Random moments for the new eigenfunctions of point scatterers on rectangular flat tori

TL;DR

This paper introduces a random model for the moments of eigenfunctions of point scatterers on rectangular tori, analyzing their distribution and conditions for Gaussian behavior in the semi-classical limit.

Contribution

It proposes a novel probabilistic framework for understanding moments of eigenfunctions and characterizes when these moments exhibit Gaussian asymptotics.

Findings

Asymptotic Gaussianity occurs if a certain multiplicity function diverges.

The model describes the distributional accumulation points of randomized moments.

Disproves the universal Gaussian conjecture in the deterministic setting.

Abstract

We define a random model for the moments of the new eigenfunctions of a point scat-terer on a 2-dimensional rectangular flat torus. In the deterministic setting,Seba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg-Uebersch{\"a}r on Diophantine tori. In our model, we describe the accumulation points in distribution of the randomized moments, in the semi-classical limit. We prove that asymptotic Gaussianity holds if and only if some function, modeling the multiplicities of the Laplace eigenfunctions, diverges to +$\infty$.