We prove a quantum version of the Sabine law from acoustics describing the location
of resonances in transmission problems. This work extends the work of the author 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.
Keywords
transmission, resonances, boundary integral operators,
transparent, scattering
