Symmetry of Bound and Antibound States in the Semiclassical Limit for a General Class of Potentials

TL;DR

This paper studies the symmetry of bound states and resonances in a semiclassical Schrödinger operator with specific potentials, showing they are symmetric up to exponentially small errors as the semiclassical parameter approaches zero.

Contribution

It demonstrates that eigenvalues and purely imaginary resonances are symmetric up to an exponentially small error for a class of semiclassical Schrödinger operators with compactly supported potentials.

Findings

Eigenvalues and resonances are symmetric up to $Ce^{-rac{ ext{const}}{h}}$ error.

Symmetry holds for potentials positive near the support endpoint.

Results apply to a broad class of potentials on the half-line.

Abstract

We consider the semiclassical Schr\"odinger operator $-h^2\partial_x^2+V(x)$ on a half-line, where $V$ is a compactly supported potential which is positive near the endpoint of its support. We prove that the eigenvalues and the purely imaginary resonances are symmetric up to an error $Ce^{-\delta/h}$.