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Other titles in the PseudoDifferential Operators series:
PseudoDifferential Operators #5: Symplectic Methods in Harmonic Analysis and in Mathematical Physicsby Maurice A. De Gosson
Synopses & ReviewsPublisher Comments: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 timefrequency 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 pseudodifferential operators, and Feichtinger's theory of modulation spaces. Several applications to timefrequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a nonstandard pseudodifferential calculus on phase space is introduced and studied, where the main role is played by "Bopp operators" (also called "Landau operators" in the literature). 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 pseudodifferential 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 timefrequency 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 selfcontained and includes an extensive list of references.
Synopsis:The novel approach to deformation quantization outlined in this text makes use of established tools in timefrequency analysis. As one of the first volumes to discuss mathematical physics using Feichtinger's modulation spaces, this is a valuable reference.
Synopsis:"Deformation quantization" is a mathematical theory which provides an alternative approach to quantum mechanics. It has ramifications in both pure mathematics and physics. This book gives a novel approach to the subject by using pseudodifferential methods ("Bopp quantization") where the theory of modulation spaces plays a central role.
Table of Contents1. Introduction. 2. Weyl Calculus. 3. Bopp Pseudodifferential Operators. 4. The Uncertainty Principle. 5. Deformation Quantization. 6. Symplectic Covariance Properties. 7. The Shubin Symbol Classes. 8. Modulation spaces. 9. Global Hypoellipticity. 10. Spectral Properties. 11. Semiclassical Theory. Bibliography. Index.
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