相空间中的调和分析

Preface

Prologue.SomeMattersofNotation

CHAPTER1.THEHEISENBERGGROUPANDITSREPRESENTATIONS

1.Backgroundfromphysics

Hamiltonianmechanics,10.Quantummechanics,12.Quantization,15.

2.TheHeisenberggroup

TheautomorphismsoftheHeisenberggroup,19.

3.TheSchr~Sdingerrepresentation

Theintegratedrepresentation,23.Twistedconvolution,25.

Theuncertaintyprinciple,27.

4.TheFourier-Wignertransform

Radarambiguityfunctions,33.

5.TheStone-yonNeumanntheorem

ThegroupFouriertransform,37.

6.TheFock-Bargmannrepresentation

Somemotivationandhistory,47.

7.Hermitefunctions

8.TheWignertransform

9.TheLaguerreconnection

10.Thenilmanifoldrepresentation

11.Postscripts

CHAPTER2.QUANTIZATIONANDPSEUDODIFFERENTIALOPERATORS

1.TheWeylcorrespondence

Covarianceproperties,83.Symbolclasses,86.Miscellaneousremarks

andexamples,90.

2.TheKohn-Nirenbergcorrespondence

3.Theproductformula

4.Basicpseudodifferentialtheory

Wavefrontsets,118.

5.TheCalder6n-VaiUancourttheorems

6.ThesharpG~rdinginequality

7.TheWickandanti-Wickcorrespondences

CHAPTER3.WAVEPACKETSANDWAVEFRONTS

1.Wavepacketexpansions

2.Acharacterizationofwavefrontsets

3.AnalyticityandtheFBItransform

4.Gaborexpansions

CHAPTER4.THEMETAPLECTICREPRESENTATION

1.Symplecticlinearalgebra

2.Constructionofthemetaplecticrepresentation

TheFockmodel,180.

3.Theinfinitesimalrepresentation

4.Otheraspectsofthemetaplecticrepresentation

Integralformulas,191.Irreduciblesubspaces,194.Dependenceon

Plancksconstant,195.Theextendedmetaplecticrepresentation,196.

TheGroenewold-vanHovetheorems,197.Someapplications,199.

5.Gaussiansandthesymmetricspace

CharacterizationsofGaussians,206.

6.Thediscmodel

7.Variantsandanalogues

Restrictionsofthemetaplecticrepresentation,216.U(n,n)asacomplex

symplecticgroup,217.Thespinrepresentation,220.

CHAPTER5.THEOSCILLATORSEMIGROUP

1.TheSchrodingermodel

Theextendedoscillatorsemigroup,234.

2.TheHermitesemigroup

3.NormalizationandtheCayleytransform

4.TheFockmodel

AppendixA.GaussianIntegralsandaLemmaonDeterminants

AppendixB.SomeHilbertSpaceResults

Bibliography

Index

　　Thephrase"harmonicanalysisinphasespace"isaconciseifsomewhatinadequatenamefortheareaofanalysisonRnthatinvolvestheHeisenberggroup,quantization,theWeyloperationalcalculus,themetaplecticrepresentation,wavepackets,andrelatedconcepts:itismeanttosuggestanalysisontheconfigurationspaceRndonebyworkinginthephasespaceRn×Rn.Theideasthatfallunderthisrubrichaveoriginatedinseveraldifferentfields--Fourieranalysis,partialdifferentialequations,mathematicalphysics,representationtheory,andnumbertheory,amongothers.Asaresult,althoughtheseideasareindividuallywellknowntoworkersinsuchfields,theirclosekinshipandthecross-fertilizationtheycanprovidehaveoftenbeeninsufficientlyappreciated.Oneoftheprincipalobjectivesofthismonographistogiveacoherentaccountofthismaterial,comprisingnotjustanefficienttourofthemajoravenuesbutalsoanexplorationofsomepicturesquebyways.