Parametric symmetries in exactly solvable real and P T symmetric complex potentials

TL;DR

This paper explores parametric symmetries in exactly solvable real and P T symmetric complex potentials, revealing new potentials and solutions, including those related to exceptional orthogonal polynomials, with implications for shape invariance and algebraic structures.

Contribution

It introduces novel parametric transformations in solvable potentials, leading to new classes of real and P T symmetric complex potentials with shape invariance and connections to exceptional orthogonal polynomials.

Findings

Parametric symmetries generate new solvable potentials.

Supersymmetric partner potentials change form under transformations.

Bound state solutions relate to exceptional orthogonal polynomials.

Abstract

In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex P T symmetric potentials. We focus our at- tention on the conventional potentials such as the generalized Poschl Teller (GPT), o Scarf-I and P T symmetric Scarf-II which are invariant under certain parametric transformations. The resulting set of potentials are shown to yield a completely dif- ferent behavior of the bound state solutions. Further the supersymmetric (SUSY) partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and P T symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature. We subse- quently take up a study of the newly discovered rationally extended SI Potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs). We discuss the transformations of the corresponding Casimir operator employing the properties of the so(2,1) algebra.