## Abstract

The non-commutative harmonic oscillator (NcHO) is a special type of selfadjoint ordinary differential operator with non-commutative coefficients. In the present note, we aim to provide a reasonable criterion that derives the simplicity of the lowest eigenvalue of NcHO. It actually proves the simplicity of the lowest eigenvalue for a large class of structure parameters. Moreover, this note describes a certain equivalence between the spectral problem of the NcHO (for the even parity) and existence of holomorphic solutions of Heun's ordinary differential equations in a complex domain. The corresponding Riemann scheme allows us to give another proof to the criterion.

Original language | English |
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Pages (from-to) | 69-73 |

Number of pages | 5 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 89 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2013 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)