Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) / / Christopher D. Sogge.
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 imp...
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Superior document: | Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 |
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Place / Publishing House: | Princeton, NJ : : Princeton University Press, , [2014] ©2014 |
Year of Publication: | 2014 |
Edition: | Course Book |
Language: | English |
Series: | Annals of Mathematics Studies ;
188 |
Online Access: | |
Physical Description: | 1 online resource (208 p.) :; 1 line illus. |
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Other title: | Frontmatter -- Contents -- Preface -- 1. A review: The Laplacian and the d'Alembertian -- 2. Geodesics and the Hadamard parametrix -- 3. The sharp Weyl formula -- 4. Stationary phase and microlocal analysis -- 5. Improved spectral asymptotics and periodic geodesics -- 6. Classical and quantum ergodicity -- Appendix -- Notes -- Bibliography -- Index -- Symbol Glossary |
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Summary: | 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. |
Format: | Mode of access: Internet via World Wide Web. |
ISBN: | 9781400850549 9783110494914 9783110665925 |
DOI: | 10.1515/9781400850549 |
Access: | restricted access |
Hierarchical level: | Monograph |
Statement of Responsibility: | Christopher D. Sogge. |