An Inverse Problem for Discontinuous Dirac Operators

This paper investigates the inverse nodal problem for a Dirac operator with a discontinuity at the midpoint. By means of an asymptotic analysis of the solutions of the associated integral equations, asymptotic formulas for the eigenvalues and nodal points are derived. These results are used to show that the potential functions of the discontinuous Dirac system can be determined from a dense subset of nodal points corresponding to the first component of the eigenfunctions. A constructive reconstruction procedure is then presented to recover the potentials from nodal data. In addition, a numerical approach based on Chebyshev polynomials of the third kind is employed to approximate the potentials. The numerical results indicate that the method yields satisfactory approximations. These results contribute to the study of inverse nodal problems for discontinuous Dirac operators.