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

TL;DR

This paper demonstrates that in three-dimensional arithmetic toral point scatterers, the eigenvalues exhibit strong level repulsion, with the probability of small gaps between consecutive eigenvalues decreasing rapidly as the system size grows.

Contribution

It establishes a quantitative level repulsion estimate for eigenvalues of 3D arithmetic toral point scatterers, a novel result in quantum chaos and spectral theory.

Findings

Eigenvalue gaps are typically large, with small gaps being rare.

The probability of small eigenvalue gaps scales as a power of the gap size.

The result applies to the set of 'new' eigenvalues in the system.

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.)