Spectral Analysis of a Quantum Waveguide with Elliptical Window

TL;DR

This paper analyzes how the shape of an elliptical window affects the spectral properties of the Dirichlet Laplacian in coupled quantum waveguides, revealing eigenvalue splitting and geometric dependence.

Contribution

It provides a detailed spectral analysis of elliptic windows in quantum waveguides, highlighting effects of anisotropy and symmetry breaking on eigenvalues.

Findings

Finite number of discrete eigenvalues below the essential spectrum

Eigenvalue splitting due to elliptic geometry

Eigenvalues vary with geometric parameters

Abstract

We investigate the Dirichlet Laplacian in two spatial waveguides coupled through an elliptic window. The elliptic geometry breaks rotational symmetry and introduces anisotropy through the semi-axes of the aperture, which modifies the coupling of transverse modes and the low-lying spectrum. We prove that the operator has a finite number of discrete eigenvalues below the threshold of the essential spectrum and study their dependence on the geometric parameters of the ellipse. In contrast to the circular case, the elliptic setting gives rise to spectral effects such as eigenvalue splitting. Numerical simulations illustrate the variation of the first eigenvalues and the ground state with the window geometry.