Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems
Abstract
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 En/E1=1/n2 scaling and gn=n2 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.
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BibTeX
@misc{zheng2026threedimensional,
title = {{Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems}},
author = {Gang Zheng and Peng Chen and Mengli Wang and Wenqi Xue and Benniu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19159},
eprint = {2609.19159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19159},
}
RIS
TY - UNPB
TI - Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems
AU - Gang Zheng
AU - Peng Chen
AU - Mengli Wang
AU - Wenqi Xue
AU - Benniu Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19159
UR - https://arxiv.org/abs/2609.19159
AB - 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.
ER -