Hangzhou Lectures on Eigenfunctions of the Laplacian

Contains the material taught during a series of lectures given at Zhejiang University in Hangzhou between 2010 and 2012.

List of contents

About the author

Christopher D. Sogge

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.

Additional text

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

Product details

• Authors: Christopher D. Sogge, Sogge Christopher D.
• Assisted by: Phillip Griffiths (Editor), John N. Mather (Editor), Elias Stein (Editor)
• Publisher: Princeton University Press
• Languages: English
• Product format: Hardback
• Released: 10.03.2014
• EAN: 9780691160757
• ISBN: 978-0-691-16075-7
• No. of pages: 208
• Series: Annals of Mathematics Studies; Annals of Mathematics Studies
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Basic principles; MATHEMATICS / Differential Equations / General, MATHEMATICS / Geometry / Differential, Differential calculus & equations, Differential & Riemannian geometry, Differential and Riemannian geometry, Differential calculus and equations