Bound States and Threshold Resonances in Quantum Wires with Circular Bends

TL;DR

This paper investigates how curvature in a two-dimensional quantum wire affects bound states and resonances, using a high-precision numerical method to analyze wave solutions, transmission, and reflection properties.

Contribution

Introduces a numerical approach for high-accuracy analysis of wave solutions in curved quantum wires, confirming analytic predictions and exploring resonance behavior near channel thresholds.

Findings

Bound state energies depend on curvature and bend angle.

Resonances occur near channel thresholds and are influenced by geometry.

Numerical results align with analytic predictions in limiting cases.

Abstract

We study the solutions to the wave equation in a two-dimensional tube of unit width comprised of two straight regions connected by a region of constant curvature. We introduce a numerical method which permits high accuracy at high curvature. We determine the bound state energies as well as the transmission and reflection matrices, ${\cal T}$ and ${\cal R}$ and focus on the nature of the resonances which occur in the vicinity of channel thresholds. We explore the dependence of these solutions on the curvature of the tube and angle of the bend and discuss several limiting cases where our numerical results confirm analytic predictions.