Smilanskys-model-of-irreversible-quantum-graphs--I

arXiv:math/0503497v1[math.SP]23Mar2005SMILANSKY’SMODELOFIRREVERSIBLEQUANTUMGRAPHS,I:THEABSOLUTELYCONTINUOUSSPECTRUMW.D.EVANSANDM.SOLOMYAKAbstract.InthemodelsuggestedbySmilansky[7]onestudiesanopera-tordescribingtheinteractionbetweenaquantumgraphandasystemofKone-dimensionaloscillatorsattachedatseveraldifferentpointsinthegraph.ThepresentpaperisthefirstoneinwhichthecaseK1isinvestigated.ForthesakeofsimplicityweconsiderK=2,butourargumentisofageneralcharacter.Inthisfirstoftwopapersontheproblem,wedescribetheabsolutelycontinuousspectrum.Ourapproachisbaseduponscatteringtheory.1.IntroductionintInthepaper[7]U.Smilanskysuggestedamathematicalmodeltowhichhegavethename“Irreversiblequantumgraph”.Inthismodelonestudiestheinteractionbetweenaquantumgraphandafinitesystemofone-dimensionaloscillatorsattachedatseveraldifferentpointsinthegraph.Recallthattheterm“quantumgraph”usuallystandsforametricgraphΓequippedwithaself-adjointdifferentialoperatoractingonL2(Γ);seethesurveypaper[4]andreferencestherein.InourcasethisoperatorwillbetheLaplacian−Δ.InSmilansky’smodeloneinitiallydealswithtwoindependentdynamicalsystems.OneofthesystemsactsinL2(Γ)anditsHamiltonianistheLaplacian.AnothersystemactsinthespaceL2(RK),K≥1andisgeneratedbytheHamiltonianHosc=PKk=1hkwherehk=ν2k2−∂2∂qk2+q2k,k=1,...,K;in[7]theoscillatorsarewritteninaslightlydifferentform;oneformreducestoanotherbyscaling.InwhatfollowsthepointsinΓaredenotedbyxandthepointsinRKbyq=(q1,...,qK).Considernowtheoperator1.1(1.1)A0=−Δ⊗I+I⊗Hosc1991MathematicsSubjectClassification.Primary:81Q10,81Q15.Secondary:35P25.Keywordsandphrases.Quantumgraphs,Absolutelycontinuousspectrum,Waveoperators.12W.D.EVANSANDM.SOLOMYAKinthespaceL2(Γ×RK).Itisdefinedbythedifferentialexpression1.2(1.2)AU=−ΔxU+12KXk=1ν2k−∂2U∂qk2+q2kU
andisself-adjointonthenaturaldomain.Thetermsin(1.1)donotinteractwitheachother.Interactionisintroducedwiththehelpofasystemof“matchingconditions”onthederivativeU′xatsomepointso1,...,oK∈Γ.Onesaysthatthek-thoscillatorisattachedtothegraphatthepointok.Theconditionatthepointokis1.4(1.3)[U′x](ok,q)=αkqkU(ok,q),k=1,...,K,where[f′x](.)standsfortheexpressionappearingintheKirchhoffcondition,wellknowninthetheoryofelectricnetworks.WhenΓ=R(whichistheonlycasewedealwithinthemainbodyofthepaper),[f′x](.)isthejumpofthederivative,1.5(1.4)[f′x](o)=f′x(o+)−f′x(o−).Therealparameterαkin(1.3)expressesthestrengthofinteractionbetweenthequantumgraphandtheoscillatorhk.Thecaseα1=...=αK=0correspondstotheoperatorA0asin(1.1).Sometimesweshalldenotebyα,νthemulti-dimensionalparametersα={α1,...,αK},ν={ν1,...,νK}.LetAα;ν=Aα1,...,αK;ν1,...,νKstandfortheoperatordefinedbythedifferentialexpression(1.2)andtheconditions(1.3).Usually,thevaluesofνkarefixedandweexcludethemfromthenotation.Ontheotherhand,weusethenotationAΓ;α;νforthisoperatorwhenitisnecessarytoreflectitsdependenceonthegraph.TheproblemtobeconsideredisthedescriptionofthespectrumofthedynamicalsystemgeneratedbytheHamiltonianAα;ν.Morespecifically,itistoconstructtheself-adjointrealizationofAα;νasanoperatorintheHilbertspaceL2(Γ×RK)andtodescribeitsspectrum.Upuntilnow,theproblemhasonlybeeninvestigatedforthesimplestcaseK=1.Thefirstresultswereobtainedinthepaper[7]bySmilansky.Thenadetailedstudyoftheproblemwascarriedoutinthepapers[8],[9]and[5].In[10],alongwithsomenewresults,adetailedsurveyofthecurrentstateoftheproblemisgiven.Onfirstsight,theproblemmightseemamenabletotheperturbationthe-oryofquadraticforms.Indeed,thespectrumσ(A0)canbeeasilydescribedbyseparationofvariablesandtheperturbationinthequadraticform,whichappearswhenpassingfromA0toAαwithα6=0,seemsnottobetoostrong.However,thisisnotso:thisperturbationturnsouttobeonlyform-boundedSPECTRUMOFIRREVERSIBLEQUANTUMGRAPHS3butnotform-compact,whichmakesitimpossibletoapplythestandardtech-niques.So,theproblemrequirescertainspecifictoolswhichweredevelopedin[8]–[10]and[5].ThemostimportantofthesetoolsisthesystematicuseofJacobimatrices.Itwasfoundintheabovementionedpapersontheone-oscillatorproblemthatthecharacterofthespectrumstronglydependsonthesizeofα:thereexistssomeα∗0suchthattheabsolutelycontinuousspectrumσa.c.(Aα)coincideswithσa.c.(A0)if|α|α∗(inparticular,itisabsentifthegraphiscompact)andfillsthewholeofRif|α|α∗.Thedependenceofthestructureofthepointspectrumσp(Aα)onαisalsowellunderstood.ThisisthefirstoftwopapersontheproblemforK1andinitwestudytheabsolutelycontinuousspectrum;inourotherpaper[1]thepointspectrumisinvestigated.Thisdivisionisnatural,sincethetechnicaltoolsusedineachpartaredifferent.Weaddressthesimplestsituation,whenΓ=RandK=2,butourargumentisofarathergeneralcharacterandwefirmlybelievethatitappliestoawideclassofgraphsandtoanyK.However,inthegeneralcase,thecalculationsbecomemorecomplicatedandthisobscuresthemainfeaturesoftheargument.Wefirstdescribeinformallythemainideaslyingbehindourapproach.TheeffectofaddingonemoreoscillatortoasystemwithKoscillatorsistwofold.Firstly,thetotaldimensionofthesetΓ×RKincreasesbyonewhichcertainlyaffectsthespectrum.Secondly,thereissomeeffectcomingfromtheadditionalmatchingcondition(1.3)atthepointoK+1.Thissecondeffectdisap-pearsifwetakeαK+1=0.Inde