Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries

TL;DR

This paper investigates how random boundary conditions in acoustic wave problems on Lipschitz domains affect the spectral properties, establishing a link between boundary eigenvalue asymptotics and Weyl's law.

Contribution

It introduces a comprehensive framework for randomizing boundary conditions via contraction operators, connecting spectral properties with boundary Laplace-Beltrami eigenvalues.

Findings

Random boundary conditions produce operators with almost surely discrete spectra.

The compactness of the resolvent relates to Weyl-type eigenvalue asymptotics.

A new randomization method based on boundary Laplace-Beltrami eigenfunctions is proposed.

Abstract

Motivated by engineering and photonics research on resonators in random or uncertain environments, we study rigorous randomizations of boundary conditions for wave equations of the acoustic-type in Lipschitz domains $\mathcal{O}$. First, a parametrization of essentially all m-dissipative boundary condition by contraction operators in the boundary $L^2$-space is constructed with the use of m-boundary tuples (boundary value spaces). We consider randomizations of these contraction operators that lead to acoustic operators random in the resolvent sense. To this end, the use of Neumann-to-Dirichlet maps and Krein-type resolvent formulae is crucial. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. For each particular applied model, one can choose a specific boundary condition from the constructed class either by means of optimization, or on the base of empirical observations. A mathematically convenient randomization is constructed in terms of eigenfunctions of the Laplace-Beltrami operator $\Delta^{\partial \mathcal{O}}$ on the boundary $\mathcal{O}$ of the domain. We show that for this randomization the compactness of the resolvent is connected with the Weyl-type asymptotics for the eigenvalues of $\Delta^{\partial \mathcal{O}}$.