Local and global analysis of eigenfunctions

Abstract

This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and $L^p$ norms of global eigenfunctions.

A survey on eigenfunctions for the Handbook of Geometric Analysis, no. 1: L. Ji, P. Li, R. Schoen and L. Simon (eds.),

References in corpus (10)

• Scarred eigenstates for quantum cat maps of minimal periods
• Nodal domains statistics - a criterion for quantum chaos
• Percolation model for nodal domains of chaotic wave functions
• On nodal sets for Dirac and Laplace operators
• L^p norms of eigenfunctions in the completely integrable case
• Patterson-Sullivan distributions and quantum ergodicity
• The morphology of nodal lines-random waves versus percolation
• The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)
• Quantum ergodicity of boundary values of eigenfunctions: A control theory approach
• On the Number of Nodal Domains of Random Spherical Harmonics

Cited by in corpus (9)

• $L^p$ norms, nodal sets, and quantum ergodicity
• Applications of small scale quantum ergodicity in nodal sets
• Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions
• Geodesics on Calabi-Yau manifolds and winding states in nonlinear sigma models
• Topologically penalized regression on manifolds
• The blowup along the diagonal of the spectral function of the Laplacian
• Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2
• Local $L^p$ norms of Schr{ö}dinger eigenfunctions on ${\mathbb{S}}^2$
• Patterson--Sullivan distributions in higher rank