2024/10/19 by Illya M. Karabash, Karabash, Illya M.
Computer Science · Mathematics · #35F45 #35P05 (Primary) 58J90 47B44 (Secondary) #47B80 #60H25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2410.15150
openalex publication_date 2024/10/19 · openalex created_date 2024/11/06 · openalex updated_date 2026/07/28
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 O. First, a parametrization of essentially all m-dissipative boundary condition by contraction operators in the boundary L2-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 Δ∂ O on the boundary 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 Δ∂ O.