Necessary and Sufficient Conditions for Solvability of Inverse Problem for Dirac Operators with Discontinuous Coefficient

TL;DR

This paper provides a complete solution to the inverse spectral problem for a class of Dirac systems with discontinuous coefficients, including necessary and sufficient conditions for solvability and an explicit reconstruction algorithm.

Contribution

It introduces new necessary and sufficient conditions for solving the inverse spectral problem for Dirac operators with discontinuous coefficients, along with an explicit reconstruction method.

Findings

Derived asymptotic formulas for eigenvalues and eigenfunctions.

Established necessary and sufficient conditions for inverse problem solvability.

Provided an algorithm for potential reconstruction from spectral data.

Abstract

In this work, a complete solution of the inverse spectral problem for a class of Dirac differential equations system is given by spectral data (eigenvalues and normalizing numbers). As a direct problem, the eigenvalue problem is solved: the asymptotic formulas of eigenvalues, eigenfunctions and normalizing numbers of problem are obtained, spectral data is defined by the sets of eigenvalues and normalizing numbers. The expansion formula with respect to eigenfunctions is obtained. Gelfand-Levitan-Marchenko equation is derived. The main theorem on necessary and sufficient conditions for the solvability of inverse spectral problem is proved and the algorithm of reconstruction of potential from spectral data is given.