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 $\Lambda :=\{{\lambda }_1<{\lambda }_2<\dots \}$ denote the ordered set of new eigenvalues. Then, given any $\gamma >0$, $$\frac{|\{i\le N:{\lambda }_{i+1}-{\lambda }_i\le \epsilon \}|}{N}=O_{\gamma }({\epsilon }^{4-\gamma })$$ as $N\to \infty$ (and $\epsilon >0$ small.)

16 pages, 2 figures