Eigenfunctions of the Laplacian on a Riemannian Manifold
About this Title
Steve Zelditch, Northwestern University, Evanston, IL
Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 2017; Volume 125
ISBNs: 978-1-4704-1037-7 (print); 978-1-4704-4344-3 (online)
MathSciNet review: MR3729409
MSC: Primary 58J50; Secondary 31C12, 35J05, 35L05, 35P20, 35R01, 58J40
Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow.
The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain.
The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.
Graduate students and researchers interested in analysis related to spectral theory and eigenfunctions of Laplacians on Riemannian manifolds.
Table of Contents
- Geometric preliminaries
- Main results
- Model spaces of constant curvature
- Local structure of eigenfunctions
- Hadamard parametrics on Riemannian manifolds
- Lagrangian distributions and Fourier integral operators
- Small time wave group and Weyl asymptotics
- Matrix elements
- $L^p$ norms
- Quantum integrable systems
- Restriction theorems
- Nodal sets: Real domain
- Eigenfunctions in the complex domain