The inverse eigenvalue problems for perturbed Bessel operator with mixed data

TL;DR

This paper investigates inverse eigenvalue problems for the perturbed Bessel operator, establishing uniqueness results for different ranges of the angular momentum quantum number and extending known results to broader cases.

Contribution

It provides new uniqueness theorems for inverse spectral problems of the perturbed Bessel operator for a wider range of the angular momentum quantum number.

Findings

Uniqueness results for $ ext{l} ext{ in } ext{N} ext{ or } ext{l} ext{ in } [-1/2, ext{N}]$

Extension of known results from $ ext{l}=0$ to $ ext{l} ext{ in } [-1/2, ext{N}]$

Corollaries based on the broader case of $ ext{l} ext{ in } [-1/2, ext{N}]$

Abstract

We consider inverse eigenvalue problems for the perturbed Bessel operator in $L^{2}(0,1)$. (1) For the case where the angular-momentum quantum number $\ell\in\mathbb{N}\cup\{0\}$, we establish a uniqueness result for the inverse spectral problem by utilizing the closedness condition of a certain function system constructed based on the eigenvalues and the norming constants. (2) For the broader case where $\ell \geq -1/2$, we provide a uniqueness result for the inverse problem by using the density condition satisfied by the eigenvalues and the norming constants, where an additional smoothness condition may be imposed on the potential. (3) In the last section of this article, we present some corollaries based on (2). The results in these corollaries have already been established for the case $\ell=0$ by Gesztesy, Simon, Wei, Xu, Hatino\v{g}lu, et al., and we extend these results to the general case $\ell \geq -1/2$.