Exactly Solvable Potentials for the One-Dimensional Stationary Dirac Equation

The Darboux transformation operator method is extended to the one-dimensional stationary Dirac equation. This permits one, starting from an exactly solvable Dirac equation, to obtain a family of matrix-valued exactly solvable Dirac potentials. New potentials can be isospectral with the initial one or their spectra differ by one or two levels. Special cases of scalar and pseudoscalar potentials for which the Dirac equation is equivalent to a supersymmetric pair of Schrodinger equations are analyzed in details. An interrelation between Schrodinger and Dirac transformation operators is found. It is shown that the Dirac case induces the transformations of either supersymmetric Schrodinger partners.