Relations Between Quantum and Classical Spectral Determinants (Zeta-Functions)

Abstract

We demonstrate that beyond the universal regime correlators of quantum spectral determinants $\Delta (\epsilon )=\det (\epsilon -\hat{H})$ of chaotic systems, defined through an averaging over a wide energy interval, are determined by the underlying classical dynamics through the spectral determinant $1/Z(z)=\det (z-\mathcal{L})$, where $e^{-\mathcal{L}t}$ is the Perron-Frobenius operator. Application of these results to the Riemann zeta function, allows us to conjecture new relations satisfied by this function.

4 pages, latex, no figures