An introduction to semiclassical and microlocal analysis
type: Item
instance of: scholarly article
title: An introduction to semiclassical and microlocal analysis
zbMATH Open document ID: 0994.35003
author: André Martinez
published in: Universitext
publication date: 2002-03-06T00:00:00Z
review text: 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 subject is semiclassical analysis, but the book can be read as well as introduction to standard microlocal analysis. Namely, the book treats operators of the form \[ Op_h(a)u(x;h)= (2\pi h)^{-n}\int e^{i(x-y)\xi/h} a(x,y,\xi) u(y)dy d \xi \] depending on the parameter \(h\to 0\), relevant examples being semi-classical differential operators \[ \sum_{|\alpha|\leq m}b_\alpha(x) (hD_x)^\alpha, \] in particular the celebrated Schrödinger operator \(-(h^2/2m) \Delta+V(x)\). The corresponding asymptotic properties, principally spectral properties as \(h\to 0\), allow to prove mathematically some typical phenomena in quantum mechanics. Roughly, this is semi-classical analysis, whereas taking \(h=1\) in the previous expressions, we have standard pseudo-differential operators and related concepts, i.e. microlocal analysis. Moving in this twofold frame, the presentation has a pedagogical character, specific contents being the following: semilinear pseudo-differential calculus, microlocalization, applications to the solutions of analytic PDE's, symplectic aspects. As useful appendix, the book presents a list of formulae. Peculiarity of the exposition is a careful treatment of the so-called FBI transform, with applications to the microlocal exponential estimates, cf. \textit{J. Sjöstrand} [Astérisque 95, 1-166 (1982; Zbl 0524.35007)], \textit{A. Martinez} [in: Microlocal Analysis and Spectral Theory, NATO ASI Ser., C, Math. Phys. Sci. 490, 349-376 (1997; Zbl 0890.35120)]. Summing up, the book collects in an original way standard results and new aspects of semiclassical microlocal analysis; the reading is suggested to non-specialists as well.
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 1716498
zbMATH Keywords: semiclassical analysis
zbMATH Keywords: microlocal analysis
zbMATH Keywords: semi-classical differential operators
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: asymptotic properties
zbMATH Keywords: spectral properties
zbMATH Keywords: semilinear pseud-differential calculus
zbMATH Keywords: microlocalization
zbMATH Keywords: FBI transform
zbMATH Keywords: microlocal exponential estimates
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description: scientific article; zbMATH DE number 1716498