Level spacing statistics for the multi-dimensional quantum harmonic oscillator: Algebraic case

We study the statistical properties of spacings between neighboring energy levels for multi-dimensional quantum harmonic oscillator that occur in a window [E, E + ΔE) fixed width ΔE as tends to infinity. This regime provides notable exception Berry–Tabor conjecture from chaos, and, reason, it was studied extensively by Berry and Tabor their seminal paper 1977. focus entirely on case (ratios of) frequencies ω1, ω2, …, ωd together with 1 form basis an algebraic number field Φ degree d 1, allowing us use tools theory. special Dyson, Bleher, Bleher–Homma–Ji–Roeder–Shen, others. Under suitable rescaling, we prove distribution behaves asymptotically quasiperiodically log E. also ratios The same holds finite words alphabet rescaled spacings. Mathematically, our work is higher dimensional version Steinhaus (three gap theorem) involving fractional parts linear more than one variable, independent interest this perspective.