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

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 wave functions 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, 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 >= 2, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator. Published by AIP Publishing.