Hangzhou Lectures on Eigenfunctions of the Laplacian

Christopher D. Sogge

• Publisher: Princeton University Press
• Year: 2014
• Print ISBN: 9780691160788, 9780691160757
• eBook ISBN: 9781400850549
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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

• Cover Page
• Title Page
• Copyright Page
• Dedication Page
• 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