Global Level Number Variance in Integrable Systems

TL;DR

This paper investigates the global level number variance in integrable systems, revealing persistent oscillations around spectral rigidity that challenge existing theoretical explanations, supported by numerical analysis of four specific systems.

Contribution

It uncovers the oscillatory behavior of global level number variance in integrable systems, which cannot be explained by the diagonal approximation of periodic orbit theory.

Findings

Global level number variance oscillates around spectral rigidity.

Oscillations cannot be explained by diagonal approximation.

Numerical analysis confirms behavior in four integrable systems.

Abstract

We study previously un-researched second order statistics - correlation function of spectral staircase and global level number variance - in generic integrable systems with no extra degeneracies. We show that the global level number variance oscillates persistently around the saturation spectral rigidity. Unlike other second order statistics - including correlation function of spectral staircase - which are calculated over energy scales much smaller than the running spectral energy, these oscillations cannot be explained within the diagonal approximation framework of the periodic orbit theory. We give detailed numerical illustration of our results using four integrable systems: rectangular billiard, modified Kepler problem, circular billiard and elliptic billiard.