A practical method for recovering Sturm-Liouville problems from the Weyl function

TL;DR

This paper introduces a direct, numerically efficient method for reconstructing Sturm-Liouville potentials from the Weyl function, applicable to various inverse spectral problems including those with multiple spectra and quantum graphs.

Contribution

The paper develops a novel direct approach using Fourier-Legendre series to solve inverse Sturm-Liouville problems from the Weyl function, extending applicability to complex spectral data.

Findings

The method effectively reconstructs potentials from partial spectral data.

It provides a unified framework for inverse problems with multiple spectra and boundary conditions.

Numerical algorithms demonstrate high accuracy and efficiency.

Abstract

In the paper we propose a direct method for recovering the Sturm-Liouville potential from the Weyl-Titchmarsh $m$-function given on a countable set of points. We show that using the Fourier-Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from arXiv:2010.15275 is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.