Classical Mechanics Exactly Yields the Full Bound-State Spectrum of the Two-Dimensional Coulomb Problem

Abstract

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 $\alpha$ 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 $\alpha$ 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.