Direct and inverse problems for the matrix Sturm-Liouville operator with the general self-adjoint boundary conditions

TL;DR

This paper studies the spectral properties and inverse problems of a matrix Sturm-Liouville operator with general self-adjoint boundary conditions and singular potentials, extending classical results to more general settings.

Contribution

It introduces a framework for analyzing spectral data and proves the uniqueness of operator recovery from spectral information using spectral mapping methods.

Findings

Spectral data characterized for the operator with singular potentials.

Proved uniqueness of inverse spectral problem solution.

Analyzed asymptotic behavior of eigenvalues and spectral data.

Abstract

The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class $W_2^{-1}$ is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.