Inverse eigenvalue problem for discrete three-diagonal Sturm-Liouville operator and the continuum limit

TL;DR

This paper presents a rigorous derivation of the inverse eigenvalue problem for a discrete Sturm-Liouville operator using a three-diagonal matrix approximation, connecting it to the continuous case through a limiting process.

Contribution

It provides a more accurate derivation of the inverse problem using a three-diagonal matrix and relates it to the Gram-Schmidt process, improving upon previous single-diagonal approaches.

Findings

Inverse problem derivation via Gram-Schmidt orthonormalization.

Connection between discrete and continuous inverse problems through limiting procedures.

Reproduction of continuous inverse problem results, including Goursat problem.

Abstract

In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schroedinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that inverse problem procedure is nothing else than well known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by the space coordinate index. All the results of usual inverse problem with continuous coordinate are reobtained by employing a limiting procedure, including the Goursat problem -- equation in partial derivatives for the solutions of the inversion integral equation.