Motivated by engineering and Photonics research on resonators in random or uncertain environments, we introduce rigorous randomizations of boundary conditions for acoustic wave equations in Lipschitz domains. First, a parametrization of essentially all m-dissipative boundary conditions in the boundary L2-space is constructed with the use of boundary tuples. Randomizations of these boundary conditions lead to acoustic operators random in the resolvent sense. We prove this using Neumann-to-Dirichlet maps and generalized Krein resolvent formulae. In order to pass to point processes of random eigenvalues, 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. Based on these results, examples of mathematically convenient randomizations are constructed in terms of eigenfunctions of the Laplace-Beltrami operator on the boundary. We show that for these special randomizations the resolvent compactness is connected with the Weyl law for the Laplace-Beltrami eigenvalues, and also discuss available Weyl-asymptotic results for the case of nonsmooth boundaries.
Moritz Kappes / JJL Velázquez