### 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 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Zeta functions for the spectrum of the non-commutative harmonic oscillators.** / Ichinose, Takashi; Wakayama, Masato.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 258, no. 3, pp. 697-739. https://doi.org/10.1007/s00220-005-1308-7

}

TY - JOUR

T1 - Zeta functions for the spectrum of the non-commutative harmonic oscillators

AU - Ichinose, Takashi

AU - Wakayama, Masato

PY - 2005/9/1

Y1 - 2005/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=23044465306&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044465306&partnerID=8YFLogxK

U2 - 10.1007/s00220-005-1308-7

DO - 10.1007/s00220-005-1308-7

M3 - Article

AN - SCOPUS:23044465306

VL - 258

SP - 697

EP - 739

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -