An inverse problem for the matrix quadratic pencil on a finite interval

TL;DR

This paper addresses the inverse problem of recovering a quadratic matrix differential pencil on a finite interval using spectral data, providing asymptotic analysis and a uniqueness theorem.

Contribution

It introduces a method to uniquely recover the matrix quadratic pencil from the Weyl matrix, advancing inverse spectral theory for matrix differential equations.

Findings

Derived asymptotic formulas for solutions of the matrix equation.

Proved the uniqueness of the inverse problem solution.

Applied spectral mapping techniques to establish theoretical results.

Abstract

We consider a quadratic matrix boundary value problem with equations and boundary conditions dependent on a spectral parameter. We study an inverse problem that consists in recovering the differential pencil by the so-called Weyl matrix. We obtain asymptotic formulas for the solutions of the considered matrix equation. Using the ideas of the method of spectral mappings, we prove the uniqueness theorem for this inverse problem.