Necessary and Sufficient Conditions for the Solvability of Inverse Problem for a Class of Dirac Operators

TL;DR

This paper establishes necessary and sufficient conditions for solving the inverse spectral problem for a class of Dirac operators, providing an explicit reconstruction algorithm based on spectral data.

Contribution

It introduces a complete characterization of solvability conditions and an explicit reconstruction method for the potential in Dirac operators from spectral data.

Findings

Derived asymptotic behaviors of eigenvalues and eigenfunctions.

Established a reconstruction algorithm for the potential.

Proved the main theorem on necessary and sufficient conditions for solvability.

Abstract

In this paper, we consider a problem for the first order Dirac differential equations system with spectral parameter dependent in boundary condition. The asymptotic behaviors of eigenvalues, eigenfunctions and normalizing numbers of this system are investigated. The expansion formula with respect to eigenfunctions is obtained and Parseval equality is given. The main theorem on necessary and sufficient conditions for the solvabilty of inverse problem is proved and the algorithm of reconstruction of potential from spectral data (the sets of eigenvalues and normalizing numbers) is given.