SCATTERING RESONANCES OF A HELMHOLTZ RESONATOR

We prove the existence of and give estimates for scattering resonances for the wave equation of an exterior domain R(n)/OMEGA approximately, where OMEGA approximately is a compact resonator with Dirichlet boundary conditions. The resonator OMEGA approximately consists of a cavity C with a tube Z(epsilon) joining it to the exterior epsilon which closes as epsilon --> 0. For any bounded region K in the complex plane and for epsilon sufficiently small, there exists a bijection between the scattering resonances in K and the interior eigenvalues coming from C union Z(epsilon) and the exterior resonances for epsilon union Z(epsilon) in K (both with Dirichlet boundary conditions). Moreover, the distance between them is shown to be O(e-(-c/epsilon)), where c > 0 is expressible in terms of the distance from C to epsilon along Z(epsilon).