f(will)i(b)r(e)g(done)g(in)f(the)h (separate)f(publication.)555 948 y(Since)37 b(the)h(constan)n(ts)e Fo(C)1370 960 y Fk(1)1444 948 y Fq(and)h Fo(C)1674 960 y Fk(2)1749 948 y Fq(in)g(\(38\))f(are)g(indep)r(enden)n(t)i(of)f Fo(m)p Fq(,)i(the)f(size)e(of)h(the)456 1048 y(matrices,)f(the)g (theorem)f(is)g(true)h(for)f(the)g(op)r(erator-v)-5 b(alued)34 b(Dirac)h(systems)f(as)h(w)n(ell.)60 b(Its)456 1147 y(statemen)n(t)37 b(is)f(v)n(ery)g(strong,)i(it)f(pro)n(v)n(es)e(a)i(certain)f(rigidit)n (y)g(of)h(the)h(Dirac)e(op)r(erator)f(\(3\))3388 1115 y Fk(1)3421 1147 y Fq(.)456 1247 y(That)27 b(mak)n(es)g(theorem)g(2.2)g (applicable)g(ev)n(en)g(to)g(some)g(PDE.)1178 1416 y(3.)41 b Fp(Mul)-6 b(tidimensional)31 b(Dira)n(c)h(opera)-6 b(tor)555 1566 y Fq(In)28 b(this)g(section,)f(w)n(e)g(consider)g(t)n(w) n(o)g(op)r(erators)1275 1700 y Fo(H)j Fq(=)23 b Fn(\000)p Fo(i\013)18 b Fn(\001)g(5)f Fq(+)h Fo(V)5 b(;)42 b(H)2030 1712 y Fm(s)2089 1700 y Fq(=)22 b Fn(\000)p Fo(i\013)c Fn(\001)h(5)e Fq(+)h Fo(V)456 1847 y Fq(The)i(\014rst)f(one)h({)f(on) 1096 1780 y Fl(\002)1130 1847 y Fo(L)1187 1817 y Fk(2)1224 1847 y Fq(\(\012\))1348 1780 y Fl(\003)1383 1797 y Fk(4)1441 1847 y Fq(with)h(b)r(oundary)f(conditions)g Fo(f)2418 1859 y Fk(3)2478 1847 y Fq(=)k Fo(f)2607 1859 y Fk(4)2667 1847 y Fq(=)g(0)c(on)h(\006.)34 b(The)20 b(second)456 1972 y(op)r(erator)25 b({)j(on)975 1905 y Fl(\002)1009 1972 y Fo(L)1066 1942 y Fk(2)1103 1972 y Fq(\()p Fj(R)1189 1942 y Fk(3)1233 1972 y Fq(\))1265 1905 y Fl(\003)1300 1922 y Fk(4)1337 1972 y Fq(.)37 b(P)n(oten)n(tial)26 b Fo(V)19 b Fq(\()p Fo(x)p Fq(\))28 b(is)g(alw)n(a)n(ys)e(assumed)h(to) g(b)r(e)h(symmetric)f(4)18 b Fn(\002)g Fq(4)456 2077 y(matrix)k(with)h(the)g(uniformly)g(b)r(ounded)g(en)n(tries.)34 b(F)-7 b(or)22 b Fo(H)7 b Fq(,)24 b(it)f(is)g(giv)n(en)f(on)g(\012,)i (for)e Fo(H)3091 2089 y Fm(s)3149 2077 y Fq({)h(on)f Fj(R)3378 2047 y Fk(3)3421 2077 y Fq(.)456 2177 y(Th)n(us,)j(w)n(e)g (ha)n(v)n(e)f(t)n(w)n(o)h(self-adjoin)n(t)g(op)r(erators)e Fi(H)i Fq(and)g Fi(H)2275 2189 y Fm(s)2309 2177 y Fq(.)36 b(W)-7 b(e)26 b(also)e(assume)h(that)g(after)g(the)456 2276 y(\\spherical)g(c)n(hange)g(of)h(v)-5 b(ariables")1569 2244 y Fk(2)1601 2276 y Fq(,)26 b(the)h(matrix)f(of)h Fo(V)45 b Fq(has)26 b(the)h(canonical)e(form)h(in)h(a)f(sense)456 2376 y(of)i(section)h(2.)39 b(The)29 b(t)n(ypical)f(example)h(of)f(suc) n(h)h(a)f(p)r(oten)n(tial)g(is)h Fo(V)19 b Fq(\()p Fo(x)p Fq(\))26 b(=)e Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)t Fq(,)31 b(where)d Fo(v)s Fq(\()p Fo(x)p Fq(\))456 2476 y(is)i(a)g(scalar)g(real-v)-5 b(alued)29 b(function)i(\(see)g([20)o(],) h(p.108)d(for)h(other)g(in)n(teractions)g(and)g(ph)n(ysical)456 2575 y(explanations\).)36 b(F)-7 b(or)27 b(simplicit)n(y)-7 b(,)27 b(w)n(e)h(will)f(deal)h(with)g(this)g(t)n(yp)r(e)g(of)f(p)r (oten)n(tials)g(only)-7 b(.)555 2675 y(F)g(or)27 b Fo(\013;)14 b(\014)t Fq(,)29 b(w)n(e)e(ha)n(v)n(e)903 2809 y Fo(\013)956 2821 y Fm(k)997 2809 y Fo(\013)1050 2821 y Fm(l)1094 2809 y Fq(+)18 b Fo(\013)1230 2821 y Fm(l)1255 2809 y Fo(\013)1308 2821 y Fm(k)1372 2809 y Fq(=)23 b(2)p Fo(\016)1539 2821 y Fm(k)q(l)1600 2809 y Fo(;)42 b(\013)1718 2821 y Fm(k)1759 2809 y Fo(\014)23 b Fq(+)18 b Fo(\014)t(\013)2016 2821 y Fm(k)2080 2809 y Fq(=)23 b(0)p Fo(;)41 b(\014)2325 2775 y Fk(2)2386 2809 y Fq(=)22 b(1;)41 b Fo(k)s(;)14 b(l)24 b Fq(=)f(1)p Fo(;)14 b Fq(2)p Fo(;)g Fq(3)456 2944 y(Notice)27 b(that)h(b)n(y)f(letting)h Fo(I)i Fq(=)23 b Fn(\000)p Fo(i\033)1569 2956 y Fk(1)1606 2944 y Fo(;)14 b(J)31 b Fq(=)22 b Fn(\000)p Fo(i\033)1948 2956 y Fk(2)1985 2944 y Fo(;)14 b(K)29 b Fq(=)22 b Fn(\000)p Fo(i\033)2350 2956 y Fk(3)2387 2944 y Fo(;)28 b Fq(w)n(e)f(ha)n(v)n(e)g(relations) 1591 3078 y Fo(I)1634 3044 y Fk(2)1694 3078 y Fq(=)c Fo(J)1836 3044 y Fk(2)1896 3078 y Fq(=)g Fo(\025)2032 3044 y Fk(2)2092 3078 y Fq(=)g Fn(\000)p Fq(1)p Fo(;)757 3213 y(I)7 b(J)31 b Fq(=)23 b Fn(\000)p Fo(J)8 b(I)f(;)41 b(I)7 b(K)28 b Fq(=)23 b Fn(\000)p Fo(K)6 b(I)h(;)41 b(K)6 b(J)30 b Fq(=)23 b Fn(\000)p Fo(J)8 b(K)q(;)41 b(I)7 b(J)31 b Fq(=)23 b Fo(K)q(;)41 b(J)8 b(K)29 b Fq(=)22 b Fo(I)7 b(;)42 b(K)6 b(I)29 b Fq(=)23 b Fo(J)456 3330 y Fq(whic)n(h)i(mak)n(es)f Fo(I)7 b(;)14 b(J)o(;)25 b Fq(and)g Fo(K)31 b Fq(quaternions.)k(The)25 b(follo)n(wing)f(algebraic) g(relations)g(are)g(easy)h(to)456 3429 y(v)n(erify)-7 b(.)456 3568 y Fv(Lemma)29 b(3.1.)40 b Fh(If)30 b Fo(\015)e Fn(2)23 b Fq(\006)p Fh(,)31 b(then)926 3697 y Fq(\()p Fo(\013)1011 3709 y Fk(1)1049 3697 y Fo(\015)1092 3709 y Fk(1)1147 3697 y Fq(+)18 b Fo(\013)1283 3709 y Fk(2)1321 3697 y Fo(\015)1364 3709 y Fk(2)1419 3697 y Fq(+)g Fo(\013)1555 3709 y Fk(3)1593 3697 y Fo(\015)1636 3709 y Fk(3)1673 3697 y Fq(\))1705 3667 y Fk(2)1766 3697 y Fq(=)k(1)926 3797 y(\()p Fo(\013)1011 3809 y Fk(1)1049 3797 y Fo(\015)1092 3809 y Fk(1)1147 3797 y Fq(+)c Fo(\013)1283 3809 y Fk(2)1321 3797 y Fo(\015)1364 3809 y Fk(2)1419 3797 y Fq(+)g Fo(\013)1555 3809 y Fk(3)1593 3797 y Fo(\015)1636 3809 y Fk(3)1691 3797 y Fq(+)g(1\))1848 3767 y Fk(2)1909 3797 y Fq(=)k(2\()p Fo(\013)2123 3809 y Fk(1)2160 3797 y Fo(\015)2203 3809 y Fk(1)2259 3797 y Fq(+)c Fo(\013)2395 3809 y Fk(2)2432 3797 y Fo(\015)2475 3809 y Fk(2)2531 3797 y Fq(+)g Fo(\013)2667 3809 y Fk(3)2704 3797 y Fo(\015)2747 3809 y Fk(3)2803 3797 y Fq(+)g(1\))926 3896 y(\()p Fo(\013)1011 3908 y Fk(1)1049 3896 y Fo(\015)1092 3908 y Fk(1)1147 3896 y Fq(+)g Fo(\013)1283 3908 y Fk(2)1321 3896 y Fo(\015)1364 3908 y Fk(2)1419 3896 y Fq(+)g Fo(\013)1555 3908 y Fk(3)1593 3896 y Fo(\015)1636 3908 y Fk(3)1691 3896 y Fq(+)g(1\))p Fo(\014)t Fq(\()p Fo(\013)1984 3908 y Fk(1)2022 3896 y Fo(\015)2065 3908 y Fk(1)2121 3896 y Fq(+)g Fo(\013)2257 3908 y Fk(2)2294 3896 y Fo(\015)2337 3908 y Fk(2)2393 3896 y Fq(+)g Fo(\013)2529 3908 y Fk(3)2566 3896 y Fo(\015)2609 3908 y Fk(3)2665 3896 y Fq(+)g(1\))23 b(=)g(0)456 4108 y Fv(Lemma)29 b(3.2.)40 b Fh(The)30 b(r)l(esolvent)g(kernel)g(of)g(the) g(fr)l(e)l(e)f(Dir)l(ac)h(op)l(er)l(ator)h Fi(H)2771 4078 y Fk(0)2770 4129 y Fm(s)2836 4108 y Fh(has)f(the)g(fol)t(lowing) 456 4208 y(form)g(\()p Fq([20)o(])p Fh(,)g(p.)39 b(39\))956 4402 y Fo(G)1021 4367 y Fk(0)1021 4422 y Fm(\025)1064 4402 y Fq(\()p Fo(x;)14 b(s)p Fq(\))24 b(=)1363 4285 y Fl(\022)1424 4402 y Fo(i)1463 4346 y(\013)18 b Fn(\001)h Fq(\()p Fo(x)g Fn(\000)f Fo(s)p Fq(\))p 1463 4383 366 4 v 1510 4459 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1744 4435 y Fk(2)1857 4402 y Fq(+)g Fo(\025)1998 4346 y(\013)h Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(s)p Fq(\))p 1998 4383 V 2064 4459 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)2392 4402 y Fq(+)g Fo(\025)2523 4285 y Fl(\023)2639 4346 y Fo(e)2678 4315 y Fm(i\025)p Fg(j)p Fm(x)p Fg(\000)p Fm(s)p Fg(j)p 2609 4383 326 4 v 2609 4459 a Fq(4)p Fo(\031)s Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)3297 4402 y Fq(\(39\))555 4586 y(Pro)r(of.)36 b(Use)27 b(the)h(iden)n(tities)1050 4720 y(\()p Fi(H)1158 4686 y Fk(0)1157 4741 y Fm(s)1212 4720 y Fn(\000)18 b Fo(\025)p Fq(\))1375 4686 y Fg(\000)p Fk(1)1488 4720 y Fq(=)23 b(\()p Fi(H)1684 4686 y Fk(0)1683 4741 y Fm(s)1738 4720 y Fq(+)18 b Fo(\025)p Fq(\)\()p Fi(H)2009 4686 y Fk(0)2008 4741 y Fm(s)2046 4669 y Fk(2)2102 4720 y Fn(\000)g Fo(\025)2233 4686 y Fk(2)2271 4720 y Fq(\))2303 4686 y Fg(\000)p Fk(1)2392 4720 y Fo(;)42 b Fi(H)2533 4686 y Fk(0)2532 4741 y Fm(s)2568 4669 y Fk(2)2629 4720 y Fq(=)22 b Fn(\000)p Fq(\001)447 b(\(40\))456 4855 y(and)27 b(the)h(explicit)g(form)n(ula)f(for)g(the)h(resolv)n(en)n (t)e(k)n(ernel)g(of)i(the)g(free)f(Laplacian.)555 4954 y(W)-7 b(e)28 b(will)g(start)f(with)h(the)g(follo)n(wing)f(theorem.)p 456 5028 499 4 v 555 5102 a Fk(1)588 5127 y Fu(This)c(rigidit)n(y)h(is) e(also)i(presen)n(t)g(for)f(p)r(olynomials)g(orthogonal)i(on)f(the)g (unit)g(circle.)555 5190 y Fk(2)588 5216 y Fu(W)-6 b(e)25 b(will)d(de\014ne)j(this)f(c)n(hange)h(of)e(v)l(ariables)h(later.)p eop %%Page: 11 11 11 10 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)221 b(11)456 450 y Fv(Theorem)46 b(3.1.)i Fh(Assume)42 b Fo(V)19 b Fq(\()p Fo(x)p Fq(\))47 b(=)f Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)t Fh(,)i(wher)l(e)43 b Fo(v)s Fq(\()p Fo(x)p Fq(\))h Fh(is)f(a)f(r)l(e)l(al-value)l(d,)47 b(uniformly)456 550 y(b)l(ounde)l(d,)30 b(sc)l(alar)h(function)f (satisfying)h(the)f(fol)t(lowing)i(c)l(ondition)1658 635 y Fl(Z)1616 873 y Fm(x)p Fg(2)p Fk(\012)1829 692 y Fo(v)1872 662 y Fk(2)1909 692 y Fq(\()p Fo(x)p Fq(\))p 1788 729 274 4 v 1788 805 a Fn(j)p Fo(x)p Fn(j)1881 781 y Fk(2)1937 805 y Fq(+)18 b(1)2095 748 y Fn(\024)23 b(1)1031 b Fq(\(41\))456 1007 y Fh(Then,)30 b Fo(\033)744 1019 y Fm(ac)815 1007 y Fq(\()p Fi(H)q Fq(\))22 b(=)h Fj(R)p Fh(.)555 1165 y Fq(Pro)r(of.)56 b(As)34 b(in)g([10)o(],)i(w)n(e)e(can)g (assume)g(without)g(loss)g(of)g(generalit)n(y)-7 b(,)34 b(that)h Fo(v)s Fq(\()p Fo(x)p Fq(\))g(=)e(0)h(in)456 1265 y(1)22 b Fo(<)h Fn(j)p Fo(x)p Fn(j)g Fo(<)g Fq(2.)42 b(Indeed,)30 b(otherwise)e(w)n(e)h(subtract)g(function)h Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)t(\037)2623 1280 y Fk(1)p Fm(<)p Fg(j)p Fm(x)p Fg(j)p Fm(<)p Fk(2)2875 1265 y Fq(\()p Fo(x)p Fq(\))h(from)e Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)t Fq(.)456 1370 y(The)h(resolv)n(en)n(t)g(of)g Fi(H)1155 1340 y Fk(0)1222 1370 y Fq(is)g(an)h(in)n(tegral)e(op)r (erator.)45 b(One)31 b(can)f(obtain)g(an)h(expression)e(for)h(its)456 1470 y(k)n(ernel)20 b(b)n(y)g(using)h(iden)n(tit)n(y)f(\(40\))h(with)g Fi(H)1739 1440 y Fk(0)1796 1470 y Fq(instead)g(of)f Fi(H)2239 1440 y Fk(0)2238 1490 y Fm(s)2275 1470 y Fq(.)35 b(Therefore,)21 b(the)g(resolv)n(en)n(t)e(of)i Fi(H)g Fq(is)456 1569 y(an)j(in)n(tegral)f(op)r(erator)f(to)r(o)j(and)f(the)h(trace-class)d (argumen)n(t)h([13)o(])i(can)f(b)r(e)g(applied.)36 b(No)n(w,)25 b(w)n(e)456 1669 y(tak)n(e)20 b(an)n(y)g(non)n(trivial)g(in\014nitely)i (smo)r(oth)f(radially-symmetric)e(function)j Fo(f)9 b Fq(\()p Fo(x)p Fq(\))22 b(with)g(supp)r(ort)456 1769 y(in)27 b Fn(f)p Fq(1)c Fo(<)f Fn(j)p Fo(x)p Fn(j)i Fo(<)e Fq(2)p Fn(g)p Fq(.)36 b(W)-7 b(e)28 b(will)g(sho)n(w)f(that)h(the)g(sp) r(ectral)f(measure)f(of)i(the)g(elemen)n(t)1572 1918 y Fo(f)9 b Fq(\()p Fo(x)p Fq(\))24 b(=)f(\()p Fo(f)9 b Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b Fq(0)p Fo(;)g Fq(0)p Fo(;)g Fq(0\))2299 1884 y Fm(t)3297 1918 y Fq(\(42\))456 2068 y(has)30 b(an)g(a.c.)46 b(comp)r(onen)n(t)30 b(whic)n(h)g(supp)r (ort)h(\014lls)g Fj(R)p Fq(.)52 b(The)30 b(pro)r(of)g(consists)g(of)h (t)n(w)n(o)e(steps.)46 b(In)456 2168 y(the)30 b(\014rst)g(step,)h(w)n (e)f(represen)n(t)f(op)r(erator)f Fi(H)i Fq(in)g(the)h(canonical)d (form)i(\(3\))g(with)h(un)n(b)r(ounded)456 2268 y(op)r(erator-v)-5 b(alued)27 b(co)r(e\016cien)n(ts.)42 b(Then,)30 b(w)n(e)g(apply)f(the)h (results)f(of)g(section)g(1)g(to)h(obtain)f(the)456 2367 y(needed)e(estimates)h(for)f(the)h(en)n(trop)n(y)e(of)i(the)g(sp)r (ectral)f(measure)f(of)i Fo(f)9 b Fq(.)555 2467 y(Let)30 b(us)f(start)g(with)h(the)g(suitable)f(c)n(hange)f(of)i(v)-5 b(ariables.)41 b(Consider)28 b(the)i(Dirac)f(op)r(erator)456 2566 y Fi(H)q Fq(.)43 b(Let)30 b(us)g(write)g(this)g(op)r(erator)f(in)h (the)g(spherical)f(co)r(ordinates)g(\(see)h([20)o(],)h(p.126)e(or)g ([21)o(],)456 2692 y(p.16\).)36 b(The)27 b(standard)g(unitary)g(op)r (erator)e Fo(f)9 b Fq(\()p Fo(x)p Fq(\))24 b Fn(2)f Fo(L)2164 2662 y Fk(2)2201 2692 y Fq(\(\012\))2365 2645 y Fe(U)2349 2692 y Fn(!)g Fo(F)12 b Fq(\()p Fo(r)r Fq(\))24 b(=)f Fo(r)r(f)9 b Fq(\()p Fo(r)r(\033)s Fq(\))p Fo(;)14 b(\033)28 b Fn(2)23 b Fq(\006)28 b(maps)456 2791 y(scalar)g(functions)j(of)g (three)f(v)-5 b(ariables)29 b(to)i(a)f(v)n(ector-v)-5 b(alued)29 b(function)i(of)f(one)h(v)-5 b(ariable.)29 b(F)-7 b(or)456 2891 y(almost)22 b(an)n(y)h Fo(r)j(>)c Fq(1,)i Fo(F)12 b Fq(\()p Fo(r)r Fq(\))25 b Fn(2)e Fo(L)1439 2861 y Fk(2)1476 2891 y Fq(\(\006\).)36 b(F)-7 b(ollo)n(wing)22 b([21)o(],)j(w)n(e)e(consider)f(the)i(follo)n(wing)e(auxiliary)456 2991 y(op)r(erators)1039 3180 y Fo(\033)1086 3192 y Fk(0)1147 3180 y Fq(=)1234 3063 y Fl(\024)1319 3129 y Fq(1)83 b(0)1319 3229 y(0)g(1)1527 3063 y Fl(\025)1585 3180 y Fo(;)27 b(\033)1682 3192 y Fm(r)1743 3180 y Fq(=)22 b Fo(r)1869 3146 y Fg(\000)p Fk(1)2016 3076 y(3)1973 3101 y Fl(X)1976 3278 y Fm(j)s Fk(=1)2107 3180 y Fo(\033)2154 3192 y Fm(j)2189 3180 y Fo(x)2236 3192 y Fm(j)2272 3180 y Fo(;)27 b(p)2364 3192 y Fm(j)2422 3180 y Fq(=)c Fn(\000)p Fo(i@)5 b(=@)g(x)2791 3192 y Fm(j)2824 3180 y Fo(;)720 3504 y(p)762 3516 y Fm(r)822 3504 y Fq(=)23 b Fn(\000)p Fo(i)p Fq(\()p Fo(@)5 b(=@)g(r)19 b Fq(+)f Fo(r)1354 3469 y Fg(\000)p Fk(1)1444 3504 y Fq(\))p Fo(;)28 b(L)1584 3516 y Fm(j)1642 3504 y Fq(=)22 b Fo(x)1776 3516 y Fm(j)s Fk(+1)1896 3504 y Fo(p)1938 3516 y Fm(j)s Fk(+2)2075 3504 y Fn(\000)c Fo(x)2205 3516 y Fm(j)s Fk(+2)2325 3504 y Fo(p)2367 3516 y Fm(j)s Fk(+1)2485 3504 y Fo(;)28 b(s)23 b Fq(=)g Fo(\033)2733 3516 y Fk(0)2789 3504 y Fq(+)2915 3400 y Fk(3)2872 3425 y Fl(X)2874 3602 y Fm(j)s Fk(=1)3006 3504 y Fo(\033)3053 3516 y Fm(j)3088 3504 y Fo(L)3145 3516 y Fm(j)456 3720 y Fq(where)28 b(indices)g Fo(j)34 b Fq(are)27 b(understo)r(o)r(d)h(mo)r (dulo)h(3,)f(as)g(usual.)39 b(Notice)29 b(that)g(the)g(op)r(erator)d Fo(p)3323 3732 y Fm(r)3389 3720 y Fq(is)456 3820 y(unitary)33 b(to)g Fn(\000)p Fo(id=dr)j Fq(under)e Fi(U)g Fq(and)f Fo(s)h Fq(is)f(indep)r(enden)n(t)i(of)f Fo(r)r Fq(.)56 b(W)-7 b(e)34 b(w)n(an)n(t)f(to)g(get)h(a)f(matrix)456 3919 y(represen)n(tation)26 b(of)i Fi(H)g Fq(con)n(v)n(enien)n(t)f(for) h(us.)38 b(T)-7 b(o)28 b(do)g(that,)h(w)n(e)e(tak)n(e)h(a)g(unitary)-7 b(,)28 b(indep)r(enden)n(t)456 4019 y(or)e Fo(r)31 b Fq(matrix)c Fo(U)1604 4163 y(U)32 b Fq(=)1780 4046 y Fl(\024)1866 4112 y Fo(\033)1913 4124 y Fk(0)2101 4112 y Fq(0)1887 4212 y(0)104 b Fn(\000)p Fo(i\033)2174 4224 y Fm(r)2252 4046 y Fl(\025)456 4345 y Fq(Then,)27 b(\([21],)h(p.)37 b(18\))958 4545 y Fo(U)1024 4511 y Fg(\003)1062 4545 y Fi(H)q Fo(U)31 b Fq(=)1313 4428 y Fl(\024)1434 4494 y Fq(0)119 b Fn(\000)p Fo(i\033)1736 4506 y Fk(0)1399 4594 y Fo(i\033)1475 4606 y Fk(0)1663 4594 y Fq(0)1814 4428 y Fl(\025)1872 4545 y Fo(p)1914 4557 y Fm(r)1969 4545 y Fn(\000)2062 4489 y Fq(1)p 2062 4526 42 4 v 2063 4602 a Fo(r)2127 4428 y Fl(\024)2213 4494 y Fq(0)83 b Fo(s)2214 4594 y(s)h Fq(0)2420 4428 y Fl(\025)2482 4545 y Fq(+)18 b Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(U)2785 4511 y Fg(\003)2825 4545 y Fo(\014)t(U)364 b Fq(\(43\))456 4750 y(Op)r(erator)20 b Fo(s)h Fq(is)h(selfadjoin)n(t)f(in)h([)p Fo(L)1510 4720 y Fk(2)1547 4750 y Fq(\(\006\)])1694 4720 y Fk(2)1731 4750 y Fq(.)35 b(Denote)22 b(its)g(orthonormal)e (eigenfunctions)h(b)n(y)g Fo( )3347 4765 y Fk(\()p Fm(n)p Fk(\))456 4875 y Fq(where)26 b Fo(n)h Fq(is)g(a)f(m)n(ultiindex.)37 b(Therefore,)26 b(in)1865 4808 y Fl(\002)1900 4875 y Fo(L)1957 4845 y Fk(2)1994 4875 y Fq(\(\006\))2118 4808 y Fl(\003)2153 4825 y Fk(4)2190 4875 y Fq(,)h(w)n(e)g(can)f(in)n(tro)r (duce)h(an)g(orthonormal)456 4975 y(basis)f(spanned)i(b)n(y)f(the)h (functions)1335 5170 y(\011)1400 5134 y Fk(+)1400 5198 y(\()p Fm(n)p Fk(\))1520 5170 y Fq(=)1607 5053 y Fl(\024)1692 5119 y Fo( )1746 5134 y Fk(\()p Fm(n)p Fk(\))1747 5219 y Fq(0)1885 5053 y Fl(\025)1943 5170 y Fo(;)14 b Fq(\011)2045 5134 y Fg(\000)2045 5198 y Fk(\()p Fm(k)q Fk(\))2160 5170 y Fq(=)2248 5053 y Fl(\024)2386 5119 y Fq(0)2333 5219 y Fo( )2387 5234 y Fk(\()p Fm(k)q Fk(\))2521 5053 y Fl(\025)p eop %%Page: 12 12 12 11 bop 456 251 a Fk(12)1084 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)456 463 y Fq(Then,)32 b(an)n(y)f(function)g Fo(f)9 b Fq(\()p Fo(x)p Fq(\))30 b Fn(2)1465 396 y Fl(\002)1499 463 y Fo(L)1556 433 y Fk(2)1593 463 y Fq(\(\012\))1717 396 y Fl(\003)1752 413 y Fk(4)1821 463 y Fq(can)h(b)r(e)h(represen)n(ted)e (as)g(a)h(sum)h Fo(f)9 b Fq(\()p Fo(x)p Fq(\))3095 416 y Fe(U)3088 463 y Fn(\030)28 b Fo(F)12 b Fq(\()p Fo(r)r Fq(\))31 b(=)456 510 y Fl(P)479 643 y Fm(n)557 572 y Fo(')611 537 y Fk(+)611 600 y(\()p Fm(n)p Fk(\))708 572 y Fq(\()p Fo(r)r Fq(\)\011)876 537 y Fk(+)876 600 y(\()p Fm(n)p Fk(\))995 572 y Fq(+)1081 510 y Fl(P)1106 647 y Fm(k)1182 572 y Fo(')1236 537 y Fg(\000)1236 600 y Fk(\()p Fm(k)q Fk(\))1329 572 y Fq(\()p Fo(r)r Fq(\)\011)1497 537 y Fg(\000)1497 600 y Fk(\()p Fm(k)q Fk(\))1591 572 y Fq(.)48 b(The)32 b(matrix)e(of)i(the)f(unp)r(erturb)r(ed)h(op)r (erator)e Fi(H)3253 584 y Fk(0)3320 572 y Fq(can)456 726 y(b)r(e)e(written)f(as)g(\([20],)h(p.128)e(and)h([21)o(],)h(p.22\)) 607 843 y Fl(2)607 989 y(6)607 1039 y(6)607 1088 y(6)607 1138 y(6)607 1188 y(6)607 1238 y(6)607 1291 y(4)845 906 y Fq(0)366 b(0)224 b Fo(:)14 b(:)g(:)83 b Fn(\000)1790 873 y Fm(d)p 1774 887 68 4 v 1774 934 a(dr)1869 906 y Fn(\000)1962 858 y Fm(\024)2001 869 y Fd(\(1\))p 1962 887 117 4 v 2004 934 a Fm(r)2346 906 y Fq(0)256 b Fo(:)14 b(:)g(:)845 1009 y Fq(0)366 b(0)224 b Fo(:)14 b(:)g(:)257 b Fq(0)g Fn(\000)2262 976 y Fm(d)p 2246 990 68 4 v 2246 1037 a(dr)2342 1009 y Fn(\000)2435 961 y Fm(\024)2474 972 y Fd(\(2\))p 2435 990 117 4 v 2477 1037 a Fm(r)2644 1009 y Fo(:)14 b(:)g(:)818 1108 y(:)g(:)g(:)311 b(:)14 b(:)g(:)196 b(:)14 b(:)g(:)229 b(:)14 b(:)g(:)376 b(:)14 b(:)g(:)229 b(:)14 b(:)g(:)730 1179 y Fm(d)p 714 1193 68 4 v 714 1240 a(dr)809 1211 y Fn(\000)902 1164 y Fm(\024)941 1175 y Fd(\(1\))p 902 1192 117 4 v 944 1240 a Fm(r)1253 1211 y Fq(0)224 b Fo(:)14 b(:)g(:)257 b Fq(0)431 b(0)256 b Fo(:)14 b(:)g(:)845 1314 y Fq(0)1138 1281 y Fm(d)p 1122 1295 68 4 v 1122 1343 a(dr)1217 1314 y Fn(\000)1310 1267 y Fm(\024)1349 1278 y Fd(\(2\))p 1310 1295 117 4 v 1352 1343 a Fm(r)1519 1314 y Fo(:)g(:)g(:)257 b Fq(0)431 b(0)256 b Fo(:)14 b(:)g(:)818 1414 y(:)g(:)g(:)311 b(:)14 b(:)g(:)196 b(:)14 b(:)g(:)229 b(:)14 b(:)g(:)376 b(:)14 b(:)g(:)229 b(:)14 b(:)g(:)2782 843 y Fl(3)2782 989 y(7)2782 1039 y(7)2782 1088 y(7)2782 1138 y(7)2782 1188 y(7)2782 1238 y(7)2782 1291 y(5)2851 818 y(2)2851 964 y(6)2851 1014 y(6)2851 1063 y(6)2851 1113 y(6)2851 1163 y(6)2851 1213 y(6)2851 1263 y(6)2851 1316 y(4)2948 881 y Fo(')3002 846 y Fk(+)3002 910 y(\(1\))3092 881 y Fq(\()p Fo(r)r Fq(\))2948 996 y Fo(')3002 960 y Fk(+)3002 1024 y(\(2\))3092 996 y Fq(\()p Fo(r)r Fq(\))3024 1108 y Fo(:)g(:)g(:)2948 1210 y(')3002 1174 y Fg(\000)3002 1238 y Fk(\(1\))3092 1210 y Fq(\()p Fo(r)r Fq(\))2948 1324 y Fo(')3002 1288 y Fg(\000)3002 1352 y Fk(\(2\))3092 1324 y Fq(\()p Fo(r)r Fq(\))3024 1436 y Fo(:)g(:)g(:)3238 818 y Fl(3)3238 964 y(7)3238 1014 y(7)3238 1063 y(7)3238 1113 y(7)3238 1163 y(7)3238 1213 y(7)3238 1263 y(7)3238 1316 y(5)456 1601 y Fq(where)25 b Fo(\024)742 1616 y Fk(\()p Fm(n)p Fk(\))865 1601 y Fq(are)g(eigen)n(v)-5 b(alues)24 b(of)i Fo(s)p Fq(.)37 b(The)26 b(b)r(oundary)f(conditions)g(at)h Fo(r)g Fq(=)c(1)k(are)f Fo(')3066 1566 y Fg(\000)3066 1629 y Fk(\()p Fm(n)p Fk(\))3163 1601 y Fq(\(1\))e(=)g(0.)555 1720 y(No)n(w,)35 b(w)n(e)f(w)n(an)n(t)f(to)g(w)n(ork)g(with)h(the)g (new)g(Hilb)r(ert)h(space)e Fi(L)p Fq(,)i(the)g Fo(L)2796 1690 y Fk(2)2866 1720 y Fq(space)e(of)h(v)n(ector-)456 1819 y(functions)i(\010\()p Fo(r)r Fq(\))i(=)f(\()p Fo(')1211 1784 y Fk(+)1211 1848 y(\(1\))1300 1819 y Fq(\()p Fo(r)r Fq(\))p Fo(;)14 b(:)g(:)g(:)h(;)f(')1643 1784 y Fg(\000)1643 1848 y Fk(\(1\))1733 1819 y Fq(\()p Fo(r)r Fq(\))p Fo(;)g(:)g(:)g(:)q Fq(\))2003 1789 y Fm(t)2032 1819 y Fq(,)38 b(with)f(the)f(norm)f(giv)n (en)g(b)n(y)h(the)g(form)n(ula)456 1928 y(b)r(elo)n(w)1063 2119 y Fn(k)p Fq(\010\()p Fo(r)r Fq(\))p Fn(k)1310 2085 y Fk(2)1370 2119 y Fq(=)1485 1990 y Fg(1)1458 2006 y Fl(Z)1465 2242 y Fk(1)1555 2041 y Fl(X)1594 2215 y Fm(n)1689 2027 y Fl(h)1728 2119 y Fn(j)p Fo(')1805 2084 y Fk(+)1805 2148 y(\()p Fm(n)p Fk(\))1902 2119 y Fq(\()p Fo(r)r Fq(\))p Fn(j)2028 2085 y Fk(2)2085 2119 y Fq(+)18 b Fn(j)p Fo(')2245 2084 y Fg(\000)2245 2148 y Fk(\()p Fm(n)p Fk(\))2343 2119 y Fq(\()p Fo(r)r Fq(\))p Fn(j)2469 2085 y Fk(2)2508 2027 y Fl(i)2561 2119 y Fo(dr)26 b(<)c Fn(1)456 2354 y Fq(So,)30 b(the)h(free)f(m)n(ultidimensional)g(Dirac)g(op)r(erator)e (can)i(b)r(e)h(represen)n(ted)e(as)h(an)g(in\014nite)h(or-)456 2454 y(thogonal)d(sum)j(of)f(one-dimensional)e(Dirac)i(op)r(erators.)42 b(Using)30 b(elemen)n(tary)f(prop)r(erties)h(of)456 2553 y(the)e(P)n(auli)e(matrices,)h(w)n(e)g(get)1603 2673 y Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(U)1823 2639 y Fg(\003)1862 2673 y Fo(\014)t(U)33 b Fq(=)22 b Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\014)456 2810 y Fq(Consider)k(the)i(m)n(ultiplication)f(b)n (y)g Fo(v)s Fq(\()p Fo(x)p Fq(\))i(in)f([)p Fo(L)1952 2780 y Fk(2)1989 2810 y Fq(\(\006\)])2136 2780 y Fk(2)2201 2810 y Fq(as)f(a)g(self-adjoin)n(t)f(b)r(ounded)i(op)r(erator)456 2910 y(with)i(matrix)g Fn(\000)p Fo(b)p Fq(\()p Fo(r)r Fq(\),)h Fo(b)p Fq(\()p Fo(r)r Fq(\))e(=)d Fn(f)p Fo(b)1515 2922 y Fm(ij)1573 2910 y Fq(\()p Fo(r)r Fq(\))p Fn(g)p Fo(;)14 b(i;)g(j)33 b Fq(=)26 b(1)p Fo(;)14 b(:)g(:)g(:)g(;)g Fn(1)p Fq(.)44 b(Since)30 b(the)h(section)f(2)f(deals)h(with)456 3009 y(an)f(in)n(terv)-5 b(al)29 b([0)p Fo(;)14 b Fn(1)p Fq(\))29 b(rather)g(than)h([1)p Fo(;)14 b Fn(1)p Fq(\),)30 b(w)n(e)f(shift)h(the)g(argumen)n(t)f Fo(r)j Fq(b)n(y)e(one.)42 b(W)-7 b(e)30 b(end)g(up)456 3109 y(with)e(the)g(canonical)f(represen)n (tation)f(\(3\),)i(where)f Fo(a)p Fq(\()p Fo(r)r Fq(\))i(is)e(an)h(un)n (b)r(ounded)g(op)r(erator)e(in)i(the)456 3209 y(diagonal)g(form)h(for)g (eac)n(h)f Fo(r)h(>)d Fq(0.)42 b(Notice)29 b(that)h(the)g(constan)n(t)e (function)i(\(1)p Fo(;)14 b Fq(0\))3006 3178 y Fm(t)3061 3209 y Fn(2)27 b Fq([)p Fo(L)3223 3178 y Fk(2)3260 3209 y Fq(\(\006\)])3407 3178 y Fk(2)456 3309 y Fq(is)k(an)h(eigenfunction)g (of)f(the)h(op)r(erator)e Fo(s)i Fq(corresp)r(onding)e(to)i(the)g (eigen)n(v)-5 b(alue)31 b(1.)49 b(F)-7 b(or)31 b(\011)3356 3274 y Fk(+)3356 3337 y(\(2\))456 3432 y Fq(and)38 b(\011)693 3396 y Fg(\000)693 3460 y Fk(\(1\))782 3432 y Fq(,)k(w)n(e)c(c)n(ho)r (ose)f(the)i(normed)g(v)n(ectors)e(collinear)g(to)i(\(1)p Fo(;)14 b Fq(0)p Fo(;)g Fq(0)p Fo(;)g Fq(0\))2812 3402 y Fm(t)2878 3432 y Fq(and)39 b(\(0)p Fo(;)14 b Fq(0)p Fo(;)g Fq(1)p Fo(;)g Fq(0\))3394 3402 y Fm(t)3421 3432 y Fq(,)456 3545 y(resp)r(ectiv)n(ely)-7 b(.)66 b(W)-7 b(e)38 b(ha)n(v)n(e)f Fo(\024)1367 3560 y Fk(\(1\))1496 3545 y Fq(=)i(1)f(and)f(the)i(function)f Fo(f)9 b Fq(,)40 b(giv)n(en)d(b)n(y)g(\(42\),)j(corresp)r(onds)456 3645 y(to)30 b(\010\()p Fo(r)r Fq(\))24 b(=)f(\(\()p Fo(r)f Fq(+)c(1\))p Fo(f)9 b Fq(\()p Fo(r)21 b Fq(+)d(1\))p Fo(;)c Fq(0)p Fo(;)g(:)g(:)g(:)e(;)i Fq(0)p Fo(;)g(:)g(:)g(:)p Fq(\))30 b(in)h(the)g(v)n(ector-v)-5 b(alued)28 b(represen)n(tation.)44 b(F)-7 b(rom)456 3745 y(no)n(w)27 b(on,)g(w)n(e)g(will)h(deal)f(with)h (the)g(represen)n(tation)e(\(3\))i(of)g(an)f(op)r(erator)f Fi(H)q Fq(.)555 3844 y(In)37 b(the)g(second)g(step)f(of)h(the)g(pro)r (of,)i(w)n(e)d(implemen)n(t)i(the)f(main)g(result)f(of)h(the)g(second) 456 3944 y(section,)28 b(theorem)g(2.2.)38 b(W)-7 b(e)29 b(can)f(not)g(do)g(that)h(directly)-7 b(,)28 b(b)r(ecause)g(w)n(e)g (are)f(dealing)h(with)h(an)456 4044 y(op)r(erator-v)-5 b(alued)25 b(co)r(e\016cien)n(t)i Fo(a)p Fq(\()p Fo(r)r Fq(\))i(un)n(b)r(ounded)e(for)g(eac)n(h)g Fo(r)f(>)c Fq(0.)37 b(T)-7 b(o)27 b(a)n(v)n(oid)e(this)j(di\016cult)n(y)-7 b(,)456 4143 y(w)n(e)40 b(consider)f(op)r(erators)f Fi(D)1376 4155 y Fm(n)1462 4143 y Fq(with)i Fo(b)1699 4155 y Fm(n)1788 4143 y Fq(=)k Fo(P)1950 4155 y Fm(n)1996 4143 y Fo(bP)2085 4155 y Fm(n)2130 4143 y Fq(,)f(where)d Fo(P)2502 4155 y Fm(n)2588 4143 y Fq(denotes)g(the)h(orthogonal)456 4243 y(pro)5 b(jection)26 b(on)n(to)h(the)h(linear)f(com)n(bination)g (of)h(the)g(\014rst)g Fo(n)f Fq(functions)h Fo( )2783 4258 y Fk(\()p Fm(k)q Fk(\))2876 4243 y Fq(.)38 b(The)27 b(matrix)h(of)456 4343 y Fo(b)492 4355 y Fm(n)536 4343 y Fq(\()p Fo(r)r Fq(\))h(is)1036 4662 y Fo(b)1072 4674 y Fm(n)1117 4662 y Fq(\()p Fo(r)r Fq(\))c(=)1332 4395 y Fl(2)1332 4542 y(6)1332 4591 y(6)1332 4641 y(6)1332 4691 y(6)1332 4744 y(4)1433 4462 y Fo(b)1469 4474 y Fk(11)1539 4462 y Fq(\()p Fo(r)r Fq(\))88 b Fn(\001)14 b(\001)g(\001)101 b Fo(b)1964 4474 y Fk(1)p Fm(n)2042 4462 y Fq(\()p Fo(r)r Fq(\))123 b(0)117 b Fn(\001)14 b(\001)g(\001)1483 4562 y(\001)g(\001)g(\001)150 b(\001)14 b(\001)g(\001)155 b(\001)14 b(\001)g(\001)154 b(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)1429 4661 y Fo(b)1465 4673 y Fm(n)p Fk(1)1543 4661 y Fq(\()p Fo(r)r Fq(\))84 b Fn(\001)14 b(\001)g(\001)97 b Fo(b)1960 4673 y Fm(nn)2046 4661 y Fq(\()p Fo(r)r Fq(\))119 b(0)e Fn(\001)14 b(\001)g(\001)1518 4761 y Fq(0)170 b Fn(\001)14 b(\001)g(\001)190 b Fq(0)209 b(0)117 b Fn(\001)14 b(\001)g(\001)1483 4860 y(\001)g(\001)g(\001)150 b(\001)14 b(\001)g(\001)155 b(\001)14 b(\001)g(\001)154 b(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)2565 4395 y Fl(3)2565 4542 y(7)2565 4591 y(7)2565 4641 y(7)2565 4691 y(7)2565 4744 y(5)2635 4662 y Fo(;)g(r)25 b(>)e Fq(0)456 5016 y(Eac)n(h)31 b(of)i Fi(D)836 5028 y Fm(n)915 5016 y Fq(can)f(b)r(e)i(written)f(as)f(an)h(orthogonal)e(sum)i(of)g (the)g(Dirac)f(op)r(erator)3150 4994 y(^)3136 5016 y Fi(D)3205 5028 y Fm(n)3283 5016 y Fq(with)456 5116 y(matrix-v)-5 b(alued)17 b(p)r(oten)n(tial)h Fo(V)1366 5128 y Fm(n)1430 5116 y Fq(of)h(size)f Fo(n)p Fn(\002)p Fo(n)f Fq(and)i(an)f(in\014nite) h(n)n(um)n(b)r(er)f(of)g(the)h(one-dimensional)456 5216 y(Dirac)28 b(op)r(erators)g(with)i(scalar)d(p)r(oten)n(tials.)42 b(Denote)30 b(b)n(y)f Fi(L)2364 5228 y Fm(n)2439 5216 y Fq(the)h(subspace)e(of)i Fi(L)f Fq(on)h(whic)n(h)p eop %%Page: 13 13 13 12 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)221 b(13)469 428 y Fq(^)456 450 y Fi(D)525 462 y Fm(n)597 450 y Fq(acts.)37 b(Matrix)27 b Fo(V)1126 462 y Fm(n)1199 450 y Fq(has)g(the)h(follo)n(wing)f(form)1342 648 y Fo(V)1390 660 y Fm(n)1458 648 y Fq(=)1546 531 y Fl(\024)1635 597 y Fn(\000)p Fo(b)1736 609 y Fm(n)1781 597 y Fq(\()p Fo(r)r Fq(\))88 b Fn(\000)p Fo(a)2081 609 y Fm(n)2126 597 y Fq(\()p Fo(r)r Fq(\))1631 697 y Fn(\000)p Fo(a)1740 709 y Fm(n)1785 697 y Fq(\()p Fo(r)r Fq(\))121 b Fo(b)2045 709 y Fm(n)2090 697 y Fq(\()p Fo(r)r Fq(\))2272 531 y Fl(\025)2329 648 y Fo(;)14 b(r)26 b(>)d Fq(0)456 846 y(with)1325 1050 y Fo(a)1369 1062 y Fm(n)1414 1050 y Fq(\()p Fo(r)r Fq(\))i(=)1629 884 y Fl(2)1629 1033 y(4)1736 897 y Fm(\024)1775 908 y Fd(\(1\))p 1736 925 117 4 v 1736 973 a Fm(r)r Fk(+1)1980 944 y Fq(0)118 b Fn(\001)14 b(\001)g(\001)148 b Fq(0)1739 1050 y Fn(\001)14 b(\001)g(\001)110 b(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)120 b(\001)14 b(\001)g(\001)1774 1152 y Fq(0)130 b Fn(\001)14 b(\001)g(\001)131 b Fq(0)2343 1105 y Fm(\024)2382 1116 y Fd(\()p Fc(n)p Fd(\))p 2343 1133 125 4 v 2347 1180 a Fm(r)r Fk(+1)2519 884 y Fl(3)2519 1033 y(5)456 1293 y Fq(Notice)34 b(that)h(the)g(function)g(\010\()p Fo(r)r Fq(\))h(lies)f(in)g Fi(L)1905 1305 y Fm(n)1985 1293 y Fq(for)f(an)n(y)g Fo(n)p Fq(.)58 b(Denote)34 b(the)h(sp)r (ectral)f(matrix-)456 1403 y(v)-5 b(alued)37 b(measure)f(of)1177 1380 y(^)1163 1403 y Fi(D)1232 1415 y Fm(n)1314 1403 y Fq(b)n(y)h Fo(\033)1486 1415 y Fm(n)1532 1403 y Fq(\()p Fo(\025)p Fq(\).)66 b(Since)38 b Fo(b)p Fq(\()p Fo(r)r Fq(\))i(=)e(0)f(on)g([0)p Fo(;)14 b Fq(1],)38 b(the)g(sp)r(ectral)f (measure)456 1502 y Fo(d\026)549 1514 y Fm(n)594 1502 y Fq(\()p Fo(\025)p Fq(\))29 b(of)e(\010\()p Fo(r)r Fq(\))i(is)f(equal) f(to)g Fn(j)p Fo(\032)p Fq(\()p Fo(\025)p Fq(\))p Fn(j)1626 1472 y Fk(2)1665 1502 y Fq(\()p Fo(d\033)1787 1514 y Fm(n)1833 1502 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1984 1514 y Fk(1)2022 1502 y Fo(;)14 b(e)2098 1514 y Fk(1)2134 1502 y Fq(\),)28 b(where)1084 1760 y Fo(\032)p Fq(\()p Fo(\025)p Fq(\))c(=)1394 1630 y Fk(1)1351 1647 y Fl(Z)1357 1882 y Fk(0)1434 1760 y Fq(\()p Fo(r)d Fq(+)d(1\))p Fo(f)9 b Fq(\()p Fo(r)21 b Fq(+)d(1\))1992 1643 y Fl(\024)2035 1760 y Fq(cos\()p Fo(\025r)r Fq(\))i(+)2411 1704 y(sin\()p Fo(\025r)r Fq(\))p 2411 1741 255 4 v 2514 1817 a Fo(\025)2675 1643 y Fl(\025)2733 1760 y Fo(dr)456 2009 y Fq(Here,)907 2172 y(cos)o(\()p Fo(\025r)r Fq(\))h(+)1283 2116 y(sin\()p Fo(\025r)r Fq(\))p 1283 2153 V 1386 2229 a Fo(\025)1547 2172 y(;)1677 2116 y(r)c Fq(cos)o(\()p Fo(\025r)r Fq(\))p 1677 2153 318 4 v 1688 2229 a Fo(\025)p Fq(\()p Fo(r)22 b Fq(+)c(1\))2023 2172 y Fn(\000)g Fq(sin\()p Fo(\025r)r Fq(\))2373 2055 y Fl(\022)2436 2172 y Fq(1)g(+)2734 2116 y(1)p 2589 2153 333 4 v 2589 2229 a Fo(\025)2637 2205 y Fk(2)2674 2229 y Fq(\()p Fo(r)k Fq(+)c(1\))2932 2055 y Fl(\023)456 2357 y Fq(are)26 b(generalized)g(eigenfunctions)i(for)f (the)h(Dirac)f(op)r(erator)f(\(3\))i(with)g(p)r(oten)n(tial)1456 2634 y Fo(V)42 b Fq(=)1633 2442 y Fl(2)1633 2588 y(6)1633 2641 y(4)1843 2549 y Fq(0)195 b Fn(\000)2225 2493 y Fq(1)p 2155 2530 183 4 v 2155 2606 a Fo(r)21 b Fq(+)d(1)1730 2723 y Fn(\000)1875 2667 y Fq(1)p 1805 2704 V 1805 2780 a Fo(r)j Fq(+)d(1)2193 2723 y(0)2389 2442 y Fl(3)2389 2588 y(7)2389 2641 y(5)456 2911 y Fq(F)-7 b(rom)27 b(the)h(pro)r(of)f (of)g(theorem)g(2.2)g(\(estimate)h(\(37\)\),)g(w)n(e)f(get)908 3043 y Fl(Z)903 3281 y Fk(\001)1004 3156 y Fq(ln)q(\()p Fo(\033)1156 3122 y Fg(0)1153 3176 y Fm(n)1198 3156 y Fq(\()p Fo(\025)p Fq(\))p Fo(e)1349 3168 y Fk(1)1387 3156 y Fo(;)14 b(e)1463 3168 y Fk(1)1500 3156 y Fq(\))p Fo(d\025)24 b Fn(\025)f Fo(C)1794 3168 y Fk(1)1850 3156 y Fq(+)18 b Fo(C)1992 3168 y Fk(2)2070 3026 y Fg(1)2043 3043 y Fl(Z)2050 3279 y Fk(0)2140 3156 y Fn(k)p Fq(\()p Fo(b)2250 3168 y Fm(n)2295 3156 y Fq(\()p Fo(r)r Fq(\))i(+)e Fo(ia)2574 3168 y Fm(n)2618 3156 y Fq(\()p Fo(r)r Fq(\)\))p Fo(e)2792 3168 y Fk(1)2831 3156 y Fn(k)2873 3122 y Fk(2)2910 3156 y Fo(dr)307 b Fq(\(44\))456 3408 y(for)27 b(an)n(y)f(\014nite)j (in)n(terv)-5 b(al)27 b(\001)c Fn(2)g Fj(R)p Fq(.)43 b(Notice)28 b(that)1386 3519 y Fg(1)1359 3536 y Fl(Z)1366 3771 y Fk(0)1456 3649 y Fn(k)p 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Fk(0)1645 4369 y Fl(Z)1604 4607 y Fm(\034)7 b Fg(2)p Fk(\006)1766 4482 y Fo(v)1809 4447 y Fk(2)1846 4482 y Fq(\(\()p Fo(r)22 b Fq(+)c(1\))p Fo(\034)9 b Fq(\))p Fo(d\034)g(dr)28 b Fn(\024)22 b Fo(C)2565 4369 y Fl(Z)2565 4607 y Fk(\012)2713 4426 y Fo(v)2756 4395 y Fk(2)2793 4426 y Fq(\()p Fo(x)p Fq(\))p 2672 4463 274 4 v 2672 4539 a Fn(j)p Fo(x)p Fn(j)2765 4515 y Fk(2)2821 4539 y Fq(+)c(1)2956 4482 y Fo(dx)456 4721 y Fq(and)27 b(w)n(e)g(clearly)g (ha)n(v)n(e)1493 4763 y Fl(Z)1489 5002 y Fk(\001)1590 4876 y Fq(ln)14 b Fo(\026)1723 4842 y Fg(0)1723 4897 y Fm(n)1768 4876 y Fq(\()p Fo(\025)p Fq(\))p Fo(d\025)25 b Fn(\025)d Fo(C)30 b(>)22 b Fn(\0001)890 b Fq(\(45\))456 5116 y(with)30 b(some)f(constan)n(t)g Fo(C)35 b Fq(indep)r(enden)n(t)c (of)e Fo(n)p Fq(.)43 b(Notice)30 b(that)g Fo(b)2454 5128 y Fm(n)2525 5116 y Fn(!)c Fo(b)k Fq(strongly)-7 b(.)41 b(So,)30 b Fo(d\026)3286 5128 y Fm(n)3331 5116 y Fq(\()p Fo(\025)p Fq(\))456 5216 y(con)n(v)n(erges)16 b(w)n(eakly)i(to)h Fo(d\026)p Fq(,)i(the)e(sp)r(ectral)g(measure)f(of)h Fo(f)27 b Fq(with)20 b(resp)r(ect)f(to)g(the)g(initial)g(op)r(erator)p eop %%Page: 14 14 14 13 bop 456 251 a Fk(14)1084 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)456 450 y Fi(H)q Fq(.)35 b(The)28 b(semicon)n(tin)n(uit)n(y)f(of)g(the)h (en)n(trop)n(y)f([8)o(])h(implies)1504 514 y Fl(Z)1500 752 y Fk(\001)1601 627 y Fq(ln)14 b Fo(\026)1734 593 y Fg(0)1757 627 y Fq(\()p Fo(\025)p Fq(\))p Fo(d\025)25 b(>)d(C)30 b(>)22 b Fn(\0001)456 860 y Ff(\004)555 959 y Fq(The)g(reduction)g(of)f Fi(H)h Fq(to)g(a)f(one-dimensional)g (system)g(with)i(the)f(op)r(erator-v)-5 b(alued)19 b(p)r(oten-)456 1059 y(tial)29 b(is,)i(probably)-7 b(,)29 b(not)h(necessary)-7 b(.)42 b(One)29 b(could)h(ha)n(v)n(e)e(in)n(tro)r(duced)i(the)g (radiativ)n(e)e(op)r(erator)456 1158 y(and)e(w)n(ork)n(ed)f(with)i (this)f(op)r(erator)f(directly)h(a)n(v)n(oiding)f(an)h(appro)n (ximation)f(b)n(y)h(Dirac)g(op)r(era-)456 1258 y(tors)c(with)h (matrix-v)-5 b(alued)22 b(p)r(oten)n(tials)g(\(see)h([16)o(]\).)36 b(In)22 b(the)i(mean)n(time,)f(this)g(reduction)g(is)f(not)456 1358 y(to)r(o)29 b(di\016cult)i(and)f(it)h(sho)n(ws)e(ho)n(w)g(v)n(ery) g(general)g(facts)h(for)f(the)i(matrix-v)-5 b(alued)29 b(orthogonal)456 1457 y(systems)e(are)f(applied)i(to)f(di\013eren)n(t)h (PDE's.)555 1557 y(W)-7 b(e)24 b(do)g(not)g(consider)e(the)i(question)g (of)f(existence)h(of)g(w)n(a)n(v)n(e)e(op)r(erators.)33 b(It)24 b(migh)n(t)g(b)r(e)g(that)456 1657 y(the)k(problem)f(can)g(b)r (e)h(solv)n(ed)e(b)n(y)i(using)f(an)g(approac)n(h)f(of)i([7)o(].)555 1756 y(In)d(the)g(next)g(theorem,)g(w)n(e)g(establish)f(an)g (asymptotics)g(of)h(the)g(Green's)g(function)g(for)f(the)456 1856 y(op)r(erator)30 b Fi(H)870 1868 y Fm(s)904 1856 y Fq(.)50 b(F)-7 b(or)32 b Fi(H)q Fq(,)g(that)g(means)g(existence)g(of) g(the)g(function)h Fo(h)f Fq(\(see)g(\(29\)\))g(satisfying)456 1955 y(homogeneous)16 b(equation)i Fo(H)7 b(h)23 b Fq(=)f Fo(\025h;)14 b(\025)24 b Fn(2)g Fj(C)1856 1925 y Fk(+)1935 1955 y Fq(with)19 b(the)g(w)n(ell-con)n(trolled)e(\\amplitude".)33 b(Due)456 2055 y(to)e(\(30\))g(and)h(\(31\),)g(w)n(e)f(ha)n(v)n(e)f(a)h (certain)g(estimate)h(on)f(the)h(en)n(trop)n(y)e(of)i(the)f(corresp)r (onding)456 2155 y(sp)r(ectral)g(measure.)48 b(That,)33 b(in)f(particular,)g(giv)n(es)e(another)h(pro)r(of)g(of)h Fo(\033)2772 2167 y Fm(ac)2842 2155 y Fq(\()p Fi(H)q Fq(\))d(=)h Fj(R)38 b Fq(for)31 b(the)456 2254 y(case)26 b(of)i(a)f(p)r(o)n(w)n(er-deca)n(y)-7 b(.)456 2384 y Fv(Theorem)42 b(3.2.)47 b Fh(Assume)39 b Fo(v)s Fq(\()p Fo(x)p Fq(\))h Fh(is)g(given)h(on)e Fj(R)2092 2354 y Fk(3)2175 2384 y Fh(and)h(satis\014es)g(an)g(estimate)g Fn(j)p Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fn(j)i Fo(<)456 2483 y(C)515 2495 y Fm(v)554 2483 y Fq(\()p Fn(j)p Fo(x)p Fn(j)22 b Fq(+)f(1\))861 2453 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)p Fg(\000)p Fm(")1086 2483 y Fo(;)33 b Fh(with)g(\014xe)l(d)g Fo(")c(>)g Fq(0)j Fh(and)i Fo(C)1980 2495 y Fm(v)2053 2483 y Fh(su\016ciently)f(smal)t(l.)50 b(Then,)35 b(the)e(r)l(esolvent) 456 2583 y(kernel)d Fo(G)768 2595 y Fm(\025)812 2583 y Fq(\()p Fo(x;)14 b Fq(0\))30 b Fh(of)g(op)l(er)l(ator)h Fo(H)1525 2595 y Fm(s)1590 2583 y Fh(at)f(a)g(p)l(oint)g Fo(\025)24 b Fq(=)e Fo(i)30 b Fh(has)g(the)g(fol)t(lowing)i(r)l(epr)l (esentation)885 2777 y Fo(G)950 2789 y Fm(i)978 2777 y Fq(\()p Fo(x;)14 b Fq(0\))24 b(=)1296 2721 y Fo(e)1335 2691 y Fg(\000j)p Fm(x)p Fg(j)p 1289 2758 186 4 v 1289 2834 a Fq(4)p Fo(\031)s Fn(j)p Fo(x)p Fn(j)1499 2660 y Fl(\024)o(\022)1604 2777 y Fo(i)1643 2721 y(\013)18 b Fn(\001)h Fo(x)p 1643 2758 161 4 v 1676 2834 a Fn(j)p Fo(x)p Fn(j)1831 2777 y Fq(+)f Fo(i)1943 2660 y Fl(\023)2018 2777 y Fi(P)2073 2789 y Fk(1)2110 2777 y Fq(\()p Fo(x)p Fq(\))i(+)e(\()p Fn(j)p Fo(x)p Fn(j)h Fq(+)f(1\))2625 2743 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)2767 2777 y Fi(P)2822 2789 y Fk(2)2859 2777 y Fq(\()p Fo(x)p Fq(\))2970 2660 y Fl(\025)3297 2777 y Fq(\(46\))456 2957 y Fh(wher)l(e)1737 3056 y Fn(j)p Fi(P)1815 3068 y Fk(2)1852 3056 y Fq(\()p Fo(x)p Fq(\))p Fn(j)24 b Fo(<)f(C)1140 b Fq(\(47\))456 3173 y Fh(uniformly)31 b(in)f Fj(R)991 3143 y Fk(3)1034 3173 y Fh(,)h Fn(k)p Fi(P)1187 3185 y Fk(1)1242 3173 y Fn(\000)18 b Fq(1)p Fn(k)23 b Fo(<)h(\016)s Fh(,)30 b Fo(\016)d Fn(!)d Fq(0)30 b Fh(as)g Fo(C)2024 3185 y Fm(v)2088 3173 y Fn(!)23 b Fq(0)p Fh(.)40 b(Positive)31 b(c)l(onstant)f Fo(C)36 b Fh(dep)l(ends)31 b(on)456 3273 y Fo(C)515 3285 y Fm(v)584 3273 y Fh(and)f Fo(")g Fh(only.)555 3402 y Fq(W)-7 b(e)36 b(need)f(the)g(smallness)f(of)h Fo(C)1591 3414 y Fm(v)1666 3402 y Fq(to)f(guaran)n(tee)f(con)n(v)n (ergence)g(of)i(a)f(certain)h(series.)57 b(In)456 3502 y(general)31 b(situation,)j(one)f(can)g(tak)n(e)f(the)h(sp)r(ectral)g (parameter)e Fo(\025)j Fq(su\016cien)n(tly)f(far)f(from)h(the)456 3602 y(sp)r(ectrum.)h(Or,)22 b(w)n(e)f(can)f(tak)n(e)g Fo(\025)k Fq(=)e Fo(i)p Fq(,)h(and)d(let)i(p)r(oten)n(tial)e(b)r(e)i (zero)e(on)g(the)i(large)d(ball)i(cen)n(tered)456 3701 y(at)h(origin.)34 b(That)22 b(w)n(ould)g(mak)n(e)f(constan)n(t)h Fo(C)1857 3713 y Fm(v)1919 3701 y Fq(as)g(small)f(as)h(w)n(e)g(w)n(an)n (t)2617 3669 y Fk(3)2672 3701 y Fq(but)h(w)n(ould)e(not)i(c)n(hange)456 3801 y(the)28 b(scattering)e(picture.)555 3901 y(Let)i(us)g(pro)n(v)n (e)d(some)i(auxiliary)g(lemmas)g(\014rst.)456 4030 y Fv(Lemma)i(3.3.)40 b Fh(The)31 b(fol)t(lowing)h(estimate)e(holds)h Fq(\(1)23 b Fo(<)g(\032)g Fn(\024)f Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(3\))1453 4094 y Fl(Z)1396 4336 y Fg(j)p Fm(y)r Fg(j)p Fk(=)p Fm(\032)1589 4207 y Fo(e)1628 4173 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j\000j)p Fm(y)r Fg(j)1975 4207 y Fo(d\034)2054 4219 y Fm(y)2118 4207 y Fo(<)h(C)6 b(\032e)2353 4173 y Fg(\000j)p Fm(x)p Fg(j)3297 4207 y Fq(\(48\))1236 4402 y Fl(Z)1179 4644 y Fg(j)p Fm(y)r Fg(j)p Fk(=)p Fm(\032)1372 4515 y Fo(e)1411 4481 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j\000j)p Fm(y)r Fg(j)1772 4515 y Fq(sin)14 b Fo(\020)6 b Fq(\()p Fo(x;)14 b(y)s Fq(\))p Fo(d\034)2201 4527 y Fm(y)2266 4515 y Fo(<)22 b(C)2418 4459 y Fn(p)p 2488 4459 43 4 v 2488 4515 a Fo(\032e)2570 4481 y Fg(\000j)p Fm(x)p Fg(j)1274 4692 y Fl(Z)1217 4934 y Fg(j)p Fm(y)r Fg(j)p Fk(=)p Fm(\032)1409 4805 y Fo(e)1448 4771 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j\000j)p Fm(y)r Fg(j)1810 4805 y Fq(sin)1912 4771 y Fk(2)1963 4805 y Fo(\020)6 b Fq(\()p Fo(x;)14 b(y)s Fq(\))p Fo(d\034)2276 4817 y Fm(y)2340 4805 y Fo(<)23 b(C)6 b(e)2532 4771 y Fg(\000j)p Fm(x)p Fg(j)456 5045 y Fh(wher)l(e)30 b Fo(\020)6 b Fq(\()p Fo(x;)14 b(y)s Fq(\))31 b Fh(is)f(an)g(angle)g(b)l(etwe)l(en) g Fo(x)g Fh(and)g Fo(y)s Fh(.)p 456 5124 499 4 v 555 5197 a Fk(3)624 5223 y Fu(With)24 b(resp)r(ect)g(to)h(a)e(smaller)f Fb(")p Fu(.)p eop %%Page: 15 15 15 14 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)221 b(15)555 450 y Fq(Pro)r(of.)50 b(Without)33 b(loss)f(of)g(generalit)n(y)-7 b(,)32 b(assume)g(that)g Fo(x)g Fq(=)e(\(0)p Fo(;)14 b Fq(0)p Fo(;)g Fn(j)p Fo(x)p Fn(j)p Fq(\).)51 b(In)n(tro)r(ducing)32 b(the)456 550 y(spherical)26 b(co)r(ordinates)g Fo(y)1284 562 y Fk(1)1344 550 y Fq(=)d Fo(\032)14 b Fq(cos)f Fo(\022)j Fq(cos)d Fo(';)h(y)1926 562 y Fk(2)1986 550 y Fq(=)23 b Fo(\032)14 b Fq(cos)f Fo(\022)j Fq(sin)d Fo(';)h(y)2558 562 y Fk(3)2619 550 y Fq(=)22 b Fo(\032)14 b Fq(sin)g Fo(\022)r(;)28 b Fq(w)n(e)f(get)748 817 y Fo(\032)791 782 y Fk(2)886 687 y Fm(\031)847 704 y Fl(Z)823 938 y Fg(\000)p Fm(\031)948 817 y Fo(d')1104 680 y Fm(\031)r(=)p Fk(2)1098 704 y Fl(Z)1040 945 y Fg(\000)p Fm(\031)r(=)p Fk(2)1234 817 y Fo(d\022)16 b Fq(cos)d Fo(\022)j Fq(exp\()p Fn(\000)p Fo(\032)i Fn(\000)1880 741 y Fl(p)p 1963 741 763 4 v 76 x Fn(j)p Fo(x)p Fn(j)2056 793 y Fk(2)2112 817 y Fq(+)g Fo(\032)2238 793 y Fk(2)2294 817 y Fn(\000)g Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(\032)c Fq(sin)g Fo(\022)r Fq(\))600 1190 y Fo(<)83 b(C)6 b(\032)856 1156 y Fk(2)893 1190 y Fo(e)932 1156 y Fg(\000j)p Fm(x)p Fg(j)1123 1053 y Fm(\031)r(=)p Fk(2)1117 1077 y Fl(Z)1060 1319 y Fg(\000)p Fm(\031)r(=)p Fk(2)1253 1190 y Fo(d\022)16 b 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y Fo(x)p 1835 1845 94 4 v 1835 1921 a Fn(j)p Fo(x)p Fn(j)1938 1744 y Fl(\014)1938 1793 y(\014)1938 1843 y(\014)1938 1893 y(\014)1989 1864 y Fo(<)c(C)d Fq(sin)14 b Fo(\020)989 b Fq(\(49\))456 2067 y(T)-7 b(ogether)28 b(with)h Fo(\020)6 b Fq(\()p Fo(x;)14 b(y)s Fq(\),)31 b(consider)d Fo(\037)p Fq(\()p Fo(x;)14 b(y)s Fq(\))30 b({)e(an)h(angle)g(b)r(et)n(w)n(een)g Fo(x)20 b Fn(\000)f Fo(y)31 b Fq(and)e Fo(x)p Fq(.)42 b(F)-7 b(rom)29 b(the)456 2167 y(sine-theorem,)d(sin)14 b Fo(\037)23 b Fq(=)g Fn(j)p Fo(y)e Fn(\000)d Fo(x)p Fn(j)1485 2137 y Fg(\000)p Fk(1)1575 2167 y Fn(j)p Fo(y)s Fn(j)c Fq(sin)f Fo(\020)6 b Fq(.)37 b(Consequen)n(tly)-7 b(,)1124 2249 y Fl(\014)1124 2299 y(\014)1124 2349 y(\014)1124 2399 y(\014)1185 2314 y Fo(x)19 b Fn(\000)f Fo(y)p 1162 2351 239 4 v 1162 2427 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1429 2370 y(\000)1545 2314 y Fo(x)p 1522 2351 94 4 v 1522 2427 a Fn(j)p Fo(x)p Fn(j)1625 2249 y Fl(\014)1625 2299 y(\014)1625 2349 y(\014)1625 2399 y(\014)1676 2370 y Fo(<)23 b(C)d Fq(sin)13 b Fo(\037)23 b Fq(=)g Fo(C)6 b Fn(j)p Fo(y)22 b Fn(\000)c Fo(x)p Fn(j)2425 2336 y Fg(\000)p Fk(1)2514 2370 y Fn(j)p Fo(y)s Fn(j)c Fq(sin)g Fo(\020)527 b Fq(\(50\))456 2568 y(By)27 b(the)h(triangle)f(inequalit)n (y)-7 b(,)1127 2646 y Fl(\014)1127 2696 y(\014)1127 2746 y(\014)1127 2796 y(\014)1188 2711 y Fo(y)p 1164 2748 90 4 v 1164 2824 a Fn(j)p Fo(y)s Fn(j)1283 2767 y(\000)1399 2711 y Fo(x)19 b Fn(\000)f Fo(y)p 1376 2748 239 4 v 1376 2824 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1624 2646 y Fl(\014)1624 2696 y(\014)1624 2746 y(\014)1624 2796 y(\014)1675 2767 y Fn(\024)1763 2646 y Fl(\014)1763 2696 y(\014)1763 2746 y(\014)1763 2796 y(\014)1823 2711 y Fo(y)p 1800 2748 90 4 v 1800 2824 a Fn(j)p Fo(y)s Fn(j)1919 2767 y(\000)2035 2711 y Fo(x)p 2012 2748 94 4 v 2012 2824 a Fn(j)p Fo(x)p Fn(j)2115 2646 y Fl(\014)2115 2696 y(\014)2115 2746 y(\014)2115 2796 y(\014)2161 2767 y Fq(+)2244 2646 y Fl(\014)2244 2696 y(\014)2244 2746 y(\014)2244 2796 y(\014)2305 2711 y Fo(x)p 2282 2748 V 2282 2824 a Fn(j)p Fo(x)p Fn(j)2404 2767 y(\000)2520 2711 y Fo(x)h Fn(\000)f Fo(y)p 2497 2748 239 4 v 2497 2824 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2746 2646 y Fl(\014)2746 2696 y(\014)2746 2746 y(\014)2746 2796 y(\014)3297 2767 y Fq(\(51\))456 2970 y(F)-7 b(or)27 b Fn(j)p Fo(y)s Fn(j)22 b Fo(<)h Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(3,)1513 3018 y Fl(\014)1513 3068 y(\014)1513 3118 y(\014)1513 3167 y(\014)1574 3082 y Fo(y)p 1551 3119 90 4 v 1551 3195 a Fn(j)p Fo(y)s Fn(j)1669 3138 y(\000)1785 3082 y Fo(x)c Fn(\000)f Fo(y)p 1762 3119 239 4 v 1762 3195 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2011 3018 y Fl(\014)2011 3068 y(\014)2011 3118 y(\014)2011 3167 y(\014)2062 3138 y Fn(\024)k Fo(C)e Fq(sin)14 b Fo(\020)917 b Fq(\(52\))456 3324 y(Let)27 b Fn(j)p Fo(x)p Fn(j)d Fo(>)f Fq(1)k(and)g(\007)c(=)g Fn(f)p Fo(y)i Fq(:)e Fn(j)p Fo(y)s Fn(j)g Fo(>)f Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(3)p Fo(;)14 b Fn(j)p Fo(x)k Fn(\000)g Fo(y)s Fn(j)23 b Fo(>)g Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(3)p Fn(g)p Fq(.)456 3484 y Fv(Lemma)29 b(3.4.)40 b Fh(The)31 b(fol)t(lowing)h(estimate)e(holds)1222 3562 y Fl(Z)1219 3800 y Fk(\007)1319 3675 y Fq(exp)o(\()p Fn(\000j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)g(\000)g(j)p Fo(y)s Fn(j)p Fq(\))p Fo(dy)26 b Fn(\024)d Fo(C)d Fq(exp\()p Fn(\000)p Fo(\015)5 b Fn(j)p Fo(x)p Fn(j)p Fq(\))619 b(\(53\))456 3928 y Fh(with)30 b Fo(\015)e(>)22 b Fq(1)p Fh(.)555 4088 y Fq(Pro)r(of.)36 b(Indeed,)28 b(in)g(\007,)1392 4242 y Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)g Fq(+)g Fn(j)p Fo(y)s Fn(j)23 b Fo(>)g Fn(j)p Fo(y)s Fn(j)p Fo(=)p Fq(5)17 b(+)h(16)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(15)456 4395 y(T)-7 b(aking)26 b Fo(\015)i Fq(=)23 b(16)p Fo(=)p Fq(15,)i(w)n(e)i(obtain)g(the)h(statemen)n(t)g(of)g(the)g(lemma.)36 b Ff(\004)555 4495 y Fq(In)c(the)h(follo)n(wing)d(three)i(lemmas,)h(w)n (e)e(will)h(b)r(e)h(estimating)e(certain)h(in)n(tegrals)e(o)n(v)n(er)g (the)456 4594 y Fj(R)510 4564 y Fk(3)553 4594 y Fq(.)37 b(Lemma)26 b(3.4)h(sho)n(ws)f(that)h(the)h(con)n(tribution)e(coming)h (from)f(the)i(in)n(tegration)e(o)n(v)n(er)f(\007)i(is)456 4694 y(small)g(and)g(can)g(b)r(e)h(neglected.)456 4854 y Fv(Lemma)h(3.5.)40 b Fh(The)31 b(fol)t(lowing)h(b)l(ound)e(is)g(true) 1216 4949 y Fl(Z)1262 5138 y Fa(R)1309 5121 y Fd(3)1372 5006 y Fo(e)1411 4976 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1364 5043 276 4 v 1364 5119 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1603 5095 y Fk(2)1778 5006 y Fo(e)1817 4976 y Fg(\000j)p Fm(y)r Fg(j)p 1660 5043 406 4 v 1660 5119 a Fn(j)p Fo(y)s Fn(j)1750 5095 y Fk(1)p Fm(:)p Fk(5+)p Fm(")1941 5119 y Fq(+)g(1)2075 5062 y Fo(dy)26 b(<)d(C)2425 5006 y(e)2464 4976 y Fg(\000j)p Fm(x)p Fg(j)p 2348 5043 327 4 v 2348 5119 a Fn(j)p Fo(x)p Fn(j)2441 5095 y Fk(1)p Fm(:)p Fk(5)2550 5119 y Fq(+)18 b(1)p eop %%Page: 16 16 16 15 bop 456 251 a Fk(16)1084 b(SER)n(GEY)22 b(A.)g(DENISO)n(V)555 450 y Fq(Pro)r(of.)36 b(By)27 b(lemma)h(3.3,)456 632 y Fl(Z)502 821 y Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)842 689 y Fo(e)881 659 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 834 726 276 4 v 834 802 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1073 778 y Fk(2)1248 689 y Fo(e)1287 659 y Fg(\000j)p Fm(y)r Fg(j)p 1130 726 406 4 v 1130 802 a Fn(j)p Fo(y)s Fn(j)1220 778 y Fk(1)p Fm(:)p Fk(5+)p Fm(")1411 802 y Fq(+)g(1)1545 745 y Fo(dy)26 b(<)d(C)1869 689 y(e)1908 659 y Fg(\000j)p Fm(x)p Fg(j)p 1818 726 274 4 v 1818 802 a Fn(j)p Fo(x)p Fn(j)1911 778 y Fk(2)1967 802 y Fq(+)18 b(1)2137 608 y Fg(j)p Fm(x)p Fg(j)2116 632 y Fl(Z)2122 868 y Fk(0)2380 689 y Fo(\032)p 2222 726 359 4 v 2222 802 a(\032)2265 778 y Fk(1)p Fm(:)p Fk(5+)p Fm(")2456 802 y Fq(+)g(1)2591 745 y Fo(d\032)23 b(<)f(C)2981 689 y(e)3020 659 y Fg(\000j)p Fm(x)p Fg(j)p 2863 726 409 4 v 2863 802 a Fn(j)p Fo(x)p Fn(j)2956 778 y Fk(1)p Fm(:)p Fk(5+)p Fm(")3147 802 y Fq(+)c(1)3282 745 y(;)619 926 y Fl(Z)665 1115 y Fg(j)p Fm(y)r Fg(\000)p Fm(x)p Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)1096 983 y Fo(e)1135 953 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1087 1020 276 4 v 1087 1096 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1326 1072 y Fk(2)1501 983 y Fo(e)1540 953 y Fg(\000j)p Fm(y)r Fg(j)p 1383 1020 406 4 v 1383 1096 a Fn(j)p Fo(y)s Fn(j)1473 1072 y Fk(1)p Fm(:)p Fk(5+)p Fm(")1664 1096 y Fq(+)g(1)1798 1039 y Fo(dy)26 b Fq(=)1996 926 y Fl(Z)2042 1115 y Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)2374 983 y Fo(e)2413 953 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 2374 1020 260 4 v 2441 1096 a Fn(j)p Fo(y)s Fn(j)2531 1072 y Fk(2)2846 983 y Fo(e)2885 953 y Fg(\000j)p Fm(y)r Fg(j)p 2654 1020 555 4 v 2654 1096 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)2893 1072 y Fk(1)p Fm(:)p Fk(5+)p Fm(")3083 1096 y Fq(+)g(1)3218 1039 y Fo(dy)1884 1351 y(<)k(C)2165 1295 y(e)2204 1265 y Fg(\000j)p Fm(x)p Fg(j)p 2047 1332 409 4 v 2047 1408 a Fn(j)p Fo(x)p Fn(j)2140 1384 y Fk(1)p Fm(:)p Fk(5+)p Fm(")2331 1408 y Fq(+)c(1)2501 1214 y Fg(j)p Fm(x)p Fg(j)2479 1238 y Fl(Z)2486 1474 y Fk(0)2636 1295 y Fo(d\032)p 2586 1332 186 4 v 2586 1408 a(\032)h Fq(+)f(1)2805 1351 y Fo(<)23 b(C)3045 1295 y(e)3084 1265 y Fg(\000j)p Fm(x)p Fg(j)p 2968 1332 327 4 v 2968 1408 a Fn(j)p Fo(x)p Fn(j)3061 1384 y Fk(1)p Fm(:)p Fk(5)3170 1408 y Fq(+)18 b(1)456 1628 y Ff(\004)456 1804 y Fv(Lemma)29 b(3.6.)40 b Fh(The)31 b(fol)t(lowing)h(r)l(elation)f(is)f(true)1181 1927 y Fl(Z)1227 2116 y Fa(R)1274 2099 y Fd(3)1329 1984 y Fo(e)1368 1954 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1329 2021 260 4 v 1339 2097 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1612 1923 y Fl(\022)1683 1984 y Fo(\013)h Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(y)s Fq(\))p 1683 2021 371 4 v 1749 2097 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2082 2040 y Fq(+)g(1)2207 1923 y Fl(\023)2383 1984 y Fo(e)2422 1954 y Fg(\000j)p Fm(y)r Fg(j)p 2292 2021 353 4 v 2292 2097 a Fn(j)p Fo(y)s Fn(j)2382 2073 y Fk(2+)p Fm(")2520 2097 y Fq(+)g(1)2654 2040 y Fo(dy)1033 2282 y Fq(=)1223 2226 y Fo(e)1262 2196 y Fg(\000j)p Fm(x)p Fg(j)p 1191 2263 237 4 v 1191 2339 a Fn(j)p Fo(x)p Fn(j)h Fq(+)f(1)1451 2165 y Fl(\024\022)1566 2226 y Fo(\013)h Fn(\001)f Fo(x)p 1566 2263 161 4 v 1599 2339 a Fn(j)p Fo(x)p Fn(j)1755 2282 y Fq(+)g(1)1880 2165 y Fl(\023)1955 2282 y Fo(')2009 2294 y Fk(1)2046 2282 y Fq(\()p Fo(x)p Fq(\))i(+)e(\()p Fn(j)p Fo(x)p Fn(j)h Fq(+)f(1\))2561 2248 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)2703 2282 y Fo(')2757 2294 y Fk(2)2794 2282 y Fq(\()p Fo(x)p Fq(\))2905 2165 y Fl(\025)456 2519 y Fh(wher)l(e)30 b Fo(')744 2534 y Fk(1\(2\))867 2519 y Fq(\()p Fo(x)p Fq(\))g Fh(ar)l(e)g (matrix-functions)g(uniformly)g(b)l(ounde)l(d)h(in)e Fj(R)2627 2489 y Fk(3)2671 2519 y Fh(.)555 2695 y Fq(Pro)r(of.)1043 2802 y Fl(Z)1090 2990 y Fa(R)1137 2974 y Fd(3)1192 2859 y Fo(e)1231 2829 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1192 2896 260 4 v 1202 2972 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1475 2798 y Fl(\022)1546 2859 y Fo(\013)h Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(y)s Fq(\))p 1546 2896 371 4 v 1612 2972 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1945 2915 y Fq(+)g(1)2070 2798 y Fl(\023)2246 2859 y Fo(e)2285 2829 y Fg(\000j)p Fm(y)r Fg(j)p 2155 2896 353 4 v 2155 2972 a Fn(j)p Fo(y)s Fn(j)2245 2948 y Fk(2+)p Fm(")2382 2972 y Fq(+)g(1)2517 2915 y Fo(dy)817 3157 y Fq(=)904 3044 y Fl(Z)950 3232 y Fa(R)997 3216 y Fd(3)1053 3101 y Fo(e)1092 3070 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1053 3138 260 4 v 1063 3214 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1336 3040 y Fl(\022)1407 3101 y Fo(\013)h Fn(\001)f Fq(\()p Fo(x)i Fn(\000)e Fo(y)s Fq(\))p 1407 3138 371 4 v 1473 3214 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1806 3157 y(\000)1899 3101 y Fo(\013)h Fn(\001)f Fo(x)p 1899 3138 161 4 v 1933 3214 a Fn(j)p Fo(x)p Fn(j)2070 3040 y Fl(\023)2246 3101 y Fo(e)2285 3070 y Fg(\000j)p Fm(y)r Fg(j)p 2155 3138 353 4 v 2155 3214 a Fn(j)p Fo(y)s Fn(j)2245 3190 y Fk(2+)p Fm(")2382 3214 y Fq(+)g(1)2517 3157 y Fo(dy)1189 3399 y Fq(+)1268 3282 y Fl(\022)1338 3343 y Fo(\013)h Fn(\001)f Fo(x)p 1338 3380 161 4 v 1372 3456 a Fn(j)p Fo(x)p Fn(j)1527 3399 y Fq(+)g(1)1652 3282 y Fl(\023)1727 3286 y(Z)1773 3474 y Fa(R)1820 3458 y Fd(3)1875 3343 y Fo(e)1914 3312 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1875 3380 260 4 v 1886 3456 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2246 3343 y Fo(e)2285 3312 y Fg(\000j)p Fm(y)r Fg(j)p 2155 3380 353 4 v 2155 3456 a Fn(j)p Fo(y)s Fn(j)2245 3432 y Fk(2+)p Fm(")2382 3456 y Fq(+)g(1)2517 3399 y Fo(dy)86 b Fq(=)c Fo(I)2870 3411 y Fk(1)2927 3399 y Fq(+)18 b Fo(I)3046 3411 y Fk(2)456 3625 y Fq(F)-7 b(ollo)n(wing)26 b(the)i(pro)r(of)f(of)h(lemma)f(3.5,)g(w)n(e)g(get) 1110 3748 y Fl(Z)1157 3937 y Fa(R)1204 3920 y Fd(3)1259 3805 y Fo(e)1298 3775 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1259 3842 260 4 v 1269 3918 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1629 3805 y Fo(e)1668 3775 y Fg(\000j)p Fm(y)r Fg(j)p 1538 3842 353 4 v 1538 3918 a Fn(j)p Fo(y)s Fn(j)1628 3894 y Fk(2+)p Fm(")1766 3918 y Fq(+)g(1)1901 3861 y Fo(dy)26 b(<)c(C)2206 3805 y(e)2245 3775 y Fg(\000j)p Fm(x)p Fg(j)p 2173 3842 237 4 v 2173 3918 a Fn(j)p Fo(x)p Fn(j)d Fq(+)f(1)2503 3861 y(\(sharp!\))456 4087 y(F)-7 b(or)27 b Fo(I)641 4099 y Fk(1)678 4087 y Fo(;)h Fq(w)n(e)f(ha)n(v)n(e) 1056 4210 y Fl(Z)1102 4398 y Fg(j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)1524 4266 y Fo(e)1563 4236 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1524 4304 260 4 v 1534 4380 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1807 4206 y Fl(\022)1878 4266 y Fo(\013)h Fn(\001)f Fq(\()p Fo(x)i Fn(\000)e Fo(y)s Fq(\))p 1878 4304 371 4 v 1944 4380 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2277 4323 y(\000)2370 4266 y Fo(\013)h Fn(\001)f Fo(x)p 2370 4304 161 4 v 2404 4380 a Fn(j)p Fo(x)p Fn(j)2541 4206 y Fl(\023)2717 4266 y Fo(e)2756 4236 y Fg(\000j)p Fm(y)r Fg(j)p 2626 4304 353 4 v 2626 4380 a Fn(j)p Fo(y)s Fn(j)2716 4356 y Fk(2+)p Fm(")2853 4380 y Fq(+)h(1)2988 4323 y Fo(dy)908 4576 y Fq(=)83 b Fn(\000)1135 4463 y Fl(Z)1180 4652 y Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)1513 4520 y Fo(e)1552 4490 y Fg(\000j)p Fm(y)r Fg(j)p 1513 4557 170 4 v 1553 4633 a Fn(j)p Fo(y)s Fn(j)1706 4459 y Fl(\022)1777 4520 y Fo(\013)19 b Fn(\001)g Fo(y)p 1777 4557 158 4 v 1811 4633 a Fn(j)p Fo(y)s Fn(j)1963 4576 y(\000)2056 4520 y Fo(\013)g Fn(\001)f Fo(x)p 2056 4557 161 4 v 2089 4633 a Fn(j)p Fo(x)p Fn(j)2227 4459 y Fl(\023)2432 4520 y Fo(e)2471 4490 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 2312 4557 502 4 v 2312 4633 a Fn(j)p Fo(x)g Fn(\000)g Fo(y)s Fn(j)2550 4609 y Fk(2+)p Fm(")2688 4633 y Fq(+)g(1)2823 4576 y Fo(dy)456 4818 y Fq(By)27 b(\(49\))g(and)h(lemma)f(3.3,)456 5005 y Fl(Z)502 5193 y Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)834 5062 y Fo(e)873 5031 y Fg(\000j)p Fm(y)r Fg(j)p 834 5099 170 4 v 874 5175 a Fn(j)p Fo(y)s Fn(j)1028 4997 y Fl(\014)1028 5047 y(\014)1028 5097 y(\014)1028 5147 y(\014)1088 5062 y Fo(y)p 1065 5099 90 4 v 1065 5175 a Fn(j)p Fo(y)s Fn(j)1184 5118 y(\000)1300 5062 y Fo(x)p 1277 5099 94 4 v 1277 5175 a Fn(j)p Fo(x)p Fn(j)1380 4997 y Fl(\014)1380 5047 y(\014)1380 5097 y(\014)1380 5147 y(\014)1553 5062 y Fo(e)1592 5031 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1432 5099 502 4 v 1432 5175 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1671 5151 y Fk(2+)p Fm(")1808 5175 y Fq(+)g(1)1943 5118 y Fo(dy)26 b(<)2210 5062 y(C)6 b(e)2314 5031 y Fg(\000j)p Fm(x)p Fg(j)p 2150 5099 357 4 v 2150 5175 a Fn(j)p Fo(x)p Fn(j)2243 5151 y Fk(2+)p Fm(")2382 5175 y Fq(+)18 b(1)2552 4981 y Fg(j)p Fm(x)p Fg(j)2530 5005 y Fl(Z)2537 5240 y Fk(0)2627 5118 y Fo(\032)2670 5083 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)2812 5118 y Fo(d\032)23 b(<)g(C)3203 5062 y(e)3242 5031 y Fg(\000j)p Fm(x)p Fg(j)p 3084 5099 409 4 v 3084 5175 a Fn(j)p Fo(x)p Fn(j)3177 5151 y Fk(1)p Fm(:)p Fk(5+)p Fm(")3369 5175 y Fq(+)18 b(1)p eop %%Page: 17 17 17 16 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)221 b(17)456 450 y Fq(W)-7 b(e)28 b(no)n(w)f(estimate)g(in)n (tegral)f(in)i Fo(I)1539 462 y Fk(1)1605 450 y 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1534 y(\014)2939 1584 y(\014)2939 1634 y(\014)2991 1548 y Fo(e)3030 1518 y Fg(\000j)p Fm(y)r Fg(j)p 2991 1585 170 4 v 3031 1661 a Fn(j)p Fo(y)s Fn(j)3170 1604 y Fo(dy)802 1916 y(<)1054 1860 y(C)6 b(e)1158 1830 y Fg(\000j)p Fm(x)p Fg(j)p 1036 1897 274 4 v 1036 1973 a Fn(j)p Fo(x)p Fn(j)1129 1949 y Fk(2)1185 1973 y Fq(+)18 b(1)1355 1780 y Fg(j)p Fm(x)p Fg(j)1333 1803 y Fl(Z)1340 2039 y Fk(0)1598 1860 y Fo(\032)p 1440 1897 359 4 v 1440 1973 a(\032)1483 1949 y Fk(1)p Fm(:)p Fk(5+)p Fm(")1674 1973 y Fq(+)g(1)1808 1916 y Fo(d\032)24 b(<)2101 1860 y(C)6 b(e)2205 1830 y Fg(\000j)p Fm(x)p Fg(j)p 2015 1897 409 4 v 2015 1973 a Fn(j)p Fo(x)p Fn(j)2108 1949 y Fk(1)p Fm(:)p Fk(5+)p Fm(")2300 1973 y Fq(+)18 b(1)456 2159 y(F)-7 b(or)27 b(the)h(region)e Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)23 b Fo(<)f Fq(2)p Fn(j)p Fo(x)p Fn(j)p Fo(=)p Fq(3,)27 b(w)n(e)g(ha)n(v)n(e)674 2246 y Fl(Z)720 2435 y Fg(j)p Fm(y)r Fg(\000)p Fm(x)p Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)1142 2303 y Fo(e)1181 2273 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1142 2340 260 4 v 1152 2416 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1425 2239 y Fl(\014)1425 2288 y(\014)1425 2338 y(\014)1425 2388 y(\014)1486 2303 y Fo(x)h Fn(\000)f Fo(y)p 1463 2340 239 4 v 1463 2416 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)1730 2359 y(\000)1846 2303 y Fo(x)p 1823 2340 94 4 v 1823 2416 a Fn(j)p Fo(x)p Fn(j)1927 2239 y Fl(\014)1927 2288 y(\014)1927 2338 y(\014)1927 2388 y(\014)1968 2359 y Fn(j)p Fo(v)s Fq(\()p Fo(y)s Fq(\))p Fn(j)2179 2239 y Fl(\014)2179 2288 y(\014)2179 2338 y(\014)2179 2388 y(\014)2240 2303 y Fo(y)p 2217 2340 90 4 v 2217 2416 a Fn(j)p Fo(y)s Fn(j)2335 2359 y(\000)2451 2303 y Fo(x)h Fn(\000)f Fo(y)p 2428 2340 239 4 v 2428 2416 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2677 2239 y Fl(\014)2677 2288 y(\014)2677 2338 y(\014)2677 2388 y(\014)2729 2303 y Fo(e)2768 2273 y Fg(\000j)p Fm(y)r Fg(j)p 2729 2340 170 4 v 2769 2416 a Fn(j)p Fo(y)s Fn(j)2908 2359 y Fo(dy)86 b Fq(=)764 2499 y Fl(Z)810 2688 y Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fk(2)p Fg(j)p Fm(x)p Fg(j)p Fm(=)p Fk(3)1142 2556 y Fo(e)1181 2526 y Fg(\000j)p Fm(x)p Fg(\000)p Fm(y)r Fg(j)p 1142 2593 260 4 v 1152 2669 a Fn(j)p Fo(x)19 b Fn(\000)f Fo(y)s Fn(j)1425 2492 y Fl(\014)1425 2542 y(\014)1425 2592 y(\014)1425 2641 y(\014)1486 2556 y Fo(y)p 1463 2593 90 4 v 1463 2669 a Fn(j)p Fo(y)s Fn(j)1581 2612 y(\000)1697 2556 y Fo(x)p 1674 2593 94 4 v 1674 2669 a Fn(j)p Fo(x)p Fn(j)1778 2492 y Fl(\014)1778 2542 y(\014)1778 2592 y(\014)1778 2641 y(\014)1819 2612 y Fn(j)p Fo(v)s Fq(\()p Fo(x)i Fn(\000)e Fo(y)s Fq(\))p Fn(j)2180 2492 y Fl(\014)2180 2542 y(\014)2180 2592 y(\014)2180 2641 y(\014)2240 2556 y Fo(y)p 2217 2593 90 4 v 2217 2669 a Fn(j)p Fo(y)s Fn(j)2335 2612 y(\000)2451 2556 y Fo(x)h Fn(\000)f Fo(y)p 2428 2593 239 4 v 2428 2669 a Fn(j)p Fo(x)h Fn(\000)f Fo(y)s Fn(j)2677 2492 y Fl(\014)2677 2542 y(\014)2677 2592 y(\014)2677 2641 y(\014)2729 2556 y Fo(e)2768 2526 y Fg(\000j)p Fm(y)r Fg(j)p 2729 2593 170 4 v 2769 2669 a Fn(j)p Fo(y)s Fn(j)2908 2612 y Fo(dy)1705 2924 y(<)1889 2868 y(C)6 b(e)1993 2838 y Fg(\000j)p Fm(x)p Fg(j)p 1803 2905 409 4 v 1803 2981 a Fn(j)p Fo(x)p Fn(j)1896 2957 y Fk(1)p Fm(:)p Fk(5+)p Fm(")2087 2981 y Fq(+)18 b(1)2257 2788 y Fg(j)p Fm(x)p Fg(j)2235 2811 y Fl(Z)2242 3047 y Fk(0)2392 2868 y Fo(d\032)p 2342 2905 186 4 v 2342 2981 a(\032)h Fq(+)f(1)2561 2924 y Fo(<)2703 2868 y(C)6 b(e)2807 2838 y Fg(\000j)p Fm(x)p Fg(j)p 2659 2905 327 4 v 2659 2981 a Fn(j)p Fo(x)p Fn(j)2752 2957 y Fk(1)p Fm(:)p Fk(5)2861 2981 y Fq(+)18 b(1)456 3161 y Ff(\004)555 3260 y Fq(Pro)r(of)26 b(of)h(the)h(theorem)f(3.2.)36 b(Let)27 b(us)g(iterate)g(the)h(second)e(resolv)n(en)n(t)g(iden)n(tit)n (y)h(to)g(get)g(the)456 3360 y(needed)g(estimate)h(for)f Fo(G)1257 3372 y Fm(i)1285 3360 y Fq(\()p Fo(x;)14 b Fq(0\))1085 3543 y Fo(G)1150 3555 y Fm(i)1178 3543 y Fq(\()p Fo(x;)g Fq(0\))23 b(=)g Fo(G)1544 3509 y Fk(0)1544 3564 y Fm(i)1581 3543 y Fq(\()p Fo(x;)14 b Fq(0\))19 b Fn(\000)1873 3430 y Fl(Z)1859 3672 y Fa(R)1906 3655 y Fd(3)1970 3543 y Fo(G)2035 3509 y Fk(0)2035 3564 y Fm(i)2073 3543 y Fq(\()p Fo(x;)14 b(s)p Fq(\))p Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))p Fo(G)2522 3555 y Fm(i)2551 3543 y Fq(\()p Fo(s;)g Fq(0\))p Fo(ds)456 3797 y Fq(Using)25 b(lemmas)g(3.5{3.7)f(and)i(explicit)g(form)n(ula)e(for)i Fo(G)2209 3767 y Fk(0)2209 3819 y Fm(i)2246 3797 y Fq(\()p Fo(x;)14 b(s)p Fq(\),)27 b(w)n(e)e(see)h(that)g(the)g Fo(n)p Fn(\000)p Fq(th)g(term)456 3897 y(in)h(the)h(corresp)r(onding)e (series)h(has)g(the)h(follo)n(wing)e(form)1048 4036 y Fo(e)1087 4006 y Fg(\000j)p Fm(x)p Fg(j)p 1015 4073 237 4 v 1015 4149 a Fn(j)p Fo(x)p Fn(j)19 b Fq(+)f(1)1276 3975 y Fl(\024\022)1391 4036 y Fo(\013)h Fn(\001)f Fo(x)p 1391 4073 161 4 v 1424 4149 a Fn(j)p Fo(x)p Fn(j)1580 4092 y Fq(+)g(1)1705 3975 y Fl(\023)1779 4092 y Fo(')1833 4049 y Fk(\()p Fm(n)p Fk(\))1833 4114 y(1)1931 4092 y Fq(\()p Fo(x)p Fq(\))h(+)f(\()p Fn(j)p Fo(x)p Fn(j)i Fq(+)e(1\))2446 4058 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)2587 4092 y Fo(')2641 4049 y Fk(\()p Fm(n)p Fk(\))2641 4114 y(2)2739 4092 y Fq(\()p Fo(x)p Fq(\))2850 3975 y Fl(\025)3297 4092 y Fq(\(54\))456 4307 y(with)30 b Fn(j)p Fo(')724 4264 y Fk(\()p Fm(n)p Fk(\))724 4335 y(1\(2\))847 4307 y Fq(\()p Fo(x)p Fq(\))p Fn(j)e Fo(<)e Fq([)p Fo(C)6 b Fq(\()p Fo(")p Fq(\))p Fo(C)1350 4319 y Fm(v)1390 4307 y Fq(])1413 4277 y Fm(n)1459 4307 y Fq(.)43 b(That)30 b(can)g(b)r(e)g(easily)f(pro)n(v)n(ed)f(b)n(y)i(the)g(induction.)44 b(Indeed,)456 4420 y(for)26 b(the)i(\014rst)f(term,)h(w)n(e)f(ha)n(v)n (e)f(the)h(represen)n(tation)f(\(39\).)36 b(Assume)28 b(that)f(\(54\))g(holds)g(for)g(the)456 4520 y Fo(n)p Fq(-th)g(term)h Fo(T)887 4532 y Fm(n)931 4520 y Fq(.)37 b(Then,)859 4698 y Fo(T)908 4710 y Fm(n)p Fk(+1)1036 4698 y Fq(\()p Fo(x)p Fq(\))25 b(=)d Fn(\000)1338 4585 y Fl(Z)1324 4827 y Fa(R)1371 4810 y Fd(3)1434 4698 y Fo(G)1499 4664 y Fk(0)1499 4719 y Fm(i)1537 4698 y Fq(\()p Fo(x;)14 b(s)p Fq(\))p Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))p Fo(T)1970 4710 y Fm(n)2017 4698 y Fq(\()p Fo(s)p Fq(\))p Fo(ds)23 b Fq(=)g Fn(\000)p Fq(\()p Fo(I)2446 4710 y Fk(1)2502 4698 y Fq(+)18 b Fo(I)2621 4710 y Fk(2)2677 4698 y Fq(+)g Fo(I)2796 4710 y Fk(3)2852 4698 y Fq(+)g Fo(I)2971 4710 y Fk(4)3009 4698 y Fq(\))456 4942 y(where)456 5112 y Fo(I)492 5124 y Fk(1)529 5112 y Fq(\()p Fo(x)p Fq(\))24 b(=)794 5056 y Fo(i)p 762 5093 92 4 v 762 5169 a Fq(4)p Fo(\031)878 4999 y Fl(Z)864 5240 y Fa(R)911 5224 y Fd(3)984 5056 y Fq(exp\()p Fn(\000j)p Fo(x)19 b Fn(\000)f Fo(s)p Fn(j)p Fq(\))p 984 5093 490 4 v 1112 5169 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1498 4995 y Fl(\022)1569 5056 y Fo(\013)h Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(s)p Fq(\))p 1569 5093 366 4 v 1635 5169 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1963 5112 y Fq(+)g(1)2088 4995 y Fl(\023)2163 5112 y Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))2374 4995 y Fl(\022)2446 5056 y Fo(\013)h Fn(\001)f Fo(s)p 2446 5093 153 4 v 2480 5169 a Fn(j)p Fo(s)p Fn(j)2627 5112 y Fq(+)g(1)2752 4995 y Fl(\023)2836 5056 y Fq(exp\()p Fn(\000j)p Fo(s)p Fn(j)p Fq(\))p 2836 5093 342 4 v 2893 5169 a Fn(j)p Fo(s)p Fn(j)g Fq(+)g(1)3201 5112 y Fo(')3255 5069 y Fk(\()p Fm(n)p Fk(\))3255 5134 y(1)3352 5112 y Fq(\()p Fo(s)p Fq(\))p Fo(ds)p eop %%Page: 19 19 19 18 bop 744 251 a Fk(ON)29 b(THE)g(ABSOLUTEL)-5 b(Y)29 b(CONTINUOUS)g(SPECTR)n(UM)g(OF)g(DIRA)n(C)h(OPERA)-5 b(TOR)221 b(19)553 487 y Fo(I)589 499 y Fk(2)626 487 y Fq(\()p Fo(x)p Fq(\))24 b(=)891 431 y Fo(i)p 859 468 92 4 v 859 544 a Fq(4)p Fo(\031)975 374 y Fl(Z)960 616 y Fa(R)1007 600 y Fd(3)1081 431 y Fq(exp\()p Fn(\000j)p Fo(x)19 b Fn(\000)f Fo(s)p Fn(j)p Fq(\))p 1081 468 490 4 v 1209 544 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1595 370 y Fl(\022)1666 431 y Fo(\013)h Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(s)p Fq(\))p 1666 468 366 4 v 1732 544 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)2060 487 y Fq(+)g(1)2185 370 y Fl(\023)2260 487 y Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))2471 370 y Fl(\022)2564 431 y Fq(exp\()p Fn(\000j)p Fo(s)p Fn(j)p Fq(\))p 2543 468 383 4 v 2543 544 a(\()p Fn(j)p Fo(s)p Fn(j)h Fq(+)f(1\))2836 520 y Fk(1)p Fm(:)p Fk(5)2935 370 y Fl(\023)3010 487 y Fo(')3064 444 y Fk(\()p Fm(n)p Fk(\))3064 510 y(2)3162 487 y Fq(\()p Fo(s)p Fq(\))p Fo(ds)480 774 y(I)516 786 y Fk(3)554 774 y Fq(\()p Fo(x)p Fq(\))24 b(=)818 718 y Fo(i)p 787 755 92 4 v 787 831 a Fq(4)p Fo(\031)902 661 y Fl(Z)888 903 y Fa(R)935 886 y Fd(3)1009 718 y Fo(\013)19 b Fn(\001)g Fq(\()p Fo(x)g Fn(\000)f Fo(s)p Fq(\))p 1009 755 366 4 v 1056 831 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1290 807 y Fk(2)1399 657 y Fl(\022)1470 718 y Fq(exp\()p Fn(\000j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)p Fq(\))p 1470 755 490 4 v 1598 831 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1970 657 y Fl(\023)2045 774 y Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))2256 657 y Fl(\022)2328 718 y Fo(\013)h Fn(\001)f Fo(s)p 2328 755 153 4 v 2361 831 a Fn(j)p Fo(s)p Fn(j)2508 774 y Fq(+)g(1)2633 657 y Fl(\023)2718 718 y Fq(exp\()p Fn(\000j)p Fo(s)p Fn(j)p Fq(\))p 2718 755 342 4 v 2774 831 a Fn(j)p Fo(s)p Fn(j)h Fq(+)f(1)3083 774 y Fo(')3137 731 y Fk(\()p Fm(n)p Fk(\))3137 796 y(1)3234 774 y Fq(\()p Fo(s)p Fq(\))p Fo(ds)624 1061 y(I)660 1073 y Fk(4)698 1061 y Fq(\()p Fo(x)p Fq(\))24 b(=)962 1004 y Fo(i)p 930 1042 92 4 v 930 1118 a Fq(4)p Fo(\031)1046 948 y Fl(Z)1032 1189 y Fa(R)1079 1173 y Fd(3)1153 1004 y Fo(\013)19 b Fn(\001)f Fq(\()p Fo(x)h Fn(\000)f Fo(s)p Fq(\))p 1153 1042 366 4 v 1200 1118 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)1434 1094 y Fk(2)1542 944 y Fl(\022)1614 1004 y Fq(exp)o(\()p Fn(\000j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)p Fq(\))p 1614 1042 490 4 v 1742 1118 a Fn(j)p Fo(x)h Fn(\000)f Fo(s)p Fn(j)2113 944 y Fl(\023)2188 1061 y Fo(\014)t(v)s Fq(\()p Fo(s)p Fq(\))2399 944 y Fl(\022)2492 1004 y Fq(exp\()p Fn(\000j)p Fo(s)p Fn(j)p Fq(\))p 2471 1042 383 4 v 2471 1118 a(\()p Fn(j)p Fo(s)p Fn(j)h Fq(+)f(1\))2764 1094 y Fk(1)p Fm(:)p Fk(5)2864 944 y Fl(\023)2939 1061 y Fo(')2993 1017 y Fk(\()p Fm(n)p Fk(\))2993 1083 y(2)3090 1061 y Fq(\()p Fo(s)p Fq(\))p Fo(ds)456 1283 y Fq(W)-7 b(e)29 b(apply)g(lemma)h(3.7)e(to)h Fo(I)1374 1295 y Fk(1)1412 1283 y Fq(,)h(lemma)f(3.6)g(to)g Fo(I)2009 1295 y Fk(2)2047 1283 y Fq(,)g(and)h(lemma)f(3.5)f(to)i Fo(I)2807 1295 y Fk(3)2874 1283 y Fq(and)f Fo(I)3073 1295 y Fk(4)3111 1283 y Fq(.)42 b(If)29 b Fo(C)3319 1295 y Fm(v)3389 1283 y Fq(is)456 1382 y(small)e(enough,)h(w)n(e)g(will)g(get)g(uniform)g (con)n(v)n(ergence)e(of)i(the)g(series)f(and)h(the)h(estimate)f (\(47\).)456 1482 y(Clearly)-7 b(,)26 b Fn(k)p Fi(P)857 1494 y Fk(1)912 1482 y Fn(\000)18 b Fq(1)p Fn(k)k Fo(<)h(\016)31 b Fq(with)d Fo(\016)e Fn(!)d Fq(0)k(if)h Fo(C)1819 1494 y Fm(v)1882 1482 y Fn(!)23 b Fq(0.)37 b Ff(\004)555 1582 y Fv(Remark.)c Fq(It)20 b(is)g(clear)f(that)i(the)f(asymptotics)f(of)h (the)g(Green's)g(function)g Fo(G)2954 1594 y Fm(i)2982 1582 y Fq(\()p Fo(x;)14 b(s)p Fq(\))p Fo(;)g Fn(j)p Fo(s)p Fn(j)24 b Fo(<)f Fq(1)456 1681 y(can)k(b)r(e)h(obtained)f(similarly:)36 b(it)28 b(is)g(close)e(to)i Fo(G)1966 1651 y Fk(0)1966 1703 y Fm(i)2003 1681 y Fq(\()p Fo(x;)14 b(s)p Fq(\))29 b(as)e Fn(j)p Fo(x)p Fn(j)c(!)h(1)p Fq(.)555 1781 y(Fix)30 b(an)n(y)e Fo(")e(>)f Fq(0.)42 b(Consider)28 b(the)h(Dirac)g(op)r (erator)f Fi(H)h Fq(with)g Fn(j)p Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fn(j)f Fo(<)d(C)2853 1793 y Fm(v)2893 1781 y Fq(\()p Fn(j)p Fo(x)p Fn(j)20 b Fq(+)f(1\))3196 1751 y Fg(\000)p Fk(0)p Fm(:)p Fk(5)p Fg(\000)p Fm(")3421 1781 y Fq(,)456 1880 y(where)32 b Fo(C)760 1892 y Fm(v)832 1880 y Fq(is)h(su\016cien)n(tly)g(small,)g(and)g(op)r(erator)e Fi(H)2170 1892 y Fm(s)2236 1880 y Fq(on)i Fj(R)2411 1850 y Fk(3)2487 1880 y Fq(with)g Fo(v)s Fq(\()p Fo(x)p Fq(\))g(=)e(0)h(on)h Fn(j)p Fo(x)p Fn(j)f Fo(<)f Fq(1.)456 1997 y(Then,)26 b(for)f(an)n(y)g(nonzero)g(function)h Fo(f)9 b Fq(\()p Fo(x)p Fq(\))23 b(=)g(\()p Fo(f)1950 2009 y Fk(1)1987 1997 y Fo(;)14 b(f)2065 2009 y Fk(2)2102 1997 y Fo(;)g(f)2180 2009 y Fk(3)2217 1997 y Fo(;)g(f)2295 2009 y Fk(4)2332 1997 y Fq(\))2364 1967 y Fm(t)2416 1997 y Fn(2)2495 1930 y Fl(\002)2529 1997 y Fo(L)2586 1967 y Fk(2)2623 1997 y Fq(\()p Fj(R)2709 1967 y Fk(3)2752 1997 y Fq(\))2784 1930 y Fl(\003)2819 1947 y Fk(4)2882 1997 y Fq(with)27 b(supp)r(ort)e(in)456 2097 y(the)j(unit)g(ball,)f(the)h(function)1508 2271 y Fo(h)p Fq(\()p Fo(x)p Fq(\))c(=)1779 2158 y Fl(Z)1765 2399 y Fa(R)1812 2383 y Fd(3)1876 2271 y Fo(G)1941 2283 y Fm(i)1969 2271 y Fq(\()p Fo(x;)14 b(s)p Fq(\))p Fo(f)9 b Fq(\()p Fo(s)p Fq(\))p Fo(ds)456 2515 y Fq(satis\014es)19 b(equation)i Fo(H)7 b(h)22 b Fq(=)h Fo(ih)d Fq(\()p Fo(h)h Fq(do)r(es)f(not)h(satisfy)f(b)r(oundary)g(condition)h(on)f(\006)h (unless)f Fo(h)j Fq(=)g(0)456 2614 y(in)k(\012\).)37 b(Therefore,)26 b(b)n(y)h(the)h(theorem)e(3.2)h(and)g(b)n(y)f(the)i (remark)e(ab)r(o)n(v)n(e,)g(w)n(e)g(can)h(con)n(trol)f(the)456 2714 y(\\amplitude")h(of)g Fo(h)p Fq(.)37 b(F)-7 b(or)27 b Fn(j)p Fo(x)p Fn(j)h Fq(large)e(enough,)927 2944 y Fo(e)966 2910 y Fg(j)p Fm(x)p Fg(j)1047 2944 y Fn(j)p Fo(x)p Fn(j)p Fo(h)p Fq(\()p Fo(x)p Fq(\))f(=)1453 2888 y Fo(i)p 1421 2925 92 4 v 1421 3001 a Fq(4)p Fo(\031)1537 2827 y Fl(\022)1608 2888 y Fo(\013)19 b Fn(\001)f Fo(x)p 1608 2925 161 4 v 1642 3001 a Fn(j)p Fo(x)p Fn(j)1797 2944 y Fq(+)g(1)1922 2827 y Fl(\023)1997 2831 y(Z)1983 3073 y Fa(R)2030 3056 y Fd(3)2094 2944 y Fq(exp)o Fn(h)2286 2888 y Fo(x)p 2262 2925 94 4 v 2262 3001 a Fn(j)p Fo(x)p Fn(j)2366 2944 y Fo(;)c(s)p Fn(i)p Fo(f)9 b Fq(\()p Fo(s)p Fq(\))p Fo(ds)19 b Fq(+)2829 2922 y(^)2811 2944 y Fo(f)9 b Fq(\()p Fo(x)p Fq(\))456 3166 y(and)1663 3268 y Fn(k)1722 3246 y Fq(^)1705 3268 y Fo(f)f Fq(\()p Fo(x)p Fq(\))p Fn(k)23 b Fo(<)g(\016)s Fn(jj)p Fo(f)9 b Fn(jj)2200 3280 y Fk(1)456 3386 y Fq(with)28 b Fo(\016)e Fn(!)d Fq(0)k(as)g Fo(C)1044 3398 y Fm(v)1107 3386 y Fn(!)c Fq(0.)36 b(F)-7 b(or)27 b(the)h(in)n(tegrals,)f(w)n(e)g(ha)n(v)n(e)970 3447 y Fl(Z)956 3689 y Fa(R)1003 3673 y Fd(3)1067 3560 y Fq(exp)p Fn(h)1259 3504 y Fo(x)p 1236 3541 V 1236 3617 a Fn(j)p Fo(x)p Fn(j)1339 3560 y Fo(;)14 b(s)p Fn(ij)p Fo(f)1511 3572 y Fm(j)1546 3560 y Fq(\()p Fo(s)p Fq(\))p Fn(j)p Fo(ds)24 b(>)f(e)1905 3526 y Fg(\000)p Fk(1)2007 3447 y Fl(Z)1993 3689 y Fa(R)2040 3673 y Fd(3)2104 3560 y Fn(j)p Fo(f)2168 3572 y Fm(j)2203 3560 y Fq(\()p Fo(s)p Fq(\))p Fn(j)p Fo(ds;)14 b Fq(\()p Fo(j)29 b Fq(=)22 b(1)p Fo(;)14 b(:)g(:)g(:)f(;)h Fq(4\))456 3800 y(The)27 b(\\amplitude")g(is)h(calculated)f(as)1074 3936 y Fi(A)p Fq(\()p Fn(j)p Fo(x)p Fn(j)1263 3902 y Fg(\000)p Fk(1)1352 3936 y Fo(x)p Fq(\))d(=)70 b(lim)1543 3994 y Fg(j)p Fm(x)p Fg(j!1)1766 3936 y Fn(j)p Fo(x)p Fn(j)14 b Fq(exp\()p Fn(j)p Fo(x)p Fn(j)20 b(\000)e Fq(1\)\()p Fo(h)2382 3948 y Fk(3)2419 3936 y Fq(\()p Fo(x)p Fq(\))p Fo(;)c(h)2615 3948 y Fk(4)2653 3936 y Fq(\()p Fo(x)p Fq(\)\))2796 3902 y Fm(t)456 4120 y Fq(T)-7 b(ak)n(e)35 b Fo(f)706 4132 y Fk(1)782 4120 y Fq(=)j Fo(f)926 4132 y Fk(2)1001 4120 y Fq(=)g(0)f(and)f Fo(f)1394 4132 y Fk(3)1431 4120 y Fo(;)14 b(f)1509 4132 y Fk(4)1583 4120 y Fq({)36 b(arbitrary)f (non-negativ)n(e.)63 b(Then,)39 b Fi(A)e Fq(is)g(w)n(ell-de\014ned)456 4219 y(as)d(an)h(elemen)n(t)h(of)f([)p Fo(L)1185 4189 y Fk(2)1222 4219 y Fq(\(\006\)])1369 4189 y Fk(2)1442 4219 y Fq(for)g Fo(v)1617 4231 y Fm(n)1698 4219 y Fq(=)h Fo(v)s Fq(\()p Fo(x)p Fq(\))p Fo(\037)2005 4234 y Fg(j)p Fm(x)p Fg(j)p Fm(