Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) [electronic resource].

Format:

Book

Author/Creator:

Sogge, Christopher D.

Series:

Annals of Mathematics Studies

Language:

English

Subjects (All):

Eigenfunctions.

Laplacian operator.

Local Subjects:

Eigenfunctions.

Laplacian operator.

Physical Description:

1 online resource (206 p.)

Place of Publication:

Princeton : Princeton University Press, 2014.

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 fi

Contents:

Cover; Title; Copyright; Dedication; Contents; Preface; 1 A review: The Laplacian and the d'Alembertian; 1.1 The Laplacian; 1.2 Fundamental solutions of the d'Alembertian; 2 Geodesics and the Hadamard parametrix; 2.1 Laplace-Beltrami operators; 2.2 Some elliptic regularity estimates; 2.3 Geodesics and normal coordinates-a brief review; 2.4 The Hadamard parametrix; 3 The sharp Weyl formula; 3.1 Eigenfunction expansions; 3.2 Sup-norm estimates for eigenfunctions and spectral clusters; 3.3 Spectral asymptotics: The sharp Weyl formula; 3.4 Sharpness: Spherical harmonics

3.5 Improved results: The torus3.6 Further improvements: Manifolds with nonpositive curvature; 4 Stationary phase and microlocal analysis; 4.1 The method of stationary phase; 4.2 Pseudodifferential operators; 4.3 Propagation of singularities and Egorov's theorem; 4.4 The Friedrichs quantization; 5 Improved spectral asymptotics and periodic geodesics; 5.1 Periodic geodesics and trace regularity; 5.2 Trace estimates; 5.3 The Duistermaat-Guillemin theorem; 5.4 Geodesic loops and improved sup-norm estimates; 6 Classical and quantum ergodicity; 6.1 Classical ergodicity; 6.2 Quantum ergodicity

Notes:

Description based upon print version of record.

OCLC:

869281847