Concentration of eigenfunctions on singular Riemannian manifolds

TL;DR

This paper studies how high-frequency eigenfunctions concentrate near the boundary of a singular Riemannian manifold, revealing their profiles and rates, with applications to acoustic modes on gas planets.

Contribution

It characterizes the concentration behavior of eigenfunctions on manifolds with boundary and singular metrics, extending understanding of spectral properties in singular geometric settings.

Findings

Eigenfunctions concentrate at the boundary in a supercritical regime

The profile and rate of concentration are explicitly identified

Application to acoustic modes on gas planets demonstrates practical relevance

Abstract

We consider a compact Riemannian manifold with boundary and a metric that is singular at the boundary. The associated Laplace-Beltrami operator is of the form of a Grushin operator plus a singular potential. In a supercritical parameter regime, we identify the rate of concentration and profile of the high-frequency eigenfunctions that accumulate at the boundary. We give an application to acoustic modes on gas planets.