Par Kurlberg: Level repulsion for arithmetic toral point scatterers

Abstract:

The Seba billiard was introduced to study the transition between integrability and chaos in quantum systems. The model seem to exhibit intermediate level statistics with strong repulsion between nearby eigenvalues (consistent with random matrix theory predictions for spectra of chaotic systems), whereas large gaps seem to have "Poisson tails" (as for spectra of integrable systems.)
 
We investigate the closely related "toral point scatterer"-model, i.e., the Laplacian perturbed by a delta-potential, on 3D tori of the form \(R^3/Z^3\). This gives a rank one perturbation of the original Laplacian, and it is natural to split the spectrum/eigenspaces into two parts: the "old" (unperturbed) one spanned by eigenfunctions vanishing at the scatterer location, and the "new" part (spanned by Green's functions). We show that there is strong repulsion between the new set of eigenvalues.