Resummation and the semiclassical theory of spectral statistics

TL;DR

This paper develops a semiclassical resummation approach that explains why chaotic quantum systems exhibit universal spectral statistics, unifying oscillatory and non-oscillatory contributions and extending to system-specific behaviors.

Contribution

It introduces a resummation formalism that preserves unitarity and unifies spectral correlation calculations, advancing understanding of quantum chaos and spectral universality.

Findings

Unified framework for spectral correlation functions

Explicit inclusion of oscillatory contributions

Extension to system-specific semiclassical behavior

Abstract

We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of Random Matrix Theory. To do so, we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits. This allows us to obtain both the non-oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework, thus overcoming a significant problem in previous approaches. In addition, our results extend beyond the universal regime to describe the system-specific approach to the semiclassical limit.