Pure and Applied Analysis

The quantum Sabine law for resonances in transmission problems

Jeffrey Galkowski

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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.

Article information

Pure Appl. Anal., Volume 1, Number 1 (2019), 27-100.

Received: 24 April 2018
Revised: 25 June 2018
Accepted: 8 August 2018
First available in Project Euclid: 4 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 35P25: Scattering theory [See also 47A40]

transmission resonances boundary integral operators transparent scattering


Galkowski, Jeffrey. The quantum Sabine law for resonances in transmission problems. Pure Appl. Anal. 1 (2019), no. 1, 27--100. doi:10.2140/paa.2019.1.27. https://projecteuclid.org/euclid.paa/1549297978

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