Symmetry of bound and antibound states in the semiclassical limit

TL;DR

This paper demonstrates that in one-dimensional scattering, bound and antibound states are nearly symmetric in the semiclassical limit when a mild positive barrier exists, supported by numerical methods for resonance computation.

Contribution

It reveals the symmetry of bound and antibound states in the semiclassical limit under specific barrier conditions, supported by a numerical scheme for resonance calculation.

Findings

Bound and antibound states are symmetric modulo exponentially small errors in 1/h.

Numerical scheme enables efficient computation of resonances in one dimension.

The symmetry holds when a mild positive barrier separates the interaction region from infinity.

Abstract

We consider one dimensional scattering and show how the presence of a mild positive barrier separating the interaction region from infinity implies that the bound and antibound states are symmetric modulo exponentially small errors in 1/h. This simple result was inspired by a numerical experiment and we describe the numerical scheme for an efficient computation of resonances in one dimension.