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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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910292497137664 |
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| author | Nakahama, Ryosuke |
| author_facet | Nakahama, Ryosuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17118 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$ and a generalization of non-commutative harmonic oscillators Nakahama, Ryosuke Mathematical Physics Classical Analysis and ODEs Representation Theory Spectral Theory 81Q10, 34M03, 22E45 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. |
| title | Representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$ and a generalization of non-commutative harmonic oscillators |
| topic | Mathematical Physics Classical Analysis and ODEs Representation Theory Spectral Theory 81Q10, 34M03, 22E45 |
| url | https://arxiv.org/abs/2310.17118 |