AI 中文总结
该研究从经典力学结合相位标度参数α出发，精确推导出二维库仑问题的完整束缚态能谱，为二维激子能谱分析提供了纯经典几何基准，无需量子近似即可匹配实验观测。
AI 中文摘要
二维半导体中的高能里德伯激子普遍表现出与三维系统不同的特征奇整数能量标度。尽管二维库仑相互作用的这一标志性特征已从量子力学解中广为人知，但其更深层的经典几何起源仍未阐明。本文中，我们表明作为二维激子物理核心模型的二维库仑问题的完整束缚态能谱结构，可从经典力学结合一个具有作用量量纲的单相位标度参数α得出精确定理。我们推导了振幅闭合准则，作为经典传播子核满足线性演化方程的充要条件，并证明奇异库仑势可通过Levi-Civita正则化逐层映射到严格服从该准则的二次哈密顿量类中。所得能谱具有奇整数模态数、1/N²能量比和N重简并，均与α无关且与高能里德伯激子的实验观测一致。本工作为二维激子能谱分析提供了纯经典几何基准，可定量区分普适库仑效应与材料特异性屏蔽效应。整个过程未引入任何半经典、短波或ℏ→0近似。我们的结果颠倒了该可积系统的通常逻辑层级：波动方程是基础经典几何的一种表示，而非独立的第一性原理。
英文摘要
High-lying Rydberg excitons in two-dimensional semiconductors universally exhibit a characteristic odd-integer energy scaling distinct from three-dimensional systems. While this hallmark of two-dimensional Coulomb interaction is well known from quantum mechanical solutions, its deeper classical geometric origin remains unclarified. Here we show that the complete bound-state spectral structure of the two-dimensional Coulomb problem---a central model for two-dimensional exciton physics---follows as an exact theorem from classical mechanics augmented by a single phase-scale parameter $α$ with dimensions of action. We derive an amplitude-closure criterion as a necessary and sufficient condition for a classical propagator kernel to satisfy a linear evolution equation, and demonstrate that the singular Coulomb potential can be mapped shell-by-shell via Levi-Civita regularization into the class of quadratic Hamiltonians that obey this criterion exactly. The resulting spectrum bears odd-integer modal numbers, $1/N^2$ energy ratios and $N$-fold degeneracies, all independent of $α$ and consistent with experimental observations of high-lying Rydberg excitons. This work provides a pure classical-geometry benchmark for two-dimensional exciton spectral analysis, allowing quantitative disentanglement of universal Coulomb effects from material-specific screening effects. No semiclassical, short-wavelength or $\hbar \to 0$ approximation is invoked at any stage. Our results invert the usual logical hierarchy for this integrable system: the wave equation emerges as a representation of the underlying classical geometry, rather than as an independent first principle.
Comments25 pages, 1 figures
