Representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$ and a generalization of non-commutative harmonic oscillators

Abstract

The non-commutative harmonic oscillator (NCHO) was introduced as a specific Hamiltonian operator on $L^2(\mathbb{R})\otimes\mathbb{C}^2$ by Parmeggiani and Wakayama. Then it was proved by Ochiai and Wakayama that the eigenvalue problem for NCHO is reduced to a Heun differential equation. In this article, we consider some generalization of NCHO for $L^2(\mathbb{R}^n)\otimes\mathbb{C}^p$ as a rotation-invariant differential equation. Then by applying a representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$, we check that its restriction to the space of products of radial functions and homogeneous harmonic polynomials is reduced to a holomorphic differential equation on the unit disk, which is generically Fuchsian.