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arxiv: 2607.01778 · v1 · pith:XZUQTA7Xnew · submitted 2026-07-02 · ✦ hep-th

Morse Bridge between Planar Kepler and Hyperbolic Landau Dynamics

Mikhail S. Plyushchay This is my paper

Pith reviewed 2026-07-03 09:04 UTC · model grok-4.3

classification ✦ hep-th
keywords Kepler-Coulomb problemLandau problemhyperbolic planeMorse Hamiltonianspectral mappinghorocyclic reductionLiouville transformation
0
0 comments X

The pith

The radial Kepler problem and fixed-horocyclic-momentum sectors of the hyperbolic Landau problem both map to the same Morse spectral problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how the planar Kepler-Coulomb problem and the Landau problem on the hyperbolic plane are linked by transforming both into the Morse Hamiltonian. On the Kepler side this happens through a Liouville transformation and coupling-constant metamorphosis that turns the radial dynamics into Morse evolution. On the Landau side a horocyclic reduction achieves the same result, including a quantum correction. A reader would care because this creates a dictionary that relates the bound states, resonances, scattering, and trajectories of these two systems, revealing shared structures in their dynamics and symmetries.

Core claim

The radial Kepler problem and the fixed-horocyclic-momentum sectors of the hyperbolic Landau problem are mapped to one Morse spectral problem, relating their bound spectra, continuum thresholds, resonances and scattering data. The Landau time evolution has a Kepler-conic form and reduces to the bound, threshold and scattering trajectories of the Morse system. The resulting dictionary connects Kepler conics with magnetic circles, horocycles and hypercycles, and turns the magnetic SL(2,R) symmetry of the Landau problem into the spectrum-generating algebraic structure of the Morse system.

What carries the argument

The Morse Hamiltonian as one-dimensional mediator, obtained from Liouville transformation plus coupling-constant metamorphosis on the Kepler radial equation and from horocyclic reduction on the Landau side.

If this is right

  • Bound spectra of the two systems are identified through the eigenvalues of the shared Morse problem.
  • Continuum thresholds, resonances, and scattering data are correspondingly related.
  • Landau time evolution reduces to Morse trajectories that correspond to Kepler conics.
  • The magnetic SL(2,R) symmetry of the Landau problem generates the spectrum of the Morse system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique could connect other central-force problems to magnetic dynamics on spaces of constant curvature.
  • Known exact solutions and algebraic properties of the Morse oscillator can be transferred directly to both the Kepler and Landau systems via the mapping.
  • The dictionary between conics, circles, horocycles and hypercycles may extend to classical trajectory correspondences in related integrable models.

Load-bearing premise

The horocyclic reduction of the hyperbolic magnetic dynamics produces exactly the same Morse Hamiltonian as the transformed Kepler radial problem, including the quantum half-density correction.

What would settle it

Compute the discrete energy levels of the radial Kepler problem at fixed angular momentum and of the fixed-horocyclic-momentum sectors of the Landau problem on H2, then check whether they coincide after the explicit change of variables to the Morse problem.

Figures

Figures reproduced from arXiv: 2607.01778 by Mikhail S. Plyushchay.

Figure 1
Figure 1. Figure 1: Schematic representation of the Kepler–Morse–Landau bridge. The planar Kepler–Coulomb [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

We show that two paradigmatic systems, the planar Kepler--Coulomb problem and the Landau problem on the hyperbolic plane $H^2$, are connected by a common one-dimensional mediator: the Morse Hamiltonian. On the Kepler side, a Liouville transformation and coupling-constant metamorphosis turn the radial dynamics into the Morse problem, with the Kepler polar angle becoming the Morse evolution parameter. On the Landau side, horocyclic reduction of the hyperbolic magnetic dynamics gives the same Morse Hamiltonian, with a quantum half-density correction. Consequently, the radial Kepler problem and the fixed-horocyclic-momentum sectors of the hyperbolic Landau problem are mapped to one Morse spectral problem, relating their bound spectra, continuum thresholds, resonances and scattering data. We further show that the Landau time evolution has a Kepler-conic form and reduces to the bound, threshold and scattering trajectories of the Morse system. The resulting dictionary connects Kepler conics with magnetic circles, horocycles and hypercycles, and turns the magnetic $SL(2,\mathbb R)$ symmetry of the Landau problem into the spectrum-generating algebraic structure of the Morse system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that the radial Kepler-Coulomb problem and fixed-horocyclic-momentum sectors of the Landau problem on H^2 are both mapped to the same one-dimensional Morse spectral problem. On the Kepler side this is achieved by a Liouville transformation combined with coupling-constant metamorphosis; on the Landau side it follows from horocyclic reduction of the magnetic Laplacian together with a quantum half-density correction. The resulting dictionary relates bound spectra, continuum thresholds, resonances and scattering data, identifies a Kepler-conic form for the Landau time evolution, and converts the magnetic SL(2,R) symmetry into the spectrum-generating algebra of the Morse system.

Significance. If the claimed operator-level equivalence holds, the work supplies a concrete spectral and dynamical bridge between two standard integrable systems, one Euclidean and one hyperbolic-magnetic. The explicit dictionary between conics and magnetic trajectories, together with the shared Morse mediator, would constitute a useful addition to the literature on hidden symmetries and reductions in curved geometries.

major comments (2)
  1. [Landau-side reduction (abstract and associated derivation)] The central claim rests on the assertion that horocyclic reduction of the hyperbolic Landau operator, after the stated quantum half-density correction, produces exactly the same Morse Hamiltonian obtained from the Kepler side. The abstract states the equivalence but does not exhibit the explicit operator identity or the resulting effective potential and measure; without this verification the asserted correspondence of bound spectra, thresholds, resonances and scattering data cannot be confirmed.
  2. [Quantum half-density correction (abstract)] The half-density correction is invoked to reconcile the reduced Landau operator with the Morse Hamiltonian, yet its precise form and its effect on the domain and self-adjointness are not spelled out. Any mismatch in the correction term would alter the continuum threshold and resonance locations, undermining the claimed spectral dictionary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicitness of the operator equivalence in the abstract and the details of the half-density correction. We respond to each comment below and will revise the manuscript to address them.

read point-by-point responses

  1. Referee: [Landau-side reduction (abstract and associated derivation)] The central claim rests on the assertion that horocyclic reduction of the hyperbolic Landau operator, after the stated quantum half-density correction, produces exactly the same Morse Hamiltonian obtained from the Kepler side. The abstract states the equivalence but does not exhibit the explicit operator identity or the resulting effective potential and measure; without this verification the asserted correspondence of bound spectra, thresholds, resonances and scattering data cannot be confirmed.

    Authors: We agree that the abstract would benefit from greater explicitness. The full derivation of the horocyclic reduction, including the explicit reduced operator, effective potential, and measure after the half-density correction, appears in Section 3. To make the operator identity immediately verifiable from the abstract, we will revise the abstract to state the explicit form of the reduced Morse Hamiltonian and the resulting effective potential. This change will also strengthen the presentation of the spectral dictionary without altering the body of the paper. revision: yes

  2. Referee: [Quantum half-density correction (abstract)] The half-density correction is invoked to reconcile the reduced Landau operator with the Morse Hamiltonian, yet its precise form and its effect on the domain and self-adjointness are not spelled out. Any mismatch in the correction term would alter the continuum threshold and resonance locations, undermining the claimed spectral dictionary.

    Authors: The precise form of the quantum half-density correction, together with its effect on the measure, domain, and self-adjointness, is derived in Section 3.2, where it is shown to produce exact agreement with the Morse Hamiltonian and the correct continuum threshold. We will revise the abstract to include the explicit expression for the correction term and add a brief clarifying sentence in the introduction on its implications for self-adjointness. This will render the reconciliation fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds via independent transformations

full rationale

The paper presents two separate derivations that each produce the Morse Hamiltonian: a Liouville transformation plus coupling-constant metamorphosis applied to the radial Kepler problem, and a horocyclic reduction (with half-density correction) applied to fixed-momentum sectors of the hyperbolic Landau problem. These steps are described as constructive mappings that relate spectra and trajectories as a consequence, without any quoted reduction of a claimed result back to a fitted parameter, self-definition, or load-bearing self-citation. No equations in the provided abstract or description equate the final spectral correspondence to its inputs by construction. The central claim therefore remains a derived equivalence rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on these two domain assumptions about the validity of the reductions and transformations in the respective systems.

axioms (2)
  • domain assumption Liouville transformation and coupling-constant metamorphosis turn the radial dynamics into the Morse problem
    This is the key step on the Kepler side as stated.
  • domain assumption horocyclic reduction gives the Morse Hamiltonian with quantum half-density correction
    This is the key step on the Landau side.

pith-pipeline@v0.9.1-grok · 5715 in / 1102 out tokens · 31738 ms · 2026-07-03T09:04:07.564691+00:00 · methodology

discussion (0)

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