Shape Invariant potentials depending on n parameters transformed by translation

TL;DR

This paper investigates shape invariant potentials with multiple parameters transformed by translation, providing a general solution to a previously unsolved problem in quantum mechanics.

Contribution

The authors derive the general solution for shape invariant potentials depending on n parameters transformed by translation, extending prior work in the field.

Findings

Derived the general solution for n-parameter shape invariant potentials

Extended the class of known shape invariant potentials

Provided insights into Riccati equation properties in this context

Abstract

Shape Invariant potentials in the sense of [Gendenshte\"{\i}n L.\'E., JETP Lett. 38, (1983) 356] which depend on more than two parameters are not know to date. In [Cooper F., Ginocchio J.N. and Khare A., Phys. Rev. {\bf 36 D}, (1987) 2458] was posed the problem of finding a class of Shape Invariant potentials which depend on n parameters transformed by translation, but it was not solved. We analyze the problem using some properties of the Riccati equation and we find the general solution.