Recovery of the matrix quadratic differential pencil from the spectral data

TL;DR

This paper addresses the inverse spectral problem for matrix Sturm-Liouville operators, providing a method to recover the operator pencil from spectral data using spectral mappings and offering a constructive solution algorithm.

Contribution

It introduces a new approach to recover matrix quadratic differential pencils from spectral data, including eigenvalues and weight matrices, via a linear equation in a Banach space.

Findings

Spectral characteristics of matrix Sturm-Liouville operators are analyzed.

A linear equation approach in Banach space is developed for inverse problems.

A constructive algorithm for recovering the operator pencil is provided.

Abstract

We consider a pencil of matrix Sturm-Liouville operators on a finite interval. We study properties of its spectral characteristics and inverse problems that consist in recovering of the pencil by the spectral data: eigenvalues and so--called weight matrices. This inverse problem is reduced to a linear equation in a Banach space by the method of spectral mappings. Constructive algorithm for the solution of the inverse problem is provided.