lectures on semiclassical analysis evans

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1、LECTURESONSEMICLASSICALANALYSISVERSION0.2LAWRENCEC.EVANSANDMACIEJZWORSKIDEPARTMENTOFMATHEMATICSUCBERKELEY12PrefaceTheselecturesoriginatewithacourseMZtaughtatUCBerkeleyduringthespringsemesterof2003,notesforwhichLCEtookinclass.Inthispresentationwehavetriedha

2、rdtoworkoutthefulldetailsformanyproofsonlysketchedinclass.Inaddition,wehavereworkedtheorderofpresentation,addedadditionaltopics,andincludedmoreheuristiccommentary.Wehaveaswellintroducedconsistentnotation,muchofwhichiscollectedintoAppendixA.Relevantfunction

3、alanalysisandotherfactshavebeenconsolidatedintoAppendicesB–D.Weshouldmentionthattwoexcellenttreatmentsofmathematicalsemiclassicalanalysishaveappearedrecently.ThebookbyDimassiandSj¨ostrand[D-S]startswiththeWKB-method,developsthegeneralsemiclassicalcalculus,

4、andthenprovides“hightech”spectralasymp-totics.ThepresentationofMartinez[M]isbasedonasystematicdevelopmentofFBI(Fourier-Bros-Iagolnitzer)transformtechniques,withapplicationstomicrolocalexponentialestimatesandpropagationestimates.Thesenotesareintendedasamore

5、elementaryandbroaderintro-duction.Exceptforthegeneralsymbolcalculus,wherewefollowedChapter7of[D-S],thereislittleoverlapwiththeseothertwotexts,orwiththeearlyandinfluentialbookofRobert[R].InhisstudyofsemiclassicalcalculusMZhasbeenprimarilyinfluencedbyhislongco

6、llaborationwithJohannesSj¨ostrandandheacknowledgesthatwithpleasureandgratitude.OurthankstoFayeYeagerfortypingafirstdraftandtoHansChris-tiansonforhiscarefulreadingofearlierversionsofthesenotes.AndthanksalsotoJonathanDorfmanforTeXadvice.Wearequiteawarethatman

7、yerrorsremaininourexposition,andsoweaskourreaderstopleasesendanycommentsorcorrectionstousatevans@math.berkeley.eduzworski@math.berkeley.edu.ThisVersion0.1representsourfirstdraft,theclarityofwhichwehopegreatlytoimproveinlatereditions.Wewillperiodicallypostim

8、provedversionstheselecturesonourwebsites.LCEissupportedinpartbyNSFgrantDMS-0500452,andMZbyNSFgrantDMS-0200732.3ContentsPreface21.Introduction42.Symplecticgeometry93.Fouriertransform,stationaryphase184.Quantiz