The critical case for the concentration of eigenfunctions on singular Riemannian manifolds

TL;DR

This paper investigates how eigenfunctions of a Laplace-Beltrami operator on a singular Riemannian manifold with boundary concentrate across scales, revealing a uniform distribution pattern in the critical case as eigenvalues grow large.

Contribution

It provides a detailed analysis of eigenfunction distribution on singular manifolds at the critical case, extending understanding of spectral behavior near boundaries with singular metrics.

Findings

Eigenfunctions are evenly distributed across all scales near the boundary in the critical case.

The distribution of eigenfunctions remains uniform over scales between $ ext{lambda}^{-1/2}$ and 1 as eigenvalues increase.

A precise asymptotic description of eigenfunction distribution as $ ext{lambda} o \infty$.

Abstract

We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus a potential. We show in the critical case that the average density of eigenfunctions for the Laplace-Beltrami operator with eigenvalues below $\lambda>0$ is distributed over all length scales between $\lambda^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $\lambda\to\infty$.