Hangzhou Lectures on Eigenfunctions of the Laplacian

Preface ix
1A review: The Laplacian and the d'Alembertian 1
1.1 The Laplacian 1
1.2 Fundamental solutions of the d'Alembertian 6
2Geodesics 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
3The 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
4Stationary 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
5Improved 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
6Classical 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'(O) and E'(O) 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