Eigenvalue fluctuations for the crushed ice problem

Abstract

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon ,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq {\mathbb{R}}^d$ ($d\ge 2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to \infty$ and $\epsilon \downarrow 0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa >0$, the eigenvalues of $-\Delta$ in $D_{\epsilon ,N}$ homogenize to those of the Schrödinger operator $-\Delta +\kappa \rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge 2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

40 pages