Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

TL;DR

This paper explores the distribution of spectral gaps in quantum square billiards and toral point scatterers, proposing a new conjecture related to sums of two squares that suggests these gaps follow a Poisson distribution, supported by numerical evidence.

Contribution

It formulates an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares and links it to the Poisson distribution of spectral gaps in quantum systems.

Findings

Spectral gaps, after removing degeneracies and rescaling, are Poisson distributed.

Level spacings of arithmetic toral point scatterers are also Poisson distributed in the weak coupling limit.

Numerical evidence supports the proposed conjecture and its implications.

Abstract

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.