Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

by Christopher D. Sogge

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Product Details

ISBN-13: 9780691160757
Publisher: Princeton University Press
Publication date: 03/10/2014
Series: Annals of Mathematics Studies , #212
Pages: 208
Product dimensions: 7.20(w) x 10.00(h) x 1.00(d)

About the Author


Christopher D. Sogge is the J. J. Sylvester Professor of Mathematics at Johns Hopkins University. He is the author of Fourier Integrals in Classical Analysis and Lectures on Nonlinear Wave Equations.

Table of Contents


Preface ix
1 A review: The Laplacian and the d'Alembertian 1
1.1 The Laplacian 1
1.2 Fundamental solutions of the d'Alembertian 6
2 Geodesics and the Hadamard parametrix 16
2.1 Laplace-Beltrami operators 16
2.2 Some elliptic regularity estimates 20
2.3 Geodesics and normal coordinates|a brief review 24
2.4 The Hadamard parametrix 31
3 The sharp Weyl formula 39
3.1 Eigenfunction expansions 39
3.2 Sup-norm estimates for eigenfunctions and spectral clusters 48
3.3 Spectral asymptotics: The sharp Weyl formula 53
3.4 Sharpness: Spherical harmonics 55
3.5 Improved results: The torus 58
3.6 Further improvements: Manifolds with nonpositive curvature 65
4 Stationary phase and microlocal analysis 71
4.1 The method of stationary phase 71
4.2 Pseudodifferential operators 86
4.3 Propagation of singularities and Egorov's theorem 103
4.4 The Friedrichs quantization 111
5 Improved spectral asymptotics and periodic geodesics 120
5.1 Periodic geodesics and trace regularity 120
5.2 Trace estimates 123
5.3 The Duistermaat-Guillemin theorem 132
5.4 Geodesic loops and improved sup-norm estimates 136
6 Classical and quantum ergodicity 141
6.1 Classical ergodicity 141
6.2 Quantum ergodicity 153
Appendix 165
A.1 The Fourier transform and the spaces S(ℝn) and S'(ℝn)) 165
A.2 The spaces D'(Ω) and E'(Ω) 169
A.3 Homogeneous distributions 173
A.4 Pullbacks of distributions 176
A.5 Convolution of distributions 179
Notes 183
Bibliography 185
Index 191
Symbol Glossary 193

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