DeepAI

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

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.

• 6 publications
• 7 publications
10/28/2020

### A direct method for solving inverse Sturm-Liouville problems

We consider two main inverse Sturm-Liouville problems: the problem of re...
10/26/2020

### Mining for Spectra - The Dortmund Spectrum Estimation Algorithm

Obtaining the energy spectra of incident particles such as neutrinos or ...
07/08/2021

### Toeplitz matrices in the Boundary Control method

Solving inverse problems by dynamical variant of the BC-method is basica...
02/12/2020

### On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Parameter identification problems for partial differential equations are...
03/24/2022

### Universal algorithms for solving inverse spectral problems

This paper explores the complexity associated with solving the inverse S...
07/26/2020

### A transmutation operator method for solving the inverse quantum scattering problem

The inverse quantum scattering problem for the perturbed Bessel equation...
08/24/2021

### Finite-difference approximation of the inverse Sturm-Liouville problem with frozen argument

This paper deals with the discrete system being the finite-difference ap...