Level Repulsion for Arithmetic Toral Point Scatterers in Dimension 3

We show that arithmetic toral point scatterers in dimension three (“Šeba billiards on ℝ^3/ℤ^3 ”) exhibit strong level repulsion between the set of “new” eigenvalues. More precisely, let Λ := {λ _1< λ _2 < …} denote the unfolded set of new eigenvalues. Then, given any γ >0 , |{ i ≤ N : λ _i+1-λ _i≤ϵ}|/N = O_γ(ϵ ^4-γ) as N →∞ (and ϵ >0 small.) To the best of our knowledge, this is the first mathematically rigorous demonstration of a level repulsion phenomena for the quantization of a deterministic system.