Level repulsion for arithmetic toral point scatterers in dimension $3$

Abstract

We show that arithmetic toral point scatterers in dimension three ("Seba billiards on $R^3/Z^3$") exhibit strong level repulsion between the set of "new" eigenvalues. More precisely, let $\Lambda := \{\lambda_{1} < \lambda_{2} < \ldots \}$ denote the ordered set of new eigenvalues. Then, given any $\gamma>0$, $$ \frac{|\{i \leq N : \lambda_{i+1}-\lambda_{i} \leq \epsilon \}|}{N} = O_{\gamma}(\epsilon^{4-\gamma})$$ as $N \to \infty$ (and $\epsilon>0$ small.)