Semiclassical analysis (Q2897331)

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.NEWLINENEWLINEAlthough 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.NEWLINENEWLINEThe 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.