On a method for solving the inverse Sturm–Liouville problem

Recieved: 03/31/2025

Accepted: 05/04/2025

Keywords: Schrödinger and telegraph equation, reaction function, inverse scattering problem, revers of difference scheme

To cite this article

Baev A.V., Mozgovykh V.V. On a method for solving the inverse Sturm–Liouville problem. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2025. N 3, p.3-10 https://doi.org/10.55959/MSU/0137–0782–15–2025–49–3–3–10.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Baev A.V. (Lomonosov Moscow State University (MSU))
Mozgovykh V.V. (Lomonosov Moscow State University (MSU))

Abstract

The inverse Sturm–Liouville problem consists in determining the coefficient (potential) in the stationary Schrцdinger equation on a segment based on a set of eigenvalues. The paper considers a numerical solution to the inverse problem based on a finite set of the first eigenvalues of two Sturm–Liouville problems. The remaining eigenvalues are set according to the classical asymptotics.

The method of solving the inverse spectral problem is based on a one-to-one correspondence between the inverse spectral problem and the nonstationary inverse problem for a telegraphic equation with a variable coefficient (potential). The reduction to a non-stationary problem is carried out analytically by inverting the Laplace transform according to the Mellin formula. An explicit formula for the reaction function in the inverse scattering problem is obtained.

The inverse scattering problem for the telegraphic equation is to determine an unknown coefficient from the reaction function. This problem is solved numerically by the inversion of a difference scheme. The paper presents the results of solving a series of inverse Sturm–Liouville problems. In conclusion, it is noted that the number of given frequencies corresponds to the number of harmonics in the expansion of a desired potential.