Shape invariant potentials in second-order supersymmetric quantum mechanics

Abstract

Shape invariance condition in the framework of second-order supersymmetric quantum mechanics is studied. Two classes of solvable shape invariant potentials are consequently constructed, in which the parameters $a_0$ and $a_1$ of partner potentials are related to each other by translation $a_1 = a_0 + \alpha$. In each class, general properties of the obtained shape invariant potentials are systematically investigated. The energy eigenvalues are algebraically determined and the corresponding eigenfunctions are expressed in terms of generalized associated Laguerre polynomials. It is found that these shape invariant potentials are inherently singular, characterized by the $1/x^2$ singularity at the origin.