Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions

TL;DR

This paper explores rational extensions of quantum oscillator and Kepler-Coulomb problems in curved spaces, revealing deformed shape invariance and mapping solutions to known shape-invariant potentials via point canonical transformations.

Contribution

It introduces rational extensions of these problems in curved spaces using deformed supersymmetry and point canonical transformations, connecting them to known shape-invariant potentials.

Findings

Rational extensions are consistent with Euclidean space results.

Deformed shape invariance is preserved in extended potentials.

Extended potentials are isospectral to original ones.

Abstract

The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schr\"odinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.