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

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