Distant perturbation asymptotics in window-coupled waveguides. I. The non-threshold case

TL;DR

This paper investigates how the discrete spectrum of coupled quantum waveguides behaves asymptotically as the distance between windows increases, focusing on the generic case where eigenvalues are separated from the threshold.

Contribution

It provides new asymptotic formulas for the discrete spectrum in window-coupled waveguides with different widths, expanding understanding of spectral behavior in non-threshold cases.

Findings

Asymptotic behavior of the discrete spectrum is characterized as window distance increases.

Results apply to waveguides with different widths and non-symmetric window configurations.

The analysis covers eigenvalues separated from the spectral threshold.

Abstract

We consider a pair of adjacent quantum waveguides, in general of different widths, coupled laterally by a pair of windows in the common boundary, not necessarily of the same length, at a fixed distance. The Hamiltonian is the respective Dirichlet Laplacian. We analyze the asymptotic behavior of the discrete spectrum as the window distance tends to infinity for the generic case, i.e. for eigenvalues of the corresponding one-window problems separated from the threshold.