This book is based on a course entitled 'Wigner measures and semiclassical limits of nonlinear Schrodinger equations,' which the author taught at the Courant Institute of Mathematical Sciences at New York University in the spring of 2007. The author's main purpose is to apply the theory of semiclassical pseudodifferential operators to the study of various high-frequency limits of equations from quantum mechanics. In particular, the focus of attention is on Wigner measure and recent progress on how to use it as a tool to study various problems arising from semiclassical limits of Schrodinger-type equations. At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is self-contained and is suitable for an advanced graduate course on the topic.
The classical WKB method Wigner measure The limit from the one-dimensional Schrodinger-Poisson to Vlasov-Poisson equations Semiclassical limit of Schrodinger-Poisson equations Semiclassical limit of the cubic Schrodinger equation in an exterior domain Incompressible and compressible limits of coupled systems of nonlinear Schrodinger equations High-frequency limit of the Helmholtz equation Global solutions to (3.14) Denseness of polynomials Global existence of a solution to (5.1) Global smooth solution to (6.1) Bibliography.