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 $α$ 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.

• The paper derives the complete two-dimensional Coulomb bound-state spectrum from classical mechanics by combining amplitude closure with Levi-Civita regularization, yielding Eₙ = −2MC²/[α²(2n−1)²].
• The construction maps each negative-energy Coulomb shell to a two-dimensional harmonic oscillator, while the Levi-Civita double cover enforces odd modal numbers, shell degeneracy gₙ = 2n−1, and exact classical–wave chirality correspondence.
• The result predicts universal energy ratios 1:(1/9):(1/25):… for high-lying two-dimensional excitons, while leaving screening corrections, scattering states, and extensions to three-dimensional hydrogen as open problems.

The paper by Zheng, Xue, Wang, Chen, and Zhang (2608.16751) addresses a foundational question: whether the complete discrete bound-state spectrum of a non-trivial system can be derived exactly from classical mechanics, without taking the quantum wave equation as an independent axiom. For the two-dimensional Coulomb problem, the authors answer affirmatively. The construction rests on two elements: an "amplitude-closure" criterion that identifies the class of classical systems whose propagation kernel satisfies a linear Schrödinger-type equation exactly, and the Levi-Civita (LC) conformal regularization, which maps each negative-energy Coulomb shell onto a harmonic oscillator belonging to that class. The resulting spectrum reproduces the odd-integer Rydberg series observed in high-lying excitons of two-dimensional semiconductors, with all structural features independent of the single phenomenological parameter $\alpha$.

The amplitude-closure criterion

The starting point is a classical propagation kernel constructed from the two-point action $S(q,t;q_0)$ and the normalized Van Vleck amplitude $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$, combined as $K_\alpha = F\,e^{iS/\alpha}$, where $\alpha$ is a phase-scale parameter with dimensions of action. The central theorem states that this kernel satisfies the linear evolution equation

$$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$

if and only if the amplitude closure condition $\nabla_q^2 F = 0$ holds identically in all non-caustic regions. The proof is by direct substitution: the $\alpha^0$-order terms cancel via the Hamilton-Jacobi equation, and the remaining terms separate into a classical transport equation for $F$ and the harmonic condition on $F$.

The significance is that quadratic Hamiltonians form an exact "closure class." For the isotropic harmonic oscillator, the two-point action is quadratic in the endpoints, so the Van Vleck amplitude depends only on time, making $S(q,t;q_0)$0 trivially satisfied. Classical propagation and linear wave evolution are then mathematically identical for all wavelengths and all times—not approximately, but exactly. Generic non-quadratic potentials fail this condition globally, which is why the Coulomb problem must be regularized to enter the closure class. The paper also extends the framework to caustics via Maslov patching in the distributional sense, and to coherent versus incoherent classical ensembles through a correlation-kernel formalism.

Levi-Civita regularization and the Coulomb–oscillator duality

On each fixed negative-energy shell, the LC map $S(q,t;q_0)$1 (a holomorphic double cover of the physical plane) converts the regularized Hamiltonian $S(q,t;q_0)$2 into an isotropic two-dimensional harmonic oscillator with shell-dependent frequency $S(q,t;q_0)$3 and constant offset $S(q,t;q_0)$4.

The paper argues that this mapping is structurally unique rather than a mathematical coincidence. Three simultaneous requirements—potential flattening ($S(q,t;q_0)$5 constant), energy-term quadraticity ($S(q,t;q_0)$6), and kinetic standardization ($S(q,t;q_0)$7 with $S(q,t;q_0)$8 holomorphic)—force $S(q,t;q_0)$9, whose unique holomorphic solution is $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$0. The Coulomb potential is therefore the only central potential that can be shell-wise mapped into a quadratic Hamiltonian via holomorphic conformal regularization. This is a strong claim, and it is what elevates the LC map from a technical device (as in Duru-Kleinert path-integral quantization) to a structural necessity.

The double-cover topology is physically consequential: each physical point corresponds to antipodal image points $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$1, so single-valuedness of physical observables restricts image-space angular momentum to even values $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$2.

The spectrum and its classical origin

Combining the oscillator eigenvalues $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$3 with the even-subspace constraint yields the modal number

$$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$4

and, after the $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$5 gauge cancels identically, the physical spectrum

$$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$6

with ratios $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$7, i.e., the sequence $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$8. The degeneracy of the $$F = \sqrt{\det(-\partial^2 S/\partial q_i \partial q_{0j})}$$9-th shell is $K_\alpha = F\,e^{iS/\alpha}$0: one non-rotating $K_\alpha = F\,e^{iS/\alpha}$1 mode plus $K_\alpha = F\,e^{iS/\alpha}$2 time-reversed chirality pairs. An exact chirality correspondence holds: the classical angular momentum satisfies $K_\alpha = F\,e^{iS/\alpha}$3 exactly, so the sign of the modal probability current matches the classical orbital orientation for all $K_\alpha = F\,e^{iS/\alpha}$4.

The quantization condition is reinterpreted as winding-number dynamics: the reduced-action increment per image period is $K_\alpha = F\,e^{iS/\alpha}$5, and the phase bookkeeping per physical Kepler period (spatial phase $K_\alpha = F\,e^{iS/\alpha}$6 for odd $K_\alpha = F\,e^{iS/\alpha}$7, combined with the Maslov phase $K_\alpha = F\,e^{iS/\alpha}$8) yields single-valuedness. All ratios and degeneracies are independent of $K_\alpha = F\,e^{iS/\alpha}$9, which sets only the absolute scale and can be calibrated from the measured ground-state binding energy via $\alpha$0. The authors make no ontological claim identifying $\alpha$1 with $\alpha$2.

Uniqueness of the evolution operator and the self-adjoint domain

The construction goes beyond the spectrum. A uniqueness theorem shows that any second-order local operator $\alpha$3 compatible, shell-wise, with the regularized oscillator must have $\alpha$4 and $\alpha$5; the two-dimensional conformal Laplacian admits no first-order correction, so no free parameters remain.

The self-adjoint extension at the origin is also fixed geometrically. For $\alpha$6 the effective centrifugal coefficient $\alpha$7 places the origin in the limit-point regime with a unique extension. For $\alpha$8 the coefficient is the critical value $\alpha$9 (fall-to-the-centre), admitting a one-parameter family of extensions; standard quantum mechanics selects the Friedrichs extension without first-principles justification. Here, since the image-space oscillator is everywhere regular and classical orbits never develop divergent density at the origin, the analytic regular branch is the unique natural choice, excluding the logarithmic solution. The paper further asserts completeness: essential spectrum $$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$0, no additional bound states beyond the discrete series, and a complete orthonormal basis of the negative-spectral subspace. Notably, the physical-space operator is determined by the pure-pullback equation-layer identity, not by measure transport or unitary conjugation.

Relation to prior work and experimental implications

The authors emphasize a reversal of the usual logical hierarchy. Semiclassical quantization (Sommerfeld-Einstein, EBK) is asymptotic and valid only for large quantum numbers; geometric quantization imposes quantization axioms externally; the Duru-Kleinert transformation operates within a postulated quantum framework. In the present construction, the wave equation is derived as a linear representation of classical geometry, and the spectrum is exact for all quantum numbers including the ground state.

Experimentally, the framework offers a benchmark for Rydberg excitons in van der Waals semiconductors, where states up to $$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$1–$$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$2 have been observed in WSe$$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$3 and WS$$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$4. Because Rytova-Keldysh screening modifies only the short-range potential, high-lying states should approach the odd-integer $$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$5 series, while deviations at low $$i\alpha\,\partial_t K_\alpha = -\frac{\alpha^2}{2M}\nabla_q^2 K_\alpha + V(q)\,K_\alpha$$6 quantify screening strength. Odd-integer ratios and degeneracy counts are attributed to universal two-dimensional Coulomb geometry; level shifts, valley and spin splittings are material-specific. This separation avoids misidentifying universal geometric effects as novel material properties.

Limitations and open questions

The paper is explicit about its boundaries. The construction covers only negative-energy bound states; extension to the continuous spectrum and scattering states remains open. Generalization to three-dimensional hydrogen via the Kustaanheimo-Stiefel transformation is formally possible but involves quaternionic coordinates and first-class constraints, with work in progress. Real screened potentials (Rytova-Keldysh) lie outside the exact closure class, and whether a controlled perturbative or variational extension of the closure framework exists is unresolved. Most fundamentally, whether analogous exact constructions are possible for non-integrable systems—for which the amplitude-closure condition will not hold—remains an open question.

Conclusion

The paper demonstrates, for the integrable two-dimensional Coulomb problem, that the full bound-state spectrum—energy ratios, degeneracies, the evolution operator, and the self-adjoint domain—follows exactly from classical Hamiltonian mechanics, LC double-cover topology, and a single calibrated phase-scale parameter, with no semiclassical approximation or independent quantum postulate. The odd-integer Rydberg structure emerges as a consequence of the even-subspace constraint of the double cover. Whether this classical-geometric derivation of quantum spectral structure extends beyond this uniquely regularizable integrable system is the central question the paper leaves open.