The Quantum Sabine Law for Resonances in Transmission Problems

TL;DR

This paper establishes a quantum analogue of the Sabine law, linking resonance locations in transmission problems to dynamical ray properties, with applications to scattering and wave equations.

Contribution

It extends previous work to a broader class of systems, providing a sharp characterization of resonance free regions via dynamical quantities.

Findings

Resonance locations are related to ray dynamics and reflectivity.

Sharp resonance free regions are characterized in terms of chord lengths.

The work applies to scattering by obstacles and boundary stabilized wave equations.

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.