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Author : Maciej Zworski
Publisher : American Mathematical Soc.
Page : 448 pages
File Size : 33,27 MB
Release : 2012
Category : 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
Page : 193 pages
File Size : 35,39 MB
Release : 2013-03-14
Category : 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 :
Page : 446 pages
File Size : 26,93 MB
Release : 2013
Category : Fourier integral operators
ISBN : 9781571462763
Author : Bernard Helffer
Publisher : World Scientific
Page : 200 pages
File Size : 24,91 MB
Release : 2002
Category : 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 : Remi Carles
Publisher : World Scientific
Page : 256 pages
File Size : 26,8 MB
Release : 2008-03-04
Category : 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 : Maciej Zworski
Publisher : American Mathematical Society
Page : 431 pages
File Size : 27,65 MB
Release : 2022-05-09
Category : Mathematics
ISBN : 1470470624
This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. —Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel–Kramers–Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.
Author : STEPHEN J. GUSTAFSON
Publisher :
Page : pages
File Size : 27,40 MB
Release : 2020
Category : 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
Page : 243 pages
File Size : 17,9 MB
Release : 1999-09-16
Category : Mathematics
ISBN : 0521665442
This book presents the basic methods and applications in semiclassical approximation in the light of developments.
Author : Spyridon Kamvissis
Publisher : Princeton University Press
Page : 280 pages
File Size : 27,99 MB
Release : 2003-08-18
Category : Mathematics
ISBN : 1400837189
This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.
Author : Elliott H. Lieb
Publisher : American Mathematical Soc.
Page : 378 pages
File Size : 42,98 MB
Release : 2001
Category : Analysis
ISBN : 0821827839
This course in real analysis begins with the usual measure theory, then brings the reader quickly to a level where a wider than usual range of topics can be appreciated. Topics covered include Lp- spaces, rearrangement inequalities, sharp integral inequalities, distribution theory, Fourier analysis, potential theory, and Sobolev spaces. To illustrate these topics, there is a chapter on the calculus of variations, with examples from mathematical physics, as well as a chapter on eigenvalue problems (new to this edition). For graduate students of mathematics, and for students of the natural sciences and engineering who want to learn tools of real analysis. Assumes a previous course in calculus. Lieb is affiliated with Princeton University. Loss is affiliated with Georgia Institute of Technology. c. Book News Inc.