Microlocal Methods in Mathematical Physics and Global Analysis

146 Pages·2013·1.114 MB·English

by  Hans Christianson, Jared Wunsch (auth.), Daniel Grieser, Stefan Teufel, Andras Vasy (eds.)

Trends in Mathematics
ResearchPerspectives
Research Perspectives collects core ideas and developments discussed at con-
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Microlocal Methods
in Mathematical Physics
and Global Analysis
Daniel Grieser
Stefan Teufel
Andras Vasy
Editors
Editors
DanielGrieser StefanTeufel
DepartmentofMathematics DepartmentofMathematics
UniversityofOldenburg UniversityofTu¨bingen
Germany Germany
AndrasVasy
DepartmentofMathematics
StanfordUniversity
Stanford,CA,USA
ISBN978-3-0348-0465-3 ISBN978-3-0348-0466-0(eBook)
DOI10.1007/978-3-0348-0466-0
SpringerBaselHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012951681
MathematicsSubjectClassification(2010):35-XX,58-XX
(cid:2)c SpringerBasel2013
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Preface
Sinceitsinventioninthemid-twentiethcenturythefieldofmicrolocalanalysiswas
characterized by a tight interplay of mathematics and physics. While in language
and rigor it is a field of mathematics, many of its ideas originate in physics: Its
playingfieldisthecotangentbundleofamanifold,themathematicalcounterpartof
thephasespaceofclassicalmechanics;itsmotivatingproblemscamefrompartial
differentialequations,particularlythosearisinginphysics,liketheLaplaceequation
andtheassociatedspectralproblem,andthewaveequation.Amongitsfundamental
tools is the WKBJ method, which was invented for quantum mechanics and also
usedingeometricaloptics;associatinganoperatortoasymbolisonemanifestation
of quantization. These ideas, often used rather informally in the physics context,
were made mathematically precise in microlocal analysis, and often triggered
further developments in mathematics, foremost in the field of partial differential
equations,butalsoinsymplecticgeometryorinsingularitytheory,tonameafew;
theywerealsousedinindextheoryasoneoftheirdeepestapplications.
In spite of this common ancestry of microlocal analysis in mathematics and
physics there are few conferences which bridge the gap between the various
communitiesthatusethissetoftechniques.Thisishowtheideafortheconference
‘Microlocalmethodsin mathematicalphysicsandglobalanalysis’arose. Our aim
wastobringtogetherresearchersatthehighestinternationallevelintheseareas,in
ordertofosterinteraction,informaboutnewdevelopmentsandgetastateoftheart
picture.
Themajorthemesoftheconferenceweretheuseordevelopmentofmicrolocal
techniques in semiclassical and adiabatic limits, singular spaces, spectral and
scattering theory, wave propagation and topological applications, and we have
organizedthiscollectionofextendedabstractsaccordingly,althoughnoteverytalk
admits a unique assignment to one of these categories. A recurrent overall theme
of many of the talks was the occurence of singular settings, that is, where the
underlying space is singular or non-compact, or where one studies a family of
operatorsorspacesapproachingasingularlimit.Thesystematicexplorationofsuch
singular problemshas been a focusof much research in microlocalanalysis since
the1980s.
v
vi Preface
The present volume is a collection of extended abstracts of most talks at the
conference.The talks were given by the first-mentionedauthorsof each extended
abstract. We believe that the format of extended abstract is a valuable means for
quick communication of current research, since it allows authors to disseminate
their results at an early stage and in a condensed form, and puts an emphasis on
conveyingtheessenceofthelectures,withoutbeingoverburdenedbytechnicalities.
Therefore,wehopethatthisformat,whichwasintroducedbytheMathematisches
ForschungsinstitutOberwolfachin2004underthenameOberwolfachReports,will
be used more frequently at other conferences as well. We are very grateful to
Birkha¨user Verlag for engaging in this new format and agreeing to publish this
volume.
The conference could not have happened without the support from various
sources: Major funding was supplied by the Deutsche Forschungsgemeinschaft
(SFB/Transregio 71, “Geometric PDEs”) and the National Science Foundation of
theU.S.A.(grantNo.1067924);wearealsogratefultotheUniversityofTu¨bingen
forhostingtheconference.
Theorganizers
D.Grieser
S.Teufel
A.Vasy
Contents
PartI SemiclassicalandAdiabaticLimits
Local Smoothing with a Prescribed Loss
fortheSchro¨dingerEquation................................................... 3
HansChristiansonandJaredWunsch
PropagationThrough Trapped Sets and Semiclassical
ResolventEstimates.............................................................. 7
KirilDatchevandAndra´sVasy
Space–AdiabaticTheoryforRandom–LandauHamiltonian:
ResultsandProspects............................................................ 11
GiuseppeDeNittis
MicrolocalAnalysisofFIOswithSingularities............................... 15
RalucaFelea
ANonlinearAdiabaticTheoremforCoherentStates........................ 19
ClotildeFermanianKammererandRe´miCarles
AdiabaticLimitsandRelatedLatticePointProblems....................... 25
YuriA.KordyukovandAndreyA.Yakovlev
TheEffectiveHamiltonianinCurvedQuantumWaveguides
andWhenItDoesNotWork.................................................... 29
DavidKrejcˇiˇr´ıkandHelenaSˇediva´kova´
TheAdiabaticLimitoftheLaplacianonThinFibreBundles.............. 33
JonasLampartandStefanTeufel
AdiabaticLimitwithIsolatedDegenerateFibres ............................ 37
RichardB.Melrose
MicrolocalAnalysisandAdiabaticProblems:The Caseof
PerturbedPeriodicSchro¨dingerOperators................................... 45
GianlucaPanati
vii
viii Contents
RecentResultsinSemiclassicalApproximationwithRoughPotentials ... 49
T.Paul
PartII Singularspaces
OntheClosureofEllipticWedgeOperators.................................. 55
JuanB.Gil,ThomasKrainer,andGerardoA.Mendoza
GeneralizedBlow-UpofCornersandFiberProducts....................... 59
ChrisKottkeandRichardMelrose
TraceExpansionsforEllipticConeOperators ............................... 63
ThomasKrainer,JuanB.Gil,andGerardoA.Mendoza
SpectralGeometryfortheRiemannModuliSpace .......................... 69
RafeMazzeo
InvariantIntegralOperatorsontheOshimaCompactification
ofaRiemannianSymmetricSpace:KernelAsymptoticsand
RegularizedTraces............................................................... 73
PabloRamacherandAprameyanParthasarathy
PseudodifferentialOperatorsonManifoldswithFoliatedBoundaries .... 77
Fre´de´ricRochon
TheDeterminantoftheLaplacianonaConicallyDegenerating
FamilyofMetrics ................................................................ 81
DavidA.Sher
PartIII SpectralandScatteringTheory
RelativelyIsospectralNoncompactSurfaces.................................. 87
PierreAlbin,ClaraAldana,andFre´de´ricRochon
MicrolocalAnalysisofScatteringDataforNestedConormal
Potentials.......................................................................... 91
SureshEswarathasan
EquidistributionofEisensteinSeriesforConvexCo-compact
HyperbolicManifolds............................................................ 95
ColinGuillarmouandFre´de´ricNaud
LowerBoundsfortheCountingFunctionofanIntegralOperator........ 99
YuriSafarov
The Identification Problems in SPECT: Uniqueness,
Non-uniquenessandStability................................................... 103
PlamenStefanov
Contents ix
Eigenvalues and Spectral Determinants on Compact
HyperbolicSurfaces ............................................................. 105
AlexanderStrohmaierandVilleUski
PartIV WavePropagationandTopologicalApplications
ASupportTheoremforaNonlinearRadiationField........................ 111
DeanBaskinandAnto´nioSa´Barreto
Propagation of Singularities Around a Lagrangian
SubmanifoldofRadialPoints................................................... 113
NickHaberandAndra´sVasy
LocalEnergy Decay for SeveralEvolutionEquations on
AsymptoticallyEuclideanManifolds .......................................... 117
DietrichHa¨fnerandJean-Franc¸oisBony
RayleighSurfaceWavesandGeometricPseudo-differentialCalculus .... 121
So¨nkeHansen
TopologicalImplicationsofGlobalHypoellipticity........................... 125
GerardoA.Mendoza
Chern-SimonsLineBundleonTeichmu¨llerSpace........................... 129
SergiuMoroianuandColinGuillarmou
A Simple Diffractive Boundary Value Problem on an
AsymptoticallyAnti-deSitterSpace........................................... 133
HaPham
QuantizationinaMagneticField .............................................. 137
RaduPurice,ViorelIftimie,andMariusMa˘ntoiu
Price’sLawonBlackHoleSpace-Times....................................... 145
DanielTataru
Part I
Semiclassical and Adiabatic Limits