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| Author | : Maciej Zworski |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 448 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821883208 |
"...A graduate level text introducing readers to semiclassical and microlocal methods in PDE." -- from xi.
| Author | : André Bach |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 193 |
| Release | : 2013-03-14 |
| Genre | : Mathematics |
| ISBN | : 1475744951 |
This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.
| Author | : Victor Guillemin |
| Publisher | : |
| Total Pages | : 446 |
| Release | : 2013 |
| Genre | : Fourier integral operators |
| ISBN | : 9781571462763 |
| Author | : Remi Carles |
| Publisher | : World Scientific |
| Total Pages | : 256 |
| Release | : 2008-03-04 |
| Genre | : Mathematics |
| ISBN | : 9814471747 |
These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
| Author | : STEPHEN J. GUSTAFSON |
| Publisher | : |
| Total Pages | : |
| Release | : 2020 |
| Genre | : Mathematics |
| ISBN | : 3030595625 |
The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content. It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include: many-body systems, modern perturbation theory, path integrals, the theory of resonances, adiabatic theory, geometrical phases, Aharonov-Bohm effect, density functional theory, open systems, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group. With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. Some of the sections could be used for introductions to geometrical methods in Quantum Mechanics, to quantum information theory and to quantum electrodynamics and quantum field theory.
| Author | : Mouez Dimassi |
| Publisher | : Cambridge University Press |
| Total Pages | : 243 |
| Release | : 1999-09-16 |
| Genre | : Mathematics |
| ISBN | : 0521665442 |
This book presents the basic methods and applications in semiclassical approximation in the light of developments.
| Author | : Leon Cohen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 302 |
| Release | : 2011-12-21 |
| Genre | : Technology & Engineering |
| ISBN | : 1441966242 |
David Middleton was a towering figure of 20th Century engineering and science and one of the founders of statistical communication theory. During the second World War, the young David Middleton, working with Van Fleck, devised the notion of the matched filter, which is the most basic method used for detecting signals in noise. Over the intervening six decades, the contributions of Middleton have become classics. This collection of essays by leading scientists, engineers and colleagues of David are in his honor and reflect the wide influence that he has had on many fields. Also included is the introduction by Middleton to his forthcoming book, which gives a wonderful view of the field of communication, its history and his own views on the field that he developed over the past 60 years. Focusing on classical noise modeling and applications, Classical, Semi-Classical and Quantum Noise includes coverage of statistical communication theory, non-stationary noise, molecular footprints, noise suppression, Quantum error correction, and other related topics.
| Author | : Vladimir F. Lazutkin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 390 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642762476 |
It is a surprising fact that so far almost no books have been published on KAM theory. The first part of this book seems to be the first monographic exposition of this subject, despite the fact that the discussion of KAM theory started as early as 1954 (Kolmogorov) and was developed later in 1962 by Arnold and Moser. Today, this mathematical field is very popular and well known among physicists and mathematicians. In the first part of this Ergebnisse-Bericht, Lazutkin succeeds in giving a complete and self-contained exposition of the subject, including a part on Hamiltonian dynamics. The main results concern the existence and persistence of KAM theory, their smooth dependence on the frequency, and the estimate of the measure of the set filled by KAM theory. The second part is devoted to the construction of the semiclassical asymptotics to the eigenfunctions of the generalized Schrödinger operator. The main result is the asymptotic formulae for eigenfunctions and eigenvalues, using Maslov`s operator, for the set of eigenvalues of positive density in the set of all eigenvalues. An addendum by Prof. A.I. Shnirelman treats eigenfunctions corresponding to the "chaotic component" of the phase space.
| Author | : Bernard Helffer |
| Publisher | : World Scientific |
| Total Pages | : 200 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 9789812380982 |
This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log-Sobolev inequality.
| Author | : Victor P. Maslov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 320 |
| Release | : 2001-11-30 |
| Genre | : Science |
| ISBN | : 9781402003066 |
This volume is concerned with a detailed description of the canonical operator method - one of the asymptotic methods of linear mathematical physics. The book is, in fact, an extension and continuation of the authors' works [59], [60], [65]. The basic ideas are summarized in the Introduction. The book consists of two parts. In the first, the theory of the canonical operator is develop ed, whereas, in the second, many applications of the canonical operator method to concrete problems of mathematical physics are presented. The authors are pleased to express their deep gratitude to S. M. Tsidilin for his valuable comments. THE AUTHORS IX INTRODUCTION 1. Various problems of mathematical and theoretical physics involve partial differential equations with a small parameter at the highest derivative terms. For constructing approximate solutions of these equations, asymptotic methods have long been used. In recent decades there has been a renaissance period of the asymptotic methods of linear mathematical physics. The range of their applicability has expanded: the asymptotic methods have been not only continuously used in traditional branches of mathematical physics but also have had an essential impact on the development of the general theory of partial differential equations. It appeared recently that there is a unified approach to a number of problems which, at first sight, looked rather unrelated.
