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Special values of the spectral zeta function for the noncommutative harmonic oscillator and modular forms

Research Project

Project/Area Number 20740021
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionUniversity of the Ryukyus

Principal Investigator

KIMOTO Kazufumi  University of the Ryukyus, 理学部, 助教 (10372806)

Project Period (FY) 2008 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords非可換調和振動子 / スペクトルゼータ関数 / 特殊値 / モジュラー形式 / スペクトルゼー関数 / パラメタ変形 / 多重ゼータ値
Research Abstract

We have studied the spectral zeta function associated to a differential operator called the noncommutative harmonic oscillator (NCHO in short). Each special value (i.e. the values at integral points) of the spectral zeta is expressed as a sum of a Riemann zeta value and certain terms (remainder terms) involving the structure parameter of NCHO, which is thought to reflect the noncommutativity of the NCHO. These remainder terms induce higher analogue of Apery-like numbers and some variants of multiple zeta values. We found a certain structure among the generating functions of these higher Apery-like numbers, as well as calculate explicitly the multiple zeta values associated to the remainder terms.

Report

(3 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • Research Products

    (13 results)

All 2009 Other

All Journal Article (3 results) (of which Peer Reviewed: 3 results) Presentation (2 results) Remarks (8 results)

  • [Journal Article] A variation of multiple L-values arising from the spectral zeta function of the non-commutative harmonic oscillator.2009

    • Author(s)
      K. Kimoto, Y. Yamasaki
    • Journal Title

      Proc. Amer. Math. Soc. 137

      Pages: 2503-2515

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] A variation of multiple L-values arising from the spectral zetafunction of the non-commutative harmonic oscillator2009

    • Author(s)
      K.Kimoto, Y.Yamasaki
    • Journal Title

      Proc.Amer.Math.Soc. 137

      Pages: 2503-2515

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A variation of multiple L-values arising from the spectral zeta function of the non-commutative harmonic oscillator2009

    • Author(s)
      K. Kimoto and Y. Yamasaki
    • Journal Title

      Proceedings of the American Mathematical Society 137

      Pages: 2503-2515

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] 非可換調和振動子のスペクトルゼータ関数の特殊値について2009

    • Author(s)
      木本一史
    • Organizer
      2009年度日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-27
    • Related Report
      2009 Final Research Report
  • [Presentation] 非可換調和振動子のスペクトルゼータ関数の特殊値について2009

    • Author(s)
      木本一史
    • Organizer
      日本数学会2009年度秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2009 Annual Research Report
  • [Remarks] プレプリント

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator. arXiv:0901.0658

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Special value formula for the spectral zeta function of the non-commutative harmonic oscillator. arXiv:0903.5165

    • Related Report
      2009 Final Research Report
  • [Remarks] 研究集会等での講演

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Arithmetic aspects of the non-commutative harmonic oscillator. K-Theory, Quadratic Forms and Number Theory Seminars, University College Dublin, 2008年10月

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Special values of the spectral zeta function of the non-commutative harmonic oscillator. Institute of Advanced Studies, University of Bologna, 2008年11月

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Arithmetics derived from the noncommutative harmonic oscillator. Casimir Force, Casimir Operators and the Riemann Hypothesis, 九大西新プラザ, 2009年11月

    • Related Report
      2009 Final Research Report
  • [Remarks] K. Kimoto: Spectral anomalies in special values of the spectral zeta function associated to the non-commutative harmonic oscillator. Zetas and Limit Laws in Okinawa 2009, 沖縄コンベンションセンター, 2009年11月

    • Related Report
      2009 Final Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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