Necessary and sufficient conditions for the solvability of inverse problem for a class of Dirac operators

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.2010 Mathematics Subject Classification. 34A55, 34L40. Key words and phrases. Dirac operator, eigenvalues and normalizing numbers, expansion formula, inverse problem, necessary and sufficient conditions.