多维量子谐振子的能级间距统计：代数情形
Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case
AI总结：
针对频率比生成代数数域的多维量子谐振子，本文利用代数数论证明重标度能级间距、相邻间距比及有限间距词的分布均随 log E 渐近准周期。
摘要：
我们研究多维量子谐振子相邻能级之间间距的统计性质；这些能级出现在当 E 趋于无穷时、固定宽度 ΔE 的窗口 [E,E+ΔE) 中。这一机制构成了量子混沌中 Berry-Tabor 猜想的一个显著例外，正因如此，Berry 和 Tabor 在其 1977 年的开创性论文中对其进行了广泛研究。我们完全聚焦于频率 ω_1,ω_2,…,ω_d（的比值）与 1 共同构成次数为 d+1 的代数数域 Φ 的一组基的情形，这使我们能够使用代数数论工具。这一特殊情形曾由 Dyson、Bleher、Bleher-Homma-Ji-Roeder-Shen 等人研究。在适当的重新标度下，我们证明间距分布在 log E 中渐近表现为准周期的。我们还证明相邻间距之比的分布在 log E 中也渐近表现为准周期的。由重新标度后的间距构成的有限字母表上有限词的分布同样如此。在数学上，我们的工作是 Steinhaus 猜想（三间隙定理）的高维版本，涉及多变量线性形式的分数部分，并且从这一角度看具有独立的研究意义。
英文摘要：
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.
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