Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces

Abstract

We introduce two new families of quasi-exactly solvable (QES) extensions of the oscillator in a $d$-dimensional constant-curvature space. For the first three members of each family, we obtain closed-form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the Bethe ansatz method. We prove that the first member of each family has a hidden sl(2,$\mathbb{R}$) symmetry and is connected with a QES equation of the first or second type, respectively. One-dimensional results are also derived from the $d$-dimensional ones with $d\ge 2$, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator.

30 pages, 8 figures, published version

References in corpus (3)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions
• An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

Cited by in corpus (7)

• Position-dependent mass charged particles in magnetic and Aharonov-Bohm flux fields: separability, exact and conditionally exact solvability
• Deformed shape invariance symmetry and potentials in curved space with two known eigenstates
• Liénard Type Nonlinear Oscillators and Quantum Solvability
• Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere
• Extended Nikiforov-Uvarov method, roots of polynomial solutions, and functional Bethe ansatz method
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
• Quasi-exactly solvable Schrödinger equations, symmetric polynomials, and functional Bethe ansatz method