Symplectic Geometry and Fourier AnalysisSymplectic Geometry and Fourier Analysis



Suitable for graduate students in mathematics, this monograph covers differential and symplectic geometry, homogeneous symplectic manifolds, Fourier analysis, metaplectic representation, quantization, Kirillov theory.

Author: Nolan R. Wallach

Publisher: Courier Dover Publications

ISBN: 9780486829623

Category:

Page: 272

View: 548

Suitable for graduate students in mathematics, this monograph covers differential and symplectic geometry, homogeneous symplectic manifolds, Fourier analysis, metaplectic representation, quantization, Kirillov theory. Includes Appendix on Quantum Mechanics by Robert Hermann. 1977 edition.

Harmonic Analysis on Homogeneous SpacesHarmonic Analysis on Homogeneous Spaces



This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus.

Author: Nolan R. Wallach

Publisher: Courier Dover Publications

ISBN: 9780486836430

Category:

Page: 384

View: 746

This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem. Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.

Symplectic Methods in Harmonic Analysis and in Mathematical PhysicsSymplectic Methods in Harmonic Analysis and in Mathematical Physics



The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency ...

Author: Maurice A. de Gosson

Publisher: Birkhäuser

ISBN: 3764399910

Category:

Page: 338

View: 549

The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Quantum statistical mechanics and lie group harmonic analysisQuantum statistical mechanics and lie group harmonic analysis



Symplectic Geometry and Fourier Analysis, by N. Wallach 6. The 1976 Ames
Research Center (NASA) Conference on: The Geometric Theory of Non-Linear
Waves. 7. The 1976 Ames Research Center (NASA) Conference on Geometric ...

Author: Norman Hurt

Publisher: Math Science Pr

ISBN: 0915692309

Category:

Page: 251

View: 615

Geometric Optics on Phase SpaceGeometric Optics on Phase Space



The appendices link the geometry thus introduced to wave optics through Lie methods. The book addresses researchers and graduate students.

Author: Kurt Bernardo Wolf

Publisher: Springer Science & Business Media

ISBN: 3540220399

Category:

Page: 376

View: 421

Symplectic geometry, well known as the basic structure of Hamiltonian mechanics, is also the foundation of optics. In fact, optical systems (geometric or wave) have an even richer symmetry structure than mechanical ones (classical or quantum). The symmetries underlying the geometric model of light are based on the symplectic group. Geometric Optics on Phase Space develops both geometric optics and group theory from first principles in their Hamiltonian formulation on phase space. This treatise provides the mathematical background and also collects a host of useful methods of practical importance, particularly the fractional Fourier transform currently used for image processing. The reader will appreciate the beautiful similarities between Hamilton's mechanics and this approach to optics. The appendices link the geometry thus introduced to wave optics through Lie methods. The book addresses researchers and graduate students.

Introduction to Symplectic Dirac OperatorsIntroduction to Symplectic Dirac Operators



This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject.

Author: Katharina Habermann

Publisher: Springer

ISBN: 9783540334217

Category:

Page: 125

View: 305

This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.

Deformation Theory and Symplectic GeometryDeformation Theory and Symplectic Geometry



The cotangent bundle T = T * G is the underlying symplectic manifold of a
geometric deformation quantization of g * ... pullback by the exponential map
from g to G and Fourier transform from densities on the Lie algebra g to functions
on g * .

Author: Daniel Sternheimer

Publisher: Springer

ISBN: UOM:39015047132207

Category:

Page: 361

View: 603

Proceedings of the Ascona Meeting, June 1996

Notices of the American Mathematical SocietyNotices of the American Mathematical Society



... Discrete geometry and convexity Mark J. Gotay , Symplectic geometry Linda
Keen , Dynamical systems ( AMS - CMS ) ... Number theory ( AMS - CMS )
Gregory Verchota , Harmonic analysis techniques in partial differential equations
( AMS ...

Author: American Mathematical Society

Publisher:

ISBN: UCSD:31822017710104

Category:

Page:

View: 781

Fourier Integral OperatorsFourier Integral Operators



Additionally, this book is designed for a one-semester introductory course on Fourier integral operators aimed at a broad audience. This book remains a superb introduction to the theory of Fourier integral operators.

Author: J.J. Duistermaat

Publisher: Springer Science & Business Media

ISBN: 0817681086

Category:

Page: 142

View: 406

This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods.

Maslov Classes Metaplectic Representation and Lagrangian QuantizationMaslov Classes Metaplectic Representation and Lagrangian Quantization



WALLACH , N . R . Symplectic Geometry and Fourier Analysis , Series " Lie
Groups history , frontiers and applications ” , Math Sci . Press , Brookline , Mass . ,
1977 . WEIL , A . Sur certains groupes d ' opérateurs unitaires , Acta Math .

Author: Maurice de Gosson

Publisher: Wiley-VCH

ISBN: UOM:39015041231484

Category:

Page: 186

View: 870

The Maslov Classes have been playing an essential role in various parts of applied and pure mathematics, and physics, since the early 70's. Their correct definition is due to V. I. Arnold and J. Leray, in the transversal case, and to P. Dazord and the author in the general case. The aim of this book is to give a thorough treatment of the theory of the Maslov classes and of their relationship with the metaplectic group. It is (among other things) shown that these classes can be reconstructed, modulo 4, using only the analytic properties of the metaplectic group. In the last chapter the author sketches a scheme for geometric quantization by introducing two new concepts, that of metaplectic half-form and that of Lagrangian catalogue, the latter generalizes and simplifies the notion of "Lagrangian function" introduced by J. Leray. A Lagrangian catalogue is a collection of metaplectic half-forms which are themselves "cohomological wave functions", whose definition is made possible by using the combinatorial properties of the Maslov classes. The transformation of Lagrangian catalogues under the metaplectic group and of Hamiltonian flows is studied, and it is shown that one thus recovers very easily the so-called "quasi-classical approximation" to the solutions of Schrödinger equation if one introduces a natural concept, that of projection of a Lagrangian catalogue. An application to geometric phase shifts, including Berry's phase, is given.

Symplectic Geometry and Quantum MechanicsSymplectic Geometry and Quantum Mechanics



This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics.

Author: Maurice A. de Gosson

Publisher: Springer Science & Business Media

ISBN: 9783764375751

Category:

Page: 368

View: 185

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

SIAM Journal on Control and OptimizationSIAM Journal on Control and Optimization



490-492 . ( 18 ) N. R. WALLACH , Symplectic Geometry and Fourier Analysis ,
Mathematical Sciences Press , Brookline , MS , 1977 . ( 19 ] A. WEINSTEIN ,
Symplectic manifolds and their Lagrangian submanifolds , Adv . Mathematics , 6 (
1971 ) ...

Author: Society for Industrial and Applied Mathematics

Publisher:

ISBN: UOM:39076001655922

Category:

Page:

View: 198