Spectral statistics for quantized skew translations on the torus

TL;DR

This paper investigates the spectral properties of quantized skew translations on the torus, revealing that certain spectral statistics do not exist in the semiclassical limit for irrational parameters.

Contribution

It provides explicit analysis showing the non-existence of level-spacing distribution and number variance for these systems in the semiclassical limit.

Findings

Level-spacing distribution does not exist in the semiclassical limit.

Number variance does not exist in the semiclassical limit.

Spectral statistics behave differently for ergodic but non-mixing systems.

Abstract

We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level--spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.