Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) / / Christopher D. Sogge.

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Sogge, Christopher D.,
• 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
• Physical Description: 1 online resource (208 p.) :; 1 line illus.

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.

Table of Contents

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