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

Yves Colin De Verdière

Description

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

Sujets

• University
• Semi-continuity
• Self-adjoint

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 42
• Langue: English

Extrait

Essentialself-adjointnessforcombinatorial Schro¨dingeroperatorsII-Metricallynon completegraphs YvesColindeVerdie`re ∗ NabilaTorki-Hamza † Franc¸oiseTruc ‡ October12,2010 Abstract Weconsiderweightedgraphs,weequipthemwithametricstructure givenbyaweighteddistance,andwediscussessentialself-adjointnessfor weightedgraphLaplaciansandSchro¨dingeroperatorsinthemetricallynon completecase. 1Introduction Thispaperisacontinuationof[To]whichcontainssomestatementsaboutes- sentialself-adjointnessofSchro¨dingeroperatorsongraphs.In[To],itwasproved thatonanymetricallycompleteweightedgraphwithboundeddegree,theLapla- cianisessentiallyself-adjointandthesameholdsfortheSchro¨dingeroperator providedtheassociatedquadraticformisboundedfrombelow.Theseresults remindthoseinthecontextofRiemannianmanifoldin[Ol]andalsoin[B-M-S], 0 Keywords:metricallynoncompletegraph,weightedgraphLaplacian,Schro¨dingeroper- ator,essentialselfadjointness. 0 MathSubjectClassification(2000):05C63,05C50,05C12,35J10,47B25. ∗ GrenobleUniversity,InstitutFourier,Unite´mixtederechercheCNRS-UJF5582,BP 74,38402-SaintMartind’He`resCedex(France); [e-mail] ; http://www-fourier.ujf-grenoble.fr/ ∼ ycolver/ † Universite´du7Novembrea`Carthage,Faculte´desSciencesdeBizerte,Mathe´matiques etApplications(05/UR/15-02),7021-Bizerte(Tunisie); [e-mail] ; [e-mail] ‡ GrenobleUniversity,InstitutFourier,Unite´mixtederechercheCNRS-UJF5582, BP74,38402-SaintMartind’He`resCedex(France); [e-mail] ; http://www-fourier.ujf-grenoble.fr/ ∼ trucfr/ 1

[Shu1],[Shu2].Therearemanyrecentindependentresearchesinlocallyfinite graphsinvestigatingessentialself-adjointness(see[Jor],[Go-Sch],[Ma]),andre- lationsbetweenstochasticcompletenessandessentialself-adjointness(see[We], [Woj2]aswellasthethesis[Woj1]).Similarresultshavebeenextendedforarbi- traryregularDirichletformsondiscretesetsin[Ke-Le-2]whichismostlyasurvey oftheoriginalarticle[Ke-Le-1].Morerecentlythepaper[Hu]isdevotedtothe stabilityofstochasticincompleteness,inalmostthesamesetupasin[Ke-Le-1]. Here,wewillinvestigateessentialself-adjointnessmainlyonmetricallynoncom- pletelocallyfinitegraphs. LetusrecallthataweightedgraphGisageneralizationofanelectricalnetwork wherethesetofverticesandthesetofedgesarerespectivelyweightedwithpos- itivefunctionsωandc.Foranygivenpositivefunctionp,aweighteddistanced p canbedefinedonG.ThuswehavetheusualnotionofcompletenessforGasa metricspace. ThemainresultofSection3statesthattheweightedgraphLaplacianΔ ω,c (see thedefinition(1)below)isnotessentiallyself-adjointifthegraphisoffinite volumeandmetricallynoncomplete(herethemetricd p isdefinedusingthe 1 − weightsp x,y =c x,y 2 ).Theproofisderivedfromtheexistenceofthesolutionfor aDirichletproblematinfinity,establishedinSection2. InSection4,weestablishsomeconditionsimplyingessentialself-adjointness. Moreprecisely,defining 1 themetricd p withrespecttotheweightsp x,y givenby p x,y =(min{ω x ,ω y })c x − ,y 2 ,andaddressingthecaseofmetricallynoncomplete graphs,wegettheessentialself-adjointnessofΔ ω,c +Wprovidedthepotential WisboundedfrombelowbyN/2D 2 ,whereNisthemaximaldegreeandDthe distancetotheboundary.Weuseforthisresultatechnicaltooldeducedfrom Agmon-typeestimatesandinspiredbythenicepaper[Nen],seealso[Col-Tr]. WediscussinSection5thecaseofstar-likegraphs.Undersomeassumptionson a,weprovethatforanypotentialW,Δ 1,a +Wisessentiallyself-adjointus- inganextensionofWeyl’stheorytothediscretecase.Intheparticularcaseof thegraph N ,thesameresulthadbeenprovedin[Ber](p.504)inthecontext ofJacobimatrices.WegivesomeexamplesinSubsection5.3toillustratethe linksbetweenthepreviousresults.Moreoverweestablishthesharpnessofthe conditionsofTheorem4.2. ThelastSectionisdevotedtoAppendixAdealingwithWeyl’slimitpoint-limit circlecriteria(see[RS])inthediscretecaseaswellasinthecontinuouscase, andtoAppendixBincludingtheunitaryequivalencebetweenLaplaciansand Schro¨dingeroperators[To]usedrepeatedlyinSubsection5.3. Letusstartwithsomedefinitions. G=(V,E)willdenoteaninfinitegraph,withV=V(G)thesetofverticesand E=E(G)thesetofedges.Wewritex∼yfor{x,y}∈E. ThegraphGisalwaysassumedtobelocallyfinite,thatisanyx∈Vhasafinite numberofneighbors,whichwecallthedegree(orvalency)ofx.Ifthedegreeis 2