Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

TL;DR

This paper develops a constructive method to recover matrix Sturm-Liouville operators with singular potentials from spectral data, extending classical inverse spectral theory to more general operators on graphs.

Contribution

It introduces a new inverse problem solution and spectral data characterization for matrix Sturm-Liouville operators with singular potentials of class W_2^{-1}.

Findings

Reduced inverse problem to a linear equation in a Banach space.

Developed a constructive algorithm for inverse problem solution.

Obtained spectral data characterization for the operator.

Abstract

The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator.