Bound states in open coupled asymmetrical waveguides and quantum wires

TL;DR

This paper investigates bound states in asymmetric waveguide and quantum wire configurations, demonstrating through numerical and qualitative analysis that certain bound states persist even as one arm's width approaches zero.

Contribution

It provides a detailed analysis of bound states in asymmetric waveguides, showing their persistence under extreme width asymmetries and offering a qualitative explanation for this behavior.

Findings

The fundamental mode remains bound for various widths studied.

Bound states persist even when one arm's width approaches zero.

The lowest mode in certain configurations remains bound up to a critical width ratio.

Abstract

The behavior of bound states in asymmetric cross, T and L shaped configurations is considered. Because of the symmetries of the wavefunctions, the analysis can be reduced to the case of an electron localized at the intersection of two orthogonal crossed wires of different width. Numerical calculations show that the fundamental mode of this system remains bound for the widths that we have been able to study directly; moreover, the extrapolation of the results obtained for finite widths suggests that this state remains bound even when the width of one arm becomes infinitesimal. We provide a qualitative argument which explains this behavior and that can be generalized to the lowest energy states in each symmetry class. In the case of odd-odd states of the cross we find that the lowest mode is bounded when the width of the two arms is the same and stays bound up to a critical value of the ratio between the widths; in the case of the even-odd states we find that the lowest mode is unbound up to a critical value of the ratio between the widths. Our qualitative arguments suggest that the bound state survives as the width of the vertical arm becomes infinitesimal.