Stability of the inverse resonance problem on the line

TL;DR

This paper investigates the stability of the inverse resonance problem for Schrödinger operators on the line, showing that small changes in spectral data lead to small changes in the potential, especially in the absence of half-bound states.

Contribution

It establishes quantitative stability results for the inverse resonance problem, including cases with and without half-bound states, based on spectral data proximity.

Findings

Potentials are stable under small spectral data perturbations.

Stability results hold for potentials without half-bound states.

Small perturbations of the zero potential are stable using eigenvalues and resonances.

Abstract

In the absence of a half-bound state, a compactly supported potential of a Schr\"odinger operator on the line is determined up to a translation by the zeros and poles of the meropmorphically continued left (or right) reflection coefficient. The poles are the eigenvalues and resonances, while the zeros also are physically relevant. We prove that all compactly supported potentials (without half-bound states) that have reflection coefficients whose zeros and poles are $\eps$-close in some disk centered at the origin are also close (in a suitable sense). In addition, we prove stability of small perturbations of the zero potential (which has a half-bound state) from only the eigenvalues and resonances of the perturbation.