Inverse uniqueness results for Schr\"odinger operators using de Branges theory

TL;DR

This paper employs de Branges space theory to establish conditions under which Schrödinger operators with singular potentials are uniquely identified by their spectral measures, advancing inverse spectral problem solutions.

Contribution

It introduces a novel application of de Branges theory to prove inverse uniqueness for Schrödinger operators with singular potentials, including perturbed spherical cases.

Findings

Spectral measure uniquely determines certain Schrödinger operators.

Established inverse uniqueness for operators with strongly singular potentials.

Extended results to perturbed spherical Schrödinger operators.

Abstract

We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness theorem for perturbed spherical Schr\"odinger operators.