Vol. 1, No. 1, 2019
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The quantum Sabine law for resonances in transmission problems

Jeffrey Galkowski
Vol. 1 (2019), No. 1, 27–100
DOI: 10.2140/paa.2019.1.27
Abstract

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.

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Keywords
transmission, resonances, boundary integral operators, transparent, scattering
Mathematical Subject Classification 2010
Primary: 35P20, 35P25
Milestones
Received: 24 April 2018
Revised: 25 June 2018
Accepted: 8 August 2018
Published: 21 November 2018
Authors
Jeffrey Galkowski
Mathematics Department
Stanford University
Stanford, CA
United States