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

TL;DR

This work analyzes the concentration of eigenfunctions of the Dirichlet Laplacian for a singular metric with critical near-boundary behavior ($\beta=2/n$). By deriving trace asymptotics via a one-dimensional model operator $P_{\mu}$ and bracketing techniques, it quantifies how the average eigenfunction density below $\lambda$ distributes across boundary-adjacent scales $X_{\lambda^{\gamma}}$ with $\gamma\in[-1/2,0]$, yielding the exact limit $\lim_{\lambda\to\infty}\frac{1}{N(\lambda)}\sum_{\lambda_j<\lambda}\int_{X_{\lambda^{\gamma}}}|\Phi_j|^2\,d\mathrm{vol}_g=2\big(\tfrac{1}{2}+\gamma\big)$. An equivalent boundary-formulation using $\log x/\log \lambda$ is provided. The results show that, in the critical regime, eigenfunction mass is distributed across a continuum of scales near the boundary, in contrast to the sharp localization observed in some supercritical cases, and they extend the understanding of spectral concentration in singular geometric settings.

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 $λ>0$ is distributed over all length scales between $λ^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $λ\to\infty$.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• proof : Proof of Theorem \ref{['th:main']}
• proof : Proof of Theorem \ref{['th:test_fct']}