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The inverse eigenvalue problems for perturbed Bessel operator with mixed data

Zeguang Liu, Xin-Jian Xu

TL;DR

This work analyzes the inverse spectral problem for the perturbed Bessel operator $L(,q)$ on $(0,1)$ with endpoint singularity in the framework $\u2264 - frac12$, focusing on recovering the real potential $q$ from mixed spectral data. The authors introduce the characteristic function $(,q)$ and the entire function $H_{}()$ to connect eigenvalues and norming constants to the potential, and develop a uniqueness theory based on the closedness of a function system $S_{}(,)$ and the operator $A_{}$. They establish three main results: (i) a sufficiency condition for uniqueness under closedness, (ii) a conditional uniqueness under an isomorphism assumption on $A_q$ with density-type inputs, and (iii) a Jensen-formula–driven strong uniqueness criterion using the counting function $m(t)$; these are then extended to corollaries for various mixed-data scenarios, including generalized $rac12$ cases. The results broaden prior $$-case conclusions to the full range $rac12$, providing practical criteria for unique recovery of $q$ from mixed spectral data and partial potentials, with implications for radial Schrödinger and spherical problems.

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, Hatinoǧlu, et al., and we extend these results to the general case $\ell \geq -1/2$.

The inverse eigenvalue problems for perturbed Bessel operator with mixed data

TL;DR

This work analyzes the inverse spectral problem for the perturbed Bessel operator on with endpoint singularity in the framework , focusing on recovering the real potential from mixed spectral data. The authors introduce the characteristic function and the entire function to connect eigenvalues and norming constants to the potential, and develop a uniqueness theory based on the closedness of a function system and the operator . They establish three main results: (i) a sufficiency condition for uniqueness under closedness, (ii) a conditional uniqueness under an isomorphism assumption on with density-type inputs, and (iii) a Jensen-formula–driven strong uniqueness criterion using the counting function ; these are then extended to corollaries for various mixed-data scenarios, including generalized cases. The results broaden prior -case conclusions to the full range , providing practical criteria for unique recovery of from mixed spectral data and partial potentials, with implications for radial Schrödinger and spherical problems.

Abstract

We consider inverse eigenvalue problems for the perturbed Bessel operator in . (1) For the case where the angular-momentum quantum number , 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 , 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 by Gesztesy, Simon, Wei, Xu, Hatinoǧlu, et al., and we extend these results to the general case .
Paper Structure (5 sections, 19 theorems, 121 equations)

This paper contains 5 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The characteristic function $\Delta(\lambda,q)$
  3. The entire function $H_{\ell}(\lambda)$
  4. Proofs of Theorems \ref{['t4.1']}--\ref{['t4.3']}
  5. Some corollaries of Theorem \ref{['t4.3']}

Key Result

Theorem 1.1

Let $\ell\in \mathbb{N}\cup\{0\}$, $p\in(1, \infty)$, $q\in L^p(0,1)$, and $a \in(0,1]$. Suppose the elements $\lambda_n$ in $\Lambda =\{\lambda_{n}\mid\lambda_{n}\in\sigma(\ell, q, \beta_n), n\in\mathbb{N}\}$ satisfy $\lambda_n \nrightarrow -\infty$, and let $S \subset \Lambda$ be a subset such tha is closed in $L^p(0, a)$, then $q$ on $(a, 1), \Lambda$, and $S$ uniquely determine $q \in L^p(0, a

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 30 more