论文标题
多维量子谐波振荡器的水平间距统计数据：代数情况
Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case
论文作者
论文摘要
我们研究了固定宽度$Δe$ a as $ e $的窗口$ [e，e+δe）$ the to to Infinity to to Infinity的多维量子谐波振荡器的相邻能量水平之间间距的统计特性。该制度提供了质量混乱的浆果猜想的一个明显例外，因此，贝里和塔博尔在1977年的开创性论文中对它进行了广泛的研究。我们完全关注频率（比率）$ω_1，ω________________________的$ 1 $ $ $ $ $ $ $ $ $ $ $ $ fra的情况。 $ d+1 $，允许我们使用代数数理论中的工具。这个特殊案例是由戴森，Bleher，Bleher-Homma-Ji-Roeder-Shen等研究的。在合适的重新缩放下，我们证明间距的分布在$ \ log e $中渐近地隔离。我们还证明，相邻间距的比率分布在$ \ log e $中渐近地表达。在有限的重新间距字母中分布有限单词的分布也是如此。 从数学上讲，我们的工作是Steinhaus猜想（三个差距定理）的较高维度版本，涉及线性形式的分数以多个变量的形式，从这个角度来看，它具有独立的兴趣。
We study the statistical properties of the spacings between neighboring energy levels for the multi-dimensional quantum harmonic oscillator that occur in a window $[E,E+ΔE)$ of fixed width $ΔE$ as $E$ tends to infinity. This regime provides a notable exception to the Berry-Tabor Conjecture from Quantum Chaos and, for that reason, it was studied extensively by Berry and Tabor in their seminal paper from 1977. We focus entirely on the case that the (ratios of) frequencies $ω_1,ω_2,\ldots,ω_d$ together with $1$ form a basis for an algebraic number field $Φ$ of degree $d+1$, allowing us to use tools from algebraic number theory. This special case was studied by Dyson, Bleher, Bleher-Homma-Ji-Roeder-Shen, and others. Under a suitable rescaling, we prove that the distribution of spacings behaves asymptotically quasiperiodically in $\log E$. We also prove that the distribution of ratios of neighboring spacings behaves asymptotically quasiperiodically in $\log E$. The same holds for the distribution of finite words in the finite alphabet of rescaled spacings. Mathematically, our work is a higher dimensional version of the Steinhaus Conjecture (Three Gap Theorem) involving the fractional parts of a linear form in more than one variable, and it is of independent interest from this perspective.
