Spectral Theory of Infinite-Area Hyperbolic Surfaces

355 Pages·2007·2.544 MB·English

by  David Borthwick (auth.)

Progress in Mathematics
Volume256
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
David Borthwick
Spectral Theory
of Infinite-Area
Hyperbolic Surfaces
Birkha¨user
Boston • Basel • Berlin
DavidBorthwick
DepartmentofMathematicsandComputerScience
EmoryUniversity
Atlanta,GA30322
U.S.A.
[email protected]
MathematicsSubjectClassification(2000):58J50,47A40,11F72,30F35
LibraryofCongressControlNumber:2007932363
ISBN-13:978-0-8176-4524-3 e-ISBN-13:978-0-8176-4653-0
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ForSarah,Julia,andBenjamin
Preface
I first encountered the spectral theory of hyperbolic surfaces as an undergraduate
physics student, through the intriguing expository article of Balazs–Voros [11] on
relations betweentheSelbergtheoryofautomorphicformsandquantumchaos.At
thetimeIwasquiteimpressedattherangeoftopicsrepresented,includingquantum
physics,discretegroups,differentialgeometry,numbertheory,complexanalysis,and
spectral theory. In my previous experience these were completely separate realms,
butheretheywereallmixedtogetherinthesamesetting.
Twentyyearslater,thesetopicsdonotseemsofaraparttome.However,Iamno
lessamazedbytherichcross-fertilizationofideasinthissubjectarea.Theprimary
motivationforthisbookistheconvictionthatthissortofmathematicsthatbridges
the divides between fields ought to be made accessible to as broad an audience as
possible—tograduatestudentsespecially,forwhomregularcourseworkoftenexag-
geratestheimpressionofboundariesbetweendisciplines.
The spectral theory of compact and finite-area Riemann surfaces is a classi-
cal subject with a history going back to the pioneering work of Atle Selberg, who
broughttechniquesfromspectraltheoryandharmonicanalysisintothestudyofau-
tomorphic forms. These cases have been thoroughly covered in various expository
sources.Inparticular,Buser[37]developsthespectraltheoryforcompactRiemann
surfaces with a concrete approach based on hyperbolic geometry and cutting and
pasting.Mosttreatmentsofthefinite-areacase,forexampleVenkov[210],empha-
sizearithmeticsurfacesandconnectionstonumbertheory.
Forinfinite-areahyperbolicsurfaces,agoodunderstandingofthespectraltheory
has emerged only recently. The assumption of infinite area changes the character
of the theory. The resolvent of the Laplacian takes on a predominant role and the
emphasisshiftsfromdiscreteeigenvaluestoscatteringtheoryandresonances.Ithas
only been through dramatic advances in geometric scattering theory that the full
developmentoftheinfinite-areatheoryhasbecomepossible.
Mygoalinthisbookistopresentarelativelyself-containedaccountofthisrecent
development.Althoughmanyoftheresultscouldbestatedingreatergenerality(e.g.,
higher dimensions), the book is restricted to the hyperbolic surface context for the
viii Preface
sakeofaccessibility.Thenotesattheendofeachchapterincludereferencestomore-
generalresults.
The book assumes basic algebra and topology, at the level of a first graduate
course. An undergraduate course on curves and surfaces should provide sufficient
background in differential geometry. Because spectral theory is the primary topic,
the analysis requirements are necessarily somewhat steeper. Beyond the basic real
andcomplexanalysis,astudentwouldneedbasicfunctionalanalysisandsomein-
troductiontotheanalysisoflinearpartialdifferentialequations.Theappendix,while
notaself-sufficientintroductiontothesetopics,ismeanttoserveasaguidetoread-
erswhoneedmorebackgroundinformation.
Iwouldliketothankmycollaborators,ChrisJudgeandPeterPerry,withwhom
Ilearnedmuchofthismaterial,andEdwardTaylor,whointroducedmetoscattering
theory on hyperbolic manifolds. Thanks also to Arthur Wightman, who supervised
the undergraduate project in which I first learned about the Selberg trace formula,
andtoRichardMelroseandRafeMazzeo,forencouragementwhenIfirstundertook
to learn some scattering theory. I am very grateful to Colin Guillarmou and Peter
Hislopforreadingpartsofthemanuscriptandofferingcorrectionsandsuggestions.
DavidBorthwick
Atlanta
February2007
Contents
1 Introduction................................................... 1
2 HyperbolicSurfaces............................................ 7
2.1 Thehyperbolicplane........................................ 8
2.2 Fuchsiangroups............................................ 13
2.3 Geometricallyfinitegroups .................................. 18
2.4 Classificationofhyperbolicends.............................. 22
2.5 Gauss–Bonnettheorem...................................... 28
2.6 LengthspectrumandSelberg’szetafunction.................... 31
3 CompactandFinite-AreaSurfaces............................... 37
3.1 Selberg’straceformulaforcompactsurfaces.................... 37
3.2 Consequencesofthetraceformula ............................ 42
3.3 Finite-areahyperbolicsurfaces ............................... 45
4 SpectralTheoryfortheHyperbolicPlane......................... 49
4.1 Resolvent ................................................. 49
4.2 Generalizedeigenfunctions .................................. 52
4.3 Scatteringmatrix........................................... 56
5 ModelResolventsforCylinders.................................. 61
5.1 Hyperboliccylinders........................................ 61
5.2 Funnels................................................... 68
5.3 Paraboliccylinder .......................................... 70
6 TheResolvent ................................................. 75
6.1 Compactification........................................... 75
6.2 AnalyticFredholmtheorem .................................. 79
6.3 Continuationoftheresolvent................................. 81
6.4 Structureoftheresolventkernel .............................. 84
6.5 Thestretchedproduct ....................................... 87
x Contents
7 SpectralandScatteringTheory.................................. 93
7.1 Essentialanddiscretespectrum............................... 93
7.2 Absenceofembeddedeigenvalues ............................ 95
7.3 Generalizedeigenfunctions .................................. 102
7.4 Scatteringmatrix........................................... 105
7.5 Scatteringmatricesforthefunnelandcylinders ................. 114
8 ResonancesandScatteringPoles................................. 117
8.1 Multiplicitiesofresonances .................................. 118
8.2 Structureoftheresolventataresonance........................ 119
8.3 Scatteringpoles ............................................ 124
8.4 Operatorlogarithmicresidues ................................ 126
8.5 Half-integerpoints ......................................... 131
8.6 Coincidenceofresonancesandscatteringpoles ................. 137
9 UpperBoundforResonances.................................... 147
9.1 Resonancesandzerosofdeterminants ......................... 148
9.2 Singularvalueestimates..................................... 151
9.3 Upperbound .............................................. 154
9.4 Estimatesonmodelterms.................................... 156
10 SelbergZetaFunction .......................................... 171
10.1 Relativescatteringdeterminant ............................... 173
10.2 Regularizedtraces.......................................... 175
10.3 Theresolvent0-tracecalculation.............................. 183
10.4 Structureofthezetafunction................................. 189
10.5 Orderbound............................................... 196
10.6 DeterminantoftheLaplacian................................. 203
11 WaveTraceandPoissonFormula ................................ 207
11.1 Regularizedwavetrace...................................... 208
11.2 Modelwavekernel ......................................... 209
11.3 Wave0-traceformula ....................................... 211
11.4 Poissonformula............................................ 215
12 ResonanceAsymptotics ......................................... 223
12.1 Lowerboundonresonances.................................. 223
12.2 Lowerboundnearthecriticalline............................. 226
12.3 Weylformulaforthescatteringphase.......................... 229
13 InverseSpectralGeometry ...................................... 237
13.1 Resonancesandthelengthspectrum........................... 238
13.2 Hyperbolictrigonometry .................................... 239
13.3 Teichmu¨llerspace .......................................... 242
13.4 Finitenessofisospectralclasses............................... 248