Semiclassical analysis. (English) Zbl 1252.58001

Zworski, Maciej

Graduate Studies in Mathematics 138. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8320-4/hbk). xii, 431 p. (2012).

The book by Zworski is an excellent and selfcontained introduction to the semiclassical and microlocal methods in the study of PDEs. It originates from a course taught by the author at UC Berkeley, and its major goal is understanding the relationships between dynamical systems and the behaviour of solutions to various linear partial and pseudodifferential equations with small parameters. The author constructs a wide variety of mathematical tools to address these issues, among them: the apparatus of symplectic geometry (to record the behaviour of classical dynamical systems), the Fourier transform (to display dependence on both the position and the momentum variables), stationary phase (to describe asymptotics of various expressions involving rescaled Fourier transforms as the parameter tends to zero), pseudodifferential operators (to microlocalize functional behaviour in phase space and to get important results about propagation of singularities for general classes of equations). Some techniques developed for PDE, such as local solvability, acquire new life when translated to the semiclassical settings. The reader will find good examples in the study of pseudospectra of nonselfadjoint operators, or in the connection between tunneling and unique continuation nicely unified by the semiclassical Carleman estimates.
Although the book is devoted to semiclassical analysis as a branch of the linear PDE theory, the ideas explored are useful in other areas such as: the study of quantum maps (where symplectic transformations on compact manifolds are quantized to give matrices), the study of nonlinear PDE with an asymptotic parameter, the paradifferential calculus of Bony, Coifman and Meyer, and the Littlewood-Paley decomposition.
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. The readers are expected to have reasonable familiarity with standard PDE theory and a basic understanding of linear functional analysis.

Reviewer: Dian K. Palagachev (Bari)

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58J40	Pseudodifferential and Fourier integral operators on manifolds
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
58J51	Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
35S05	Pseudodifferential operators as generalizations of partial differential operators
35S30	Fourier integral operators applied to PDEs
35Q40	PDEs in connection with quantum mechanics
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81S10	Geometry and quantization, symplectic methods

Keywords:

semiclassical analysis; microlocal analysis; harmonic analysis; symplectic geometry; semiclassical and microlocal methods for PDE