Approximate shape-invariant potentials in quantum mechanics

We introduce and explore the concept of approximate shape invariance of parametrized potential functions for non-relativistic one-dimensional quantum systems. The supersymmetric partner V2 of an approximate shape-invariant potential V1 does not exhibit the same shape as the original potential. However, by an appropriate choice of the values of the relevant parameters, V2 can still be approximated by V1. We also propose a measure for the degree of shape invariance exhibited by a parametrized potential function. In order to illustrate these ideas we consider (a) the Lillo-Mantegna potentials admitting exact analytic power-law ground-state wavefunctions, and (b) a family of potentials whose ground-state eigenfunctions are given by the trial wavefunctions employed by Cooper, Dawson and Shepard in the SUSY-based variational method. (author)