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1、LectureNotesinMathematics1862Editors:J.--M.Morel,CachanF.Takens,GroningenB.Teissier,ParisBernardHelfferFrancisNierHypoellipticEstimatesandSpectralTheoryforFokker-PlanckOperatorsandWittenLaplacians123AuthorsBernardHelfferLaboratoiredeMathematiques´UMRCNRS8628UniversiteParis-Sud´Batimen
2、t425ˆ91404OrsayFrancee-mail:bernard.helffer@math.u-psud.frFrancisNierIRMARUMRCNRS6625UniversitedeRennes1´CampusdeBeaulieu35042RennesCedexFrancee-mail:francis.nier@univ-rennes1.frLibraryofCongressControlNumber:2004117183MathematicsSubjectClassification(2000):35H10,35H20,35P05,35P15,58J
3、10,58J50,58K65,81Q10,81Q20,82C05,82C31,82C40ISSN0075-8434ISBN3-540-24200-7SpringerBerlinHeidelbergNewYorkDOI:10.1007/b104762Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustration
4、s,recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
5、areliableforprosecutionundertheGermanCopyrightLaw.SpringerisapartofSpringerScience+BusinessMediahttp://www.springeronline.comcSpringer-VerlagBerlinHeidelberg2005PrintedinGermanyTheuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsen
6、ceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Typesetting:Camera-readyTEXoutputbytheauthors41/3142/du-543210-Printedonacid-freepaperForewordThistextisanexpandedversionofinformalnotespreparedbythefirstauthorforamini
7、courseofeighthours,reviewingthelinksbetweenhypoelliptictech-niquesandthespectraltheoryofSchr¨odingertypeoperators.TheselecturesweregivenatRennesfortheworkshop“Equationscin´etiques,hypoellipticit´eetLaplaciendeWitten”organizedinFebruary2003bythesecondauthor.Theircontenthasbeensubstanti
8、allyc
