Random acoustic boundary conditions and Weyl's lawOberseminar Analysis

by Dr Illia Karabash (IAM Funktionalanalysis)

Thursday Jan 23, 2025, 2:15 PM → 4:00 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

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.