Small gaps in the spectrum of the rectangular billiard

TL;DR

This paper investigates the minimal gaps between eigenvalues of the Laplacian in rectangular billiards with irrational aspect ratios, revealing Poisson-like behavior for certain irrationals but violations at finer scales, using number theory techniques.

Contribution

It provides new results on eigenvalue gap sizes for irrational rectangular billiards, connecting spectral statistics with Diophantine approximation and continued fractions.

Findings

Minimal gaps are about 1/N for quadratic irrationals like square roots of rationals.

Poisson statistics hold at a coarse scale for a full measure set of irrationals.

Poisson statistics are violated at a fine scale for all irrational aspect ratios.

Abstract

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.