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

TL;DR

This paper introduces two new families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces, providing explicit solutions and revealing hidden symmetries, thus expanding the class of solvable quantum models.

Contribution

The paper presents novel families of QES oscillator extensions in curved spaces, with explicit solutions and symmetry analysis, connecting to nonlinear oscillators.

Findings

Explicit energy and wavefunction expressions for some parameters

Hidden sl(2,R) symmetry in the first family

Connections to nonlinear oscillators

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.