Level Repulsion for Arithmetic Toral Point Scatterers in Dimension 3

Abstract

We show that arithmetic toral point scatterers in dimension three (“Šeba billiards on {mathbb R}^3/{mathbb Z}^3”) exhibit strong level repulsion between the set of “new” eigenvalues. More precisely, let Lambda := { lambda _{1}< lambda _{2} < ldots } denote the unfolded set of new eigenvalues. Then, given any gamma >0, |{i≤N:λi+1-λi≤ϵ}|N=Oγ(ϵ4-γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{|\\{ i \\le N : \\lambda _{i+1}-\\lambda _{i} \\le \\epsilon \\}|}{N} = O_{\\gamma }(\\epsilon ^{4-\\gamma }) \\end{aligned}$$\\end{document}as N rightarrow infty (and epsilon >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.