3D Helmholtz resonator with two close point-like windows: Regularisation for Dirichlet case

Abstract

In this paper, a model of 3D Helmholtz resonator with two close point-like windows is considered. The Dirichlet condition is assumed at the boundary. The model is based on the theory of self-adjoint extensions of symmetric operators in Pontryagin space. The model is explicitly solvable and allows one to obtain the equation for resonances (quasi-eigenvalues) in an explicit form. A proper choice of the model parameter leads to the coincidence of the model solution with the main term of the asymptotics (in the window width) of the realistic solution, corresponding to small windows. A regularization is suggested to obtain a realistic limiting result for two merging windows.