The Absence Of Saturation Of The Level Number Variance In A Rectangular Box

TL;DR

This paper investigates the level number variance in a rectangular box, revealing persistent oscillations at large intervals due to shortest periodic orbits, challenging the expectation of saturation.

Contribution

It demonstrates that the level number variance does not saturate but oscillates indefinitely, explained by a superposition of a few fundamental harmonics.

Findings

Variance exhibits large oscillations for wide intervals.

Oscillations are linked to shortest periodic orbits.

Analytical and numerical results agree well.

Abstract

The variance of the number of levels in an energy interval around a level with large quantum numbers (semiclassical quantization) is studied for a particle in a rectangular box. Sampling involves changing the ratio of the rectangle's sides while keeping the area constant. For sufficiently narrow intervals, one finds the usual linear growth with the width of the interval. For wider intervals, the variance undergoes large, non-decaying oscillations around what is expected to be the saturation value. These oscillations can be explained as a superposition of just a few harmonics that correspond to the shortest periodic orbits in the rectangle. The analytical and numerical results are in excellent agreement.