Exponential splitting of bound states in a waveguide with a pair of distant windows

TL;DR

This paper analyzes how the bound states in a waveguide with two distant windows split exponentially as the distance between the windows increases, revealing detailed asymptotic behavior of the spectrum.

Contribution

It provides the first detailed asymptotic analysis of the eigenvalue splitting in a waveguide with two distant Neumann windows, including cases with threshold resonances.

Findings

Eigenvalues around each single-window eigenvalue split exponentially as distance increases.

Distances between eigenvalues vanish exponentially as the windows become distant.

Threshold resonances turn into isolated eigenvalues due to the presence of the second window.

Abstract

We consider Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann ``windows'' of the same length the centers of which are $2l$ apart, and study the asymptotic behaviour of the discrete spectrum as $l\to\infty$. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit $l\to\infty$. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.