Tunneling resonances in systems without a classical trapping

TL;DR

This paper investigates quantum tunneling resonances in a waveguide with finite barriers, showing how these resonances approach eigenvalues of an infinite barrier system as the barrier lengths increase.

Contribution

It provides an analysis of how finite barrier systems exhibit resonances converging to eigenvalues of the infinite barrier case, including asymptotic expansion derivations.

Findings

Resonances approach eigenvalues as barrier lengths increase

Asymptotic expansion of resonance positions derived

Finite barriers produce resonances converging to the infinite barrier eigenvalues

Abstract

In this paper we analyze a free quantum particle in a straight Dirichlet waveguide which has at its axis two Dirichlet barriers of lengths $\ell_\pm$ separated by a window of length 2a. It is known that if the barriers are semiinfinite, i.e. we have two adjacent waveguides coupled laterally through the boundary window, the system has for any a>0 a finite number of eigenvalues below the essential spectrum threshold. Here we demonstrate that for large but finite $\ell_\pm$ the system has resonances which converge to the said eigenvalues as $\ell_\pm\to\infty$, and derive the leading term in the corresponding asymptotic expansion.