### Abstract

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.

Original language | English |
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Pages (from-to) | 697-739 |

Number of pages | 43 |

Journal | Communications in Mathematical Physics |

Volume | 258 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2005 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Ichinose, T., & Wakayama, M. (2005). Zeta functions for the spectrum of the non-commutative harmonic oscillators.

*Communications in Mathematical Physics*,*258*(3), 697-739. https://doi.org/10.1007/s00220-005-1308-7