Uniqueness Results for Schroedinger Operators on the Line with Purely Discrete Spectra

TL;DR

This paper develops an abstract framework for one-dimensional Schrödinger operators with purely discrete spectra, demonstrating how their spectra and norming constants uniquely determine the operators, with applications to quantum harmonic oscillators.

Contribution

It introduces a new abstract approach using singular Weyl-Titchmarsh theory to establish uniqueness results for Schrödinger operators with discrete spectra, including perturbed harmonic oscillators.

Findings

Proves a new uniqueness theorem for certain Schrödinger operators.

Extends Weyl-Titchmarsh theory to singular operators.

Establishes a Hochstadt-Liebermann type result for these operators.

Abstract

We provide an abstract framework for singular one-dimensional Schroedinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove a new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Liebermann type result for these operators. Our approach is based on the singular Weyl-Titchmarsh theory which is extended to cover the present situation.