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

Christopher D. Sogge

Published by Princeton University Press 2014

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

Christopher D. Sogge

Volume 188 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400850549

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About this book

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.

Author information

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.

Reviews

The book is very well written. . . . I would definitely recommend it to anybody who wants to learn spectral geometry.---Leonid Friedlander, Mathematical Reviews

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Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Requires Authentication Unlicensed Licensed Download PDF	ix
1. A review: The Laplacian and the d'Alembertian Requires Authentication Unlicensed Licensed Download PDF	1
2. Geodesics and the Hadamard parametrix Requires Authentication Unlicensed Licensed Download PDF	16
3. The sharp Weyl formula Requires Authentication Unlicensed Licensed Download PDF	39
4. Stationary phase and microlocal analysis Requires Authentication Unlicensed Licensed Download PDF	71
5. Improved spectral asymptotics and periodic geodesics Requires Authentication Unlicensed Licensed Download PDF	120
6. Classical and quantum ergodicity Requires Authentication Unlicensed Licensed Download PDF	141
Appendix Requires Authentication Unlicensed Licensed Download PDF	165
Notes Requires Authentication Unlicensed Licensed Download PDF	183
Bibliography Requires Authentication Unlicensed Licensed Download PDF	185
Index Requires Authentication Unlicensed Licensed Download PDF	191
Symbol Glossary Requires Authentication Unlicensed Licensed Download PDF	193

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