The trace formula for quantum graphs with general self adjoint boundary conditions

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

8

2

y

a

M

2

]

h

p

-h

ta

m

[

1

v

1

1

1

3

.

5

8

:0v

i

X

r

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’.

13

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

本文来源：https://www.bwwdw.com/article/wj31.html