Pure and Applied Analysis

The quantum Sabine law for resonances in transmission problems

Jeffrey Galkowski

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.paa/1549297978

Digital Object Identifier
doi:10.2140/paa.2019.1.27

Mathematical Reviews number (MathSciNet)
MR3900029

Zentralblatt MATH identifier
07027485

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

Keywords
transmission resonances boundary integral operators transparent scattering

Citation

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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