Resonance asymptotics for quantum waveguides with semitransparent multi-perforated wall

Abstract

A pair of coupled quantum waveguides with a common semitransparent wall is considered. The wall has a finite number of small windows. We consider resonance states localized near each window. The presence of several windows forces one to describe their common influence differently from that of the single-window case. Using the "matching of asymptotic expansions" method, we derive formulas for resonances and resonance states.

Two-dimensional Helmholtz resonator with two close point-like windows: regularization for the Neumann case

“…But it was not until 1860 that Helmholtz formulated the first mathematical theory of Helmholtz resonators [9]. One of the directions in the study of Helmholtz resonators is the study of resonances that arise in many problems of mathematics, physics and technology [4,12,14,16,20,24,25]. Although the study of the resonances of the Helmholtz resonator was carried out by Rayleigh [22], it is still relevant.…”