Inverse spectral problems for Dirac operators on a finite interval

TL;DR

This paper studies the spectral properties of Dirac operators on a finite interval, providing a complete description of spectral data and an algorithm for reconstructing the matrix-valued potential from spectral information.

Contribution

It offers a comprehensive characterization of spectral data and introduces a reconstruction algorithm for the potential in Dirac operators with matrix-valued functions.

Findings

Complete spectral data description for Dirac operators.

Algorithm for potential reconstruction from spectral data.

Analysis of eigenvalues and norming matrices.

Abstract

We consider the direct and inverse spectral problems for Dirac operators that are generated by the differential expressions $$ \mathfrak t_q:=\frac{1}{i}[I&0 0&-I]\frac{d}{dx}+[0&q q^*&0] $$ and some separated boundary conditions. Here $q$ is an $r\times r$ matrix-valued function with entries belonging to $L_2((0,1),\mathbb C)$ and $I$ is the identity $r\times r$ matrix. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest an algorithm of reconstructing the potential $q$ from the corresponding spectral data.