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1、LecturesatStanfordGeometricscatteringtheoryRichardB.MelroseMassachusettsInstituteofTechnologyCAMBRIDGEUNIVERSITYPRESSCambridgeNewYorkPortChesterMelbourneSydneyPrefaceThesenotesarebasedonlecturesdeliveredatStanfordUniversityin1January1994andthenrepeatedatMITint
2、heSpringsemester.IamverygratefultothemembersoftheMathematicsDepartmentatStan-ford,andinparticularRalphCohen,fortheinvitationandhospitality.Myespecialthankstothosewhoattendedthelecturesandcontributedinonewayoranother.Iamparticularlypleasedtoacknowledgethein
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3、ceonmythinkingoftwoofthemembersoftheaudience,RalphPhillipsandJoeKeller.RafeMazzeoencouragedmetowriteupthelec-tures,providedmewithhisownnotesand,asifthatwerenotenough,mademanyhelpfulcommentsonthemanuscript.IshouldalsoliketoextendmythankstoSangChin,DanielGrieser
4、,AndrewHassell,MarkJoshi,OlivierLa tte,EckhardMeinrenken,EdithMooersandAndras2VasywhoattendedthesecondhearingofthelecturesatMITandto-gethermademanyusefulremarksAndrasVasywasparticularlyhelpfulinreadingandcorrectingthenotesastheydribbledout.Iwouldalsoliketothan
5、kTanyaChristiansenandGuntherUhlmannfortheirassis-tanceandLarsHormander,GeorgiVodevandMaciejZworskifortheir3commentsonlaterversionsofthemanuscript.Itismyhopethatthesenotesmayserveasanintroductiontoanactiveandgrowingareaorresearch,althoughIfeartheyrepresentarath
6、ersteeplearningcurve.1ItwasahorriblemonthinCambridgeIamtold,verypleasantindedinPaloAlto.Thisfootnoteisanindicationofthingstocomeinthebodyofthenotes.Ifyoucan'tstandit,stopnow!2OfcourseIhadreallywantedtodothingsintheotherorderbutdidnotmanagetogetmythoughtstogeth
7、erintime.3Ofcourse,Iclaimsolecreditforallremainingerrors.iiiContentsListofIllustrationspage1Introduction21EuclideanLaplacian31.1TheLaplacian31.2Spectralresolution41.3Scatteringmatrix61.4Resolventfamily81.5Limitingabsorptionprinciple91.6Analyticcontinuation111.
8、7Asymptoticexpansion131.8Radialcompacti cation15n2PotentialscatteringonR172.1Theresolventof+V172.2Polesoftheresolvent202.3Boundarypairing212.4Formalsolutions232.5Uniquecontinuatio
