Nearest Neighbor Distances on a Circle: Multidimensional Case

TL;DR

This paper investigates the behavior of spacings between energy levels in a quantum harmonic oscillator, revealing bounds, distribution patterns, and unbounded cases depending on the properties of the spring constant.

Contribution

It extends the understanding of energy level spacings by analyzing cases with algebraic number field components and badly approximable vectors, providing new bounds and distribution insights.

Findings

Number of distinct spacings is uniformly bounded for certain cases.

Spacings form a finite set when normalized for algebraic number field components.

Spacing distribution exhibits quasiperiodic behavior in log E for specific cases.

Abstract

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.