6 research outputs found
Random moments for the new eigenfunctions of point scatterers on rectangular flat tori
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-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 +∞
On the eigenvalue spacing distribution for a point scatterer on the flat torus
We study the level spacing distribution for the spectrum of a point scatterer
on a flat torus. In the 2-dimensional case, we show that in the weak coupling
regime the eigenvalue spacing distribution coincides with that of the spectrum
of the Laplacian (ignoring multiplicties), by showing that the perturbed
eigenvalues generically clump with the unperturbed ones on the scale of the
mean level spacing. We also study the three dimensional case, where the
situation is very different.Comment: 25 page