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arXiv · 1612.00682

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

C. Quesne
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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.

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C. Quesne. 2016-12-02. Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces. https://doi.org/10.1063/1.4983563

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