An Introduction to Semiclassical and Microlocal Analysis

ISBN-13: 9781441929617 / Angielski / Miękka / 2011 / 191 str.

Andre Martinez

Opis

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.

Recenzje

From the reviews:

"The book is very clearly written, and is indeed much simpler than most of others. ... this is an excellent book, suitable even for students with a poor background in the subject. In my opinion, every department offering courses on partial differential equations or asymptotic analysis should have it in the library. It will be very useful to all mathematicians working in semi-classical or microlocal analysis ... ." (Yuri Safarov, Bulletin of the London Mathematical Society, Issue 35, 2003)

"This is a concise and accessible book on the basic techniques of semiclassical and microlocal analysis. ... An appendix provides a useful summary of the major formulas used in the text. Each chapter is followed by a collection of provocative and well chosen exercises. The exercises are essential for understanding the work and provide important applications. ... the text does provide a very readable, clear, and concise introduction to semiclassical and microlocal analysis." (Peter D. Hislop, Mathematical Reviews, 2003 b)

"The contents of the book correspond to a course at Ph. D. Level, given by the author at the Universities of Bologna and Paris-Nord. ... the book collects in an original way standard results and new aspects of semi-classical microlocal analysis; the reading is suggested to non-specialists as well." (L. Rodino, Zentralblatt MATH, Vol. 994 (19), 2002)

"This book presents most of the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics. ... The originality lies in the fact that the main features of analytic microlocal analysis are derived from a single and elementary a priori estimate. Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. Applications to the study of the Schrödinger operator are also discussed, to further the understanding of new notions ... ." (L' Enseignement Mathematique, Vol. 48 (1-2), 2002)

"The book under review consists of lecture notes corresponding to a course taught for several years at the universities of Paris-Nord (France) and Bologna (Italy). It is addressed mainly to non-specialists in the subject, and the prerequisites are essentially reduced to the basic notions of the theory of distributions. ... The author tries to make clear how the things work and show examples where they can be applied. It is nicely written ... . the book can be highly recommended." (J. Synnatzschke, Zeitschrift für Analysis und ihre Anwendungen - ZAA, Vol. 21 (3), 2002)

Spis treści

Introduction * Semiclassical Pseudodifferential Calculus * Microlocalization * Applications to the Solutions of Analytic Linear PDEs * Complements: Symplectic Aspects * Appendix: List of Formulae * Bibliography * Index * List of Notations 

O autorze

Semi-classical analysis has become a mathematical tool for dealing with a variety of asymptotic problems. This book provides a systematic and extremely useful presentation of the theory in its traditional setting motivated by quantum mechanics. Both the standard C8 pseudodifferential calculus and the analytic microlocal analysis are developed, while the author replaces the delicate local theory with a more straightforward global theory. The only prerequisite required is a basic knowledge of the theory of distributions.

Blurb

This book presents most of the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics. Both the standard C? pseudodifferential calculus and the analytic microlocal analysis are developed, in a context which remains intentionally global so that only the relevant difficulties of the theory are encountered. The originality lies in the fact that the main features of analytic microlocal analysis are derived from a single and elementary a priori estimate.
Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. Applications to the study of the Schrödinger operator are also discussed, to further the understanding of new notions or general results by replacing them in the context of quantum mechanics. This book is aimed at non-specialists of the subject and the only required prerequisite is a basic knowledge of the theory of distributions.
André Martinez is currently Professor of Mathematics at the University of Bologna, Italy, after having moved from France where he was Professor at Paris-Nord University. He has published many research articles in semiclassical quantum mechanics, in particular related to the Born-Oppenheimer approximation, phase-space tunneling, scattering theory and resonances.