发表机构
Chongqing Jiaotong University（重庆交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个基于辛几何和相空间约束的几何框架，通过Kustaanheimo-Stiefel正则化与三个几何约束，无需算子对角化即可精确恢复三维库仑束缚态的离散能级和简并度，并应用于杂质态、莫尔超晶格等凝聚态系统。
AI 中文摘要
类库仑局域束缚态中的离散能级结构通常通过直接对角化薛定谔方程或半经典近似获得，但后者在低能下失效。本文发展了一个几何相空间框架，从经典辛几何和全局边界约束中恢复三维库仑束缚态的精确离散能级和壳层简并度，绕过了显式算子对角化。利用Kustaanheimo--Stiefel正则化，我们将负能量开普勒轨道映射为四维各向同性谐振子。辛流的群合成律构造了一个由经典两点作用量和Van Vleck振幅构成的积分核，其无穷小生成元满足精确的线性演化方程——我们将此性质称为“二次闭包”。施加三个几何约束——无穷远处衰减、原点处正则性以及Hopf纤维化下的纤维不变性——我们恢复了精确的$E_n/E_1=1/n^2$标度关系和$g_n=n^2$简并度。相位尺度$\u03b1$设定作用量单位；所有谱结构特征与其取值无关。该框架为谱数据提供了一条对称透明的路径：对于二次哈密顿量，经典核即使在基态能量下也产生零半经典误差，几何约束直接识别对称保护的简并度而无需对角化。我们将该方法应用于浅杂质态、莫尔超晶格和半经典输运，展示了几何筛选规则如何简化凝聚态环境中的谱分析。
英文摘要
Discrete energy-level structures in Coulomb-like localized bound states are conventionally obtained by direct diagonalization of the Schrödinger equation or by semiclassical approximations that fail at low energies. Here we develop a geometric phase-space framework that recovers the exact discrete energy levels and shell degeneracy of three-dimensional Coulomb bound states from classical symplectic geometry and global boundary constraints, bypassing explicit operator diagonalization. Using Kustaanheimo--Stiefel regularization, we map negative-energy Kepler orbits to four-dimensional isotropic harmonic oscillators. The group composition law of the symplectic flow determines an integral kernel built from the classical two-point action and Van Vleck amplitude, whose infinitesimal generator satisfies an exact linear evolution equation---a property we term \textit{quadratic closure}. Imposing three geometric constraints---decay at infinity, regularity at the origin, and fiber invariance under the Hopf fibration---we recover the exact $E_n/E_1=1/n^2$ scaling and $g_n=n^2$ degeneracy. The phase scale $α$ sets the action unit; all spectral structural features are independent of its value. This framework offers a symmetry-transparent route to spectral data: for quadratic Hamiltonians, the classical kernel incurs zero semiclassical error even at ground-state energies, and the geometric constraints directly identify symmetry-protected degeneracies without diagonalization. We apply the method to shallow impurity states, moiré superlattices, and semiclassical transport, demonstrating how geometric screening rules simplify spectral analysis in condensed-matter environments.
Comments49 pages
