The trace formula for quantum graphs with general self adjoint boundary conditions
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We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
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aThetraceformulaforquantumgraphswithgeneralselfadjointboundaryconditionsJensBolte1DepartmentofMathematicsRoyalHolloway,UniversityofLondonEgham,TW200EX,UnitedKingdomSebastianEndres2Institutf¨urTheoretischePhysikUniversit¨atUlm,Albert-Einstein-Allee11D-89069Ulm,GermanyAbstractWeconsidercompactmetricgraphswithanarbitraryselfadjointrealisationofthe
di erentialLaplacian.AfterdiscussingspectralpropertiesofLaplacians,weproveseveralversionsoftraceformulae,relatingLaplacespectratosumsoverperiodicorbitsonthegraph.Thisincludestraceformulaewith,respectively,absolutelyandconditionallyconvergentperiodicorbitsums;theconvergencedependingonproper-tiesofthetestfunctionsused.Wealsoproveatraceformulafortheheatkernelandprovidesmall-tasymptoticsforthetraceoftheheatkernel.
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
1Introduction
SometenyearsagoKottosandSmilansky[KS97,KS99b]introducedquantumgraphsasconvenientmodelsinthe eldofquantumchaos[Haa01,St¨o99],whereamajorgoalistounderstandtheconnectionbetweendynamicalpropertiesofaquantumsystemanditsassociatedclassicalcounterpart[BGS84].Previouslyintroducedmodelsthatpossessclassicalcounterpartswithchaoticdynamicsincludequantumbilliards,motionsonRie-mannianmanifoldswithnegativecurvaturesandquantummaps.Thesemodelshavebeenstudiedwithconsiderablesuccess,however,theyhaveoftenturnedouttobearunwantedcomplications.Quantumgraphsareconstructedalongthelinesofmanyofthesemod-elsinthattheyaremainlyconcernedwithspectralpropertiesofLaplacians.Inasensetheyaremaximallyreducedversionsofsuchmodelsinthattheunderlyingcon gurationspaceisonedimensional.Thenontrivialtopologyofthegraph,however,introducessu -cientcomplexitysuchthatthequantumsystembehavesliketypicalquantumsystemswithchaoticclassicalcounterparts,see[KS97,KS99b].Ontheotherhand,manydetailsoftheclassicaldynamicsareconsiderablysimplerandquantumspectraaregenerallyknowntosomemoredetail,sothatquantumgraphmodelsprovedtobeveryusefulinthe eldofquantumchaos[GS06].
Traceformulaeprovideadirectconnectionbetweenclassicalandquantumdynamicsinthattheyrelatequantumspectratoclassicalperiodicorbits.Ingeneral,thisconnec-tionarisesintheformofanasymptoticrelation,validforlargewavenumbers.Thereareonlyfewexceptionalcasesinwhichtraceformulaeareidentities.Amongtheseare,mostnotably,Laplacianson attoriandonRiemannianmanifoldswithconstantnegativesectionalcurvatures.Inbothcasesthecon gurationmanifoldsareRiemanniansymmetricspaces,allowingfortheapplicationofpowerfulmethodsofharmonicanalysisinprovingtherelevanttraceformulae,i.e.,thePoissonsummationformulaandtheSelbergtraceformula
[Sel56],respectively.Generically,however,thetoolsofharmonicanalysisarenotavail-able,andtraceformulaehavetobeprovenusingsemiclassicalormicrolocaltechniques,whichnaturallyinvolveasymptoticmethods.SemiclassicaltraceformulaewereintroducedbyGutzwiller[Gut71]forthespectraldensityofquantumHamiltonians.Subsequently,BalianandBloch[BB72]setupanalogoustraceformulaeforLaplaciansondomainsinRninashortwavelengthapproximation.The rstmathematicalproofs,forLaplaciansonRiemannianmanifolds,areduetoColindeVerdi`ere[CdV73]aswellasDuistermaatandGuillemin[DG75].Later,proofsforthesemiclassicalcasefollowed[Mei92,PU95].
OneofthevirtuesofquantumgraphmodelsthatledKottosandSmilanskytointro-ducethemtothe eldofquantumchaosisthattheirtraceformulaeareidentities,verymuchinanalogytotheSelbergtraceformula.The rstquantumgraphtraceformula,however,isduetoRoth[Rot83],whoexpressedthetraceoftheheatkernelforaLaplacianwithKirchho boundaryconditionsintheverticesofthegraphasasumoverperiodicorbits.KottosandSmilansky[KS99b]thenintroducedatraceformulaforthespectraldensityoftheLaplacian,andlaterKostrykin,Pottho andSchrader[KPS07]extendedRoth’straceformulatomoregeneralboundaryconditions.InthesecasestheboundaryconditionscharacterisingthedomainoftheLaplacianwereofanon-Robintypeinthat
2
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
theydonotmixboundaryvaluesoffunctionsandtheirderivatives.ThisleadstoperiodicorbitsumsinthetraceformulaethatcloselyresemblethoseoccurringintheSelbergtraceformula.InthispaperourprincipalgoalnowistoconsidergeneralselfadjointrealisationsofLaplaciansoncompactmetricgraphsandtoproveassociatedtraceformulaewithfairlygeneraltestfunctions.Inparticular,wedonotrequiretheirFouriertransformstobecompactlysupported.AllowingforRobin-typeboundaryconditionsleadstotraceformu-laethatarestillidentities,yettheamplitudesmultiplyingtheoscillatingfactorsineachperiodicorbitcontributiondependonthewavenumberinanontrivialway.Inprinciple,theseamplitudefunctionsareknownandpossessasymptoticexpansionsforlargewavenumbers.Therefore,inasensethesetraceformulaeareintermediatebetweenSelbergandGutzwiller/Duistermaat-Guillemintraceformulae,whereinthelattercaseonlytheasymptoticexpansionsoftheamplitudefunctionsareknown,andthetestfunctionsmusthavecompactlysupportedFouriertransforms.
Thispaperisorganisedasfollows:Insection2webrie yreviewtheconstructionofquantumgraphs,includingparametrisationsofselfadjointrealisationsoftheLaplaciandevelopedbyKostrykinandSchrader[KS99a]aswellasKuchment[Kuc04].FollowingthisweinvestigatevariouspropertiesofquantumgraphedgeS-matricesasintroducedin[KS99b],focussingontheiranalyticproperties.Insection4wethendiscussgeneralpropertiesofLaplacespectraoncompactgraphs.Themainpartofthispapercanbefoundinsection5where,aftersomenecessarypreparations,we nallyproveseveralversionsofquantumgraphtraceformulae:Theorem5.8containsatraceformulawithaconditionallyconvergentsumoverperiodicorbitsandallowsforalargeclassoftestfunctions.InTheorem5.9,however,werestricttheclassoftestfunctionswiththee ectthattheperiodicorbitsumsconvergeabsolutely.AsuitablechoiceofatestfunctionthenallowstoestablishthetraceoftheheatkernelforarbitraryselfadjointrealisationsoftheLaplacian,seeTheorem5.10.We nallysummariseanddiscussourresultsinsection6.Someoftheresultspresentedherehavebeenannouncedin[BEar].
2Preliminaries
Webeginwithreviewingtherelevantconceptsunderlyingtheconstructionofquantumgraphs.
2.1Metricgraphs
Inthesequelweshallconsider nite,metricgraphsΓ=(V,E,l).HereVisa nitesetofvertices{v1,...,vV}andEisa nitesetofedges{e1,...,eE}.Whenanedgeeconnectstheverticesvandw,thesearecallededgeends.Twoedgesareadjacent,iftheyshareanedgeend;loopsandpairsofmultiplyconnectedverticesshallbeallowed.Thedegreedvofavertexvspeci esthenumberofedgeswithvasoneoftheiredgeends.Ametricstructurecanbeintroducedbyassigningintervals[0,li]toedgesei,alongwithcoordinatesxi∈[0,li].TheE-tuplel=(l1,...,lE)thencollectsalledgelengths.Throughthisproceduretheedge
3
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
endsaremappedtotheendpointsoftheintervalsinaspeci edmanner.Theedgeendcorrespondingtoxi=0isthencalledinitialpointand,correspondingly,theotherendpointistheterminalpointoftheedgeei.Hencenotonlytheconnectednessofthegraphisspeci ed,butalsoanorientationoftheedgesisintroduced.Weemphasisethatthespeci cchoiceoftheorientationthusmadedoesnotimpacttheresultsofthispaper,infactthechoiceoftheinitialandterminalpointsoftheedgesarearbitrary.TheintervalsareonlyusedtoconstructtheLaplaceoperatorasadi erentialoperator.
Itisusefultoarrangethe2Eedgeendsinaparticularway:welisttheinitialpointsintheorderastheyoccurinthelistofedges,followedbytheterminalpointsinthesameorder.Forthetraceformulawe,moreover,requirethefollowingnotions.
De nition2.1.AclosedpathinΓisa nitesequenceofedges(ei)n
i=1,suchthattheedges
eiandei+1for∈{1,...,n 1}andtheedgesenande1areadjacent.Aperiodicorbitisanequivalenceclassofclosedpathsmodulocyclicpermutationsoftheedges.
Thenumbernofedgesinaperiodicorbitpisitstopologicallength,whereasthesumlp=le1+···+lenofthemetriclengthsofitsedgesisthemetriclength,orsimplylength,
ofp.Aperiodicorbitisprimitive,ifitisnotamultiplerepetitionofanotherperiodicorbit.Furthermore,thesetofperiodicorbitsofthegraphiscalledP,andPnisitssubsetoforbitswithtopologicallengthn.
2.2Quantumgraphs
i QuantummechanicsonametricgraphcanbestudiedintermsoftheSchr¨odingerequation
fj(xj)gj(xj)dxj.
4
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
Asadi erentialexpressionthe(negative)Laplacianissimplygivenby
′′′′ F:=( f1,..., fE),
wheredashesdenotederivatives.Thisexpressionmaynowserveasawaytointroduceaclosed,symmetricoperator( ,D0)withdomain
D0=E j=12H0(0,lj).
HereeachtermintheorthogonalsumconsistsofanL2-Sobolevspaceoffunctionswhich,togetherwiththeirderivatives,vanishattheedgeends.Thede ciencyindicesofthisoperatorare(2E,2E),andthusitpossessesselfadjointextensionsthatcanbeclassi- edaccordingtovonNeumann’stheory(see,e.g.,[RS75]).AnalternativeapproachhasbeendevelopedindetailbyKostrykinandSchrader[KS99a],whichprovidesaconvenientparametrisationthatisparticularlyusefulforlaterpurposes.Inthiscontextoneintroducestheboundaryvalues
TFbv=f1(0),...,fE(0),f1(l1),...,fE(lE),(2.2) ′ T′′′′Fbv=f1(0),...,fE(0), f1(l1),..., fE(lE),
offunctionsandtheirderivatives,wherebythesignsensurethatinwardderivativesareconsideredatalledgeends.Noticethattheorderofthetermsfollowstheconventionofarrangingedgeendsintroducedpreviously.Boundaryconditionsonthefunctionsinthedomainofagivenselfadjointoperatorarespeci edthroughalinearrelationbetweenboundaryvalues;theyareoftheform
′AFbv+BFbv=0,(2.3)
see[KS99a].HereA,B∈M(2E,C)aretwomatricessuchthat
thematrix(A,B),consistingofthecolumnsofAandB,hasmaximalrank2E, AB isselfadjoint.
Theseconditionsthenimplytheselfadjointnessoftheoperator,andeveryselfadjointextensioncanbeachievedinthismanner.Occasionally,weshalldenoteaparticularsuchselfadjointrealisationoftheLaplacianas (A,B,l).Thisparametrisation,however,isobviouslynotuniquebecauseamultiplicationof(2.3)withC∈GL(2E,C)fromtheleftdoesnotchangetheboundaryconditions.Ontheotherhand,if (A,B,l)= (A′,B′,l),thereexistsC∈GL(2E,C)withA′=CAandB′=CB,see[KS99a].Thus,foranyC∈GL(2E,C)bothA,BandA′=CA,B′=CBprovideanequivalentcharacterisationofthesameoperator.
Thelinearrelations(2.3)caninprinciplerelateboundaryvaluesatanysetofedgeends.Wewish,however,theoperatortorespecttheconnectednessofthegraphandthereforewe
5
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
restrictourselvestolocalboundaryconditions.Thesearecharacterisedbytheconditionthat(2.3)onlyrelatesedgeendsthatformasinglevertex.Tothisendwenowgrouptheedgeendsin(2.2)accordingtotheverticestheybelongto.LocalboundaryconditionsthenleadtoablockstructureofthematricesAandB,
A=AvandB=Bv,(2.4)
v∈Vv∈V
whereonlythenonvanishingmatrixentriesareindicated.Hereµvmustbereal;whenµv=0theusualKirchho conditionsarerealised.
Thenon-uniquenessinthechoiceofthematricesAandBcanbeovercomebyparametris-ingtheselfadjointrealisationsoftheLaplacianintermsofprojectorsontosubspacesofthe2E-dimensionalspacesofboundaryvalues.TothisendKuchment[Kuc04]introducedtheprojectorPontothekernelofBaswellastheprojectorQ=1 Pontotheorthogonalcomplement(kerB)⊥=ranB inC2E.Hethende nedthe(selfadjoint)endomorphism
1 AQ(2.6)L:=B|ranB
ofranB ,andshowedthattheboundaryconditions(2.3)areequivalentto
PFbv=0and′LQFbv+QFbv=0.suchthateachblock,representedbyAvandBv,exactlyrelatestheboundaryvaluesoffunctionsandtheirderivativesatthevertexv.Inthiscontextselfadjointnessofthe Laplacianisachieved,ifforallv∈Vtherankof(Av,Bv)isdvandAvBvisselfadjoint.Tomentionafewexamples,avertexwithDirichletboundaryconditionscanbechar-acterisedbyAv=1dv,Bv=0,whereasforNeumannboundaryconditionsonewouldchooseAv=0,Bv=1dv.Moreover,thegeneralisedKirchho boundaryconditionsusedbyKottosandSmilansky[KS99b]canbeachievedbychoosing 1 1 .... .. , (2.5)Av= andBv= 1 1 1······1µv(2.7)
Moreover,thereexistsaC∈GL(2E,C)suchthat
A′=CA=P+L
implyingthatandB′=CB=Q,(2.8)
L=A′B′ .
Are nementofthisconstructioncanbefoundin[FKW07].From(2.7)oneconcludesthatincaseswhereL=0,theboundaryconditionsdonotmixboundaryvaluesofthefunctionsthemselveswiththoseoftheirderivatives.Wecallthesenon-Robinboundaryconditions,andallothercasesRobinboundaryconditions.
6
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
3TheS-matrix
InquantumgraphmodelsLaplaceeigenvaluescanbeconvenientlycharacterisedintermsofzerosof nitedimensionaldeterminants,andthusthesemodelsareamenabletopow-erfulanalyticalaswellasnumericalmethods.InquantumbilliardsarelatedmethodwaspioneeredbyDoronandSmilansky[DS92]asthescatteringapproachtoquantisation.Ingeneral,thismethodreliesonsemiclassicalapproximations.As rstdemonstratedbyKot-tosandSmilansky[KS99b],however,inquantumgraphsthescatteringapproachallowstodetermineLaplaceeigenvaluesexactlyfroma nitedimensionalsecularequation.
Thescatteringapproachbearsitsnamefromthefactthatitisbasedonscatteringprocessesoccurringwhenoneopensupagivenclosedquantumsystemappropriately.Inquantumgraphstheprocedureofopeningupconsistsofreplacingeachvertexanditsattachededgesbyanin nitestargraph.Thisisthesingle,givenvertexvwithdvin nitehalflinesattachedthatreplacetheedgesof nitelengths.Carryingoverthelocalboundaryconditionsattheverticesfromtheoriginalclosedquantumgraph,onethusobtainsVopenquantumsystems,whicheachpossessanon-shellscatteringmatrixσv(k).IntermsoftheparameterisationoftheboundaryconditionsdescribedinSection2.2onethen ndsthat
σv(k)= (Av+ikBv) 1(Av ikBv),fork∈R\{0}.(3.1)TheconditionsimposedonAvandBvinordertoachieveselfadjointboundaryconditionsensurethatAv±ikBvareinvertibleandthatthevertexS-matrixσv(k)isunitaryforallk∈R\{0},see[KS99a].
ThelocalscatteringmatrixassociatedwithavertexwithDirichletorNeumannbound-aryconditionsisσv= 1dvorσv=1dv,respectively.Incontrast,accordingto(2.5)
generalisedKirchho boundaryconditionsleadtoavertexS-matrixwithelementsoftheform[KS99b]
σv1dvk iµ
v
ei,ej(k)= δij+dv
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
Furthermore,usingtheparametrisation(2.7)ofboundaryconditionsandutilising(2.8)aswellasthefactthat(L+ik) 1commuteswithL ik,oneobtainstherepresentation
S(A,B;k)= P Q(L+ik) 1(L ik)Q,k∈R\{0},(3.4)
fortheedgeS-matrix,see[KS06a].
Fromtheexpressions(3.3)and(3.4)itappearsthattheS-matrixgenerallydependsonthewavenumberkinanon-trivialway.However,certainboundaryconditionsleadtok-independentS-matrices.ObviousexamplesareDirichletandNeumannboundaryconditions,aswellastheusualKirchho boundaryconditions,i.e.,(3.2)withµv=0.Ageneralcharacterisationofsuchboundaryconditionswasprovidedin[KPS07]intermsofthefollowingequivalentconditions:
S(A,B;k)isk-independent.
S(A,B;k)isselfadjointforsome,andhenceforall,k>0.
S(A,B;k)=1 2Pforsome,andhenceforall,k>0.
AB =0,i.e.,L=0.
Thelastpointshowsthatk-independentS-matricesariseexactlyinthecaseofnon-Robinboundaryconditions.
BelowwearegoingtoprovesomepropertiesofS-matricesthatarerelevantforthetraceformula.Inthiscontext,forRobinboundaryconditionsanimportantrolewillbeplayedbythespectrumσ(L)oftheselfadjointmatrixL(2.6).
WeshallmakeextensiveuseoftheS-matrixextendedtocomplexwavenumberskandthereforeneedthefollowing.
Lemma3.1.LetAandBspecifyselfadjointboundaryconditionsfortheLaplacianonthegraph.ThentheS-matrix(3.3)hasthefollowingproperties:
1.S(A,B;k)canbecontinuedintothecomplexk-planeasameromorphicfunction,andhassimplepolesatthepointsofthesetiσ(L)\{0}.
2.S(A,B;k)isunitaryforallk∈R.
3.S(A,B;k)isinvertibleforallk∈C\[±iσ(L)\{0}],anditsinverseisS(A,B; k).Proof.WehenceforthextendtheselfadjointendomorphismLofranB ,see(2.6),toanendomorphismofC2Ebysettingittozeroon(ranB )⊥=kerB.WethendiagonaliseLutilisinganappropriateunitaryW,anddenotethenon-zeroeigenvalues(countedwiththeirmultiplicities)by{λ1,...,λd}.Thisleavestheeigenvaluezerowithamultiplicityof2E d.Therearer:=dimranB d(orthonormal)eigenvectorsofLinranB ands:=dimkerB=2E dimranB eigenvectorsinkerB,respectively,correspondingtotheeigenvaluezero.
8
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
Employingthisdiagonalisationintherepresentation(3.4)oftheS-matrixthenleadstotheexpression
S(A,B;k)=W λ1 ik
SincetheunitaryWisindependent λkr 1s W.(3.5)d+i1 ofkthe rststatementofthe lemmaisobvious.
TheunitarityofS(A,B;k)forrealkalsofollowsimmediatelybyobservingthatthediagonalentriesin(3.5)areallofunitabsolutevalue.
Thethirdstatementfollowsinacompletelyanalogousfashionfromtherepresentation(3.5).
KnowingthattheS-matrixisanalyticink,onewouldliketocalculateitsderivative.Thisinfactisrequiredintheproofofthetraceformulabelow.ItisevenpossibletorelatethederivativeofS(k)totheS-matrixitself.
Lemma3.2.UnderthesameassumptionsasinLemma3.1oneobtainsfork∈C\
[±iσ(L)\{0}],
d
2k S(A,B;k) S(A,B,k) 1 S(A,B;k).(3.6)
WeremarkthatforrealktheunitarityoftheS-matrixcanbeinvokedtoobtainfrom(3.6)thatitisindependentofk,i itisselfadjoint.
Proof.Letus rstassumethatk∈R\{0}andabbreviateS(A,B;k)asS(k).Wealsode-notederivativesw.r.t.kbyadashandusetherelationd
2 S(k) 1 andC(k)B= 1
2k S(k)+1S(k) 1
= 1
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
whichprovesthestatementfork∈R\{0}.
FromLemma3.1weinferthatS(k)isanalyticinaneighbourhoodofk=0andthattheright-handsideof(3.9)hasaremovablesingularityatk=0;hence(3.9)extendstoallrealk.Since,moreover,S(k)isunitaryonRandanalyticonC\[±iσ(L)],and1
kn(iL)n.(3.10)
2.For|k|<λmin,
S(A,B;k)= 1+2P 2 ∞kn
n=1
iL n,(3.11)wherePandL emergefromPandL,respectively,byreplacingA,Bwith B,A.
Proof.Forthe rstexpansionwerefertotherepresentation(3.5)oftheS-matrixandemploytheexpansion
λα ikkλα=1+2
kλα ∞n=1 iλα
kn(iL)nonranB and
1ForonkertheBsecond.SinceexpansionL=0onwekerremarkB,thethatrelationfrom(3.10)(3.3)onefollows.canreadilydeducetherelationS(A,B;k)= S( B,A;1
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
Lemma3.3alsoprovideslimitingexpressionsfortheedgeS-matrixas|k|→∞and|k|→0,respectively, ,S∞=1 2PandS0= 1+2P
whichweshallusesubsequently.
LaterweshallintegrateexpressionscontainingtheS-matrixalongcontoursintheuppercomplexhalfplaneand,therefore,weneedtoestimatethenormoftheS-matrixalongthecontours.Tothisendweintroduce
λ+ min{λ∈σ(L);λ>0},if λα>0
min:=∞,else,
andobtainthefollowing.
Lemma3.4.Letk∈Rand0<κ<λ+min,then
S(k+iκ) , S(k iκ) 1 ≤max 1,λ+min+κ(3.12)
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
wherek∈C.InadditiontotheedgeS-matrixthisquantityisthenusedtointroduce
U(k):=S(A,B;k)T(l;k).(4.2)
ThetopologicalandthemetricdataenteringU(k)arehenceclearlyseparated.
ForrealktheendomorphismsS(k),T(k)andU(k)ofC2Eareobviouslyunitary.WethereforedenotetheeigenvaluesofU(k)byeiθ1(k),...,eiθ2E(k).FollowingLemma3.1weconcludethatU(k)canbeextendedintothecomplexk-planeasameromorphicfunctionwithpolesatiσ(L)\{0}.Thedeterminantfunction
F(k):=det 1 U(k) ,(4.3)
onwhichthescatteringapproachisbased(see[KS99b]),henceisalsomeromorphic.Itspolesareiniσ(L)\{0},butdonotnecessarilyexhausttheentireset.
Proposition4.1(Kostrykin,Schrader[KS06b]).Thedeterminantfunction(4.3)ismero-morphiconthecomplexplanewithpolesinthesetiσ(L)\{0}.Furthermore,letknC\[iσ(L)∪{0}]withImkn≥0,thenk2nisaneigenvalueof ,i knisazeroof∈thefunction(4.3),i.e.,F(kn)=0.Moreover,thespectralmultiplicitygnoftheLaplaceeigenvaluek2n>0coincideswiththemultiplicityoftheeigenvalueoneofU(kn).
Proposition4.1establishesacloseconnectionbetweenzerosofthedeterminantfunction(4.3)andLaplaceeigenvalues.NoticethatalthoughLaplaceeigenvaluesoccurassquares,k2,thefunction(4.3)isnotinvariantunderachangeofsigninitsargument.Thereexists,however,afunctionalequationunderthesubstitutionk→ k.
Lemma4.2.ForallC\[±iσ(L)\{0}]thefollowingidentityholds:
F(k)=( 1)Me2ikL dλα ik
α=1
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
rstcaseKostrykinandSchradershowedthatthenumberofnegativeLaplaceeigenvaluesis nite;infact,itsnumberisboundedbythenumberofpositiveeigenvaluesofL[KS06b].
Thatimplies,inparticular,thatLaplacianswithnon-Robinboundaryconditions,whereL=0,possessanon-negativespectrum.Moreover,theyfoundthefollowinglowerbound,
≥ s2.
Heres≥0istheuniquesolutionof
stanh slmin(4.5)
λ+ik
Eventually,oneobtains
F(k)=1 e2ikl
0eeikl0λ ikikl .2=λ.
Thisconditionisequivalentto(4.6),demonstratingthatthebound(4.5)issharpforthis‘quantumgraph’.
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We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
4.2Theeigenvaluezero
Ingeneral,zeroisaLaplaceeigenvalueaswellasazeroofthedeterminant(4.3),andinsofarProposition4.1alsoappliestok0=0.Thespectralmultiplicityg0,however,typicallyisdi erentfromthedegreeofk0=0asazeroofF(k).ForKirchho boundaryconditionsithasbeenshownin[KN05]thatthedegreeofthezeroisE V+2,whereasthezeroLaplaceeigenvalueisnon-degenerate,i.e.,g0=1.Kurasov[Kur08]subsequentlylinkedthisdi erenceinthemultiplicitiestothetopologyofthegraphbynoticingthatasuitabletraceformulacontainsthequantity
1 1
2(V E),
andhencetheEulercharacteristicofthegraph.Thisobservationwasgeneralisedtoyieldanindextheoremforanyquantumgraphwithnon-RobinboundaryconditionsbyFulling,
(V E)KuchmentandWilson[FKW07].Onecanviewthet-independentterm12inthetraceformulafortheheatkernel(dueto[Rot83]forKirchho boundaryconditions
and[KPS07]forgeneralnon-Robinconditions)asapredecessorofthisresult.Seealso
[BEar]foramoredetaileddiscussion.
WeherewishtogiveafurthercharacterisationofthespectralmultiplicityofthezeroeigenvalueinthecaseofageneralselfadjointrealisationoftheLaplacian.Tothisendwe rstintroduce,fork∈R\{0},thematrix
i
C(l;k):= 2i 2i l1+l1k2..2i.lE+lEk
l1
+l1k...
k
+lEk
inwhichallmatrixentriesnotindicatedarezero.Thisnowenablesustoformulatethefollowing.
Proposition4.4.ForanygivenselfadjointrealisationoftheLaplacianspeci edthroughA,B,zeroisaLaplaceeigenvalue,i oneisaneigenvalueofS(A,B;k)C(l;k)forone,andhenceany,k∈R\{0}.Moreover,themultiplicityofthiseigenvalueonecoincideswiththespectralmultiplicityg0ofthezeroLaplaceeigenvalue.
Proof.EigenfunctionsoftheLaplaciancorrespondingtotheeigenvaluezeromustbeoftheformF=(f1,...,fE)Twith
fj(x)=αj+βjx,
14x∈[0,lj].(4.8) k +lE k , lE k(4.7)+lE
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
Hence,theboundaryvalues(2.2)taketheform TFbv=α1,...,αE,α1+β1l1,...,αE+β1lE, T′Fbv=β1,...,βE, β1,..., βE.
Wenowemploytheboundaryconditions(2.3)byusingtheexpressions(3.8)forapossiblechoiceofAandB.Theresultcanberearrangedtoyield
αα=C (l;k),(4.9)S(k)C+(l;k)ββ
where,foranyk∈R\{0},wehaveintroduced
1E±i
1+D1(l)kE l1 ..withD1(l)= .
.C±(l;k):=lE
Wealsousetheabbreviations
ThematricesC±(l,k)areinvertibleforallk∈R\{0},with
ii
C±(l;k) 1 = ± ±2i±±2ikα1 . α:= .. αE β1 . andβ:= .. .βE
1
l1k lE lEk...
1
lEk 1 l1k... 1 l1k
Wenowsubstitute
in(4.9)andobtain αv(k):=C (l;k)β k l1 .k (4.10)
S(k)C+(l;k)C (l;k) 1v(k)=v(k).
ItisstraightforwardtocheckthatC+(l;k)C (l;k) 1=C(l;k),compare(4.7).Thelin-earlyindependenteigenvectorsofSCcorrespondingtotheeigenvalueonethenyield,via(4.10)and(4.8),coe cientsαjandβjforasmanylinearlyindependentLaplaceeigen-functionsinL2(Γ).
15
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
whichisinnoobviouswayrelatedtothemultiplicityoftheeigenvalueoneofS(k)C(l;k)thatappearsinProposition4.4.Inthecaseofnon-Robinboundaryconditions,wheretheedgeS-matrixisindependentofk,however,Fulling,KuchmentandWilson[FKW07]wereabletorelatethedi erentmultiplicitiesintheformofanindextheorem.Theyshowed,inparticular,thatthen1trS.g0 4
Asmentionedabove,thistermwillreappearinthetraceformula.WeremarkthattheorderNofk0=0asazeroofthefunction(4.3)isthemultiplicityoftheeigenvalueoneof 01E,U(0)=S01E0
5Thetraceformula
AtraceformulaexpressescountingfunctionsofLaplaceeigenvaluesintermsofsums
2overperiodicorbits.Ideally,onewouldliketocountLaplaceeigenvalueskn,withtheir
multiplicitiesgn,inintervalsIintheform
TrχI( )=gn.(5.1)
2∈Ikn
Thesharpcut-o providedbythecharacteristicfunctionχIoftheintervalI,however,cannotbedealtwith.Onethereforereplaces(5.1)withasmoothcut-o and,moreover,performsthiscountintermsoftheassociatedwavenumberskn,i.e.,oneseeksto ndarepresentationfor gnh(kn)(5.2)
n
intermsofsumsoverperiodicorbits.Oneambitionthenisto ndasu cientlylargeclassoftestfunctionsh.
5.1Arougheigenvaluecount
AnestimateforthenumberofLaplaceeigenvaluesinintervalscanbeobtainedbyconsid-eringtheeigenphasesθ(k)ofU(k),see(4.2),de nedthrough
U(k)v(k)=eiθ(k)v(k)with v(k) =1.(5.3)
WerecallthatU(k)=S(A,B;k)T(l;k)isanalyticinC\[iσ(L)\{0}].Accordingtoanalyticperturbationtheory(see,e.g.,[Kat95])itseigenvalues,forwhichwekeepthenotationeiθ(k),arecontinuousonthissetanddi erentiableapartfrompossiblyisolatedpoints.Since,however,U(k)isreal-analyticandnormalforallk∈R,wecanapplyasharpenedversionofanalyticperturbationtheory(see[LT85])toconcludethatthe
16
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
eigenvaluesarerealanalyticforallk∈Randthatthereexistsachoiceofeigenvectorswiththesameproperty.
Thefollowingstatementisageneralisationofaresultfoundin[KS99b,BW08]thatis
validfork-independentS-matrices.
Lemma5.1.Letθ(k)beaneigenphaseofU(k),k∈R,withassociatednormalisedeigen-vectorv(k)=(v1(k),...,v2E(k))T.Then
d
L2+k2v(k) .
C2E
Proof.TakingLemma3.2intoaccountwe rstobservethat
d
2k
+iS(k) S(k) S (k)T(k)D(l)v(k) v(k)+S(k)T(k)v′(k),
where
D(l):= D1(l)0
0D1(l)
Wethenformascalarproductwithv(k),employingthe .relation
1
L2+k2
thatfollowsfrom(3.4).Thisyields
v(k),d
L2+k2v(k)+i U (k)v(k),D(l)v(k)
+ U (k)v(k),v′(k)
= 2ieiθ(k) v(k),L
dk[eiθ(k)v(k)] =iθ′(k)eiθ(k)+eiθ(k) v(k),v′(k) .
Comparing(5.6)and(5.7)provesthestatement(5.4).
17(5.4)(5.5)(5.7)
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
inglmax/mintodenotethelargestandthesmallestedgelength,respectively,andintroducing min{|λ|;λ∈σ(L)∩R },if λα<0 ,λmin:=∞,else
inanalogyto(3.12),
we
immediately
getthefollowing.
Corollary5.2.Thederivativeθ′(k)ofaneigenphaseθ(k)isboundedfromaboveandbelowaccordingto2.(5.8)lmin λ min
Inparticular,iflmin>2/λ+minthederivativesofalleigenphasesarealwayspositive.Proof.Obviously,
lmin≤2E i=1li|vi(k)|2≤lmax,
sincetheeigenvectorissupposedtobenormalised.Moreover,afteradiagonalisationofL,whenWLW isdiagonalwiththeeigenvaluesλαonthediagonal,oneobtains
Lv(k),,2λ2+kα
wherew(k)=Wv(k).This rstyields
1
L2+kv(k)2 ≤1
iθi(k)1 e= iθ′(k)eiθi(k)=0,theclaimfollowsfromCorollary5.2andPropo-dksition4.1.
18
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
WeremarkthatsincethescatteringapproachtotheproofofthetraceformulaisbasedoncountingzerosofthefunctionF(k)onthereallinewiththeirmultiplicities,therequirementlmin>2/λ+minisessential.OtherwiseonemightcountLaplaceeigenvalueswithincorrectmultiplicities.WheneverLhasnopositivepart,however,theconditionisempty.Thisis,e.g.,thecasefornon-Robinboundaryconditions.
AsafurtherapplicationofLemma5.1wenowperformaroughcountofLaplaceeigenvalues,whichisreminiscentofWeyl’slawforthecountofLaplaceeigenvaluesonmanifolds.
Lemma5.4.Assumethatlmin>2/λ+min,thenthereexistconstantsC+≥C >0suchthat
2thenumberN(K)ofLaplaceeigenvalueskn>0(includingmultiplicities)with0<kn≤Kful ls
C K≤N(K)≤C+K.(5.9)
Proof.AccordingtoProposition4.1thepositiveLaplaceeigenvaluesderivefromzerosofthedeterminantfunctionF(k).These,inturn,correspondtotheexistenceofα∈{1,...,2E},j∈Zandkα,j>0,suchthat
θα(kα,j)=2πj.(5.10)
Sincetheeigenphasesaredi erentiable,anapplicationofthemeanvaluetheoremallowstoconcludethat′θα(kα,j+1)=θα(kα,j)+(kα,j+1 kα,j)θα(k0),
wherekα,j≤k0≤kα,j+1.WithCorollary5.2andeq.(5.10)thisyields
2π
λ min≤kα,j+1 kα,j≤2πλ+min.
Hence,thenumberNα(K)ofsolutionsto(5.10)with0≤kα,j≤Kcanbeestimatedas KK≤Nα(K)≤+1.(5.11) λ+λminmin
AccordingtoLemma5.3,thebounds(5.9)followfromsumming(5.11)overallα.
Oncethetraceformulaisavailable,theresult(5.9)canbere ned,seeCorollary5.11below.Thislemma,however,providesuswithasu cientaprioriestimaterequiredintheproofofthetraceformula.It,moreover,immediatelyimpliesthefollowingwellknownresult.
Corollary5.5.ThediscreteLaplacespectrumconsistsofin nitelymanyeigenvaluesthatonlyaccumulateatin nity.
Asexplainedabove,thetraceformulacountseigenvalueswithasmoothcut-o .Wehenceconsideraparticulartypeofcut-o functions.
19
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
De nition5.6.Foreachr≥0thespaceHrconsistsofallfunctionsh:C→Csatisfyingthefollowingconditions:
hiseven,i.e.,h(k)=h( k).
Foreachh∈Hrthereexistsδ>0suchthathisanalyticinthestripMr+δ:={k∈C;|Imk|<r+δ}.
1 Foreachh∈Hrthereexistsε>0suchthath(k)=O
F′
2πi Cε,K
∞l
2πi +trΛ(k)U(k)h(k)dk,
∞
20(5.14)
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi
where2LΛ(k)= i
+∞ F′
2πi(k+iε)h(k+iε)
∞F dk.(5.16)
Beginningwiththeleft-handside,wehavetoshowthatin(5.13)thelimitK→∞canbetaken,followedbyε→0.Tothisendwenoticethattheright-handsideof(5.13)isexplicitlyindependentofε,intherangedescribedabovethatequation.Furthermore,from(5.12)wealreadyknowthatthesumoverknconvergesabsolutelyinthelimitK→∞.Inordertoperformtheselimitsontheleft-handsideof(5.13),andthusproducing(5.16),wehavetoestimatethecontributiontotheintegralcomingfromtheverticalpartsofthecontour.
Lemma3.3impliesthatfork∈Cwith|k|>λmaxtheapproximationF(k)=FO(|k| 1)holds,where∞(k)+
F∞(k):=det
d 1 S∞T(k) =1+ bdneiβnk,(5.17)n=1withn∈C,βn>0andb∈N.Thelastexpressionis rstde nedfork∈R,butcanbereadilyextendedtocomplexk.SinceS′(k)=O(|k| 2),seeLemma3.2andeq.(5.5),wealso ndF′(k)=F∞′(k)+O(|k| 2).Forsu cientlylarge|k|onecanthereforeapproximateF′(k)/F(k)byF∞′(k)/F∞(k).Inordertoestimatethelatterweemploy(5.17)toobtain F∞′(k) ≤bdmaxβmaxe βminImk,(5.18)withdmax:=max{|dn
k.Furthermore,|}andweβmax/min:=max/min{βn
dentofRepickk(0)
thefactthatF∞∈(kR),suchk∈Rthat},isF.a(Notequasi-periodick(0))=that0asthiswellboundasF
function.∞is(kindepen-(0))=0,
andtakeadvantageofOnecanhenceconstructa(strictlyincreasing)series{k(j);j∈N0}with
|F∞(k(j)) F∞(k(0))|<|F∞(k(0))|
4(5.20)
when|κ|issu cientlysmall.Therefore,whenεissmallenough,thefunction|F∞(k(j)+iκ)|, suchε≤thatκ≤ε,isuniformlyboundedfrombelowawayfromzero.ThusthereexistsCε>0
k(j)+iε
F
∞′(k)
k(j) iε
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