The Quantum Sabine Law for Resonances in Transmission Problems

We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the author's previous work to a broader class of systems. Our main applications are to scattering by transparent obstacles, scattering by highly frequency dependent delta potentials, and boundary stabilized wave equations. We give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances or generalized eigenvalues to the chord lengths and reflectivity coefficients for the ray dynamics, thus proving a quantum version of the Sabine law.