# Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems

## TL;DR

This paper derives a universal, parameter-free formula for the power-spectrum of energy level fluctuations in quantum chaotic systems with broken time-reversal symmetry, using random matrix theory and Painlevé transcendents.

## Contribution

It provides the first exact, non-perturbative integral representation and universal prediction for the power-spectrum in quantum chaotic systems, challenging traditional assumptions.

## Key findings

- Derived an exact integral representation of the power-spectrum.
- Established a universal prediction expressed via Painlevé V transcendent.
- Numerical results support the theoretical predictions.

## Abstract

We present a non-perturbative analysis of the power-spectrum of energy level fluctuations in fully chaotic quantum structures. Focussing on systems with broken time-reversal symmetry, we employ a finite-$N$ random matrix theory to derive an exact multidimensional integral representation of the power-spectrum. The $N\rightarrow \infty$ limit of the exact solution furnishes the main result of this study -- a universal, parameter-free prediction for the power-spectrum expressed in terms of a fifth Painlev\'e transcendent. Extensive numerics lends further support to our theory which, as discussed at length, invalidates a traditional assumption that the power-spectrum is merely determined by the spectral form factor of a quantum system.