Toward semiclassical theory of quantum level correlations of generic chaotic systems

TL;DR

This paper develops a semiclassical approach to analyze quantum level correlations in generic chaotic systems, using spectral zeta functions and probabilistic methods, providing an independent derivation of the two-point correlation function.

Contribution

It introduces a novel semiclassical derivation of quantum spectral correlations based on spectral zeta functions and probabilistic reasoning, applicable to generic chaotic systems.

Findings

Derived the two-point correlation function using periodic orbit theory.

Established relations between diagonal and off-diagonal parts of correlations.

Results applicable to systems with broken time reversal symmetry.

Abstract

In the present work we study the two-point correlation function $R(\epsilon)$ of the quantum mechanical spectrum of a classically chaotic system. Recently this quantity has been computed for chaotic and for disordered systems using periodic orbit theory and field theory. In this work we present an independent derivation, which is based on periodic orbit theory. The main ingredient in our approach is the use of the spectral zeta function and its autocorrelation function $C(\epsilon)$. The relation between $R(\epsilon)$ and $C(\epsilon)$ is constructed by making use of a probabilistic reasoning similar to that which has been used for the derivation of Hardy -- Littlewood conjecture. We then convert the symmetry properties of the function $C(\epsilon)$ into relations between the so-called diagonal and the off-diagonal parts of $R(\epsilon)$. Our results are valid for generic systems with broken time reversal symmetry, and with non-commensurable periods of the periodic orbits.