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Annals of Mathematics Studies #188: Hangzhou Lectures on Eigenfunctions of the Laplacian


Annals of Mathematics Studies #188: Hangzhou Lectures on Eigenfunctions of the Laplacian Cover


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Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.

Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

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

Product Details

Sogge, Christopher D.
Princeton University Press
Mathematical Analysis
Mathematics-Analysis General
Edition Description:
Trade paper
Annals of Mathematics Studies
Publication Date:
1 line illus.
10 x 7 in

Related Subjects

Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » Differential Geometry

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