Deformed shape invariance symmetry and potentials in curved space with two known eigenstates

TL;DR

This paper explores deformed shape invariance in curved space oscillators, establishing conditions for known eigenstates and developing a general method to construct potentials with explicit ground and first excited states.

Contribution

It introduces a method to construct deformed shape invariant potentials with known low-lying eigenstates in curved space, extending previous supersymmetric quantum mechanics frameworks.

Findings

First two members of extension families are conditionally deformed shape invariant.

Compatibility conditions enable construction of potentials with known ground and first excited states.

A general method using generating functions is devised for broader potential construction.

Abstract

We consider two families of extensions of the oscillator in a $d$-dimensional constant-curvature space and analyze them in a deformed supersymmetric framework, wherein the starting oscillator is known to exhibit a deformed shape invariance property. We show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build potentials with known ground and first excited states. To extend such results to any members of the two families, we devise a general method wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function $W_+(r)$ (and its accompanying function $W_-(r)$).