Arithmetic aspects of the non-commutative harmonic oscillator

**Speaker: **Professor Kazufumi Kimoto (University of Ryukyus / IHES)

**Time: **3:00PM

**Date:** Wed 15th October 2008

**Location:** Mathematical Sciences Seminar Room

**Abstract**

The eigenvalues of the ordinary (quantum) harmonic oscillator H=-D^2/2+x^2/2 (D=d/dx) is given by Spec(H)={1/2, 3/2, 5/2, ...}, so that its spectral zeta function sum_{lambdain Spec(H)}lambda^{-s} is essentially given by the Riemann zeta function. In the talk, we introduce the spectral zeta function of a differential operator called "the non-commutative harmonic oscillator", which can be regarded as a parametric deformation and/or higher dimensional generalization of H, and focus on its special values. We will show that these special values are written by certain integrals, which are interesting from an arithmetic viewpoint.

(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)

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