圆上的最近邻距离：多维情形
Nearest Neighbor Distances on a Circle: Multidimensional Case
摘要：
我们研究了量子谐振子的相邻能级对之间的距离，称为间距。具体来说，我们考虑落在E和E+1之间的所有能级，并研究这些能级之间的间距如何随着E的各种选择而变化，特别是当E趋于无穷大时。首先，我们研究的情况下，弹性常数是一个很难逼近的向量。我们首先给出Boshernitzan-Dyson的证明，证明了不同间距的个数有一个与E无关的一致界。然后，如果弹性常数的组件形成的代数数域的基础上，我们表明，当规范化到一个单位，间距是从一个有限的集合。此外，在场具有一个基本单位的特定情况下，这些间距的概率分布在log E中表现出准周期性。我们的结论是通过研究间距的情况下，弹性常数是不是很好的近似，提供的例子，不同的间距的数量是无限的。
英文摘要：
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.
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