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The Spectral Zeta Function

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1992)

Abstract

In the first section of this chapter we will briefly recall, addressing the reader to Robert’s paper [63] (see also Aramaki [1]), the construction of the spectral zeta function ζA(s) = TrA-s, where 0 < A = A∗ε OPScl(mζ, g;MN) is an elliptic N×N system of GPDOs in Rn of order ζ∈ N. We will then prove a theorem about the meromorphic continuation of the spectral zeta function of an elliptic NCHO A = A* > 0 in Rn (Theorem 7.2.1) by using the parametrix approximation of the heat-semigroup e-tA constructed in Chapter 6, Section 6.1, from which we immediately deduce a corollary (Corollary 7.2.8) for our NCHO Qw (α,β)(x,D) (α,β >0 and αβ > 1) which gives part of the result of Ichinose and Wakayama (Theorem 7.3.1 below; see [31]), whose proof we will sketch in the final section.

Keywords

  • Zeta Function
  • Pseudodifferential Operator
  • Simple Pole
  • Riemann Zeta Function
  • Complex Power

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Correspondence to Alberto Parmeggiani .

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Parmeggiani, A. (2010). The Spectral Zeta Function. In: Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction. Lecture Notes in Mathematics(), vol 1992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11922-4_7

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