Inverse Eigenvalue Problems for Perturbed Spherical Schroedinger Operators

TL;DR

This paper studies how small perturbations affect the eigenvalues of spherical Schrödinger operators and demonstrates that spectral data can uniquely identify the perturbation function.

Contribution

It provides a detailed analysis of eigenvalue shifts due to perturbations and establishes uniqueness results for the inverse spectral problem.

Findings

Eigenvalues are approximated by unperturbed eigenvalues with decaying error.

Spectral data uniquely determine the perturbation function q(x).

Results depend on the behavior of q(x) near x=0.

Abstract

We investigate the eigenvalues of perturbed spherical Schr\"odinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to an decaying error depending on the behavior of $q(x)$ near $x=0$. Furthermore, we provide sets of spectral data which uniquely determine $q(x)$.