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

Ichinose, Takashi; Wakayama, Masato

doi:10.1007/s00220-005-1308-7

Cited by 42 publications

(70 citation statements)

References 11 publications

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“…The aim of this note is to give a survey of our recent results in [1] (cf. [2]) on the spectral zeta function for -the zeta function associated with the spectrum of -the non-commutative harmonic oscillator Q := Q(x, D x ), improving a previous result there on the lower and upper bounds of the eigenvalues of Q, and also to mention some recent results.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…We have shown one of the basic analytic properties of the spectral zeta function ζ Q (s) for the non-commutative harmonic oscillator Q = Q(x, D x ) in (1). There still remain a number of open questions.…”

Section: Some Remarks and Open Questionsmentioning

confidence: 99%

“…The following theorem gives an improvement of the result in [1], where we treated only the case for the first eigenvalue λ 1 by use of a Lieb-Thirring inequality [8].…”

Section: Theoremmentioning

confidence: 99%

“…We develop a method used in [1], Remark 3, p.714, to give an alternative proof of the assertion for λ 1 , not appealing to a Lieb-Thirring inequality.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…A similar technique we developed in [1] and [2] allows us to state a conjecture that the Hurwitz type spectral zeta function defined by…”

Section: Proof Of Theoremmentioning

confidence: 99%

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On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

Ichinose

Wakayama²

2007

Reports on Mathematical Physics

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The spectral zeta function for the so-called non-commutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ(s) has, and possesses a trivial zero at each non-positive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the non-commutative harmonic oscillator.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Some Remarks and Open Questionsmentioning

confidence: 99%

“…The following theorem gives an improvement of the result in [1], where we treated only the case for the first eigenvalue λ 1 by use of a Lieb-Thirring inequality [8].…”

Section: Theoremmentioning

confidence: 99%

“…We develop a method used in [1], Remark 3, p.714, to give an alternative proof of the assertion for λ 1 , not appealing to a Lieb-Thirring inequality.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…A similar technique we developed in [1] and [2] allows us to state a conjecture that the Hurwitz type spectral zeta function defined by…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations