Nearest Neighbor Distances on a Circle: Multidimensional Case

• Published in: Journal of statistical physics 2012-01, Vol.146 (2), p.446-465
• Main Authors: Bleher, Pavel M., Homma, Youkow, Ji, Lyndon L., Roeder, Roland K. W., Shen, Jeffrey D.
• Format: Article
• Language: English
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-011-0367-8

We study the distances, called spacings , between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E +1, and study how the spacings between these levels change for various choices of E , particularly when E goes to infinity. Primarily, we study the case in which the spring constant is a badly approximable vector. We first give the proof by Boshernitzan-Dyson that the number of distinct spacings has a uniform bound independent of E . Then, if the spring constant has components forming a basis of an algebraic number field , we show that, when normalized up to a unit, the spacings are from a finite set. Moreover, in the specific case that the field has one fundamental unit, the probability distribution of these spacings behaves quasiperiodically in log E . We conclude by studying the spacings in the case that the spring constant is not badly approximable , providing examples for which the number of distinct spacings is unbounded.