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

TL;DR

This paper explores the connection between quantum spectral determinants of chaotic systems and classical dynamics, revealing that classical spectral determinants influence quantum correlators and proposing new relations for the Riemann zeta function.

Contribution

It establishes a link between quantum and classical spectral determinants and introduces conjectures about the Riemann zeta function based on this relationship.

Findings

Quantum correlators are determined by classical spectral determinants.

Classical dynamics influence quantum spectral statistics.

New conjectured relations for the Riemann zeta function.

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 $1/Z(z)=\det (z- {\cal L})$, where $e^{-{\cal L}t}$ is the Perron-Frobenius operator. Application of these results to the Riemann zeta function, allows us to conjecture new relations satisfied by this function.