Level repulsion for arithmetic toral point scatterers

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


Ingemar Bengtsson and Edwin Langmann