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The non-commutative harmonic oscillator (NCHO) was introducedNon-commutative harmonic oscillator 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 equationHeun 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 theoryRepresentation 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 genericallyFuchsian Fuchsian.

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Representation Theory of  \(\mathfrak {sl}(2,\mathbb {R})\simeq \mathfrak {su}(1,1)\) and a Generalization of Non-commutative Harmonic Oscillators

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  • 发表日期：

    2026年

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  • Ryosuke Nakahama

摘要

The non-commutative harmonic oscillator (NCHO) was introducedNon-commutative harmonic oscillator 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 equationHeun 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 theoryRepresentation 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 genericallyFuchsian Fuchsian.