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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

Ryosuke Nakahama

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.

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 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 as a rotation-invariant differential equation. Then by applying a representation theory of , 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.
Paper Structure (4 sections, 5 theorems, 112 equations)

This paper contains 4 sections, 5 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Usual harmonic oscillator and harmonic polynomials
  3. Representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{su}(1,1)$
  4. A generalization of NCHO

Key Result

Theorem 4.1

Let $h(x)\in\mathcal{H}\mathcal{P}_k(\mathbb{R}^n)$, $\{u_m\}_{m=0}^\infty\subset\mathbb{C}^p$, and let Then $\psi(x)\in L^2(\mathbb{R}^n)\otimes\mathbb{C}^p$ and (formula_NCHO_a) hold if and only if hold.

Theorems & Definitions (10)

  • Remark 3.1
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • Remark 4.5
  • Proposition 4.6
  • Corollary 4.7
  • Example 4.8