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en collaboration avec Yves Colin de Verdière et Nabila Torki Mathematical Physics Analysis and Geometry pp

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Niveau: Supérieur, Doctorat, Bac+8
ar X iv :1 00 6. 57 78 v2 [ ma th. SP ] 11 O ct 20 10 Essential self-adjointness for combinatorial Schrodinger operators II- Metrically non complete graphs Yves Colin de Verdiere? Nabila Torki-Hamza † Franc¸oise Truc‡ October 12, 2010 Abstract We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrodinger operators in the metrically non complete case. 1 Introduction This paper is a continuation of [To] which contains some statements about es- sential self-adjointness of Schrodinger operators on graphs. In [To], it was proved that on any metrically complete weighted graph with bounded degree, the Lapla- cian is essentially self-adjoint and the same holds for the Schrodinger operator provided the associated quadratic form is bounded from below. These results remind those in the context of Riemannian manifold in [Ol] and also in [B-M-S], 0 Keywords: metrically non complete graph, weighted graph Laplacian, Schrodinger oper- ator, essential selfadjointness. 0 Math Subject Classification (2000): 05C63, 05C50, 05C12, 35J10, 47B25. ?Grenoble University, Institut Fourier, Unite mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d'Heres Cedex (France); yves.

  • essential self

  • self-adjointness mainly

  • adjoint laplacians

  • self-adjointness

  • finite graphs

  • lower semi-continuous

  • graphs investigating essential

  • laplacian ∆?

  • self-adjoint

  • self


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Essentialself-adjointnessforcombinatorialSchro¨dingeroperatorsII-MetricallynoncompletegraphsYvesColindeVerdie`reNabilaTorki-HamzaFranc¸oiseTrucOctober12,2010AbstractWeconsiderweightedgraphs,weequipthemwithametricstructuregivenbyaweighteddistance,andwediscussessentialself-adjointnessforweightedgraphLaplaciansandSchro¨dingeroperatorsinthemetricallynoncompletecase.1IntroductionThispaperisacontinuationof[To]whichcontainssomestatementsaboutes-sentialself-adjointnessofSchro¨dingeroperatorsongraphs.In[To],itwasprovedthatonanymetricallycompleteweightedgraphwithboundeddegree,theLapla-cianisessentiallyself-adjointandthesameholdsfortheSchro¨dingeroperatorprovidedtheassociatedquadraticformisboundedfrombelow.TheseresultsremindthoseinthecontextofRiemannianmanifoldin[Ol]andalsoin[B-M-S],0Keywords:metricallynoncompletegraph,weightedgraphLaplacian,Schro¨dingeroper-ator,essentialselfadjointness.0MathSubjectClassification(2000):05C63,05C50,05C12,35J10,47B25.GrenobleUniversity,InstitutFourier,Unite´mixtederechercheCNRS-UJF5582,BP74,38402-SaintMartind’He`resCedex(France);yves.colin-de-verdiere@ujf-grenoble.fr;http://www-fourier.ujf-grenoble.fr/ycolver/Universite´du7Novembrea`Carthage,Faculte´desSciencesdeBizerte,Mathe´matiquesetApplications(05/UR/15-02),7021-Bizerte(Tunisie);nabila.torki-hamza@fsb.rnu.tn;torki@fourier.ujf-grenoble.frGrenobleUniversity,InstitutFourier,Unite´mixtederechercheCNRS-UJF5582,BP74,38402-SaintMartind’He`resCedex(France);francoise.truc@ujf-grenoble.fr;http://www-fourier.ujf-grenoble.fr/trucfr/1
[Shu1],[Shu2].Therearemanyrecentindependentresearchesinlocallyfinitegraphsinvestigatingessentialself-adjointness(see[Jor],[Go-Sch],[Ma]),andre-lationsbetweenstochasticcompletenessandessentialself-adjointness(see[We],[Woj2]aswellasthethesis[Woj1]).Similarresultshavebeenextendedforarbi-traryregularDirichletformsondiscretesetsin[Ke-Le-2]whichismostlyasurveyoftheoriginalarticle[Ke-Le-1].Morerecentlythepaper[Hu]isdevotedtothestabilityofstochasticincompleteness,inalmostthesamesetupasin[Ke-Le-1].Here,wewillinvestigateessentialself-adjointnessmainlyonmetricallynoncom-pletelocallyfinitegraphs.LetusrecallthataweightedgraphGisageneralizationofanelectricalnetworkwherethesetofverticesandthesetofedgesarerespectivelyweightedwithpos-itivefunctionsωandc.Foranygivenpositivefunctionp,aweighteddistancedpcanbedefinedonG.ThuswehavetheusualnotionofcompletenessforGasametricspace.ThemainresultofSection3statesthattheweightedgraphLaplacianΔω,c(seethedefinition(1)below)isnotessentiallyself-adjointifthegraphisoffinitevolumeandmetricallynoncomplete(herethemetricdpisdefinedusingthe1weightspx,y=cx,y2).TheproofisderivedfromtheexistenceofthesolutionforaDirichletproblematinfinity,establishedinSection2.InSection4,weestablishsomeconditionsimplyingessentialself-adjointness.Moreprecisely,defining1themetricdpwithrespecttotheweightspx,ygivenbypx,y=(min{ωxy})cx,y2,andaddressingthecaseofmetricallynoncompletegraphs,wegettheessentialself-adjointnessofΔω,c+WprovidedthepotentialWisboundedfrombelowbyN/2D2,whereNisthemaximaldegreeandDthedistancetotheboundary.WeuseforthisresultatechnicaltooldeducedfromAgmon-typeestimatesandinspiredbythenicepaper[Nen],seealso[Col-Tr].WediscussinSection5thecaseofstar-likegraphs.Undersomeassumptionsona,weprovethatforanypotentialW1,a+Wisessentiallyself-adjointus-inganextensionofWeyl’stheorytothediscretecase.IntheparticularcaseofthegraphN,thesameresulthadbeenprovedin[Ber](p.504)inthecontextofJacobimatrices.WegivesomeexamplesinSubsection5.3toillustratethelinksbetweenthepreviousresults.MoreoverweestablishthesharpnessoftheconditionsofTheorem4.2.ThelastSectionisdevotedtoAppendixAdealingwithWeyl’slimitpoint-limitcirclecriteria(see[RS])inthediscretecaseaswellasinthecontinuouscase,andtoAppendixBincludingtheunitaryequivalencebetweenLaplaciansandSchro¨dingeroperators[To]usedrepeatedlyinSubsection5.3.Letusstartwithsomedefinitions.G=(V,E)willdenoteaninfinitegraph,withV=V(G)thesetofverticesandE=E(G)thesetofedges.Wewritexyfor{x,y}∈E.ThegraphGisalwaysassumedtobelocallyfinite,thatisanyxVhasafinitenumberofneighbors,whichwecallthedegree(orvalency)ofx.Ifthedegreeis2