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

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 Schrödinger 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.

32 pages, no figure; published version

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• Infinitely many shape invariant potentials and new orthogonal polynomials
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Cited by in corpus (11)

• Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
• Deformed shape invariance symmetry and potentials in curved space with two known eigenstates
• Classical ladder functions for Rosen-Morse and curved Kepler-Coulomb systems
• Supersymmetry and Shape Invariance of exceptional orthogonal polynomials
• So (2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation
• Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere
• Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra $\mathfrak{osp}\left(1|2\right)$: exact solution for the quantum harmonic oscillator with a position-dependent mass
• Mapping of Two-Dimensional Schrödinger Equation under the Point Transformation
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
• Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of $\hslash$
• Superintegrability, symmetry and point particle T-duality