The Quantum Sabine Law for Resonances in Transmission Problems

Abstract

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.

75 pages, 10 figures. A portion of the semiclassical preliminaries section is taken from arXiv:1204.1305 with the authors' permission

References in corpus (1)

• Quantum ergodicity for restrictions to hypersurfaces

Cited by in corpus (5)

• Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index
• Schrödinger operators with oblique transmission conditions in ${\mathbb{R}}^2$
• Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
• Mathematical Study of Scattering Resonances
• High-frequency approximation of the interior dirichlet-to-neumann map and applications to the transmission eigenvalues