M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces

Abstract The paper studies the properties of acoustic operators in bounded Lipschitz domains $$\Omega $$ Ω with m-dissipative generalized impedance boundary conditions, which are used to model open acoustic and optical cavities. We prove that such acoustic operators have compact resolvent if and only if the impedance operator from the trace space $$H^{1/2} (\partial \Omega )$$ H 1 / 2 ( ∂ Ω ) to the other trace space $$H^{-1/2} (\partial \Omega )$$ H - 1 / 2 ( ∂ Ω ) is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is partially based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations and to models involving conservative or lossy resonators. In order to study impedance boundary conditions with $$L^q$$ L q -coefficients, we consider the classes of pointwise multipliers between fractional Sobolev spaces on Lipschitz boundaries $$\partial \Omega $$ ∂ Ω . Examples of m-dissipative boundary conditions that lead to acoustic operators with non-empty essential spectra are also provided.