Random Moments for the New Eigenfunctions of Point Scatterers on Rectangular Flat Tori

Abstract

We define a random model for the moments of the new eigenfunctions of a point scatterer 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-Ueberschär 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 +∞.

Quantum Transport in a Crystal with Short-Range Interactions: The Boltzmann–Grad Limit

“…Assumptions of this kind have previously been used in modeling spectral correlations of diffractive systems, see for instance [10,11,32]. To formulate Assumption 1 in precise terms, define…”