; TeX output 2002.06.21:1719y?^gwKNff cmbx12Theffessentialsps3ectrumofrelativistici;Qone-electronffionsintheJansen-Hessmos3del3aXQ cmr12D.H.Jakubaa-AmrundsenpK`y cmr10MathematicsUUInstitute,UniversityUUofMunich 80333UUMunich,Germany]=!Kt : cmbx9Abstractctdo cmr9Itissho9wnthattheessentialspAectrumofthepseudo-relativisticDirac WopAeratoraccordingtoJansenandHesswhic9hincludestheCoulombpAo-Wten9tial+uptosecondorder,qjextendsfrom5" cmmi9mc-=Aacmr62Vstoin nitywhenthenu-Wclearec9hargeisbAelowthecriticalv|ralueZe-=2 cmsy9X1:006:ThereisalsonoWsingularcon9tinuousspAectruminthatcase,"andforsmallZnoembAeddedWeigen9v|ralues.This]1workisanextensionofinvestigationsbyEv|rans,o(PerryWandYSieden9topontheBrown-RavenhallopAeratorwhichisof rstorderinWthepAoten9tial.ZItisbasedonthefact,urecentlyprovenbyBrummelhuis,WSieden9topUandStoAckmeyer,| thatthespAectrumoftheJansen-HessoperatorWisTbAoundedfrombelo9wforsubcriticalc9hargesZ.1*y?>1VLIntros3duction>TheDiracopGeratorofanelectronwithmass b> cmmi10mandmomentum&"V cmbx10p 瀲inanexternal > eldUUV9isgivenby(inrelativisticunits,$ msbm10~=c=1)5HH^.=0)DF cmmib10 )pp+ m+Vqò(1.1)>where Pand -$aretheDiracmatricesandinthecoGordinaterepresentationof>HA;hphastobGeidenti edwith !", cmsy10i@8=@x:[7HoweverH,actingontheHilbGertspace>LٓRcmr72|s(R^3)8C^4|s;misUUnotbGoundedfrombelow.qInthecaseofaCoulombpGotential,V8(x)0=<$K Kwfe (֍@x ;) UP:=Ze2qò(1.2)⍑>(withx:=2jxj;Z8thenuclearchargenumbGeroftheionande^2=(137:04)^ O!cmsy71>theY nestructureconstant),wheretheexacteigenfunctionsareknown,thisdif-> cultycmaybGecircumventedbyintroGducingthepro8jectionoperatorP+ Sontothe>pGositivespectralsubspaceofH뉲andconsideringtheboundedoperatorP+HzP+>insteadUUwhichhasthesamepGositive-energyeigenstatesasH%S[15 ].pMF*ornoncoulombiccentralpGotentialstheeigenfunctionsandhenceP+ 7^areun->known.YuAn ]approximationtoP+HzP+ wasconsideredbyBrownandRavenhall>[3]UUwhointroGducedtheoperator(seealso[15 ])lŵBG:=0+Hz+;)+(p):=<$c1cwfe (֍2fW^u cmex10˲1+<$3+ kp+8 m3+wfe+ (֍>E 0ercmmi7p3[^qò(1.3)@>whereU,+pro8jectsontothepGositivespGectralsubspaceofthe/': cmti10fr}'eeiDiracU,operator,>EpbGeing6theenergyoftheelectron.RByconstruction, =Bvconsistsofazero-order>andaa rst-orderterminthepGotentialV8,handonemayde neanopGerator>b0m +Ib1mactingnonthetwo-dimensionalnspaceL2|s(R^3)C^2 bynusingthefact>that}anyfour-spinor <ݸ2DG(Bq)canberepresentedintermsofaPaulispinor>u2L2|s(R^3)8C^2|s:mOneUUidenti es[6]Cn( [;B+ )0=:(u;(b0m +8b1m )u)q(1.4)>whereb0mandb1mdenotethecorrespGondingzero-and rst-ordercontributions>inUUV8,respGectively +Wb0m\&:=0Epfj:=p1fe#{ Dpr2S+8mr2b1m\&:=0A(p)[V8ܲ+RV8R]A(p)q(1.5)I>withmA(p):=Ϋs Ήfe#h2<$33Ep2+8m33wfe! (֍J2Ep0/-; R߲:=h(p)(p^:p 5);h(p):=<$"pKwfe! (֍Ep2+8m|>where^UUp :=0p=p;qp:=jpjmandUU SisUUthevectorofthePaulimatrices.pMAmethoGdtoconstructanoperatorwhichapproximatesP+HzP+ 3`tohigher>orderinVAҲistheF*oldy-Wouthuysentransformationtechnique[7].XOItconsistsof2y?>a3seriesofunitarytransformationssuccessivelyappliedtoHwhichcastHinto >ajbloGck-diagonalformtoanygivenorderinV8,p$leading(withinthisorder)toa>decouplingUUofthepGositiveandnegativespGectralsubspacesofH.pMF*ollowingUUDouglasandKroll[5],thetransformedopGeratorisde nedbyx3(Unq~:::U1|sU0)H(Un:::U1U0)1J=:0H(n)v+OG(V8n+1e)q(1.6)>where~H^(n)yhasbloGck-diagonalstructure,eachbloGckactingonL2|s(R^3).C^2|s:FThe > rst;transformation,5U0|s,isthefree-particleF*oldy-Wouthuysentransformation>[2]|-U0 =0A(p)(1+ R0|s);)R0C:=h(p)( ^@p ̲)q(1.7)>whichcaststhezero-ordertermofHintobloGck-diagonalformwhilethenon->(bloGck)diagonal$remainderisof rstorderinV8.aThetransformationsU1|s;:::;Un>haveUUtheformsUi|=0(1+Wc2፴i)1=2²+WiTL;)i=1;:::;nqò(1.8)>whereDtheanti-hermiteanopGeratorsWi#aresuccessivelyconstructedfromthe >requirement1thatthenon-(bloGck)diagonaltermsoforderivqanish.\Anexplicit>expressionforH^(2)_[wasprovidedbyDouglasandKroll[5],Mbutwaslatercor->rected'byJansenandHess[11 ].bItsuppGerblock,0termedbm,whichcorrespGonds>tothepGositivespectralsubspace,Gagreeswithb0m +tb1mCfrom(1.4)upto rst>orderUUinV8,buthasanadditionalsecond-ordertermbm &˲=0b0m+b1m+b2meb2m\&:=<$c1cwfe (֍2(w1m O1 |kO1|sw1m)q(1.9)\ >where6O1:=~XA(p)(RV2NV8R)A(p);v'and6w1m isanintegralopGeratorlinear>inV8, =de nedbyw1m Ep8+[EpRw1m=MO1|s:\ItshouldbGenotedthatduetothe>particularOjstructure(1.8)ofthetransformationU2ݲcombinedwiththelinearity>ofUUW2ȲinV9;8bm isuna ectedbyU2|s:MTheTCoulombcase(1.2)iswellsuitedforinvestigatingthequalityofthe>abGove4Iapproximationssinceonecancomparewiththeexactsolutions.fF*orthis>case,the֚spGectralpropertiesoftheBrown-Ravenhall֚operatorb0m z+;lb1mkwerein>detailstudiedbyEvqansandcoworkers[6,1],andthebGoundednessfrombelow>ofWtheJansen-HessopGeratorbm forsubcriticalcharges wasrecentlyprovenby>Brummelhuisitetal[4].$Inthepresentworkwewanttoprove,n|fortheCoulomb>pGotentialUU(1.2),p>TheoremT1.1L}'etPthecriticalcouplingconstant c.ݸ1:006bede nedasthe>smallersolutionof18 8=2([=2+2=)+ 8^2 =8(=22=)^2 =0:ZIf UP< c;pFr(i)Wthetessentialsp}'ectrumtofbm ^isgivenby0t}\cmti7ess w(bm)0=ess(b0m )=[m;1);3y?Cb(ii)Wthesingularc}'ontinuousspectrumofbm ,isempty.pMEvqansetal[6]provedTheorem1.1fortheopGeratorb0m +]b1m ;thecritical >couplingsconstantbGeing2=([=2G+M42=)Ը124:16e^2|s;mandsBalinskyandEvqans>[1]jdshowedthatabGovemaxڸfm;m(2 " z1z&fes2ò)gTtherearenoembGeddedeigenvqalues>intheessentialspGectrum.Byusingsimilartechniquesweextendtheirresults>toUUthesecond-ordertermb2m :)4>2VLPreliminaries>Let7',92S(R^3|s). C^2 AbGeaspinorintheSchwartzspaceofsmoGothstrongly>loGcalisedfunctions.Thenonecande netheenergyoftheelectronastheex->pGectationUUvqalueofbm;SEm('):=0(';bm')= ۱2 X tLi=0(';bim l'):qò(2.1);>ThestrategyofthisworkistobaseproGofsontheresultsforthecaseofmassless>particles,ni.e.Othe?indexmifreferenceismadetothem=0?case).ʅWhenm=0;b0=p;>andUUsimplescalingpropGertiesarefoundtohold.pMItwasprovenbyBrummelhuisetal[4]thatthedi erencebGetweenthe>energiesUUinthemassiveandinthemasslesscaseisbGounded,i.e.᭸jEm(')8E(')j0mdk'k2|s;)d>0:qò(2.2)MF*rom-thisitiseasilyshownthatbm TisformbGoundedfrombelowbecauseof>theUUpGositivityofb,i.e.q(';b')00mforUU UP ch6[4]:1Em(')0mdk'k2 |k+(';b')0qmdk'k2qò(2.3)>with cfromTheorem1.1.Sincebm oissymmetric(bGeingafunctionofpand >its+conjugatex),4;(2.3)allowsfortheF*riedrichsextensionofbm ďtoaself-adjoint>opGerator'ontheHilbertspaceL2|s(R^3)C^2|s.=In'thefollowing,thenotationbm>willUUalwaysimplyitsF*riedrichsextension.pMIntroGducing,theresolvents(bm _+nĵ)^1band(b0mҲ+)^1 t,btheirbGoundedness>forjasuitablychosen>1jfollowsfromthestrictpGositivityof(bm <3+)and>(b0m +t)whichistrivialforthelatteropGeratorandisaconsequenceof(2.3)>forUUbm {+8:J(';(bm {+8)')0md(';')+(';')0>0UWfor&md8+1:qò(2.4)4&.y?>F*orX^theproGofofTheorem1.1(i)itsucestoshow[6]thattheresolventdi erencet Rm():=0(bm {+8)1l(b0m +)1"˲isUUcompactq(2.5)>for;from(2.4)..jItthenfollowsthattheessentialspGectraofbm #ֲandb0m Icoincide >[14 ,UUp.112].pMThejproGofofthecompactness(2.5)isbasedonLemma2.6ofHerbst[10 ]>whereUUitisshownthat5(b0m +8)1<$<1wfe s 3ፍpUWfe3荵x5KisUUcompactq(2.6)ƍ>forallm0and1:_AnotheringredientoftheproGofof(2.5)istheb0m -form>bGoundednessUUofb1m +8b2mcwithrelativeboundclessthanone,i.e.Jj(';(b1m +8b2m )')j0c(';b0m ')+C~4(';')q(2.7)K8'2Q2|s;with260whereUUQ2C:=H:1=2 ʲ(R^3|s)8C^2 istheformdomainofb0m#>(withH:1=2 ʲ(R^3|s):= o;>f'l2L2|s(R^3)l: -īR ɍ%qymsbm7RZcmr53ոj'(p)j^2o߲(1J+p^2|s)^1=26dpص<1gcZinothep-spacerepresentation).qɍ>Withv(2.7),3bm maybGede nedasaformsumofb0m and(b1md+Vb2m )with >coincidingUUformdomainQ2Ȳforbm andb0m .MTheEformbGoundedness(2.7)canbeobtainedfromtherespectiveformbound->edness>winthemasslesscase.j(T*oshowthelatter,C thefollowinglemmaisneeded:p>LemmaT2.1zxL}'etebA:=b0Lٲ+fb1+b2 #\theeJansen-Hessop}'eratoreformA=0:If> UP4=[,G(';b1'')0(';b2'')8'2Q2|s:qò(2.8)>DuejtothescalingpropGertiesofbinthemasslesscase(discussedinthebeginning>of{section4),YtheproGofcanreadilybecarriedoutinMellinspaceusingthe>techniquesfrom[4]andisgiveninAppGendixA.Itshouldbenotedinpassing>that4pacorrespGondinginequalityforthekernels,l7b1|s(p;p^0 m)uܸ:b2(p;p^0)o^with>biTL(p;p^0 m)>de nedin(3.2)withmG=0,xdoGes>nothold;infactonecanshowthat>for ޵p^0Q=0;qp6=0;8b2|s(p;p^0 m)isdominatingoverb1|s(p;p^0 m)foranypGositivevqalue>ofthecouplingconstant V(while,e.g.Pforp=p^06=0;8ظb1|s(p;p^0 m)islargerthan>b2|s(p;p^0 m)):pMInthemasslesscase,theformbGoundedness(2.7)iseasilyderivedwiththe>helpof(2.8)andthepGositivityofb:T$IntheMellinspacerepresentation(A.4)of ƍ>(';b')?onexcanshow[4]thatfor UP< c;8ز(b1ɍ]xݍ0+۟Pp2Pfe E2b:(1)]6l `s#j+۟Pp2Pfe E2b:(2)]6l `s )(t۸i=2)0>>0mandUUhence (';b')0(';b0|s')q(2.9)53=y?>whereasUU=0for UP= c:mTherefore׍(';(b1S8b2|s)')0(1)(';b0'')}q(2.10)>whichUUproves(2.7)inthem=0casefor UP< c:pMAnLadditionalelementintheproGofofboundednessofoperatorsnecessaryto >show6thecompactness(2.5)istheestimatederivedbyLiebandY*au[13 ]which>weUUgiveinaslightlygeneralisedform:ML}'etK(p;p^0 m):=qK(p^0 1ŵ;p)q0bb}'easymmetrickernel, 8p;p^0M2R^3|s;bandlet>f(p)>0forp>0asmo}'othconvergenceinducingfunction.ThenӍ?` ?` ?` ?` BhcZG yR3 R3]LJdpdp0 mLщfeU/'(p)!:=p^ 7withUUasuitable В>0toinvestigateUUtheconvergenceUUofI.MF*or&theproGofofTheorem1.1(ii)dilationanalyticityisused[6].bLOnede nes>forUU5:=e^withjuǸj<0|s;q<߸2R;mtheunitarydilationgrouponL2|s(R^3)8C^2Ȳby׍,dMO'(p):=0'7(p):=G3=2'(p=G)}q(2.12)GMThenoneextends賲tothedomainD0¢:=F/fL=e^ڲ:^2C;׸juǸj<0|sgŲwith>0C>0UUtobGechosenlater,andde nesthedilatedoperatorspbm;:=0dMObm _d䍷1K[ t:}q²(2.13)>F*orUU52D0|s;8dیisnolongerunitary,neitherisbm;T'self-adjoint.pMW*eYhavetoshowthatbm;+isananalyticopGeratorinD0|s.AsSincewewantbm;>tobGede nedasaformsumonH:1=2 ʲ(R^3|s)}7C^2;-ɲanalyticityrequiresthat([6],>[14 ,UUp.20])ETb1m;=+b2m;)PisUUb0m;E-formbGounded}q(2.14)۴>withWrelativebGoundlessthanone W82D0|s;#xwhereWbim;:=d0ߵbim d䍷1K[ t; i=>0;1;2:>Moreover,*for׵bm;de nedasanopGeratoronL2|s(R^3)C^2|s;onemustshowthatVHH.N(b0m +8)1=2(b1m;4%+b2m;E)(b0m+)1=2 '!isUUananalyticfamilyin`D0|s:}q²(2.15)>T*aking!'2SRVC^2suchthatd7'isanalyticinD0|s,+onecanextendtheformula@Aq;K(';<${1۟wfe3A (֍bm {8z!DO')0=(d7';<$*1۟wfe#x (֍bm;78z*qd');)z72CnR}q²(2.16)6C*y?>whichforq2)D0\-4Risbasedontheunitarityofd7,stoD0 \)CbGecause >analyticityofther.h.s._allowsapplicationoftheidentitytheoremfromthe>theoryebofcomplexfunctions.SinceS%^isdenseinH:1=2 ʲ(R^3|s)[8,ep.192],(2.16)>holdsforall'2H:1=2 ʲ(R^3|s)wC^2u(althoughH:1=2(R^3|s)itselfisnotinvqariantunder>complexUUdilations).pMF*urthermoreUUweneedtoshowthat򍍑]Rm; Ҳ():=0(bm;7+8)1l(b0m;4%+)1"˲isUUcompact}q(2.17)>for#Zde nedin(2.4).Then,Vfollowingtheargumentationof[6],Vonecanuse >Lemma3of[14 ,p.111]togetherwiththestrongspGectralmappingtheorem([14,>p.109])T9toproveT9thattheessentialspGectraofb0m;O~andbm;S coincide.nr(Note>that(QtheadditionalconditionforLemma3tohold,1Ranonemptyresolventsetof>(bm;ݖ+ĵ)^1 仲containing(GinnerpGointsinC,1JfollowsfromthebGoundednessofthis>opGerator ginD0ڲasshowninsection5.2).`#Sinceess dH(b0m;E)andhenceess (bm; Ҳ)>isUUacurveinthecomplexplaneintersectingRonlyinthepGointm,wehave\!olims32ImSzI{!0/9ImvW(';<${1۟wfe3A (֍bm {8z$ g')0<1}q²(2.18)>by,meansof(2.16)exceptwhen:Re^zUòcoincideswithisolatedpGointsofR+>(namelyܵmoreigenvqaluesofbm):~(2.18)impliesthatthesingularcontinuous>spGectrumUUofbm isabsentin[m;1)m([14 ,p.137,186];[10]).){d>3VLRepresentationsffof1gff cmmi12b(m>W*ebstartbyselectingthemomentumrepresentationandde neintegralopGera->torsUUbim l(p;p^0 m);8i=1;2mbymeansofkx(';bm')0=cZ j yR3wherekwehaveindicatedexplicitlythep-depGendenceofb0m .v Theseoperators>werebcalculatedfrom(1.5)byEvqansetal[6]andfrom(1.9)byBrummelhuiset>alUU[4],ZƵb1;m (p;p0 m):=0<$. 33wfe (֍2[ٟr2<$&X1Cwfe%ܟ (֍jp8pr0 jjr2?j[1+ h^ :pHE^~p0&_h(p)h(p09)]A(p)A(p09) tru9b2m (p;p0 m):=<$c1cwfe (֍2fW`<$zi nwfe (֍2[ٟr2)Q3`/Jݱ25q2cZ:l yR3Fidp00<$! Ӳ1zIwfe(' (֍jp8pr00 ޸jr2<$OɃ1<ܟwfe*N (֍jpr00 ޸8pr0 jjr2+^<$-1$wfe+i (֍Ep 0ncmsy50 +8Ep00:0+<$E&13+wfe)# (֍Ep2+8Ep000T^|:ݵA(p)A(p09)A2|s(p00r)q(3.2)7Sy?KˀbTh2|s(p00r)8 P^ sp00/*^p0(Eh(p00)h(p09) P^ sp00/*^p% h(p00)h(p)+ P^ sp-^fp0#ӵh(p)h(p09)b?ύ>wherethefactorsofthetypGejp\ p^0 ϸj^2Fresultfromthemomentum-spacerep- >resentationxofthepGotential(1.2).W*enotethatform=0; h(p)=1xand>A(p)= '1ɟ&fe <č3pɟ3W sM2~whileninthegeneralcase,h(p)2[0;1]andA(p)2[ |133&fe <č3pɟ3W sM2 ;1]arealso bg>bGounded.pMF*or?theproGofofthecompactness(2.5)itisadvqantageoustochoGoseanx-space ؍>representation.6Identifyingagainpwithi@8=@x ?(andϵpwith() 33133xW g P2M);b1m>andUUb2mccanbGewritteninthefollowingway1|tхb1m#>=0 UPA(p)q^<$ G1 wfe (֍xxA(p)+h(p) S^ p<$1wfe (֍x ?^xp(h(p)A(p)^!O.Tb2m#>=0`<$d Gwfe  (֍2`}ݱ2'㰵A(p)q^<$ G1 wfe (֍x`A2|s(p)h2(p)W10;my+W10;mA(p)h2(p)<$133wfe (֍xA(p)zq޸<$Uٲ1Kwfe (֍x A2|s(p)h(p) S^ p,eW11;myW11;mA(p)<$133wfe (֍xA(p) S^ ph(p)Q{`- S^ p,eh(p)<$133wfe (֍x^X+pϵh(p)A2|s(p)W10;my h^ :peEh(p)W11;mA(p)<$133wfe (֍xA(p)ߍP+ S^ p,eh(p)<$133wfe (֍xA2|s(p)W11;my+ h^ :peEh(p)W10;mA(p)<$133wfe (֍xA(p) S^ p,eh(p)^m۵:qò(3.3)>Intheexpressionforb2m{wehaveintroGducedintegralopGeratorsW10;mand>W11;m˲which"Jarecloselyrelatedtow1mXasde nedbGelow(1.9).اSincelatera>factorisationwillbGeusedforeachtermof(3.3),calloperatorsbut1=xmaybe>analysedvintermsoftheirmomentumrepresentation.1}Inp-spacerepresentation,>W10;mFֲandUUW11;marede nedby8o(W10;m)')(p):=0cZ j yR3^>9p0I8h(p09)A(p0)<$ 1Kwfe& (֍Ep2+8Ep0.'(p0 1Ų)q(3.4)>ItMisreadilyveri edthatF*ouriertransformingb1m òandb2mleadstotheequations>(3.2).)w >4VLTheffessentialsps3ectrumofb(m>In+thissectionweshowtheb0m -formbGoundedness(2.7)ofb1m zȲ+庵b2mϲaswellas>the%compactnessofRm()from(2.5)inordertoprove%thatess dH(bm)=[m;1)>(Theorem1.1(i)).ManyingredientsoftheseproGofswouldhavetobGerepeated>whenvsdilationanalyticityandthecompactnessofRm; Ҳ()isshown. Therefore>we.formulatetheproGofsforthegeneralisedoperatorsbm;-Ȳandconsider(2.7)>andUU(2.5)asthespGecialcasesfor5=1:8 dy?MW*e52R^+ weUUhaveݦmgEm(')0=(';bm CC')=(d7';(d0ߵbm CCd䍷1K[ t)d')q(4.1)>andUUmakingin(3.1)thesubstitutionq:=Gp;8q^0:=p^0OoneUUobtainstEm(')0=cZ j yR3<+UPcZ㊟ yR3XߵdqGP 33333xW g P2LLщfe\s/'(q=G)01cZ5I yR3?4dq0 G3 9(ȵb1m (q=G;q0=)+8b2m (q=;q0=))}GP 33333xW g P2L'(q0=G)MUsings8thede nition(2.12)ofd7'weobtainupGonidenti cationwiththe>r.h.s.qofUU(4.1)ѵb0m;E(p):=0dMOb0m (p)d䍷1K[J=b0m (p=G)w֕=05p15fe4= ˍpr2|s=Gr2 È+8mr2IZL=<$c1cwfe (֍5p5fe2 ˍpr2 |k+8(mG)r2Qu=<$c1cwfe (֍b0mE(p)q(4.3)܍Ijbim;(p;p0 m):=0dMObim l(p;p0)d䍷1K[J=0G3 ʩbim l(p=G;p0=)0=<$c1cwfe (֍bim(p;p0);>i0@=1;2;ĭwheremthelastequalityresultsfrominspGectionoftheexplicitexpres->sionsU(3.2)forbim l(p;p^0 m);mimplyingthatbim(p;p^0)resultsfrom(3.2)bymeans>ofUUthesubstitutionsI Ep -7!E7(p):=0p1fe/= Dpr2 |k+8mr2|sGr2>;h(p)07!h(p):=<$/=+8m(q{A(p)07!A7(p):=\ <$8p8fe/= pr2 |k+8mr2|sGr2L+8mwfeU|  H28p 8fe/= pr2 |k+8mr2|sGr2eݾ\!mjٱ1=2|ܵ:qò(4.4):>Thede nition(4.3)oftheopGeratorsbim;Ӳandbimisreadilyextendedtocom->plex֊卸2pD0|s:Notethesimplescalingwith1=ofthecorrespGondingoperators>bi;H(i=0;1;2)minUUthemasslesscasewhichfollowsfrom(4.3).$<>6N cmbx124.1\The8g cmmi12b7|{Ycmr8092cmmi8mò-formb`oundednessč>LetUUustakeageneral52D0|s:mUsingthescalingpropGerty(4.3)wehavej(';(b1m;4%+8b2m;E)')j:C0 0 0 0 <$ 1 wfe (֍I I I I L[j(';(b1mb1|s)')jd+j(';(b2mLb2|s)')j+j(';(b1+Lb2|s)')j]q(4.5)>F*romd theexpGonentialform,g&=ߞe^`;onederivestheestimateforj1=Gj,gvqalidfor>':=0jReUUuǸj<1;ݮ#_180e1  <$ 1 wfe (֍I I I I -q=eRe <=e 18+2Ҫ:qò(4.6)9 wy?>F*orP$52D0̗onehasjRe#uǸj0<0C,Q.suchthatbyrequiring0C<1onecanreplace >8byUU0Ȳin(4.6).qThesameestimatesalsoholdfortheinverse,jGj:pMUsing@theb0|s-formbGoundedness(2.10)ofb1Q+޵b2Gwe ndforthelasttermof>(4.5)|ū | | | <$JM1JMwfe (֍} } } } 8j(';(b1S+b2|s)')j0(18+20|s)(1)(';b0'')q(4.7)植>whereUU0ȲcanbGechosensucientlysmallsuchthat(18+20|s)(1)=:c<1:>Providedcthe rsttwotermsin(4.5)arebGounded,thisprovestheb0|s-formbGound->ednessUUofb1m;4%+8b2m;PwithformbGoundsmallerthan1.MIn/ thissectionweareconcernedwiththeb0m -formbGoundednessinthecase>ځ=d1cM(thegeneralcasebGeingdeferredtosection5.1).Thenwecanusethe>resultsofTix[16 ,֫Theorem1]andBrummelhuisetal[4,Lemma5]whohave>shown,(by1comparingmassiveandmasslessopGerators,thebGoundednessofthe> rst-UUandsecond-orderterm,respGectively+qgj';(b1m 8b1|s)')j0(j'j;jb1m8b1|sjj'j)0md1Ck'k2nLj(';(b2m 8b2|s)')j0(j'j;jb2m8b2|sjj'j)0md2Ck'k2:qò(4.8)服>Hence,ٵb1m3+b2misb0|s-formbGounded.8Theb0m -formboundedness(2.7)isan ꍑ>immediateUUconsequencesinceb0 =p08p 8fe#{ pr2S+8mr25VIJ=b0m :$I⍍>4.2\Thecompactnessoftheresolventdi erenceRmĻ()č>W*ewillshowlaterthat(b0m;+)^1 'isbGoundedforDвinasuitabledomainD0|s: >Then,UUfollowingEvqansetal[6]weusethesecondresolventidentitytowrite@[(bm;7+8)1l(b0m;4%+)1J=0(b0m;+)1 t(b1m;+b2m;E)(bm;7+)1hvE=0bT(b0m;4%+8)1 t(b0m +)bq'8f(b0m +)1 t(b1m;4%+b2m;E)(b0m+)1=2˸g楍`hc(b0m +8)1=2 ʲ(bm;7+)1 t`iqò(4.9)Xi>W*ewillshowthattheopGeratorincurlybracketsiscompactwhilethetwoopGer->atorsNinsquarebracketsNarebGounded..Thentheproductofallthreeoperators>isUUcompact.pMF*orX thecasec=vF1theleftopGeratorinsquarebracketsX isunity.yForthe>proGofEoftheboundednessoftherightmostoperatorfor5=1weusebm +]+µ>0>tode nethebGoundedsquarerootoperator(bm ʲ+7/)^1=2andwemakethe>decompGositionK\(b0m +8)1=2 ʲ(bm {+)1J=0(b0m+)1=2 ʲ(bm {+)1=2˲(bm+)1=2˵:}q²(4.10)>Let'+2H:1=2 ʲ(R^3|s)aC^2 9²andde ne ~:=+(bm +)^1=2˵':OThen,makinguseof o;>theself-adjointnessof(bm +\Y)^1=2[(bymeansofitsF*riedrichsextension),the10 y?>requirementforthebGoundednessof(b0m Lϲ+)^1=2 ʲ(bm P\+)^1=2canbeexpressed >inUUthefollowingwayKm_k(b0m +8)1=2 ʲ(bm {+)1=2˵'k2C=k(b0m +)1=2 ʵ "k2C=0( [;(b0m+) [ٲ)I0c03k'k2C=c0( [;(bm {+8) )}q(4.11)>forUUasuitableconstantc0C>0:pMThisRymeansthattherequiredbGoundednessisprovenRyprovidedthefollowing>inequalityUUholdsbGc0C( [;(b0m +8b1m+b2m+) [ٲ)( ;(b0m +8) )00}q(4.12)>However,recallingthatfromtheb0m -formbGoundedness(2.7)onehastheesti->mate( [;(b1m +8b2m ) )0c( ;b0m ? )C~4( ; )}q(4.13)>withg1G>c:=1lA;andnotingthatb0m7uandarenonnegative,(4.12)holds>trueUUifonechoGosesc0Cq1=mand>max(1;UPc0|sCo=(c0S81)):pMNext;weshowthecompactnessoftheopGeratorincurlybracketsfrom(4.9).>The.compactnessofthe rst-orderterm(b0m^i+[)^1 tb1m;+(b0m+)^1=2ڲwas>alreadyUUshownbyEvqansetal[6].qSoweconcentrateonthesecond-ordertermۍqw(b0m +8)1 tb2m;](b0m+)1=2_=:<$K1Kwfe (֍ f_`<$  dwfe  (֍2I`-ݱ2'8 !HX t"i=11㹵 im}q²(4.14)Lt>whereUU im(sisobtainedwiththehelpof(3.3)andthescaling(4.3),C:8 >X t>i=1Nq im6=(b0m +통)1 ̵A7(p)^<$߲1zQwfe (֍x dA2፴(p)h2፴(p)W10;m+8W10;mWA(p)h2፴(p)<$133wfe (֍xA(p)%W<$Uٲ1Kwfe (֍x A2፴7(p)h(p) 7^QpW11;mWW11;mWA(p)<$133wfe (֍xA(p) 7^Qph(p)}q(4.15)P3 S^ ph7(p)<$133wfe (֍x^X+pf_h(p)A2፴(p)W10;mW h^ :pHյh(p)W11;mWA(p)<$133wfe (֍xA(p)\>+p^:pHݵh7(p)<$133wfe (֍xA2፴(p)W11;m+p^:ph(p)W10;mWA(p)<$133wfe (֍xA(p)p^:ph(p)^B~(b0m ,[+M)P 33133xW g P2>with۵W10;mandW11;mfrom(3.4)withthereplacements(4.4).XFirstwe>note Fthatalleightterms imdfrom(4.15)containthecoGordinatexintheform>1=x(anddi erintheirmomentumdepGendenceonlybythebGoundedoperators>Ep^E:pK5;qA7(p)#Worh(p)o(theirbGoundednessisshowninthenextsection).aW*eshall>onlypresenttheproGofofcompactnessforthe rstterm 1mݲindetail.One>can$readilycarrythroughtheproGoffortheotherseventermsusingthesame>techniques,c together`Kwithkp^:pHݵ'k^2 /-=](';(p^:p 5)^2|s')=k'k^2:9In`Kparticular,c each11 'y?>of6thetermsisfoundtocontainthecompactopGerator(b0m ٲ+`˵)^1 gx^1=2cfrom >(2.6)UUwhichassurescompactnessprovidedtheremainingfactorsarebGounded.pMW*eusethecommutativityofmultiplicationopGeratorsdependingonlyon>momentumUUp(suchasA7;qh;b0m +8)mtoUUdecompGose 1mPintowV: 1mu=0A7(p)f(b0m +8)1<$u1 履wfe s 3ፍpUWfe3荵x/Mg^<$ 9j11wfe s 3ፍpUWfe3荵xA2፴(p)h2፴(p)(b0m +)1=2˟^wJ^`hc(b0m +8)1=2XW10;mв(b0m+)1=2˟`i}q²(4.16)>F*orz=1;theprefactorA(p)isbGoundedby1.5Wearelefttoprovezthatthe>twoUUopGeratorsinsquarebracketsUUarebounded.MLetUUusconcentrateonthe rstopGerator.qThen,de ning ":=0A^2v7(p)h^2v(p)>(b0m _+q)^1=2˵'5weobtain,))usingtheinequalityofKato[12 ,))p.307],1=x KK&feox~2 ŕp;>andUUtheself-adjointnessof1=x-:cڸk<$1Kwfe s 3ፍpUWfe3荵x9A2፴7(p)h2፴(p)(b0m +8)1=2|s'k2C=0( [;<$9i1۟wfe (֍x * )<$ccwfe (֍2( ;p [ٲ)q<$ccwfe (֍2( [;(b0m +8) )0=<$ccwfe (֍2kA2፴7(p)h2፴(p)'k2<$ccwfe (֍2kA2፴7(p)k2Ckh2፴(p)k2Ck'k2 =:c1k'k2}q²(4.17)p>withUUfrom(2.4).qF*or5=1;8h(p)isbGoundedby1suchthatc1C<1.MNextUUweshowthebGoundednessoftheoperatorw93Wc፱10;mв:=0(b0m +8)RW10;m(b0m+8)}q²(4.18)>for=1;_ 33133&fes2 Fand F1F&fes2*whichareourcasesofinterest(actually*,bGoundedness>canUUbGeshownforjjq< K3K&fes2 )):mDe ningthenonnegativekerneljjx䍑@~>KG5U(p;p0 m):=0(b0m (p)+)<$1 8wfe%ܟ (֍jp8pr0 jjr23 jA7(p09)jq q q q <$&1Hwfe@uO (֍E(p)8+E7(pr0)Jʫ J J J Ri߲(b0m (p0)+)}q²(4.19)>and9Wtheconvolution9WK(p^0 1ŵ;p^00 *):=mīR ɍ4R3 gdpx䍑S~K (p;p^0 m)x䍑Ȳ~K 69(p;p^00)owhich9WissymmetricvY>inp^0 andp^00Yβandalsononnegative,weobtainwiththehelpoftheLiebandY*au>formulaUU(2.11)m(with ":='andthechoicef(p):=p^ 7)w]kWc፱10;m`'k2C=0cZ j yR3a9ڱ2 VֻcZUR yR3ʧdp^ cZV yR3 dp0x䍑 Q~ mK²(p;p0 m)j'(p0 1Ų)j^s7۱2z{0cZ j yR3Thus. W^cv10;mIJisbGoundedifI(p^0 1Ų)is niteforallp^0&2Cfe8R [3{withasuitablychosen > z:pMLetPBusturntothecase5=1again.pSincetheintegrandofI(p^0 1Ų)isnonneg->ative,R`weQcanestimatex䍑~K (p;p^0 m)andhenceI(p^0 1Ų)fromabGoveQbyreplacingA(p^09)>byUU1.qInadditionweneedtheestimates(with>1UUand>0)u<$Q1}wfe& (֍Ep2+8Ep0v=<$81cwfehI ;8p 8fe#{ pr2S+8mr2/:\+8p 8fe%Bps0sr2 iƲ+8mr2vs<$1cwfe (֍p8+pr0#ŵ;ЦX`Բ(b0m (p)8+) k=0(p fe#{ Dpr2S+mr2/:\+)0(p+)0(p+1) ~;;卑hq(b0m (p)8+) j0(p+m+)0((p+1)+m8++1)V:S0((p8+1)+(p8+1)(m++1)) j=(p+1)(m++2)}q²(4.21)>sinceUUp8+11:mWiththisweestimatex䍑9~K (p;p^0 m)x䍑V1~SѵK\&(p;p0 m)0(m8++2)jj R(p+1)<$1 8wfe%ܟ (֍jppr0 jjr2<$>Jj14>ߟwfe (֍p+pr0QPA(p0+1)}q²(4.22),ۍ>Apartֺfroma niteconstant, ther.h.s.Gof(4.22)isjustthecorrespGondingkernel >infthemasslesscase(sinceform҇=0; OEp qٲ=b0|s(p)=p;fandonecantake>=1): ThismeansthattheintegralI(p^09)from(4.20)canbGeestimatedbythe>correspGondingZintegralinthemasslesscase.]The nitenessofI(p^0 1Ų)form=0Zis>shownUUinAppGendixB.) >5VLDilationffanalyticity>Inthemasslesscase,dilationanalyticityistrivialbGecausefrom(4.3),dilation>of$bm AisequivqalenttomultiplicationbythebGounded,analyticfactor1=:F*or>the5صm6=0case,<$westartbyshowingthattheopGeratorsA7(p);qh(p);(E(p)+>E7(p^09))^1Zare bGoundedfor; 2D0VwithD0aneighbGourhood ofunityinthe>complexplane(de nedbGelow(2.12)),.andwederivebGoundswhicharerelated>to͵therespGectiveoperatorsfor5=1:ͲSuchboundsweregivenbyEvqansetal[6]>forXE7(p):uIndeed,fromtheestimateofjGjgivenbelow(4.6),!1\0 B:jj>18+20|s;mweUUgetforjImGuǸj00 1=2(using182x^2 cosO2x01)Pҍ(180|s)Ep -0jE7(p)j(18+20)EpR:qò(5.1)>W*edemonstratethetechniquesinthecaseofE7(p)+m5whichisneededto >estimateUUA7(p)andh(p):mTheuppGerestimateisobtainedfrom]r=jE7(p)8+mGj0jE(p)j+mjGj0(18+20|s)Ep J+m(1+20)13ؠy?.=0(18+20|s)(Ep2+m)q(5.2)pZ>F*orwzthelowerwzbGoundwede nethephases'p̲and'0ofE7(p)=8p 8fe,vҟ pr2S+8mr2|sr2 >andUU5=e^`;respGectively3Z'pfj:=<$c1cwfe (֍2?arctan<$88m^2|se^2Re <sin"] (2UUImsuDz)-KWwfeo\g (֍pr2S+8mr2|ser2Re <cos#y(2UUImsuDz);'0C:=0ImNqò(5.3) >WitheAtherestrictionFjImGuǸjᡵ<[=4weeAassurecos(2UUImsuDz)ᡵ>0i.e.pGositivityeAof>thebjdenominatorof'pR:Sincearctan isanoGdd,monotonicallyincreasingfunction>we canestimatej'pRjfromabGovebydroppingp^2\inthedenominator,j'pRjr ?331?33&fes2F arctana۾(tanr2jImGuǸj)P=jImuǸj:%TW*ethereforegetj'07'pRjP 2jImuǸj%Tand>hence\cos/('0:'pR)xcos(2UUImsuDz)x12jImGuǸj^2 c(10|s)^2 Gfor\jImGuǸj<>0 1=2:mThusUUweestimateDEjE7(p)8+mGj0=  ㅸjE(p)j+mjGjei('0 'O \cmmi5p2Ա)"sA "sA "sA |=jE(p)j+mjGjcos7('0S8'pR)#ߍxa0(180|s)Ep J+m(180)3 0(10)3C(Ep2+m)q(5.4)>where:inthe rstinequalitywehavedroppGedtheimaginarypartandusedthat>itsr.h.s.*isnonnegative.F*romthiswe nd,usingthede nition(4.4)ofA7(p)>andUUh7(p)g<$(180|s)^3wfe&X (֍Ȳ18+20A2|s(p)0jA7(p)j2 <$c18+20cwfeT (֍180)A2(p)s<$m1ßwfeT (֍18+20bh(p)0jh7(p)j<$mղ1cwfe&X (֍(180|s)r31h(p)q(5.5)MЍ>InUUasimilarwayUUwe nd'j<$|N1\cwfeF (֍jE7(p)8+E(pr09)j(v<$mղ1cwfe&X (֍(180|s)r3<$C?12Gşwfe& (֍Ep2+8Ep0_<$mղ1cwfe&X (֍(180|s)r3<$T*akingUU0C1=2massuresthat(5.5)and(5.6)arevqalidforall52D0|s:$>5.1\Theb0m;-formb`oundednessč>ReferingF>to(4.5)and(4.7)itremainstoproveF>thebGoundednessofj(';(bimѸ>biTL)')j;8i=1;2;Ųaswellastheestimateofb0 byb0m;since(4.7)onlyprovides>theUUb0|s-formbGoundedness.pMInVordertoshowtheseconditemwestartbyestimatingtherealpartof>b0mf?e²ReK3pU3fe,vҟ Dpr2S+8mr2|sGr2q=bIJ(p2S+8m2|se2Re \td:cos (2ImG#uDz))2+(m2|se2Re \td:sinX(2ImG#uDz))b|d 1d xW g P4)cos7>'pl0pcosߵ'p -pcos7(ImGuDz)p(18jImuǸj)0p(180|s)q(5.7)144y?>forjImGuǸj8<Z0֏[=4;where'pLݲisde nedin(5.3)andwehavefollowedthe >argumentationbGelow(5.3)andusedthatcosٵxڸQ1phUjxjforjxjQ1:Then>withUU(5.7)ڍ^j(';b0m;')j0=  <$ 1 wfe (֍I I I I !8j(';b0m')j(180|s)jRe#(';b0m')jh*0(180|s)2C(';p')=(18or})2C(';b0'')q(5.8)(K>Hence,Kthe`r.h.s. 'of(4.7)canbGeestimatedbyc0'j(';b0m;')j Lwithc0 ᚲ:=>(1+20|s)(1)(10|s)^2 z<1forsucientlysmall0|s,whichprovidesthe>b0m;E-formUUbGoundednesswithformbound<1:pMThef rstitemisprovenfbyestimatingeverytermofj(';(bimDbiTL)')jKby>the-correspGondingterminthev=/1case.Sincealltheseterms(for=/1)>haveseparatelybGeenshowntobeboundedbyBrummelhuisetal[4]andTix>[16 ]UUduringthecourseoftheirproGofsof(4.8),wearedone.MW*eiusethepartialwaveiexpansion(A.2)oftheenergyEm(')andnotethat>bimeandGbl `sm0areobtainedfrombimF.andbl `sm ,CrespGectively*,byGattachingto c>every7$oGccuringmthemultiplicationfactorG.3Withtheexplicitformofb:(1)6l `sm>fromUU(A.2)wegetd_j(';(b1m4%8b1|s)')j0<$ k% cwfe  (֍2]X ]$ΟcZi.Ϸ1@*Y08\dpjaɲ(p)jqcZiq1@ 0Ndp0Qja(p09)j&>\(  8Ibs9Ibfe,Va18+<$Tmlwfe (֍E7(p)F+IbsP+Ibfe/$18+<$ NpmlwfeU (֍E7(pr09)PE81  YqlȲ(p=p09)$++q q q q q IbsIbfe,Va18<$Tmlwfe (֍E7(p)@䏟IbsJ䐟Ibfe/$18<$ NpmlwfeU (֍E7(pr09) "81  6ql `+2s(p=p09)\)qò(5.9)>where'foreachtermofb1mJb1|s;thetriangleinequalitywasused.W*enow2>de nez|f(m):= _r _fe,Va918<$Tmlwfe (֍E7(p)>Q_r Q_fe/$918<$ NpmlwfeU (֍E7(pr09)Aandusethemeanvqaluetheorem >inUUthefollowingform(itisapplicabletotherealfunctionsRexf+ andImsf+)Tjf+(m)f+(0)j0=Fq1FfeOr rRe 92J(f+(m)8f+(0))+Im2n4(f+(m)8f+(0)) }1=05s15feæPˍ^\tm8Re<$dUUf+wfe (֍Cdm"Ղ(~m1 D*)^Taf۱2]Ѳ+^ \lm8Im<$1dUUf+1wfe (֍Cdm"(~m2 D*)^Y۱2}q²(5.10)#čPO0m4s 4feu̍  <$dUUf+wfe (֍Cdmi(~m1 D*)  -/ڱ26q+    <$ dUUf+ wfe (֍Cdma(~m2 D*)  2.ڱ2sImq^      <$~dUUf+~wfe (֍Cdm!N_(~m1 D*)  >+    <$ dUUf+ wfe (֍Cdma(~m2 D*)  4^>with90~UUm1;~m2D'someUUvqaluesbGetweenUU0andm.qWithz7:=m8rweUUobtain׍<$bȝdUUf+bȝwfe (֍Cdmx=0<$AhdUUf+Ahwfe (֍cdzy=<$dAhwfe K (֍dz+𓌫s,𓌉fe<(lt18+<$zlwfe)P 8p 8feP pr2S+zpr2Ywl𓌫scwm𓌉fe?lt18+<$/zlwfe,V ;p fe"UBps0sr2 iƲ+zpr2}q²(5.11)15y?9>K6=05𓌫s 6𓌉fe<(lt18+<$zlwfe)P 8p 8feP pr2S+zpr2<$P1Pwfe (֍2<$e[pZq0s^2Z@wfe! (֍pr02 +8zpr2<$.H1ٟwfesޟpXq pXfeiݟ (ps0sr2 iƲ+8zpr2 )+zpp pfe"UBps0sr2+zpr2+(p$p09)%B>wherethesymbGol(p$p^09)standsforthe rsttermwithpandp^0.interchanged, >and]dUUf=dzxisgivenby(5.11)withzxreplacedbyzp.jWiththeestimates(4.6)>andUU(5.1)forjGjandjE7(p)j;aswellas(5.2)and(5.4)wededuceabԫ b b b <$f\dUUf+f\wfe (֍Cdmv=(~mǷ)  C0(18+N~G)q\ # # # # # $ dUUf:(1)+ fe?" (֍!}dm)a(~mǷ)  =,r=1OU+     $ dUUf:(2)+ fe?" (֍!}dmղ(~mǷ)  2,r=1?џ\!}q²(5.12) ꍑ>forcasuitablex~r6,fwheredUUf:(1)+ =dmdenotesthe rstterminthelastlineof(5.11)@>andUUdUUf:(2)+ =dmisthistermwithpandp^0#interchanged.pMW*eenowfollowTix[16 ]toestimate(5.12)byanexpressionpropGortionalto>theinversemomentumandindepGendentof~mo?byusingthatfor5=1;8z7=~mU>0p˫ p p p p $SdUUf:(1)+Sfe?" (֍!}dmk(~mǷ)  Oҟ,r=1Ը0=Vp ㇟=Vfeª2h<$l1lwfe (֍2 '<$lpZq0s^2lwfe 8 (֍3pr02 <$ 1lwfe/ ;p fe%Bps0sr2 iƲ+~8mr29e<$,1cwfe9 (֍pr0}q²(5.13)%0>andUUlikewisem m m m m $ dUUf:(2)+ fe?" (֍!}dm J(~mǷ)  3t,r=1Fv<$c1cwfe (֍p :,MUpGon5substitutionof(5.10)with(5.12)and(5.13)into(5.9),moneobtains>integralsEwhichTix[16 ]hasproventobGe nitewiththehelpoftheformula>(2.11)UUofLiebandY*au.MTheoqsamemethoGd,i.e.themodi edmeanvqaluetheorem(5.10)together>withanestimateoftypGe(5.12)toeachofthetermsappGearinginthederivqative>ofC=thecorrespGondingfunctiongivenin[4],Fcanbeappliedtoshowboundedness>ofqb2m]b2byrelyingontherespGectiveproGofbyBrummelhuisetal[4]forthe>5=1UUcase.$č>5.2\Analyticityoftheop`erator\(b0mp+)2;K cmsy81=2܂(b1m;u+b2m;ͻ)(b0m+)21=2Mʍ>InthissubsectionweshowthatT2|s(G) :=(b0mO+)^1=2|sb2m;(b0m+)^1=2>isananalyticfamily*.Forthe rst-ordertermT1|s(G)relatingtob1m; thiswas>alreadyJprovenbyEvqansetal[6].Accordingto[14 ,p.14]thefollowingitems>areUUrequired:>(i) /T2|s(G)isclose}'dfor52D0:MThisQRrequirestheproGofofboundednessofT2|s()becauseT2|s()isde nedon>theHilbGertspaceL2|s(R^3)C^2|s:nF*oracompletedomain,vboundednessimplies>closure.16y?MThebGoundednessofT2|s()isshownbythesamemeansascompactnessof >(b0mu\+N)^1 gb2m;(b0m+)^1=2=i(b0m+)^1=2|sT2|s(G):$ReferingPztosection>4.2thelatterrequirestheproGofofcompactnessoftheeightoperators im>oftypGe(4.16). 7`F*orim=1andA7(p)andh(p)estimatedbythebGounded>opGeratorsA(p)andh(p),2qcf.(5.5),thel.h.s.of(4.17)isbGoundedalsofor>56=1:IntheproGofoftheboundednessof(b0m +)^1=2F2\)D0|s;Ʋwecanbymeansoftheestimates(5.5)and(5.6)fortheG-dependent>quantitiesprovideanuppGerboundforthekernelx䍑~K ((p;p^0 m)from(4.19),>which>isq'propGortionaltothecorrespondingkernelinthemv=0q'case.=Boundednessof>(b0m ~+Vp)^1=2wayasforthe~=71case.otherUU im;8i=2;:::;8:pMF*or>RthebGoundednessofT2|s()wede ne\q7~ imײ:=(b0miA+3)^1=2 ʵ im:The>basic=di erenceisthatinthedecompGositionsoftype(4.16)thecompactop->eratorN~(b0m+)^1 gx^1=2 Iisnowreplacedby(b0m+)^1=2|sx^1=2 Iandthat>(b0m t+f)^1 gW10;m`(b0m+)isreplacedby(b0m t+)^1=2|sW10;m`(b0m+)^1=2 ʵ:>However,@wehavealreadyincludedthecase/=1=2intheearlierproGofof>bGoundednessofW^cv10;m=from(4.18), soitonlyremainstoproveboundednessv>of(b0m +y)^1=2|sx^1=2˵:ThisisdonebymeansofKato'sinequality[12 ,pp.307]>inUUtheinverseUUform,1=p0 KK&feox~2 ŕx;andintroGducing ":=<$ G1cwfe s 3ፍpUWfe3荵x!'monehas,i<0k(b0m +8)1=2<$RO1wfe s 3ፍpUWfe3荵x'd'k2C=0( [;(b0m+8)1 g )0( [;<$۲1۟wfe (֍p ö )}%<$ccwfe (֍2( [;x )0=<$ccwfe (֍2(';')}q(5.14) >(ii) /Tc(G):=0T1|s()8+T2(G)hasanonemptyr}'esolventset%(Tc())fore}'achXW 52D0|s:MSincey]the(opGerator-)boundednessy]ofTc(G)impliesitsformboundednessbe->cause"ofj(';Tc(G)')j:Pk'kkT(G)'k;?the"expectationvqalueofTc()and>hencethespGectrumofTc()isbounded,*wi.e._6=C:Thismeansthat%(Tc())6=;:>(iii) NF;or`Levery0C2D0|s;8Tc(G)isananalyticfunctionofiinaneighb}'ourhoodZsU(0|s)of0:MThis=istruebGecauseD0isopenandb1m;E;hb2m;~dependanalyticallyon>hforZK#2zD0|s:Q(Notethatbim;*=(1=G)bim; i=1;2;Qandasseenfrom>(3.2)8and(1.5),qthem&̸G-dependence8entersanalyticallythroughE7(p)and>E7(p)ى+m1theFTmoGduliofwhichareboundedawayFTfromzeroforݸ2XD0>andX"mvk6=0accordingto(5.1)and(5.4)).z-T*oobtainTc(G); ~b1m;୲+hb2m;Sgis>only}multipliedbybGoundedfactorswhichareindepGendentofG,henceTc()is>analyticUUinD0|s:>(iv) F;orevery0C2D0 Nther}'eisa02%(Tc(0|s))whichisalsointher}'esolventZsetofthe'neighb}'ouring'operatorTc(G)for52U(0|s):17 ry?MThisfollowsfromtheG-independenceoftheformbGoundofTc()forall >52D0|s:F*or-j(';Tc(G)')j0M(';');every-zwithjzpj0>MEisintheresolvent>setUUofTc(G)forall52D0|s,hencealsoforT(0|s)andforT(G)with52U(0|s):$>5.3\ThecompactnessofRm;ɻ()č>Our=startingpGointis(4.9)for52D0|s.EThecompactnessoftheoperatorincurly>bracketsVewasproveninthelastsection,VandthebGoundednessofthe rstfactor>inOsquarebracketsOiseasilyshown.aWithmreplacedbyin(5.4)andwith>>1UUwehaveqэcj(b0m;4%+8)1 (b0m +)j0=  ㅵ<$Ah8pAi8fe#{ pr2S+8mr23{IJ+8Ahwfe?@ (֍E7(p)8+D{ D{ D{ D{ D{ )k0jGjq q q q q <$_l8p_m8fe#{ pr2S+8mr2?Ȳ+8HwfeN ' (֍(180|s)r3'(Ep2+)X8 X8 X8 X8 X8 a'<$ c+18+20cwfe&X (֍(180|s)r3}q²(5.15) >which(provesitsbGoundedness(andsimultaneouslythebGoundednessof(b0m; +>)^1ɲsinceUUEp2+80>1for>1):pMItremainstoshowthebGoundednessof(b0m+!)^1=2(bm; ʲ+)^1 t: вThe>bGoundedness7of(bm;+˵)^1 >for52D0=followsfromthepositivity(2.4)ofthis>opGeratorefor5=1andtheanalyticityofbm;7inD0|s:Thisassuresthatthereisa>neighbGourhoodof1inCsuchthatRe˧(bm;<+j)0>0o1andhencejbm;+jj0>0:>F*orUU0Ȳsucientlysmall,D0isasubsetofthisneighbGourhood.MF*urtherC%wenotethattheopGerator(b0m;IJ+)^1=2existsbecausefor52D0|s;>ReM#(b0m;d+it)>@Redb0m;|Y@m(10|s)@0:޽This3followsfroma(5.7)-typGe 6 >sequenceofinequalitiesbGecauseforb0m;i=Hs0p Hs0fe1vӟЍpr2|s=r2p+8mr2Baonlypandmhave>to3bGeinterchanged3in(5.7),:LasignreversalofQplayingnorolefortherealpart>ofUUthisopGerator.qWiththiswedecomposeq̍>(b0m L+X>)/ 33133xW g P2M(bm;W+)1 =`h(b0m +8)/ 33133xW g P2(b0m;4%+)P 33133xW g P2 `iyF`h}|(b0m;4%+8)/ 33133xW g P2(bm;7+)1 t`i}q²(5.16)>The%{bGoundednessofthe rstoperatorinsquarebrackets%{followsfrom(5.15).>InfGordertoprovefGbGoundednessofthesecondoperatorinsquarebracketsfGletus> rsttake52R^+Y\aD0|s:-ThenfromunitarityofthedilationopGeratordLonehas>k(b0m ++)1=2X(bm / +)1 g'k=kd0߲(b0m ++)1=2With4thechoiceof'2SqYuC^2|s;4i.e.e' kananalyticvectorin_2D0andusing>thatH@bm;G(aswellasb0m;E)isanalyticinD0|s,ther.h.s.Jof(5.17)isanalytic>inJ D0|s.OF*romtheidentityJ theoremweinferthat(5.17)holdsforall 2^D0:18!y?>However,thei'l.h.s.>of(5.17)isbGoundedby*,say*,c1嚲asi'showninsection4.2. >HenceZk(b0m;4%+8)1=2X(bm;7+)1 g'MOk0c1C(';')=c1C('7;')}q(5.18)>whichRprovesthedesiredbGoundedness.GNotethatthelastequalityin(5.18)also>isUUaconsequenceoftheidentityUUtheorem.)4>6VLAbsenceffofembs3eddedeigenvalues>W*ek concludethisworkbyshowingthatform6=0k thereisanm-depGendent>bGound- abovewhichtherearenoeigenvqaluesofbm ŧembGeddedintheessential>spGectrum.qF*orUUm=0weprovetheabsenceofeigenvqalues.p>TheoremT6.1L}'et2thecriticalcouplingconstant casinTheorem1.1.XIfm=0>and UP< c;thesp}'ectrumofbisabsolutelyc}'ontinuous.MF*orJtheproGofweonlyhavetoshowthatbhasnoeigenvqalues.RlThenthe>spGectrumisgivenby[ٲ(b).=Cess dH(b)=ac u(b)bGecausesc9A(b)=;asstatedin>TheoremX1.1(ac=absolutecontinuousXandsc=singularcontinuous).F*ollowing>EvqansUUetal[6]weproGceedintwostepspGq(i)WAssumeOE6=Lc0isaneigenvqalueofb,Ni.e.athereexists'2H:1=2 ʲ(R^3|s)n1C^2Wsuchthatb'f=E': ~Asdemonstratedin[6]fortheopGeratorb0ٲ+fb1WthisJleadstoacontradictionsinceforeach 2=D0K\4R^+; $d7'isanWeigenfunction ofbtotheeigenvqalueGE5because ofthescalingofbwithWEβ(inthemasslesscase),)Ȳ(d7bd䍷1K[ t)d'¶=[ȵb=Ų(d')=[ȵE>5d':WHowever,theJexistenceofanuncountablesetof(orthonormal)eigenvectorsWofCa(self-adjoint)opGeratorintheHilbertspaceH:1=2 ʲ(R^3|s)nC^2lcontradictsWseparabilityUUoftheHilbGertspace.D(ii)WAssumeME=e0isaneigenvqalueofb,i.e.Zthereexists'6=0suchthatWb'M=0:Using7thepartialwave7decompGositionofband'asintroducedWinUUAppGendixAwehavefrom(A.4)inMellinspace%d!00=(';b')0=E(')=X  0cZi1@81+idtja]ɲ(t8+i=2)j2Cb1ɍ]vl `s(ti=2)q(6.1)!ʍWwhereb1ɍ]vl `s := $b1ɍ]xݍ0+wPp YΟPfe E2,b:(1)]6l `s^+wPp YΟPfe E2b:(2)]6l `s : J˲However,pGositivityofbor 獑Wequivqalently*,ofԵb1ɍ]vl `s(tXi=2)for x<@ cimpliesthatther.h.s.Dof(6.1)canWonlyUUbGezeroifforeach;>sja]ɲ(t8+i=2)j0=0UWalmostUUeverywhereforz귵t2R:qò(6.2)195Sy?WIfC'N2SP%)C^2Tthena^] isananalyticfunctionof inthestripf2C: W1ҵ<it=Re4<1;40ImЇ 1&fes2 Bg:SVF*romtheidentitytheoremitWfollowsUUthatja^]ɲ(t)j0=0mandunitarityoftheMellintransformgives퍍00=cZi11@ j1dtja]ɲ(t)j2 =cZi11@ j0dpjaɲ(p)j2qò(6.3)WhenceNiaɲ(p)f7=0inR^+ andthus'=0:However,sinceSeisdenseinWH:1=2 ʲ(R^3|s)wehave'=0inH:1=2 ʲ(R^3|s)C^2UwhichisacontradictiontoourWassumptionUU'6=0:LCMF*orUUthem6=0UUcase,wehavex8>TheoremT6.2L}'etֻ `<@( cwith casinThe}'oremֻ1.1.bThentheeigenvalues>ofbm ,ar}'econ nedtoqm(1+8s( 8))Zwithȍs( 8):=0max*f0;UPs0|s(m1 8m0S+m2|s 82 )g>wher}'es0 :=5;Gm0:=0:3058;Gm1:= Cұ2Cҟ&fes5[6andm2:=2:253: Inp}'articular, >for4 <SͲ0:29(i.e.~Z <40)theessentialsp}'ectrum4ofbm hasnoemb}'edded>eigenvalues.MThe nproGofproceedsalongthelinesprovidedbyBalinskyandEvqans[1]in>thecaseoftheBrown-RavenhallopGeratorb0mM+b1m :^However,Dare nement>of/etheestimatesismandatorytoshowtheabsenceofembGeddedeigenvqaluesfor>smallUUcouplingconstants.pMStarting9{pGointisthevirialtheorem[1,? Lemma2.1].h~If'isaneigenfunction>to@bm ~۲witheigenvqalueanduseismadeofthescalingpropGerty(4.3)ofbm;>withUUG,thevirialtheoremtakestheform2OlimCд7!1?_('7;<$bm78bmwfe*e (֍ 811')0=k'k2qò(6.4)}>for_!2R^+ )sand';=d7'from(2.12).qInordertointerchangethelimit_!!1>with&Ptheintegration,/theuniformabsoluteconvergenceoftheformonthel.h.s.>ofUU(6.4)isneeded.xMSincem\jg2R^+;theproGofs[16 ,4]offormboundednessof    <$ 3dbim 3wfeR: (֍(dm8& & & & +;i=qύ>1;2;also2guaranteebGoundednessfortheo -diagonalformifuseismadeofthe>generalisedUULiebandY*auformula(2.11),Ѝk(j'7j;UP UP UP UP <$ 5ֵdbm؟wfe- (֍dm8#> #> #> #> + j'j)j0ck'k8k'k0=ck'k2|s;c2R:qò(6.5)>HenceByfromthemeanvqaluetheorem,F?with@somevaluebGetweenBymin'fm);mg >andUUmax˸fm8G;mg,[ . [ . [ . [ . ^b('7;<$bm78bmwfe*e (֍ 811')  ;͸0(j'7j;UPm  <$ 5ֵdbm؟wfe- (֍dm8#>(uDz)  7j'j)mck'k2qò(6.6)20DLy?>suchy thatthedominatedconvergencetheoremapplies.(`W*ethereforeobtainfrom >(6.4)yk'k2 =0m2'cZ U yR3Rdpj'(p)j2<$ ڲ1ZNwfe  (֍Ep8[+mcZ8 yR3 R3!5dpdp0 ݟLщfeU/'(p)$^<$,)db1m (p;p^0 m),)wfe0^ (֍sdmbB+<$db2m (p;p^0 m)wfe0^ (֍sdm7,L^1p'(p0 1Ų)q(6.7)qύ>Duetotheself-adjointnessofb1mandhenceofdb1m =dm;Žtheinterchangeof>pgandp^0intheexpGectationvqalueleadstocomplexconjugation.Thereforethe>termUUlinearinthecouplingconstantcanbGewritteninthefollowingway:3ğcZx yR3 R3Qdpdp0 mLщfeU/'(p)<$# db1m (p;p^0 m)# wfe0^ (֍sdmWLF'(p0 1Ų)#q֍l*.=RecZ? yR3 R39Hdpdp0 ݟLщfeU/'(p)%^<$10%1-wfe  (֍Ep>m<$õmwfe qǟ (֍Er2\np^cb1m (p;p0 m)'(p0 1Ų) 1aAS'+<$. 33wfe (֍2[ٟr2cZ*ڟ yR3 R3+*dpdp0 1şLщfeU/'(p)<$/1Mwfe%ܟ (֍jp8pr0 jjr2F&\A(p)A(p09)p^:p 5*^p0 h(p)h(p0)^<$ ۲1 :Owfe  (֍Ep{+<$1lwfe  (֍Er0\np_^;'(p0 1Ų):1Zqò(6.8)>F*ollowing[1],Uthe rsttermin(6.8)carryingthenegativesignofb1m;iselimi->natedUUwiththehelpoftheeigenvqalueequationintheform@?( [;bm')=cZUR yR3dpLщfe / (p) EpR'(p)8+8cZr yR3 R3qյdpdp0 1şLщfe / (p)["b1m (p;p0 m)+b2m(p;p0 m)]'(p0 1Ų)qύ|=0( [;')@ [ٲ(p):=0^<$c1ןwfe  (֍Ep R<$õmwfe qǟ (֍Er2\np^F"'(p):qò(6.9)!>ThiswproGcedureofeliminatinganegative rst-ordertermattheexpenseofad->ditional8:zero-orderterms(forwhichnofurtherestimateisneeded)andsecond->order!terms(whicharesmallforsmall 8)ismandatoryforthedesiredestimate>onUUtheeigenvqalue:mWith(6.8)and(6.9),(6.7)resultsin<$DECKwfeǷ (֍mNk'k2C=cZUR yR3ʧdpj'(p)j2'^<$SsmŸwfe  (֍Ep#+8(<$I33wfe  (֍Ep_1)(1<$ĵmlwfe  (֍Ep)^8+<$T lwfe (֍2[ٟr2ՀcZc yR3 R32cdpdp0 mLщfeU/'(p)FOA(p)A(p09)^<$g1$wfe%ܟ (֍jp8pr0 jjr2/16x^6kp>yFU^Ep0NSh(p)h(p0)^<$ 31wfe  (֍EpӲ+<$F1lwfe (֍Ep0Tҟ^<+<$T lwfe (֍4[ٟr2T2|s(p;p0 m)^'(p0 1Ų)8y}q(6.10)>wherethelengthyexpressionforT2|s(p;p^0 m)isgiveninAppGendixC.Applyingthe>Lieb+andY*auformula(2.11)with'(:= q:='Ah+and'A,respGectively*,to+the> rst-orderUUandsecond-orderterm,oneobtainswithjp^:p 5*^p^0 jq1g>^<$HsG&CwfeǷ (֍mSZ 81^icZoD yR3xdpj'(p)j2|s^ 18<$ĵmlwfe  (֍Ep?+<$lm^2lwfe D* (֍i2Er2\npp^O՗cZ: yR3dpj'(p)j2<$(Ep28m)(2Epm)wfeV (֍%Er2\np21U0y? C8ʲ+<$F Kwfe (֍2[ٟr2ccZ yR3'dpj'(p)j2'A(p)23\(nh2|s(p)cZ8 yR3Xߵdp0<$|1 wfe%ܟ (֍jp8pr0 jjr27o^<$C1@wfe  (֍EpQ4+<$ 1wfe (֍Ep0z^yɶ yɶ yɶ yɶ <$[f(p)~R>wfe= (֍f(pr09)    qڱ2$7(C+<$ ! &wfe (֍4[ٟr2;cZ zu yR3+rdp0 ݸjT2|s(p;p0 m)jq q q q <$ aef(p)Hwfe= (֍f(pr09)ø ø ø ø # ڱ2'\)}q²(6.11);>The\lasttermin(6.11)canbGefurtherestimatedbybreakingT2|s(p;p^0 m)into >its^constituentsandestimatingeachcontributionseparately*. Notethatthe>convergence8inducingfunctioncanbGechosendi erentlyforeachintegral.?Apart>fromYtheconventionalYchoicef(p)3=sp^3=4[#[1,4Z],Zwealsoallowforfunctionsof>theltypGef(p)=p^3=4theěestimates.F*urther,lthefollowingestimateisusedintheevqaluationofthe>integralsUUoverp^09,)$<${Q1[wfeF\ ɍs0p s0fe*%SЍ(q[pr09)r2S+818+8c<08 0>0>0>0< 0>0>0>0:@i<$O1wfe (֍18+c+d;8+p^0Q1=qŸ?<$O1ޟwfe (֍q[pr0&;8+p^0Q>1=qd ;c0;qq"0}q(6.12)>Anoutlineoftheevqaluationofthesecond-ordertermin ޲isgiveninAppGendix >C.dDe ningq=:=p=m;ddenotingtheestimateofīR ɍR3by2(4[ٟ^2L)^2|sq[ٟ^2M2(q[ٲ);9dand2takingf(p)y2:=p^3=4Qin2thetermlinearin 8,suchthat>(withUU(B.1),thesubstitutionq[ٟ^0*:=p^09=mq.and(6.12))xH"cZ\ yR3Ydp0<$|1 wfe%ܟ (֍jp8pr0 jjr2<$9z15wfe  (֍Er0\npF&^<$PypO͟wfe9 (֍pr0X9^_x۱3=2q04[ٟ2 z(q[ٲ)}q(6.13)"⅍> z(q[ٲ):=q1y+<$411wfe (֍Dҟ^F2HpUWHfeҪq}Qln}P }P }P }P <$ز18+q؟wfe (֍18q6D 6D 6D 6D =}q+2(q81)UParctan<$&h1"hwfe ( (֍pUWfeҪ;q4@$(q+81)UPln UO  UO UO UO ײ1+p 7feҪ;qן)fe` (֍1p 7feҪ;q1q 1q 1q 1q 7W^cz>weUUarriveatthefollowingestimate"K400m3ßcZ _ yR3dqj'(mq)j23\ ߲1㈸<$ڹ1wfe$ 8p 8fe q[ٟr2,+81.s+<$ڹ1wfe (֍q[ٟr2,+81 \!.m^1㈸<$wfeǷ (֍m-+(q[ٲ)^ }q²(6.14)gPO(q[ٲ):=<$'@q^2cwfeKHן qr2,+828p 88fe q[ٟr2+1<$_1W7wfe (֍f0|s(q)r=bwmg0|s(q)+ 8g1(q[ٲ)+ 82 g2(q[ٲ)bO_f0|s(q)u>whereWƍqsKg0|s(q[ٲ):=<$c28p 8fe qr2,+81&ݸ81cwfe:ݟ 8p 8fe qr2,+81)@+81Bs;g1(q):=<$cq+8 z(q)8p 8fe qr2,+1cwfeHj  mǟ8pmȟ8fe qr2,+810.+81bkg2|s(q[ٲ):=0(q2,+81+p 8fe Dq[ٟr2+1&)M2|s(q[ٲ);f0(q):=<$ acq+8ccwfeb (֍aq+8b}q²(6.15)>are2nonnegativebGoundedfunctions._Theauxiliaryfunctionf0Lwitha;b;c>0>hasbGeenintroducedtoimproveontheestimateof:}-Itfollowsfrom(6.14)that22iʠy?>forb3<0;eWithEjm0ӫ:=W8ming0|sf0;m1:=W8supg1f0ݲandEjm2:=W8maxVg2f0ݲforEj0W8q<1;>thisconditionisful lledform0*+m1|s <IJ+m2 8^2 <E0;i.e.O {}< 0;زsay*.OFor>a̱:=5;wYb:= 1&fes5 /;c:=1:1;Kweobtainm0I$=̱0:3058;m1= 2&fes5 /;m2=̱2:253;Kand>hence% 0=y0:29:9BThisimproves%onthevqalue 0=y0:159obtainedforf0=y1>(wherem0C= K1K&fes2 );qsup*g1=2;qsupg2= K29K&fe:4 &d): Denotingbys0xhthesupremumofthe>prefactorof(q[ٲ)inqr2xR^+;#As0 :=sup1|q^2L=(q^2X,+28p 8fe qr2,+81')f䍑10 (q)#A=x5;8owe>canUUestimate(q[ٲ)for UP> 0Ȳtoobtainfrom(6.14)cS0qm(1+(q[ٲ))m(1+s0|s(m1 q8m0_+m2|s 82 )):}q²(6.16)zMIn7theBrown-Ravenhall7case(g2C0)0(q[ٲ)canbGewrittenas 6g0|s=g1(q)>multipliedybyanonnegativefactor,andoneobtainsfortheeigenvqalues\q~of>b0m"˲+b1mitheԢestimate\q?M~ E;<Bmfor )z<~ 0[:=0:973owhichistheminimumof>g0|s=g1زineR^+:BThiscoversethewholerangeofbGoundedness(frombelow)ofthe>Brown-Ravenhall|opGerator( <έ2=([=2L+O2=)=0:906|[6])andimproves|on>theUUresultofBalinskyandEvqans[1]( UP3=4obtainedfor z(q[ٲ)=1:)"c>Ac9knowledgmentpMIshouldliketothankH.Kalf,]K.W*ol hardtandE.StoGckmeyerforvqaluable>discussions,~mandCP*.A.AmundsenforassistancewithMaple.:IBamparticularly>gratefultoH.Siedentopforstimulatingthispro8jectandforhiscontinualhelp>and8adviceduringthecourseofthiswork.4SuppGortbytheDeutscheF*orschungs->gemeinschafteaswellasbytheEuropGeanUnionthroughitsT*raining,iResearch>andUUMobilityprogramisgratefullyacknowledged.1W>App`endixA>ProQofTofLemma2.1MItUUisconvenientUUtointroGducethepartialwaveexpansions[6]{'(p)0=X  0p1 gaɲ(p) (Q^pc)(j=fl2`;M;sg$<$Es1wfe%ܟ (֍jp8pr0 jjr2sD=<$2cwfe 9 (֍ppr0DX bl `M)aqlȲ(<$Pp33wfe9 (֍pr0 <)Lщfek/Yl `M r$(Q^pc)$7Yl `M r$(Q^p0 1Ų)(A.1)rǍ>where ɲ(Q^pc)aretheDiracangularmomentumeigenstates(thevectorspherical>hamonics[2]), Yl `M r$(Q^pc)aresphericalharmonics,l=n0;1;:::;[Mβ=lK 1&fes2ĵ;l+ ?331?33&fes2Dbٵ;:::;l+ y1y&fes2;?sQ= 33133&fes2bٵ; ۱1۟&fes2 ;andnqlȲ(x)isrelatedtotheLegendrefunctionQl(x)of>thesecondkindbyqlȲ(x):=0Ql( 33133&fes2bٵx!I+ ֱ1d&feW2x E):Thentheenergy(3.1)canbGewritten>inthefollowingway[6,4],makinguseoforthonormalityoftheset ɲ(Q^pc)and23y?>likewiseUUofYl `M r$(Q^pc);vwpzEm(')0=X  0cZi1@8۱0)/dpLщfe/aɲ(p)ןcZi%ط1@!0/edp0Qbl `sm (p;p09)aɲ(p0)5rŵbl `sm (p;p09):=0b0m\&`(p8p0)+b:(1)6l `sm (p;p0)+b:(2)6l `sm (p;p0)(A.2)[>whereCQ^Kb:(1)6l `sm (p;p09)0=<$\ 33wfe (֍ D[aqlȲ(p=p0)+h(p)h(p0)ql `+2s(p=p0)]TA(p)A(p0)a>b:(2)6l `sm (p;p09)0=<$c1cwfe (֍2fW`<$/ nwfe (֍Կ`%ͣݱ2.֟cZi8׷1@4J0Bddp00 2^<$#t1wfe+i (֍Ep0 +8Ep00B+<$E&13+wfe)# (֍Ep2+8Ep000T^gA(p)A(p0)A2|s(p00r)EC[qŵqlȲ(p00r=p)h(p00)ql `+2s(p00r=p)h(p)] [*qlȲ(p09=p00r)h(p00)ql `+2s(p0=p00r)h(p0)]MAs"anextstep,,XtheMellinspacerepresentationisintroGducedbecauseinthe >m=0icase,ito ersanintegralrepresentationofE(')withapGositiveintegrand.>F*orUUafunctionfڧ2L2|s(R^+)theMellintransformf^]2L2|s(R)isde nedasr8f](t):=<$ 1cwfedv EPpUWPfe E2cZi'1@"01'Bdpf(p)pit1=2ò(A.3) >Since7theMellintransformisunitary*,E(')Rx=*2 \FPt)bGeUUcastintothefollowingform[4]獍nܵE(')0=X  0cZi1@81+idt")fe ׍a1ɍ]ɲ(t)a^$;՟cZi.;ַ1@)08_cdp0Qbl `s(;p09)aɲ(p0)^-۴](t)(A.4)%MSY=0X  0cZi1@81+idtq q q q a]ɲ(t8+<$3ilwfe (֍2 G)  4b8ڱ2=Pk`CIOb1ɍ]xݍ0 |k+=Vp UO=Vfe ª2b:(1)]6l `s߲+=Vp UO=Vfe ª2b:(2)]6l `s `r(t8<$3ilwfe (֍2 G)b>withPb1ɍ]xݍ0|s(t8<$3ilwfe (֍2 G)0=1;b:(1)]6l `s (t<$3ilwfe (֍2)0=<$ 33wfe  (֍2 -`hq1ɍ[ٴ]vlK(t8i=2)+q1ɍ[ٴ]vl `+2s(t8i=2)`if9b:(2)]6l `s (t8<$3ilwfe (֍2 G)0=<$cPpPfe E2cwfedv (֍2;2̟`<$) &wfe  (֍2395`92ݱ2B L`hF0q1ɍ[ٴ]vlK(ti=2)q1ɍ[ٴ]vl `+2s(t8i=2)`isܟݱ2E֍MSinceYq1ɍ[ٴ]vlK(tOi=2)Ÿ0 8l%2N0 8[4]onehasb:(1)]6l `s (tOi=2) Ų0andv>b:(2)]6l `s (t8i=2)00UUandthereforealsob1C0andb2C0.qW*eshowthat΍Qb:(1)]6l `s (t8<$3ilwfe (֍2 G)b:(2)]6l `s(t8<$3ilwfe (֍2 G)00(A.5)!>which thusprovesb1пTLb2C0;i.e.KLemma2.1.W*eproGceedintwosteps.KFirst>weKshowtheexistenceofasucientlylargel1suchthat(A.5)holdsforlj8\l1>andsF6= 33133&fes2bٵ:ѲSubsequentlyweusearecurrencerelationtoprovethatif(A.5)>holdsUUforagivenlitalsoholdsforlk@81:24y?GqDz(i)WF*romUU[9,p.937] jlimꪍdjzI{j!1" " " " <$(zw+8a)wfe$K (֍(zp):@zpa      $ڱ2Dz=01UWfor&z72Cn(Z[8f0g);8a2Rò(A.6)wWW*e'takea:= 33133&fes2bٵ;8z7:= lK&fes2 Β+݉1 it&fe٢2 Rand'introGducetheexplicitexpressionsWforUUq1ɍ[ٴ]vlK(t8i=2)mintermsofGammafunctions[4]!?~q1ɍ[ٴ]vlK(t8<$3ilwfe (֍2 G)0=<$՟p,fe3荵cwfeUY E2PpUWPfeE2     Ӎ#D7( 艴l33&fes2+ l1l&fes2 ԙ litl&fe٢2 x)#D7 _fe;~ l( 艴l33&fes2+1 litl&fe٢2 x)_ _ _ _ _ cKTر2ò(A.7) WThenejfrom(A.6)followstheexistenceofl0^Y2Nsuchthatforanygiven WwithUU0<<1;u-aN(18)<$1Kwfe2H    =޴l&fes2 +1 litl&fe٢2 x  x =z<02Idr Idfeu<$²233wfe (֍q1ɍ[ٴ]vlK(t<$3ilwfe (֍2 G)<(1+)<$1Kwfe2H    =޴l&fes2 +1 litl&fe٢2 x  x ò(A.8)$Wforl>sgl0|s: cTF*romthisitfollowsthattheuppGerandlowerbGoundsof cWq1ɍ[ٴ]vlK(t[i=2)andhenceofb:(1)]6l `s (t[i=2)decreaseasl2`^1 կforlx!1;whilethevWbGoundsAofb:(2)]6l `s (tc}i=2)AareoforderO(l2`^2 Բ)makingthattermnegligibleWwith^respGecttob:(1)]6l `s (tai=2)^forsucientlylargel2`:HencethereexistsWl1C2NUUsuchthat=t(b:(1)]6l `s߸8b:(2)]6l `s )(t<$3ilwfe (֍2 G)00UWforUUall4"lxl1|s;qs=<$33133wfe (֍2fg:ò(A.9)\ D(ii)WItwasshownbyBrummelhuisetal[4]thatfor Eָ4=A(andhenceforW UP< c)ƍc-n1+=Vp UO=Vfe ª2+(b:(1)]l `1;1=2 +b:(2)]l `1;1=2)01+=Vp UO=Vfe ª2(b:(1)]l `;1=2+b:(2)]l `;1=2)z(A.10)W8-l_|2N0|s;whichbymeansofb:(i)]l `+1;1=2(=β=-b:(i)]l `;1=2;i=1;2;l_|2N0 <(which ΍Wfollowsfrom(A.4))holdsalsofors=1=2:Hereandinthefollowingthe ƍWargumentUU(t8i=2)ofb:(i)]6l `s&issuppressed.qHence9⍑\(b:(1)]6l `s߸b:(2)]6l `s0b:(1)]6l `1;stnb:(2)]6l `1;stv;(l2`;s)2f(N0|s;<$۲1۟wfe (֍2 )[(N;<$33133wfe (֍2fg)g\ z²(A.11)WSettingin(A.11)lS = 0fors= S߱1Sߟ&fes2/EandlS =1fors= 33133&fes2 andcontinuing䑍Wthechainofinequalitiestotheleftuntill1\Visreached,`provesthatb:(1)]6l `sFWb:(2)]6l `s0mforUUalll2`;s:25yy?>App`endixBp>ProQofToftheboundednessofW^cl10 N:inthemasslesscaseMInordertoprovethe nitenessofI(p^0 1Ų)asde nedin(4.20)westartby >showingthatI(p^0 1Ų)onlydepGendsonp^09.r]Choosingsphericalcoordinatesforp^0 ܡ,>theUUangularintegrationscanbGeperformedbymeansof[9,p.58]%M)cZS8c ySa2^2d![ٟ0<$˲1$]wfe%ܟ (֍jp8pr0 jjr23=<$2cwfe 9 (֍ppr0Dln#D #D #D #D <$'p8+p^0'wfe (֍p8pr0B` B` B` B` H3Ͳ=<$2cwfe 9 (֍ppr0D8 D>D< D>D: Z([Q2p([Q);feşp06A+OG(p^2|s);mp!0 5 P([Q2pr0([Q);fe 8Xsp9^+OG( 83133&ferp2 lز);mp!1\m(B.1)>WithVkA(p^09)1andm=0R(andthechoice:=1)Rinthede nition(4.19)forx䍑@~>KG5U(p;p^0 m)UUwehave&I(p0 1Ų)0=cZ j yR3 R3%qdp00Gdpx䍑o~K m(p;p0 m)x䍑o~K(p;p00 *)<$pZq0s^2 Kwfe; (֍ps00r2 %"W>cZUR yR3 R3dp00 dp(pQ+1)2<$N1 mwfe%ܟ (֍jp8pr0 jjr2<$<12"wfe (֍p8+pr0Ll(p0 +1)<$D1 nwfe(' (֍jp8pr00 ޸jr2<$Bq17>wfeeP (֍p8+pr00Sl(p00nV+1)<$ ˨pZq0s^2 nwfe; (֍ps00r2 &F:=:04[ٟ2x䍑 ܲ~LI (p09)\m(B.2)>InJordertoseparatethevqariableswepGerformscalingtransformationsq":=p=p^0>andUUq[ٟ^00?c:=p^00r=q[p^0ꦲofthevqariablespandp^00,respGectively[4],toobtainx䍑eg~d4EIib}(p09)0=(p0+81) (cZi(1@0!Lqdq"q[ٟ12 (q[p0+1)2<$1 4wfe (֍q+1(tln2s 2s 2s 2s <$7mq+17mwfe (֍q1N N N N #z9cZi 1@806dq[ٟ00?c(q[ٟ00xK)12 (q[p09q00++81)<$1xwfe' (֍1+q[ٟr000Etln:Es :Es :Es :Es <$>1+q[ٟ^00>wfe' (֍1q[ٟr00[)) [)) [)) [)) \m(B.3)z>showingthat,iapartfromthefactors(q[p^0퉲+P1)^2 鬲and(qp^09q^00+P1)^-:thetwo>integralsUUarealikeanddecouple.p>Case(a):`p^0Q!1MThis casegivesthemostsevererestrictionstotheexpGonent ofthecon- >vergenceminducingfunction.AssumeqB6=i0;Eq[ٟ^00 6=0 (andmnotethatife.g.>q"!03wouldbGetakenbGeforethelimitp^0Q!1wascarriedout,:theq[ٲ-integrand>wouldBbGehavelikeq[ٟ^22 nearzero,Fleadingtotherestriction22 >1; i.e.> В<3=2UUforconvergence).qThenl~z(p0+81)ES(q[p0+1)2 Ų(q[p09q00++1) k(!0qDqq00D\m(B.4)>whichmakesx䍑r~I(p^09)indepGendentofp^0cinthelimitp^0Q!1[andleadstoasplitting>intoxtheproGductoftwointegrals,limqƍp0s!1x䍑%G~I!(p^09)=:I1x()I1():zW*exthushave26y?>to0p nd 9suchthatI1x()is niteforthetwocasesofinterest,7ѵ= K1K&fes2 );1:The >integrandUUofI1x()bGehaveslikepRq[ٟ1+2 <$+{1##wfe (֍q+81>3lnH2 H2 H2 H2 <$M\q+81M\wfe (֍q81dv dv dv dv m~sq!0^d12q[ٟ^2+2 )l;Mq"!0 12q[ٟ^1+2 %im;Mq"!1\m(B.5)Ս>Convergenceatthelowerlimitrequires2,o+2 >13andattheuppGerlimit>(18+2 z)0<1msuchUUthat nitenessofx䍑~I(p^09)forp^0Q!1isachievedUUif5=<$3333wfeW (֍j2 < <<$K3Kwfe (֍2 f_<$llwfeW (֍j2\m(B.6)Ѝ>where] eitheralluppGeroralllower] signsmustbetaken.F*or=1;] oneobtains>theWintervqalf 33133&fes2 -< H<2g*\f 33133&fes2< <1g=f 33133&fes2< H<1g"]whileWforβ=1=2>one5getsf 33133&fes4 f<@ < s7s&fes4g\f 33133&fes4 f< < s5s&fes4g=f 33133&fes4< J< s5s&fes4g:²Both5vqaluesof>areUUcoveredbytheintervqal'jf В2R+ j:<$c1cwfe (֍2< <1g\m(B.7)\ >InUUparticular,(B.7)satis esthecondition В<3=2impGosedaboveUU(B.4).M2>Case(b):`p^0Q=0pMThiscaserendersx䍑T~I(p^09)indepGendentof;givenbytheproGductofidentical>integralsx䍑 ~u}I(p^0=0)=^(I1x(0))^2|s:r,F*romu}(B.6)oneobtains nitenessfor0< )<>3=2mwhereofUU(B.7)isasubset.>Case(c):`0whichrequires -<ᳲ3=2:F*orq[ٟ^00 Y!1;itbGehaveslikeq[ٟ^(q[ٟ^00xK)^12 - for>qwY6=0andlike(q[ٟ^00xK)^12 IJforq=0,resultingintherestrictions $>=2and> >R0,respGectively*..Hence,forthevqaluesofunderconsideration,one nds ?331?33&fes2G)< В< K3K&fes2F^inUUtotal.MT*urningktotheq[ٲ-integral,$itbGehavesforq7=0likeq[ٟ^22 C asnotedbGefore,>and~forq"!1likeq[ٟ^12 +*(wherewehaveincludedthefactorq[ٟ^contained>inthesecondintegral)leadingto Ǟ>$=2:IntotalweobtainthepGermitted>intervqal 1&fes4 <T < 䇱3䇟&fes2 -:4Sincef 33133&fes2< < 䇱3䇟&fes2gG\f 33133&fes4 -< < 䇱3䇟&fes2gbf 33133&fes2< <1g;>thetcases(b)and(c)givenofurtherrestrictionontheintervqal(B.7).]'Hencewe>haveUUproventhatI(p^0 1Ų)is nitefore.g.qǵ В=3=4when2f 33133&fes2bٵ;1g:27|y?>App`endixCpMW*eMpresentguidelinestotheproGofoftheboundednessfromaboveMofthe >pGointUUspectrumofbm.MTheUUopGeratorT2Ȳde nedin(6.10)isgivenexplicitlybyVzT2|s(p;p0 m):=0cZ j yR3̟ (֍2Ep00 8(Ep2+8m)l6^<$vmrzwfe (֍Ep00 ^<$'91|wfeBU (֍Ep0SŲ(Ep0 +8Ep00 8)~+<$! 1wfe<- (֍EpR(Ep2+8Ep00 8)C6^5+UP^<$Q+1 wfe+i (֍Ep0 +8Ep00<+<$(1wfe)# (֍Ep2+8Ep00/m^~\ <$YL1wfe  (֍Ep+<$ 'ߵmwfe (֍E:2`p003\!K\#@O S^ pe^Jp0<$=*Npp^09(Ep00 +8m)%Z>wfej0 (֍2Ep00 8(Ep2+8m)(Ep0 +m)/<^<$mwfe (֍Ep00^<$|1wfe (֍Ep0<$ބ1rPwfe+i (֍Ep0 +8Ep00at+<$G1wfe  (֍Ep<$':ݲ1(wfe)# (֍Ep2+8Ep00? ^g+^<$1 :Owfe+i (֍Ep0 +8Ep009~˲+<$~1lwfe)# (֍Ep2+8Ep00-<^xA\ <$1. wfe  (֍EpL+<$F1lwfe (֍Ep0<$ a1lwfe (֍Ep00%+<$}7mlwfe (֍E:2`p00E\!n\#gԇ\)!]qMW*eRdemonstratetheproGcedureofestimatingtheintegraloverT2|s(p;p^09)in- >troGducedUUin(6.11)foroneparticularterm,m]sVI:=0cZ j yR3 R3%qdp09dp00jp^:p00eO^p0$ѽj<$p^0p^00KwfeAg? (֍2Ep00 8(Ep0 +8m)<$TmJwfe (֍Ep00 8Ep<$~1lwfe)# (֍Ep2+8Ep00!΍<$ƽ133wfe(' (֍jp8pr00rjr2<$Am1.Ɵwfe*N (֍jpr00UR8pr09jr2_" _" _" _" <$ef(p)cwfe= (֍f(pr09)zs zs zs zs }Tڱ2o:8(C.2)ۍ>W*etakef^2(p)D:=p^5=2 ʵ=(Ep=+m)andmakethesubstitutionsq^00߶:=<$p^00wwfe a (֍mq[Wand<>q^0Q:=<$ &'p^0Kwfe} (֍mq[qr00!for{p^00\andp^09,frespGectively*.)OWithq:=p=m{wepGerformtheangularR`>integrationsUUwiththehelpof(B.1)andestimatejp^:p^00eO^p^0$ѽjby1suchthat: kqIW02[ٟ2Lq[ٟ4<$1җwfe$ 8p 8fe q[ٟr2,+81<$J12wfe5ܟ 8p 8fe q[ٟr2,+81&+81mcZiw1@s|0odq[ٟ0<$ r>1wfe ԣ iq[ٟ0өC1CxW g P2ln# # # # <$( 18+q[ٟ^0( wfeŸ (֍18q[ٟr0B B B B 28y?[cZi 1@80#dq[ٟ00xKq[ٟq00/ w3wxW g P2ln    <$18+q[ٟ^00wfe' (֍18q[ٟr00:纫 : : : <$W1Cwfe,>6 (֍(q[qr00)r2S+81<$M?1uMwfef 8p 8fe q[ٟr2,+81&+8s0p 8s0fe,>6Ѝ(q[qr00)r2S+818(C.3)ญ>W*eestimatethelastfactorwiththehelpofs0p s0fe,>6Ѝ(q[qr00xK)r2S+81>I1andthenusethe >estimateUU(6.12)toobtain _dIW04[ٟ3Lq[ٟ4<$z%1'wfe$ 8p 8fe q[ٟr2,+81<$O11kwfeBm (8p 8fe q[ٟr2,+81&+81)r2y\"~YcZiZ1=q@x0dq[ٟ00xKq[ٟq00/ w3wxW g P2畲ln甫    <$p18+q[ٟ^00pwfe' (֍18q[ٟr009J 9J 9J 9J )+<$!1Kwfe O (֍q[ٟr2'CcZi'D1@}1=q$Jѵdq[ٟ00<$ ڱ1V&wfe  iq[ٟ00өw1wxW g P2翲ln&羫 & & & <$+pF18+q[ٟ^00+pFwfe' (֍18q[ٟr00Gt Gt Gt Gt M\#"m=08[ٟ3Lq[ٟ2<$1җwfe$ 8p 8fe q[ٟr2,+81<$Q H12wfeBm (8p 8fe q[ٟr2,+81&+81)r2z1r^x[Ѹ<$ C43+wfe 1 i5qө 1 xW g P217ln#16 #16 #16 #16 <$'18+q'wfe (֍18q>z >z >z >z !4S +q^<$ g2q[ٟ^2 gwfeO (֍5 g@82^:arctan<$\1Xwfe ( (֍pUWfeҪ;ql G+^ \l1㈸<$q[ٟ^2wfe O (֍'5 ^7QlnAQ AQ AQ AQ Ei18+p 7feҪ;qEi)fe` (֍18p 7feҪ;qen| en| en| en| mɲ+<$,43+wfe  (֍15f`HpHfeҪq!Uy^8(C.4)v>DueatotheabGoveachoiceoff(p);ther.h.s.=vof(C.4)q  5 xW g P2 aforq"!0;c assuringthat>its-contributiontoM2|s(q[ٲ)de nedbGelow(6.12)is nite.NTheintegralsoGccuring>here=andintheremainingcontributionstoT29canbGefoundin[9,wp.205,206] ؍>afterUUsubstitutionsofthetypGex:=1=q[;qx:=q  1 xW g P2_:)4>ReferencesC[1]R[10]R[11]R[12]R[13]R[14]R[15]R[16]R cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10