Solvable potentials for the 1D Dirac equation using the tridiagonal matrix representations

Abstract

The aim of this research is to extend the class of solvable potentials of Dirac equation as a continuation to the work in [1]. We expand the spinor wavefunction in a square integrable spinor basis functions in which the expansion coefficients are functions of energy and potential parameters. Requiring the wave operator, J = H – E, to be tridiagonal and symmetric, this transforms the wave equation to a three-term recursion relation for the expansion coefficients which can be solved using known mathematical results on orthogonal polynomials. For illustration, we restricted ourselves here to the so-called Laguerre basis and considered situations where the obtained recursion relations can be easily compared to the ones associated with a well-known class of orthogonal polynomials.