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Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators

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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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa, 920-1192, Japan

    Takashi Ichinose

  2. Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan

    Masato Wakayama

Authors
  1. Takashi Ichinose
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  2. Masato Wakayama
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Correspondence to Takashi Ichinose.

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Communicated by P. Sarnak

Work in part supported by Grant-in Aid for Scientific Research (B) No. 16340038, Japan Society for the promotion of Science

Work in part supported by Grant-in Aid for Scientific Research (B) No. 15340012, Japan Society for the promotion of Science

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Ichinose, T., Wakayama, M. Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators. Commun. Math. Phys. 258, 697–739 (2005). https://doi.org/10.1007/s00220-005-1308-7

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  • DOI: https://doi.org/10.1007/s00220-005-1308-7

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