Non-gaussian waves in Seba's billiard

TL;DR

This paper proves that Seba's billiard with irrational aspect ratio exhibits non-Gaussian eigenfunction behavior, challenging the random wave conjecture and highlighting the complex quantum-classical transition in such systems.

Contribution

It confirms that Seba billiards with irrational aspect ratios violate the random wave conjecture by constructing a large subsequence of eigenfunctions with non-Gaussian moments.

Findings

Eigenfunctions of irrational Seba billiards are non-Gaussian.

A subsequence of eigenfunctions with full density is constructed.

The semiclassical moments of these eigenfunctions do not follow Gaussian distribution.

Abstract

The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [KU] which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the semiclassical moments of the eigenfunctions ought to be Gaussian. We prove a conjecture of Keating, Marklof and Winn who suggested that Seba billiards with irrational aspect ratio violate the random wave conjecture. More precisely, in the case of diophantine tori, we construct a subsequence of eigenfunctions of essentially full density and show that its semiclassical moments cannot be Gaussian.