%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: ModelPr.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips ModelPr.dvi -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.11.02:1649 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 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/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 dup 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 dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 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}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr 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130EA21B167F951E>118 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 380 246 a Fq(A)19 b(MODEL)f(OF)h(PER)-5 b(TURBA)g(TION)19 b(THEOR)-5 b(Y)820 374 y(D.Y)g(afaev)670 434 y(Univ)n(ersit)o(\023)-27 b(e)17 b(de)i(Rennes)832 595 y Fp(Abstract)141 696 y Fo(W)l(e)c(consider)g(a)f(mo)q(del)i(of)e(the)g(p)q(erturbation)h (theory)f(where)h(the)f(Hamiltonian)i Fp(H)e Fo(is)h(a)f(sum)73 753 y(of)f(a)h(m)o(ultiplication)i(op)q(erator)c Fp(H)672 760 y Fn(0)705 753 y Fo(b)o(y)i Fm(j)p Fl(x)p Fm(j)819 736 y Fn(2)p Fk(l)862 753 y Fo(in)h(the)e(space)h Fl(L)1141 760 y Fn(2)1161 753 y Fo(\()p Fj(R)1211 736 y Fk(d)1229 753 y Fo(\))f(and)g(of)h(an)f(in)o(tegral)h(op)q(erator)73 809 y Fp(V)f Fo(with)f(oscillating)i(k)o(ernel.)20 b(W)l(e)12 b(\014nd)h(conditions)g(for)f(the)g(coincidence)j(of)d(the)g(essen)o (tial)h(sp)q(ectra)73 866 y(of)f(the)f(op)q(erators)g Fp(H)435 873 y Fn(0)466 866 y Fo(and)h Fp(H)g Fo(and)f(for)g(the)h (existence)h(and)f(completeness)g(of)g(the)f(w)o(a)o(v)o(e)g(op)q (erators)73 922 y(relating)16 b Fp(H)281 929 y Fn(0)316 922 y Fo(and)f Fp(H)p Fo(.)22 1053 y Fq(1.)36 b Fi(Our)22 b(aim)e(here)h(is)g(to)g(in)o(tro)q(duce)g(a)h(new)f(mo)q(del)f(of)i (the)f(p)q(erturbation)h(theory)l(.)36 b(Let)22 b Fq(H)1821 1060 y Fn(0)1862 1053 y Fi(b)q(e)-49 1113 y(m)o(ultiplication)14 b(b)o(y)i Fh(j)p Fg(x)p Fh(j)389 1095 y Fn(2)p Fk(l)419 1113 y Fg(;)23 b(l)15 b(>)g Fi(0)p Fg(;)h Fi(in)h(the)f(space)h Fg(L)899 1120 y Fn(2)919 1113 y Fi(\()p Fj(R)971 1095 y Fk(d)988 1113 y Fi(\))g(and)g(let)f Fq(V)i Fi(b)q(e)f(an)g(in)o (tegral)f(op)q(erator)i(de\014ned)-49 1173 y(b)o(y)609 1246 y(\()p Fq(V)q Fg(u)p Fi(\)\()p Fg(x)p Fi(\))13 b(=)849 1187 y Ff(Z)872 1281 y Fe(R)896 1272 y Fd(d)922 1246 y Fg(v)r Fi(\()p Fh(h)p Fg(x;)8 b(y)r Fh(i)p Fi(\))p Fg(u)p Fi(\()p Fg(y)r Fi(\))p Fg(dy)r(;)593 b Fi(\(1\))-49 1347 y(where)16 b Fg(v)i Fi(is)e(a)h(real)e(p)q(erio)q(dic)h(function) 502 1483 y Fg(v)r Fi(\()p Fg(t)p Fi(\))d(=)687 1429 y Fc(1)674 1442 y Ff(X)648 1531 y Fk(n)p Fn(=)p Fc(\0001)768 1483 y Fg(v)792 1490 y Fk(n)815 1483 y Fg(e)838 1463 y Fk(int)886 1483 y Fg(;)57 b(v)981 1490 y Fc(\000)p Fk(n)1045 1483 y Fi(=)16 b(\026)-26 b Fg(v)1121 1490 y Fk(n)1144 1483 y Fg(;)57 b(v)1239 1490 y Fn(0)1272 1483 y Fi(=)14 b(0)p Fg(:)488 b Fi(\(2\))-49 1627 y(The)17 b(op)q(erator)g Fq(V)g Fi(can)g(also)g(b)q(e)f(view)o(ed)f(as)i(a)f(F)l (ourier)g(in)o(tegral)g(op)q(erator.)22 b(F)l(or)16 b(example,)e(if)583 1737 y Fg(v)r Fi(\()p Fg(t)p Fi(\))f(=)h Fg(\015)d Fi(cos)e Fg(t)48 b Fi(or)h Fg(v)r Fi(\()p Fg(t)p Fi(\))13 b(=)h Fg(\015)d Fi(sin)d Fg(t;)569 b Fi(\(3\))-49 1847 y(then)18 b(\(1\))h(is)f(a)g(linear)f(com)o(bination)g(of)h(the)g(classical)g(F)l (ourier)f(op)q(erator)j(and)e(of)h(its)f(adjoin)o(t.)26 b(In)18 b(this)-49 1907 y(pap)q(er)j(w)o(e)f(study)g(sp)q(ectral)h (prop)q(erties)f(of)g(the)g(Hamiltonian)f Fq(H)h Fi(=)h Fq(H)1326 1914 y Fn(0)1359 1907 y Fi(+)14 b Fq(V)21 b Fi(in)f(the)g(space)g Fg(L)1790 1914 y Fn(2)1810 1907 y Fi(\()p Fj(R)1862 1889 y Fk(d)1879 1907 y Fi(\).)-49 1967 y(The)c(case)g(of)g(a)h(constan)o(t)f(function)g Fg(v)r Fi(\()p Fg(t)p Fi(\))f(corresp)q(onds)i([1])e(to)i(the)e(p)q (oin)o(t)h(in)o(teraction)f(and)i(is)e(excluded)-49 2028 y(b)o(y)i(the)g(condition)h Fg(v)345 2035 y Fn(0)380 2028 y Fi(=)d(0.)25 b(P)o(assing)19 b(to)e(the)h(spherical)e(co)q (ordinates,)i(considering)f(the)g(space)h Fg(L)1804 2035 y Fn(2)1824 2028 y Fi(\()p Fj(R)1876 2010 y Fk(d)1893 2028 y Fi(\))-49 2088 y(as)g Fg(L)45 2095 y Fn(2)64 2088 y Fi(\()p Fj(R)116 2095 y Fn(+)143 2088 y Fi(;)8 b Fg(L)198 2095 y Fn(2)218 2088 y Fi(\()p Fj(S)263 2070 y Fk(d)p Fc(\000)p Fn(1)325 2088 y Fi(\)\))17 b(and)g(making)e(the)h(c)o(hange)h (of)g(v)m(ariables)f Fg(\025)f Fi(=)f Fh(j)p Fg(x)p Fh(j)1302 2070 y Fn(2)p Fk(l)1332 2088 y Fi(,)j(w)o(e)f(can)g(\014t)h(the)f(op)q (erator)i Fq(H)-49 2148 y Fi(in)o(to)i(the)g(framew)o(ork)e(of)j(the)e (F)l(riedric)o(hs)g(mo)q(del.)31 b(Ho)o(w)o(ev)o(er,)19 b(the)g(results)h(of)h(the)e(pap)q(er)i([5])e(are)i(not)-49 2208 y(applicable)e(to)h(p)q(erturbations)g(\(1\))g(\(ev)o(en)e(in)i (the)f(case)g Fg(d)h Fi(=)g(1\))g(since)e(the)i(k)o(ernels)e(obtained)i (do)g(not)-49 2268 y(satisfy)d(its)f(assumptions)g(either)f(for)i Fg(\025)d Fi(=)g(0)i(or)h(for)f Fg(\025)f Fh(!)e(1)j Fi(\(or)h(for)g(b)q(oth)g(these)f(p)q(oin)o(ts\).)22 2349 y(F)l(or)i(the)g(case)h(\(3\),)f(the)g(mo)q(del)f(w)o(e)h (consider)g(\014rst)h(app)q(eared)g(in)f([4].)26 b(It)18 b(w)o(as)h(c)o(hec)o(k)o(ed)d(in)i([4])g(that)-49 2409 y Fg(\033)-21 2416 y Fk(ess)30 2409 y Fi(\()p Fq(H)p Fi(\))c(=)g Fg(\033)206 2416 y Fk(ess)256 2409 y Fi(\()p Fq(H)319 2416 y Fn(0)339 2409 y Fi(\))g(for)h(arbitrary)g Fg(l)f(>)g Fi(0)h(and)g(that)h(the)e(w)o(a)o(v)o(e)g(op)q(erators)i Fg(W)1429 2416 y Fc(\006)1458 2409 y Fi(\()p Fq(H)p Fg(;)8 b Fq(H)1587 2416 y Fn(0)1606 2409 y Fi(\))15 b(exist)f(and)h(are)-49 2470 y(complete)f(if)i(2)p Fg(l)f(>)f(d)p Fi(.)21 b(The)16 b(discrete)g(sp)q(ectrum)f(of)h Fq(H)g Fi(for)h(suc)o(h)f(functions)g Fg(v)r Fi(\()p Fg(t)p Fi(\))f(w)o(as)i(studied)f(in)g([8].)22 2551 y(W)l(e)g(sho)o(w)h(that)f(under)h(the)f(assumption)759 2633 y Fc(1)746 2645 y Ff(X)743 2736 y Fk(n)p Fn(=1)818 2687 y Fh(j)p Fg(v)856 2694 y Fk(n)879 2687 y Fh(j)p Fg(n)922 2666 y Fc(\000)p Fk(d=)p Fn(2)1019 2687 y Fg(<)d Fh(1)730 b Fi(\(4\))p eop %%Page: 2 2 2 1 bop -49 67 a Fi(the)17 b(op)q(erator)g Fq(V)h Fi(is)f(b)q(ounded)g (in)f Fg(L)631 74 y Fn(2)651 67 y Fi(\()p Fj(R)703 49 y Fk(d)720 67 y Fi(\))h(and)g Fg(\033)879 74 y Fk(ess)930 67 y Fi(\()p Fq(H)p Fi(\))d(=)g Fg(\033)1106 74 y Fk(ess)1157 67 y Fi(\()p Fq(H)1220 74 y Fn(0)1240 67 y Fi(\))i(for)h(arbitrary)g Fg(l)e(>)f Fi(0.)23 b(Moreo)o(v)o(er,)-49 128 y(if)714 150 y Fc(1)702 163 y Ff(X)699 253 y Fk(n)p Fn(=1)774 204 y Fh(j)p Fg(v)812 211 y Fk(n)834 204 y Fh(j)p Fg(n)877 184 y Fc(\000)p Fk(d=)p Fn(2+1+)p Fk(")1063 204 y Fg(<)14 b Fh(1)685 b Fi(\(5\))-49 322 y(for)16 b(some)f Fg(")e(>)h Fi(0,)i(then)f(the)g(w)o(a)o(v)o(e)g(op)q(erators)i Fg(W)862 329 y Fc(\006)891 322 y Fi(\()p Fq(H)p Fg(;)8 b Fq(H)1020 329 y Fn(0)1040 322 y Fi(\))15 b(exist)g(and)h(are)f(complete)f(pro)o (vided)g(2)p Fg(l)h(>)f Fi(1.)-49 383 y(Note)i(that)h(b)q(oth)g (conditions)f(\(4\))h(and)g(\(5\))f(are)h(getting)f(less)g(restrictiv)o (e)e(as)j Fg(d)f Fi(increases.)22 461 y(W)l(e)11 b(found)h(also)g (necessary)g(and)g(su\016cien)o(t)e(conditions)i(for)f(the)h(negativ)o (e)e(sp)q(ectrum)h(of)h(the)f(op)q(erator)-49 521 y Fq(H)18 b Fi(to)g(b)q(e)g(\014nite)f(and)h(calculated)f(explicitly)e(the)i (total)h(n)o(um)o(b)q(er)e(of)i(its)g(negativ)o(e)e(eigen)o(v)m(alues.) 25 b(These)-49 582 y(results)16 b(will)f(b)q(e)i(published)f (elsewhere.)22 683 y Fq(2.)21 b Fi(Let)c(us)f(\014rst)h(of)f(all)g (rewrite)f(\(1\))i(in)f(op)q(erator)h(notation.)22 b(Let)17 b(\010,)540 800 y(\(\010)p Fg(u)p Fi(\)\()p Fg(x)p Fi(\))c(=)h(\(2)p Fg(\031)r Fi(\))864 779 y Fc(\000)p Fk(d=)p Fn(2)955 741 y Ff(Z)978 835 y Fe(R)1002 826 y Fd(d)1028 800 y Fg(e)1051 779 y Fc(\000)p Fk(i)p Fc(h)p Fk(x;y)q Fc(i)1168 800 y Fg(u)p Fi(\()p Fg(y)r Fi(\))p Fg(dy)r(;)-49 916 y Fi(b)q(e)j(the)f(F)l(ourier)f(op)q(erator)j(and)f(let)e Fq(T)671 923 y Fk(n)695 916 y Fi(,)692 1018 y(\()p Fq(T)750 1025 y Fk(n)773 1018 y Fg(u)p Fi(\)\()p Fg(x)p Fi(\))e(=)h Fg(n)980 997 y Fk(d=)p Fn(2)1036 1018 y Fg(u)p Fi(\()p Fg(nx)p Fi(\))p Fg(;)-49 1120 y Fi(b)q(e)j(the)f(unitary)g(op)q(erator) h(of)g(dilations)f(in)g Fg(L)814 1127 y Fn(2)834 1120 y Fi(\()p Fj(R)886 1101 y Fk(d)903 1120 y Fi(\).)21 b(Then)16 b(equalit)o(y)f(\(1\))i(can)f(b)q(e)g(rewritten)g(as)521 1250 y Fq(V)f Fi(=)f(\(2)p Fg(\031)r Fi(\))722 1230 y Fk(d=)p Fn(2)801 1196 y Fc(1)789 1209 y Ff(X)786 1299 y Fk(n)p Fn(=1)861 1250 y Fg(n)890 1230 y Fc(\000)p Fk(d=)p Fn(2)973 1250 y Fq(T)1012 1257 y Fk(n)1046 1250 y Fi(\()p Fg(v)1089 1257 y Fk(n)1112 1250 y Fi(\010)1147 1230 y Fc(\003)1178 1250 y Fi(+)f(\026)-26 b Fg(v)1251 1257 y Fk(n)1274 1250 y Fi(\010\))p Fg(:)508 b Fi(\(6\))-49 1387 y(T)l(o)25 b(b)q(e)f(precise,)h(w)o(e)e(tak)o(e)h(\(6\))g(for)g (the)g(de\014nition)g(of)g Fq(V)q Fi(.)44 b(Under)24 b(assumption)g(\(4\),)h(series)f(\(6\))g(is)-49 1447 y(con)o(v)o(ergen)o(t)d(in)h(norm)g(so)h(that)g(the)f(op)q(erator)i Fq(V)f Fi(is)f(b)q(ounded.)41 b(This)23 b(implies)c(of)k(course)f(that) h(the)-49 1507 y(op)q(erator)18 b Fq(H)e Fi(is)g(self-adjoin)o(t)g(on)h (the)f(domain)f Fh(D)q Fi(\()p Fq(H)935 1514 y Fn(0)955 1507 y Fi(\))i(of)f(the)g(op)q(erator)i Fq(H)1371 1514 y Fn(0)1390 1507 y Fi(.)22 1586 y(As)e(a)h(side)e(remark,)g(w)o(e)g (note)i(that)g(the)f(op)q(erator)420 1701 y(~)411 1714 y Fq(V)f Fi(=)f(\010)p Fq(V)q Fi(\010)633 1693 y Fc(\003)667 1714 y Fi(=)g(\(2)p Fg(\031)r Fi(\))811 1693 y Fk(d=)p Fn(2)890 1660 y Fc(1)878 1672 y Ff(X)874 1763 y Fk(n)p Fn(=1)949 1714 y Fg(n)978 1693 y Fc(\000)p Fk(d=)p Fn(2)1061 1714 y Fq(T)1100 1721 y Fn(1)p Fk(=n)1170 1714 y Fi(\()p Fg(v)1213 1721 y Fk(n)1236 1714 y Fi(\010)1271 1693 y Fc(\003)1302 1714 y Fi(+)f(\026)-26 b Fg(v)1375 1721 y Fk(n)1398 1714 y Fi(\010\))-49 1850 y(has)18 b(the)f(structure)g (similar)e(to)j(that)f(of)h Fq(V)q Fi(.)24 b(In)17 b(particular,)g(if)f (\(2\))i(con)o(tains)f(only)g(a)h(\014nite)e(n)o(um)o(b)q(er)g(of)-49 1910 y(terms,)g(then)223 1897 y(~)214 1910 y Fq(V)i Fi(can)g(b)q(e)f (written)f(as)i(an)g(in)o(tegral)e(op)q(erator)j(\(1\))e(with)g(k)o (ernel)h(~)-26 b Fg(v)q Fi(\()p Fh(h)p Fg(x;)8 b(y)r Fh(i)p Fi(\))17 b(where)i(~)-26 b Fg(v)q Fi(\()p Fg(t)p Fi(\))17 b(is)-49 1970 y(again)i(a)f(p)q(erio)q(dic)f(function)h(of)g Fg(t)f Fi(\(but)h(of)g(course)g(its)f(p)q(erio)q(d)h(is)g(not)g (necessarily)e(2)p Fg(\031)r Fi(\).)25 b(Th)o(us,)18 b(all)g(our)-49 2031 y(results)e(can)h(b)q(e)f(reform)o(ulated)e(in)i (terms)f(of)i(the)f(Hamiltonian)1172 2018 y(~)1162 2031 y Fq(H)e Fi(=)1281 2018 y(~)1271 2031 y Fq(H)1315 2038 y Fn(0)1346 2031 y Fi(+)1404 2018 y(~)1395 2031 y Fq(V)j Fi(where)1605 2018 y(~)1595 2031 y Fq(H)1639 2038 y Fn(0)1673 2031 y Fi(=)c(\()p Fh(\000)p Fi(\001\))1842 2012 y Fk(l)1855 2031 y Fi(.)22 2109 y(Let)j Fg(Q)g Fi(b)q(e)h(m)o(ultipli)o(cation)d (in)i(the)g(space)g Fg(L)848 2116 y Fn(2)868 2109 y Fi(\()p Fj(R)920 2091 y Fk(d)937 2109 y Fi(\))h(b)o(y)e(some)h(b)q(ounded)h (function)f Fg(q)r Fi(\()p Fg(x)p Fi(\))f(whic)o(h)h(tends)-49 2169 y(to)k(0)h(as)f Fh(j)p Fg(x)p Fh(j)f(!)g(1)p Fi(.)32 b(Then)20 b(the)f(op)q(erator)i Fg(Q)p Fi(\010)p Fg(Q)e Fi(is)h(compact.)30 b(It)19 b(follo)o(ws)h(that)g(op)q(erators)h Fg(Q)p Fq(T)1815 2176 y Fk(n)1838 2169 y Fi(\010)p Fg(Q)-49 2230 y Fi(and)g Fg(Q)p Fq(T)128 2237 y Fk(n)151 2230 y Fi(\010)186 2211 y Fc(\003)206 2230 y Fg(Q)e Fi(are)h(compact)e(for)i (all)f Fg(n)p Fi(.)32 b(According)19 b(to)h(\(6\),)g(under)g (assumption)f(\(4\),)h(this)g(implies)-49 2290 y(compactness)15 b(of)h(op)q(erators)h Fg(Q)p Fq(V)q Fg(Q)e Fi(and,)h(in)f(particular,)g (of)h(\()p Fq(H)1159 2297 y Fn(0)1188 2290 y Fi(+)9 b Fg(cI)t Fi(\))1301 2272 y Fc(\000)p Fn(1)1348 2290 y Fq(V)q Fi(\()p Fq(H)1454 2297 y Fn(0)1483 2290 y Fi(+)h Fg(cI)t Fi(\))1597 2272 y Fc(\000)p Fn(1)1643 2290 y Fi(.)21 b(Let)16 b(us)g(no)o(w)-49 2350 y(use)h(the)f(resolv)o(en)o(t)e (iden)o(tit)o(y)14 2452 y(\()p Fq(H)d Fi(+)g Fg(cI)t Fi(\))203 2431 y Fc(\000)p Fn(1)260 2452 y Fh(\000)g Fi(\()p Fq(H)373 2459 y Fn(0)404 2452 y Fi(+)g Fg(cI)t Fi(\))519 2431 y Fc(\000)p Fn(1)579 2452 y Fi(=)j Fh(\000)670 2404 y Ff(\020)694 2452 y Fi(\()p Fq(H)d Fi(+)g Fg(cI)t Fi(\))883 2431 y Fc(\000)p Fn(1)930 2452 y Fi(\()p Fq(H)993 2459 y Fn(0)1023 2452 y Fi(+)g Fg(cI)t Fi(\))1138 2404 y Ff(\021)d(\020)1196 2452 y Fi(\()p Fq(H)1259 2459 y Fn(0)1289 2452 y Fi(+)j Fg(cI)t Fi(\))1404 2431 y Fc(\000)p Fn(1)1451 2452 y Fq(V)q Fi(\()p Fq(H)1557 2459 y Fn(0)1587 2452 y Fi(+)g Fg(cI)t Fi(\))1702 2431 y Fc(\000)p Fn(1)1749 2404 y Ff(\021)1774 2452 y Fg(;)62 b Fi(\(7\))-49 2554 y(where)18 b Fg(c)e(>)h Fi(0)h(is)f(su\016cien)o(tly)f(large.)26 b(The)18 b(\014rst)g(factor)g(in)f(the)h(righ)o(t-hand)g(side)g(is)f(b) q(ounded)i(b)q(ecause)-49 2614 y Fq(V)f Fi(is)e(b)q(ounded)h(and)g(the) g(second)f(is)g(compact.)21 b(Hence)15 b(op)q(erator)j(\(7\))f(is)f (also)h(compact)e(whic)o(h,)g(b)o(y)h(the)-49 2674 y(W)l(eyl)h (theorem,)e(ensures)j(that)g(the)f(essen)o(tial)f(sp)q(ectra)i(of)g (the)f(op)q(erators)h Fq(H)1413 2681 y Fn(0)1450 2674 y Fi(and)g Fq(H)g Fi(coincide.)23 b(This)-49 2734 y(yields)15 b(the)h(\014rst)h(result.)919 2859 y(2)p eop %%Page: 3 3 3 2 bop -49 67 a Fq(Theorem)17 b(1)24 b Fb(L)n(et)18 b(c)n(ondition)g Fi(\(4\))h Fb(hold.)24 b(Then)18 b(the)g(op)n(er)n (ator)f Fq(V)i Fb(is)f(b)n(ounde)n(d)g(and)g(the)g(op)n(er)n(ator)e Fq(H)i Fb(is)-49 128 y(self-adjoint)h(on)f(the)g(domain)f(of)g(the)h (op)n(er)n(ator)e Fq(H)903 135 y Fn(0)923 128 y Fb(.)22 b(Mor)n(e)n(over,)16 b Fg(\033)1217 135 y Fk(ess)1268 128 y Fi(\()p Fq(H)p Fi(\))e(=)g([0)p Fg(;)8 b Fh(1)p Fi(\))p Fb(.)22 246 y Fq(3.)20 b Fi(The)12 b(op)q(erators)i Fq(H)438 253 y Fn(0)469 246 y Fi(and)f Fq(V)g Fi(are)g(rotationally)f (in)o(v)m(arian)o(t.)19 b(This)12 b(allo)o(ws)g(us)h(to)g(separate)f(v) m(ariables)-49 306 y(in)17 b(the)f(spherical)f(co)q(ordinates)j(whic)o (h)e(is)g(necessary)g(for)h(a)g(more)e(detailed)g(study)i(of)g(the)f (op)q(erator)i Fq(H)p Fi(.)-49 366 y(Let)d Fa(h)59 373 y Fk(m)107 366 y Fi(b)q(e)g(the)f(subspace)i(of)f(spherical)f (functions)g Fg(Y)949 373 y Fk(m)983 366 y Fi(\()p Fg(!)r Fi(\))p Fg(;)22 b(!)16 b Fh(2)e Fj(S)1208 348 y Fk(d)p Fc(\000)p Fn(1)1270 366 y Fi(,)h(of)g(order)g Fg(m)p Fi(.)20 b(Let)15 b Fh(K)g Fi(b)q(e)g(the)g Fg(L)1876 373 y Fn(2)1896 366 y Fi(-)-49 426 y(space)g(with)f(w)o(eigh)o(t)f Fg(r)364 408 y Fk(d)p Fc(\000)p Fn(1)444 426 y Fi(of)i(functions)f (de\014ned)g(on)h Fj(R)971 433 y Fn(+)1012 426 y Fi(and)g(let)e Fa(H)1203 433 y Fk(m)1250 426 y Fi(=)h Fh(K)8 b(\012)f Fa(h)1416 433 y Fk(m)1448 426 y Fi(.)21 b(T)l(o)15 b(put)f(it)g (di\013eren)o(tly)l(,)-49 487 y Fa(H)-19 494 y Fk(m)28 487 y Fh(\032)g Fj(R)114 469 y Fk(d)147 487 y Fi(is)i(the)g(subspace)h (spanned)g(b)o(y)f(functions)327 597 y Fg(u)355 604 y Fk(m)388 597 y Fi(\()p Fg(x)p Fi(\))e(=)g Fh(j)p Fg(x)p Fh(j)576 576 y Fc(\000)p Fk(\016)621 597 y Fg(f)5 b Fi(\()p Fh(j)p Fg(x)p Fh(j)p Fi(\))p Fg(Y)772 604 y Fk(m)806 597 y Fi(\()s(^)-27 b Fg(x)o Fi(\))p Fg(;)60 b Fi(^)-27 b Fg(x)13 b Fi(=)h Fg(x)p Fh(j)p Fg(x)p Fh(j)1119 576 y Fc(\000)p Fn(1)1166 597 y Fg(;)56 b(\016)15 b Fi(=)f(\()p Fg(d)e Fh(\000)e Fi(1\))p Fg(=)p Fi(2)p Fg(;)315 b Fi(\(8\))-49 706 y(where)16 b Fg(f)j Fh(2)c Fg(L)216 713 y Fn(2)235 706 y Fi(\()p Fj(R)287 713 y Fn(+)314 706 y Fi(\))h(and)h Fg(Y)472 713 y Fk(m)519 706 y Fh(2)d Fa(h)588 713 y Fk(m)621 706 y Fi(.)22 b(Then)571 845 y Fg(L)604 852 y Fn(2)624 845 y Fi(\()p Fj(R)676 825 y Fk(d)693 845 y Fi(\))14 b(=)798 791 y Fc(1)784 804 y Ff(M)777 894 y Fk(m)p Fn(=0)862 845 y Fa(H)892 852 y Fk(m)925 845 y Fg(;)57 b Fa(H)1026 852 y Fk(m)1073 845 y Fi(=)14 b Fh(K)d(\012)g Fa(h)1246 852 y Fk(m)1279 845 y Fg(;)557 b Fi(\(9\))-49 987 y(and)17 b(ev)o(ery)d(subspace)i Fa(H)407 994 y Fk(m)456 987 y Fi(is)f(in)o(v)m(arian)o(t)g(with)h(resp)q(ect)f(to)h(the)g(F)l(ourier) f(op)q(erator)i(\010,)f(whic)o(h)f(reduces)g(to)-49 1047 y(the)h(F)l(ourier-Bessel)f(transform)h(on)h Fa(H)672 1054 y Fk(m)705 1047 y Fi(.)k(More)16 b(precisely)l(,)e(set)466 1170 y(\(\010)520 1177 y Fk(m)554 1170 y Fg(f)5 b Fi(\)\()p Fg(r)q Fi(\))14 b(=)729 1112 y Ff(Z)770 1125 y Fc(1)752 1206 y Fn(0)808 1170 y Fi(\()p Fg(r)q(s)p Fi(\))892 1150 y Fn(1)p Fk(=)p Fn(2)947 1170 y Fh(I)974 1178 y Fk(m)p Fn(+\()p Fk(d)p Fc(\000)p Fn(2\))p Fk(=)p Fn(2)1161 1170 y Fi(\()p Fg(r)q(s)p Fi(\))p Fg(f)5 b Fi(\()p Fg(s)p Fi(\))p Fg(ds;)428 b Fi(\(10\))-49 1297 y(where)16 b Fh(I)119 1305 y Fk(m)p Fn(+\()p Fk(d)p Fc(\000)p Fn(2\))p Fk(=)p Fn(2)322 1297 y Fi(is)g(the)g(Bessel)g(function.)21 b(Then,)15 b(for)i(function)f(\(8\),)524 1407 y(\(\010)p Fg(u)606 1414 y Fk(m)640 1407 y Fi(\)\()p Fg(x)p Fi(\))d(=)h Fg(i)807 1387 y Fc(\000)p Fk(m)867 1407 y Fh(j)p Fg(x)p Fh(j)923 1387 y Fc(\000)p Fk(\016)969 1407 y Fi(\(\010)1023 1414 y Fk(m)1057 1407 y Fg(f)5 b Fi(\)\()p Fh(j)p Fg(x)p Fh(j)p Fi(\))p Fg(Y)1227 1414 y Fk(m)1260 1407 y Fi(\()s(^)-27 b Fg(x)o Fi(\))p Fg(:)486 b Fi(\(11\))-49 1517 y(The)17 b(op)q(erator)g(\010)283 1524 y Fk(m)333 1517 y Fi(is)f(of)g(course)h (unitary)f(on)h Fg(L)860 1524 y Fn(2)879 1517 y Fi(\()p Fj(R)931 1524 y Fn(+)958 1517 y Fi(\))f(and)h(\010)1123 1524 y Fk(m)1170 1517 y Fi(=)d(\010)1257 1499 y Fc(\003)1257 1529 y Fk(m)1291 1517 y Fi(.)21 b(Clearly)l(,)558 1627 y(\()p Fq(T)616 1634 y Fk(n)639 1627 y Fg(u)667 1634 y Fk(m)700 1627 y Fi(\)\()p Fg(x)p Fi(\))14 b(=)f Fh(j)p Fg(x)p Fh(j)906 1606 y Fc(\000)p Fk(\016)952 1627 y Fi(\()p Fg(T)1000 1634 y Fk(n)1023 1627 y Fg(f)5 b Fi(\)\()p Fh(j)p Fg(x)p Fh(j)p Fi(\))p Fg(Y)1193 1634 y Fk(m)1226 1627 y Fi(\()s(^)-27 b Fg(x)p Fi(\))p Fg(;)519 b Fi(\(12\))-49 1737 y(where)16 b Fg(T)121 1744 y Fk(n)144 1737 y Fi(,)714 1797 y(\()p Fg(T)762 1804 y Fk(n)785 1797 y Fg(f)5 b Fi(\)\()p Fg(r)q Fi(\))15 b(=)f Fg(n)990 1776 y Fn(1)p Fk(=)p Fn(2)1045 1797 y Fg(f)5 b Fi(\()p Fg(r)q Fi(\))p Fg(;)-49 1884 y Fi(is)16 b(the)g(unitary)g(op)q(erator)i(of)e (dilations)h(in)e Fg(L)796 1891 y Fn(2)816 1884 y Fi(\()p Fj(R)868 1891 y Fn(+)895 1884 y Fi(\).)22 1965 y(Set)311 2042 y Fg(v)335 2049 y Fk(m)368 2042 y Fi(\()p Fg(t)p Fi(\))e(=)h(\(2)p Fg(\031)r Fi(\))581 2021 y Fk(d=)p Fn(2)660 1988 y Fc(1)648 2000 y Ff(X)645 2091 y Fk(n)p Fn(=1)719 2042 y Fg(n)748 2021 y Fc(\000)p Fk(\016)795 2042 y Fi(\()p Fg(v)838 2049 y Fk(n)861 2042 y Fg(i)878 2021 y Fk(m)922 2042 y Fi(+)f(\026)-26 b Fg(v)995 2049 y Fk(n)1018 2042 y Fg(i)1035 2021 y Fc(\000)p Fk(m)1095 2042 y Fi(\)\()p Fg(nt)p Fi(\))1199 2021 y Fn(1)p Fk(=)p Fn(2)1254 2042 y Fh(I)1281 2049 y Fk(m)p Fn(+\()p Fk(d)p Fc(\000)p Fn(2\))p Fk(=)p Fn(2)1468 2042 y Fi(\()p Fg(nt)p Fi(\))-49 2161 y(and)614 2221 y(\()p Fg(V)661 2228 y Fk(m)694 2221 y Fg(f)5 b Fi(\)\()p Fg(r)q Fi(\))15 b(=)869 2162 y Ff(Z)911 2175 y Fc(1)892 2257 y Fn(0)957 2221 y Fg(v)981 2228 y Fk(m)1013 2221 y Fi(\()p Fg(r)q(s)p Fi(\))p Fg(f)5 b Fi(\()p Fg(s)p Fi(\))p Fg(ds:)-49 2325 y Fi(According)16 b(to)h(\(10\))470 2414 y Fg(V)498 2421 y Fk(m)546 2414 y Fi(=)c(\(2)p Fg(\031)r Fi(\))689 2393 y Fk(d=)p Fn(2)769 2360 y Fc(1)756 2372 y Ff(X)753 2463 y Fk(n)p Fn(=1)828 2414 y Fg(n)857 2393 y Fc(\000)p Fk(d=)p Fn(2)940 2414 y Fi(\()p Fg(v)983 2421 y Fk(n)1006 2414 y Fg(i)1023 2393 y Fk(m)1067 2414 y Fi(+)g(\026)-26 b Fg(v)1140 2421 y Fk(n)1163 2414 y Fg(i)1180 2393 y Fc(\000)p Fk(m)1240 2414 y Fi(\))p Fg(T)1288 2421 y Fk(n)1311 2414 y Fi(\010)1346 2421 y Fk(m)1380 2414 y Fg(:)431 b Fi(\(13\))-49 2533 y(This)17 b(equalit)o(y)d(should)j(b)q(e)f(considered)g(as)h(the)f (precise)f(de\014nition)h(of)h(the)f(op)q(erator)h Fg(V)1593 2540 y Fk(m)1627 2533 y Fi(.)22 2614 y(Ob)o(viously)l(,)e Fa(H)291 2621 y Fk(m)340 2614 y Fi(is)h(in)o(v)m(arian)o(t)g(with)g (resp)q(ect)g(to)g(the)g(op)q(erator)i Fq(H)1256 2621 y Fn(0)1275 2614 y Fi(.)j(It)16 b(follo)o(ws)g(from)f(\(8\),)h(\(11\))h (and)-49 2674 y(\(12\))22 b(that)f(ev)o(ery)f(subspace)h Fa(H)541 2681 y Fk(m)595 2674 y Fi(is)g(also)g(in)o(v)m(arian)o(t)g (with)f(resp)q(ect)h(to)g(op)q(erator)h(\(6\).)36 b(It)20 b(reduces)h(on)-49 2734 y Fa(H)-19 2741 y Fk(m)35 2734 y Fi(to)f(the)g(op)q(erator)i(\012)p Fg(V)450 2741 y Fk(m)484 2734 y Fi(\012)519 2716 y Fc(\003)552 2734 y Fh(\012)14 b Fg(I)627 2741 y Fk(m)680 2734 y Fi(where)20 b(\012)g(:)g Fg(L)947 2741 y Fn(2)967 2734 y Fi(\()p Fj(R)1019 2741 y Fn(+)1046 2734 y Fi(\))g Fh(!)h(K)g Fi(is)f(a)g(unitary)g(op)q(erator)i(de\014ned)e(b)o(y)919 2859 y(3)p eop %%Page: 4 4 4 3 bop -49 67 a Fi(\(\012)p Fg(g)r Fi(\)\()p Fg(r)q Fi(\))15 b(=)f Fg(r)200 49 y Fc(\000)p Fk(\016)246 67 y Fg(g)r Fi(\()p Fg(r)q Fi(\))j(and)g Fg(I)466 74 y Fk(m)515 67 y Fi(is)f(the)g(iden)o(tit)o(y)e(op)q(erator)j(in)f(the)g(space)g Fa(h)1317 74 y Fk(m)1350 67 y Fi(.)21 b(Let)c Fg(H)1513 74 y Fn(0)1549 67 y Fi(b)q(e)f(m)o(ultiplication)-49 128 y(b)o(y)g Fg(x)47 110 y Fn(2)p Fk(l)94 128 y Fi(and)h Fg(H)233 110 y Fn(\()p Fk(m)p Fn(\))307 128 y Fi(=)d Fg(H)399 135 y Fn(0)430 128 y Fi(+)d Fg(V)507 135 y Fk(m)557 128 y Fi(in)16 b(the)g(space)h Fg(L)862 135 y Fn(2)882 128 y Fi(\()p Fj(R)933 135 y Fn(+)960 128 y Fi(\).)k(Then)672 266 y Fq(H)14 b Fi(=)802 213 y Fc(1)789 225 y Ff(M)782 316 y Fk(m)p Fn(=0)867 266 y Fi(\012)p Fg(H)946 246 y Fn(\()p Fk(m)p Fn(\))1007 266 y Fi(\012)1042 246 y Fc(\003)1073 266 y Fh(\012)d Fg(I)1145 273 y Fk(m)1177 266 y Fg(;)634 b Fi(\(14\))-49 408 y(and)16 b(hence)e(the)h(sp)q(ectral)g(analysis)h (of)f(the)g(op)q(erator)h Fq(H)f Fi(reduces)g(to)g(that)h(for)f(the)g (family)e(of)j(op)q(erators)-49 468 y Fg(H)-5 450 y Fn(\()p Fk(m)p Fn(\))56 468 y Fi(.)22 574 y Fq(4.)21 b Fi(No)o(w)16 b(w)o(e)g(can)h(c)o(hec)o(k)d(that)j(the)f(w)o(a)o(v)o(e)f(op)q (erators)452 684 y Fg(W)498 691 y Fc(\006)528 684 y Fi(\()p Fq(H)p Fg(;)8 b Fq(H)657 691 y Fn(0)676 684 y Fi(\))14 b(=)f Fg(s)f Fh(\000)32 b Fi(lim)844 712 y Fk(t)p Fc(!\0061)963 684 y Fi(exp\()p Fg(i)p Fq(H)p Fg(t)p Fi(\))8 b(exp)o(\()p Fh(\000)p Fg(i)p Fq(H)1356 691 y Fn(0)1375 684 y Fg(t)p Fi(\))-49 808 y(exist)14 b(and)i(are)e(complete,)f(that)i(is)f(their)g (ranges)i(coincide)d(with)i(the)f(absolutely)h(con)o(tin)o(uous)f (subspace)-49 868 y(of)j(the)f(op)q(erator)h Fq(H)p Fi(.)k(According)16 b(to)g(\(9\),)g(\(14\),)h(to)f(that)h(end)f(it)g(su\016ces)f(to)i(v)o (erify)d(the)i(same)f(facts)i(for)-49 928 y(all)j(pairs)g Fg(H)186 935 y Fn(0)206 928 y Fg(;)8 b(H)272 910 y Fn(\()p Fk(m)p Fn(\))333 928 y Fi(,)20 b Fg(m)g Fi(=)g(0)p Fg(;)8 b Fi(1)p Fg(;)g(:)g(:)g(:)p Fi(.)32 b(By)19 b(the)h(Birman-Krein)d (theorem)i([2])g(\(see)h(also)g([7]\),)g(w)o(e)f(need)-49 988 y(only)d(to)h(sho)o(w)g(that)g(op)q(erator)-19 1098 y(\()p Fg(H)44 1078 y Fn(\()p Fk(m)p Fn(\))116 1098 y Fi(+)11 b Fg(cI)t Fi(\))231 1078 y Fc(\000)p Fn(1)289 1098 y Fh(\000)g Fi(\()p Fg(H)398 1105 y Fn(0)429 1098 y Fi(+)g Fg(cI)t Fi(\))544 1078 y Fc(\000)p Fn(1)604 1098 y Fi(=)j Fh(\000)695 1050 y Ff(\020)720 1098 y Fi(\()p Fg(H)783 1078 y Fn(\()p Fk(m)p Fn(\))855 1098 y Fi(+)d Fg(cI)t Fi(\))970 1078 y Fc(\000)p Fn(1)1016 1098 y Fi(\()p Fg(H)1075 1105 y Fn(0)1106 1098 y Fi(+)g Fg(cI)t Fi(\))1221 1050 y Ff(\021)d(\020)1279 1098 y Fi(\()p Fg(H)1338 1105 y Fn(0)1369 1098 y Fi(+)j Fg(cI)t Fi(\))1484 1078 y Fc(\000)p Fn(1)1530 1098 y Fg(V)1558 1105 y Fk(m)1592 1098 y Fi(\()p Fg(H)1651 1105 y Fn(0)1682 1098 y Fi(+)g Fg(cI)t Fi(\))1797 1078 y Fc(\000)p Fn(1)1844 1050 y Ff(\021)1869 1098 y Fg(;)-49 1208 y Fi(b)q(elongs)17 b(to)g(the)f(trace)g(class)g Fa(S)540 1215 y Fn(1)577 1208 y Fi(or)g(that)606 1318 y(\()p Fg(H)665 1325 y Fn(0)696 1318 y Fi(+)11 b Fg(I)t Fi(\))790 1298 y Fc(\000)p Fn(1)837 1318 y Fg(V)865 1325 y Fk(m)898 1318 y Fi(\()p Fg(H)957 1325 y Fn(0)988 1318 y Fi(+)g Fg(I)t Fi(\))1082 1298 y Fc(\000)p Fn(1)1143 1318 y Fh(2)j Fa(S)1224 1325 y Fn(1)1244 1318 y Fg(:)567 b Fi(\(15\))22 1449 y(According)13 b(to)h(\(13\))g(it)f(su\016ces)g(to) h(c)o(hec)o(k)d(that)j(the)g(op)q(erators)h(\()p Fg(H)1259 1456 y Fn(0)1284 1449 y Fi(+)5 b Fg(I)t Fi(\))1372 1431 y Fc(\000)p Fn(1)1419 1449 y Fg(T)1448 1456 y Fk(n)1471 1449 y Fi(\010)1506 1456 y Fk(m)1540 1449 y Fi(\()p Fg(H)1599 1456 y Fn(0)1624 1449 y Fi(+)g Fg(I)t Fi(\))1712 1431 y Fc(\000)p Fn(1)1772 1449 y Fi(b)q(elong)-49 1509 y(to)17 b Fa(S)45 1516 y Fn(1)81 1509 y Fi(and)g(the)f(corresp)q(onding)h (trace)f(norms)g(satisfy)423 1591 y Fc(1)411 1604 y Ff(X)408 1694 y Fk(n)p Fn(=1)483 1645 y Fh(j)p Fg(v)521 1652 y Fk(n)544 1645 y Fh(j)p Fg(n)587 1625 y Fc(\000)p Fk(d=)p Fn(2)669 1645 y Fh(jj)p Fi(\()p Fg(H)756 1652 y Fn(0)787 1645 y Fi(+)11 b Fg(I)t Fi(\))881 1625 y Fc(\000)p Fn(1)928 1645 y Fg(T)957 1652 y Fk(n)980 1645 y Fi(\010)1015 1652 y Fk(m)1048 1645 y Fi(\()p Fg(H)1107 1652 y Fn(0)1139 1645 y Fi(+)g Fg(I)t Fi(\))1233 1625 y Fc(\000)p Fn(1)1279 1645 y Fh(jj)1307 1652 y Fn(1)1340 1645 y Fg(<)j Fh(1)p Fg(:)369 b Fi(\(16\))-49 1787 y(Since)550 1847 y(\()p Fg(H)609 1854 y Fn(0)640 1847 y Fi(+)11 b Fg(I)t Fi(\))734 1826 y Fc(\000)p Fn(1)780 1847 y Fg(T)809 1854 y Fk(n)846 1847 y Fi(=)j Fg(n)927 1826 y Fn(2)p Fk(l)958 1847 y Fg(T)987 1854 y Fk(n)1010 1847 y Fi(\()p Fg(H)1069 1854 y Fn(0)1100 1847 y Fi(+)d Fg(n)1178 1826 y Fn(2)p Fk(l)1209 1847 y Fg(I)t Fi(\))1254 1826 y Fc(\000)p Fn(1)1300 1847 y Fg(;)-49 1934 y Fi(condition)16 b(\(16\))h(is)f(equiv)m(alen)o(t)f (to)392 2019 y Fc(1)379 2031 y Ff(X)376 2122 y Fk(n)p Fn(=1)451 2073 y Fh(j)p Fg(v)489 2080 y Fk(n)512 2073 y Fh(j)p Fg(n)555 2052 y Fc(\000)p Fk(d=)p Fn(2+2)p Fk(l)694 2073 y Fh(jj)p Fi(\()p Fg(H)781 2080 y Fn(0)811 2073 y Fi(+)c Fg(n)889 2052 y Fn(2)p Fk(l)920 2073 y Fg(I)t Fi(\))965 2052 y Fc(\000)p Fn(1)1012 2073 y Fi(\010)1047 2080 y Fk(m)1080 2073 y Fi(\()p Fg(H)1139 2080 y Fn(0)1170 2073 y Fi(+)g Fg(I)t Fi(\))1264 2052 y Fc(\000)p Fn(1)1311 2073 y Fh(jj)1339 2080 y Fn(1)1372 2073 y Fg(<)j Fh(1)p Fg(:)337 b Fi(\(17\))22 2235 y(Next)15 b(w)o(e)h(use)g(the)g(follo)o (wing)-49 2329 y Fq(Lemma)g(2)24 b Fb(L)n(et)14 b(the)h(op)n(er)n(ator) e Fi(\010)585 2336 y Fk(m)633 2329 y Fb(b)n(e)i(de\014ne)n(d)g(by)g (formula)f Fi(\(10\))h Fb(and)f(let)i Fg(P)22 b Fb(and)14 b Fg(Q)g Fb(b)n(e)h(multiplic)n(ations)-49 2389 y(by)21 b(functions)h Fg(p)f Fb(and)f Fg(q)r Fb(.)31 b(Then)22 b(the)f(op)n(er)n(ator)d Fg(P)7 b Fi(\010)927 2396 y Fk(m)961 2389 y Fg(Q)20 b Fb(acting)i(in)f(the)g(sp)n(ac)n(e)f Fg(L)1479 2396 y Fn(2)1499 2389 y Fi(\()p Fj(R)1551 2396 y Fn(+)1577 2389 y Fi(\))h Fb(b)n(elongs)h(to)e(the)-49 2449 y(tr)n(ac)n(e)d(class)h(and)81 2559 y Fh(jj)p Fg(P)7 b Fi(\010)182 2566 y Fk(m)215 2559 y Fg(Q)p Fh(jj)282 2539 y Fn(2)282 2571 y(1)315 2559 y Fh(\024)14 b Fg(C)414 2501 y Ff(Z)456 2514 y Fc(1)437 2595 y Fn(0)501 2559 y Fh(j)p Fg(p)p Fi(\()p Fg(r)q Fi(\))p Fh(j)614 2539 y Fn(2)635 2559 y Fi(\(1)d(+)g Fg(r)761 2539 y Fn(2)781 2559 y Fi(\))800 2539 y Fk(\033)824 2559 y Fg(dr)880 2501 y Ff(Z)922 2514 y Fc(1)904 2595 y Fn(0)968 2559 y Fh(j)p Fg(q)r Fi(\()p Fg(s)p Fi(\))p Fh(j)1081 2539 y Fn(2)1099 2559 y Fi(\(1)h(+)f Fg(s)1226 2539 y Fn(2)1246 2559 y Fi(\))1265 2539 y Fk(\033)1288 2559 y Fg(ds;)58 b(C)17 b Fi(=)d Fg(C)t Fi(\()p Fg(m;)8 b(\033)r Fi(\))p Fg(;)127 b Fi(\(18\))-49 2686 y Fb(pr)n(ovide)n(d)17 b(the)h(right-hand)g(side)f(is)h(\014nite)g(for)f(some)h Fg(\033)d(>)f Fi(1)p Fg(=)p Fi(2)p Fb(.)919 2859 y Fi(4)p eop %%Page: 5 5 5 4 bop 22 67 a Fi(A)16 b(pro)q(of)h(can)f(b)q(e)g(obtained)g (similarly)d(to)k(the)e(case)h(when)h(the)e(role)h(of)g(\010)1398 74 y Fk(m)1448 67 y Fi(is)g(pla)o(y)o(ed)e(b)o(y)i(the)g(usual)-49 128 y(F)l(ourier)21 b(transform)f(\(see)g([3])g(or)i([6)o(]\),)f(so)h (it)e(will)g(b)q(e)h(giv)o(en)f(in)g(a)h(short)g(form.)34 b(Actually)l(,)20 b(w)o(e)g(follo)o(w)-49 188 y(closely)e(the)g(pro)q (of)i(of)f(T.)f(Kato)h(published)f(in)h([6].)27 b(Let)19 b Fg(G)g Fi(b)q(e)g(m)o(ulptiplic)o(ation)d(b)o(y)i(\(1)13 b(+)g Fg(r)1708 170 y Fn(2)1728 188 y Fi(\))1747 170 y Fn(1)p Fk(=)p Fn(2)1802 188 y Fi(.)28 b(W)l(e)-49 248 y(ha)o(v)o(e)16 b(that)h Fg(P)7 b Fi(\010)243 255 y Fk(m)276 248 y Fg(Q)14 b Fi(=)f Fg(K)421 255 y Fn(1)442 248 y Fg(LK)516 255 y Fn(0)552 248 y Fi(where)166 357 y Fg(K)207 364 y Fn(1)241 357 y Fi(=)h Fg(P)7 b(G)369 337 y Fk(\033)393 357 y Fi(\010)428 364 y Fk(m)461 357 y Fg(G)499 337 y Fc(\000)p Fk(\033)551 357 y Fg(;)56 b(L)14 b Fi(=)g Fg(G)758 337 y Fk(\033)782 357 y Fi(\010)817 364 y Fk(m)850 357 y Fg(G)888 337 y Fc(\000)p Fk(\033)940 357 y Fi(\010)975 364 y Fk(m)1008 357 y Fg(G)1046 337 y Fc(\000)p Fk(\033)1098 357 y Fi(\010)1133 364 y Fk(m)1166 357 y Fg(G)1204 337 y Fk(\033)1228 357 y Fg(;)57 b(K)1340 364 y Fn(0)1374 357 y Fi(=)14 b Fg(G)1464 337 y Fc(\000)p Fk(\033)1515 357 y Fi(\010)1550 364 y Fk(m)1583 357 y Fg(G)1621 337 y Fk(\033)1645 357 y Fg(Q:)-49 467 y Fi(Clearly)l(,)h(the)h(Hilb)q (ert-Sc)o(hmidt)d(norm)-32 590 y Fh(jj)p Fg(K)37 597 y Fn(0)56 590 y Fh(jj)84 569 y Fn(2)84 602 y(2)118 590 y Fi(=)169 531 y Ff(Z)211 544 y Fc(1)192 626 y Fn(0)257 531 y Ff(Z)298 544 y Fc(1)280 626 y Fn(0)335 590 y Fi(\(1)f(+)f Fg(r)462 569 y Fn(2)482 590 y Fi(\))501 569 y Fc(\000)p Fk(\033)552 590 y Fg(r)q(s)p Fh(I)629 569 y Fn(2)625 602 y Fk(m)p Fn(+\()p Fk(d)p Fc(\000)p Fn(2\))p Fk(=)p Fn(2)812 590 y Fi(\()p Fg(r)q(s)p Fi(\)\(1)h(+)f Fg(s)1023 569 y Fn(2)1042 590 y Fi(\))1061 569 y Fk(\033)1085 590 y Fh(j)p Fg(q)r Fi(\()p Fg(s)p Fi(\))p Fh(j)1198 569 y Fn(2)1216 590 y Fg(dr)q(ds)k Fh(\024)f Fg(C)1427 531 y Ff(Z)1468 544 y Fc(1)1450 626 y Fn(0)1514 590 y Fh(j)p Fg(q)r Fi(\()p Fg(s)p Fi(\))p Fh(j)1627 569 y Fn(2)1645 590 y Fi(\(1)e(+)f Fg(s)1772 569 y Fn(2)1792 590 y Fi(\))1811 569 y Fk(\033)1834 590 y Fg(ds;)-49 718 y Fi(since)16 b(the)g(function)g Fg(t)p Fh(I)395 700 y Fn(2)391 732 y Fk(m)p Fn(+\()p Fk(d)p Fc(\000)p Fn(2\))p Fk(=)p Fn(2)577 718 y Fi(\()p Fg(t)p Fi(\))g(is)g(b)q(ounded.)22 b(Quite)16 b(similarly)l(,)573 853 y Fh(jj)p Fg(K)642 860 y Fn(1)661 853 y Fh(jj)689 833 y Fn(2)689 866 y(2)723 853 y Fh(\024)d Fg(C)822 795 y Ff(Z)864 808 y Fc(1)845 889 y Fn(0)909 853 y Fh(j)p Fg(p)p Fi(\()p Fg(r)q Fi(\))p Fh(j)1022 833 y Fn(2)1042 853 y Fi(\(1)f(+)f Fg(r)1169 833 y Fn(2)1189 853 y Fi(\))1208 833 y Fk(\033)1231 853 y Fg(dr)o(:)-49 980 y Fi(T)l(o)16 b(conclude)f(the)g(pro)q(of)i(of)e(\(18\),)h(it)f (remains)f(to)i(c)o(hec)o(k)d(that)j(the)f(op)q(erator)i Fg(L)d Fi(=)g Fg(L)1536 987 y Fk(\033)1575 980 y Fi(is)h(b)q(ounded.)22 b(Let)-49 1040 y Fq(G)17 b Fi(b)q(e)f(m)o(ultiplic)o(ation)e(b)o(y)i (\(1)11 b(+)g Fh(j)p Fg(x)p Fh(j)618 1022 y Fn(2)637 1040 y Fi(\))656 1022 y Fn(1)p Fk(=)p Fn(2)728 1040 y Fi(in)k(the)h(space)h Fg(L)1032 1047 y Fn(2)1052 1040 y Fi(\()p Fj(R)1104 1022 y Fk(d)1121 1040 y Fi(\))f(and)638 1150 y Fq(L)672 1157 y Fk(\033)709 1150 y Fi(=)e Fq(G)805 1129 y Fk(\033)829 1150 y Fi(\010)p Fq(G)908 1129 y Fc(\000)p Fk(\033)959 1150 y Fi(\010)994 1129 y Fc(\003)1014 1150 y Fq(G)1058 1129 y Fc(\000)p Fk(\033)1109 1150 y Fi(\010)p Fq(G)1188 1129 y Fk(\033)1211 1150 y Fg(:)-49 1259 y Fi(By)i(a)g(direct)f(calculation,)g(one)h(easily)f(v)o(eri\014es)g (that)h(the)g(op)q(erator)h Fq(L)1271 1266 y Fn(2)1307 1259 y Fi(is)f(b)q(ounded)g(in)g Fg(L)1644 1266 y Fn(2)1664 1259 y Fi(\()p Fj(R)1716 1241 y Fk(d)1733 1259 y Fi(\).)21 b(Then,)-49 1319 y(b)o(y)f(the)f(complex)f(in)o(terp)q(olation,)i(this) g(results)g(extends)f(to)h(all)g Fg(\033)h Fh(2)f Fi(\(0)p Fg(;)8 b Fi(2\).)33 b(The)20 b(restriction)f(of)h(the)-49 1379 y(op)q(erator)e Fq(L)182 1386 y Fk(\033)222 1379 y Fi(on)e(the)h(subspace)f Fa(H)607 1386 y Fk(m)657 1379 y Fi(equals)g Fg(i)822 1361 y Fc(\000)p Fk(m)883 1379 y Fi(\012)p Fg(L)951 1386 y Fk(\033)974 1379 y Fi(\012)1009 1361 y Fc(\003)1041 1379 y Fh(\012)10 b Fg(I)1112 1386 y Fk(m)1145 1379 y Fi(.)22 b(Hence)15 b(the)h(op)q(erator)i Fg(L)1640 1386 y Fk(\033)1680 1379 y Fi(is)e(b)q(ounded)-49 1440 y(in)g Fg(L)41 1447 y Fn(2)61 1440 y Fi(\()p Fj(R)113 1447 y Fn(+)140 1440 y Fi(\).)22 1520 y(It)d(follo)o(ws)h(from)f(Lemma) e(2)j(that,)h(in)e(the)h(case)f(2)p Fg(l)i(>)f Fi(1,)g(all)f(op)q (erators)j(\()p Fg(H)1402 1527 y Fn(0)1428 1520 y Fi(+)6 b Fg(n)1501 1502 y Fn(2)p Fk(l)1531 1520 y Fg(I)t Fi(\))1576 1502 y Fc(\000)p Fn(1)1623 1520 y Fi(\010)1658 1527 y Fk(m)1691 1520 y Fi(\()p Fg(H)1750 1527 y Fn(0)1777 1520 y Fi(+)g Fg(I)t Fi(\))1866 1502 y Fc(\000)p Fn(1)-49 1581 y Fi(b)q(elong)17 b(to)g(the)f(trace)g(class)g(and)283 1690 y Fh(jj)p Fi(\()p Fg(H)370 1697 y Fn(0)401 1690 y Fi(+)11 b Fg(n)479 1669 y Fn(2)p Fk(l)509 1690 y Fg(I)t Fi(\))554 1669 y Fc(\000)p Fn(1)601 1690 y Fi(\010)636 1697 y Fk(m)670 1690 y Fi(\()p Fg(H)729 1697 y Fn(0)760 1690 y Fi(+)g Fg(I)t Fi(\))854 1669 y Fc(\000)p Fn(1)900 1690 y Fh(jj)928 1697 y Fn(1)961 1690 y Fh(\024)j Fg(C)t(n)1082 1669 y Fc(\000)p Fn(2)p Fk(l)p Fn(+1+)p Fk(")1229 1690 y Fg(;)56 b(C)17 b Fi(=)d Fg(C)t Fi(\()p Fg(m;)8 b(")p Fi(\))p Fg(;)-49 1800 y Fi(for)15 b(an)o(y)g Fg(")f(>)f Fi(0.)22 b(Therefore,)14 b(under)g(assumption)h(\(5\),)g(the)f(series)g (\(17\))i(con)o(v)o(erges)e(whic)o(h)g(implies)e(\(15\).)-49 1860 y(Let)17 b(us)f(form)o(ulate)f(the)h(result)g(obtained.)-49 1953 y Fq(Theorem)h(3)24 b Fb(L)n(et)15 b Fg(l)f(>)g Fi(1)p Fg(=)p Fi(2)i Fb(and)g(let)g(c)n(ondition)f Fi(\(5\))h Fb(hold.)21 b(Then)16 b(the)g(wave)g(op)n(er)n(ators)d Fg(W)1638 1960 y Fc(\006)1668 1953 y Fi(\()p Fq(H)p Fg(;)8 b Fq(H)1797 1960 y Fn(0)1816 1953 y Fi(\))15 b Fb(ex-)-49 2013 y(ist)g(and)g(ar)n(e)f(c)n(omplete.)22 b(In)15 b(p)n(articular,)g (the)g(absolutely)h(c)n(ontinuous)g(p)n(art)d(of)i Fq(H)f Fb(is)h(unitarily)g(e)n(quivalent)-49 2073 y(to)j(the)g(op)n(er)n(ator) e Fq(H)327 2080 y Fn(0)346 2073 y Fb(.)22 2166 y Fi(Compared)f(to)h ([4])f(w)o(e)g(allo)o(w)g(su\016cien)o(tly)f(more)g(general)h (functions)h Fg(v)r Fi(\()p Fg(t)p Fi(\))e(and)i(relax)f(the)g (condition)-49 2226 y(2)p Fg(l)g(>)f(d)j Fi(imp)q(osed)e(in)h([4])g(b)q (ecause)g(separation)h(of)g(v)m(ariables)f(w)o(as)g(not)h(tak)o(en)f (in)o(to)g(accoun)o(t)g(there.)-49 2378 y Fq(REFERENCES)-24 2512 y Fi([1])24 b(F.)f(A.)g(Berezin,)h(L.)g(D.)g(F)l(addeev,)h(A)e (remark)g(on)h(Sc)o(hr\177)-24 b(odinger's)24 b(equation)f(with)h(a)h (singular)52 2572 y(p)q(oten)o(tial,)15 b(So)o(viet)g(Math.)h(Dokl.)g Fq(2)g Fi(\(1961\),)i(372-375.)-24 2674 y([2])24 b(M.)15 b(Sh.)h(Birman)f(and)i(M.)e(G.)i(Krein,)e(On)h(the)g(theory)g(of)h(w)o (a)o(v)o(e)e(op)q(erators)j(and)f(scattering)f(op)q(er-)52 2734 y(ators,)g(So)o(viet)g(Math.)g(Dokl.)f Fq(3)i Fi(\(1962\),)g (740-744.)919 2859 y(5)p eop %%Page: 6 6 6 5 bop -24 67 a Fi([3])24 b(M.)d(Sh.)h(Birman)f(and)i(M.)f(Z.)g(Solom) o(y)o(ak,)f(Estimates)h(for)g(the)g(singular)h(n)o(um)o(b)q(ers)e(of)i (in)o(tegral)52 128 y(op)q(erators,)17 b(Russian)g(Math.)f(Surv)o(eys,) f Fq(32)p Fi(,)g(no.)i(1)f(\(1977\),)i(15-89.)-24 229 y([4])24 b(M.)d(Sh.)g(Birman)f(and)j(D.)f(R.)f(Y)l(afaev,)h(On)g(the)g (trace-class)g(metho)q(d)f(in)h(p)q(oten)o(tial)f(scattering)52 290 y(theory)l(,)15 b(J.)h(So)o(viet)f(Math.)h Fq(56)p Fi(,)g(no.)g(2)h(\(1991\),)g(2285-2299)q(.)-24 391 y([5])24 b(L.)14 b(D.)g(F)l(addeev,)f(On)i(the)e(F)l(riedric)o(hs)g(mo)q(del)g (in)g(the)h(theory)g(of)h(p)q(erturbations)g(of)f(the)g(con)o(tin)o (uous)52 451 y(sp)q(ectrum,)g(Amer.)g(Math.)i(So)q(c.)g(T)l(ransl.)g (Ser.2)g Fq(62)h Fi(\(1967\).)-24 553 y([6])24 b(M.)13 b(Reed)h(and)g(B.)g(Simon,)f(Metho)q(ds)h(of)h(mo)q(dern)e (mathematical)e(ph)o(ysics,)j(v.3,)g(Academic)d(Press,)52 613 y(1979.)-24 715 y([7])24 b(D.)16 b(R.)f(Y)l(afaev,)h(Mathematical)e (scattering)i(theory)l(,)g(AMS,)f(1992.)-24 817 y([8])24 b(D.)19 b(R.)h(Y)l(afaev,)f(The)h(discrete)f(sp)q(ectrum)f(in)i(the)g (singular)g(F)l(riedric)o(hs)e(mo)q(del,)h(Pr)o(\023)-23 b(epublication)52 877 y(98-06,)17 b(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(Rennes)g(1,)g(1998.)919 2859 y(6)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF