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We prove a quantum version of the Sabine law from acoustics describing thelocation of resonances in transmission problems. This work extends the author'sprevious work to a broader class of systems. Our main applications are toscattering by transparent obstacles, scattering by highly frequency dependentdelta potentials, and boundary stabilized wave equations. We give a sharpcharacterization of the resonance free regions in terms of dynamicalquantities. In particular, we relate the imaginary part of resonances orgeneralized eigenvalues to the chord lengths and reflectivity coefficients forthe ray dynamics, thus proving a quantum version of the Sabine law.

The Quantum Sabine Law for Resonances in Transmission Problems