Softcover ISBN:  9780821847015 
Product Code:  CLN/17 
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Electronic ISBN:  9781470431174 
Product Code:  CLN/17.E 
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Book DetailsCourant Lecture NotesVolume: 17; 2008; 197 ppMSC: Primary 35;
This book is based on a course entitled “Wigner measures and semiclassical limits of nonlinear Schrödinger 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 highfrequency 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 Schrödingertype equations.
At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is selfcontained and is suitable for an advanced graduate course on the topic.ReadershipGraduate students and research mathematicians interested in PDE's of Schrödinger and kinetic type.

Table of Contents

Chapters

Chapter 1. The classical WKB method

Chapter 2. Wigner measure

Chapter 3. The limit from the onedimensional SchrödingerPoisson to VlasovPoisson equations

Chapter 4. Semiclassical limit of SchrödingerPoisson equations

Chapter 5. Semiclassical limit of the cubic Schrödinger equation in an exterior domain

Chapter 6. Incompressible and compressible limits of coupled systems of nonlinear Schrödinger equations

Chapter 7. Highfrequency limit of the Helmholtz equation

Appendix A. Global solutions to (3.14)

Appendix B. Denseness of polynomials

Appendix C. Global existence of a solution to (5.1)

Appendix D. Global smooth solution to (6.1)


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This book is based on a course entitled “Wigner measures and semiclassical limits of nonlinear Schrödinger 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 highfrequency 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 Schrödingertype equations.
At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is selfcontained and is suitable for an advanced graduate course on the topic.
Graduate students and research mathematicians interested in PDE's of Schrödinger and kinetic type.

Chapters

Chapter 1. The classical WKB method

Chapter 2. Wigner measure

Chapter 3. The limit from the onedimensional SchrödingerPoisson to VlasovPoisson equations

Chapter 4. Semiclassical limit of SchrödingerPoisson equations

Chapter 5. Semiclassical limit of the cubic Schrödinger equation in an exterior domain

Chapter 6. Incompressible and compressible limits of coupled systems of nonlinear Schrödinger equations

Chapter 7. Highfrequency limit of the Helmholtz equation

Appendix A. Global solutions to (3.14)

Appendix B. Denseness of polynomials

Appendix C. Global existence of a solution to (5.1)

Appendix D. Global smooth solution to (6.1)