Table of contents for Wigner measure and semiclassical limits of nonlinear Schrodinger equations / Ping Zhang.

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Preface vii
Chapter 1. The Classical WKB Method 1
1.1. Introduction 1
1.2. WKB Approximation to the Linear Schrodinger Equation 2
1.3. The WKB Approximation to Nonlinear Schrodinger Equation 5
1.4. References and Remarks 10
Chapter 2. Wigner Measure 11
2.1. Semiclassical Pseudodifferential Operators and the FBI Transform 11
2.2. Wigner Measure 14
2.3. Semiclassical Limit of the Linear Schrodinger Equation 30
2.4. References and Remarks 34
Chapter 3. The Limit from the One-Dimensional Schrodinger-Poisson
to Vlasov-Poisson Equations 37
3.1. Introduction 37
3.2. Uniform Estimates 42
3.3. The Limit to the Vlasov-Poisson Equations 46
3.4. The Main Theorem 73
3.5. Mixed State Case 76
3.6. References and Remarks 79
Chapter 4. Semiclassical Limit of Schrodinger-Poisson Equations 81
4.1. Introduction 81
4.2. Preliminaries 84
4.3. Modulated Energy Estimate 89
4.4. The Main Theorem 95
4.5. References and Remarks 104
Chapter 5. Semiclassical Limit of the Cubic Schrodinger Equation
in an Exterior Domain 105
5.1. Introduction 105
5.2. Local Existence of a Smooth Solution to (5.7)-(5.8) 109
5.3. Uniform and Modulated Energy Estimates 115
5.4. Semiclassical Limit of the Cubic Schrodinger Equation 125
5.5. References and Remarks 127
Chapter 6. Incompressible and Compressible Limits of Coupled Systems
of Nonlinear Schrodinger Equations 129
6.1. Introduction 129
6.2. Conservation Laws 131
6.3. Incompressible Limit of (6.1) 136
6.4. Compressible Limit of (6.2) 146
6.5. References and Remarks 152
Chapter 7. High-Frequency Limit of the Helmholtz Equation 153
7.1. Introduction 153
7.2. Uniform Estimate for Solutions of (7.1) 154
7.3. Limit to the Liouville Equation 159
7.4. References and Remarks 171
Appendix A. Global Solutions to (3.14) 173
Appendix B. Denseness of Polynomials 177
Appendix C. Global Existence of a Solution to (5.1) 179
Appendix D. Global Smooth Solution to (6.1) 189
Bibliography 193

Library of Congress Subject Headings for this publication:

Schráeodinger equation.
WKB approximation.
Differential equations, Nonlinear.
Nonlinear theories.