Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window

TL;DR

This paper derives asymptotic estimates for the eigenvalues of Laplacians in planar waveguides with a narrow window, revealing how the eigenvalues shift as the window size diminishes.

Contribution

It provides precise asymptotic bounds for the eigenvalues in quantum waveguides coupled through a small window, extending understanding of spectral behavior in such geometries.

Findings

Eigenvalue shifts are proportional to the fourth power of the window size a.

The results apply to both Neumann and Dirichlet boundary conditions.

Asymptotic bounds are established for small window lengths.

Abstract

Consider the Laplacian in a straight planar strip of width $\,d\,$, with the Neumann boundary condition at a segment of length $\,2a\,$ of one of the boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has a single eigenvalue $\,\epsilon(a)\,$; we show that there are positive $\,c_1,c_2\,$ such that $\,-c_1 a^4 \le \epsilon(a)- \left(\pi/ d\right)^2 \le -c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length $\,2a\,$ in the common boundary.