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The Kepler bound-state problem and the attractive hyperbolic Landau problem are encoded in a common Morse spectral equation.

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2026-08-02 09:00 UTC pith:XZUQTA7X

load-bearing objection A careful, mostly correct unification of Kepler and hyperbolic Landau via Morse; the advertised 'encoding' overreaches at ℓ=0 but the gaps are fixable and the paper deserves serious refereeing. the 1 major comments →

arxiv 2607.01778 v2 pith:XZUQTA7X submitted 2026-07-02 hep-th

Morse Bridge between Planar Kepler and Hyperbolic Landau Dynamics

Mikhail S. Plyushchay This is my paper
classification hep-th
keywords Morse HamiltonianKepler–Coulomb problemhyperbolic Landau problemcoupling-constant metamorphosishorocyclic reductionthreshold resonanceshape invariancespectral duality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes a common one-dimensional mediator—the Morse Hamiltonian—between two paradigmatic systems: the planar Kepler–Coulomb problem and the Landau problem on the hyperbolic plane. On the Kepler side, a radial logarithmic transformation combined with a coupling-constant metamorphosis converts the bound-state radial equation into the Morse spectral problem, with the Kepler energy fixing the Morse scale and the angular momentum becoming the spectral parameter. On the Landau side, reduction at fixed horocyclic momentum yields the same Morse Hamiltonian, with the magnetic field playing the role of the Morse well-depth parameter. The paper shows that the bound-state Kepler spectrum and the attractive fixed-momentum sectors of the hyperbolic Landau spectrum are organized by the same Morse spectral equation, that each Kepler shell selects an integer Morse family, and that the zero-angular-momentum member of each shell maps to the Morse threshold resonance. If correct, this provides a concrete spectral-geometric metamorphosis linking flat electric dynamics to curved magnetic dynamics.

Core claim

The central claim is that the same one-dimensional Morse Hamiltonian, H_M = -d^2/dX^2 + C^2 e^{-2X} - 2Cλ e^{-X}, arises from both parent systems. For planar Kepler, the logarithmic coordinate r = e^{-X} and the identifications C^2 = -2E_K, λ = γ/C turn the radial bound-state equation into the Morse equation with spectral parameter E_M = -ℓ^2; classically, the Kepler polar angle becomes proportional to the Morse time. For hyperbolic Landau, fixing the horocyclic momentum p_y = C > 0 and transforming to half-density normalization gives the same Hamiltonian with λ = B, the magnetic field, and the spectral relation E_M = E_L - B^2 - 1/4. The paper concludes that physical Kepler bound states cor

What carries the argument

The load-bearing object is the Morse Hamiltonian H_M(X) = -d^2/dX^2 + C^2 e^{-2X} - 2Cλ e^{-X}, with λ = A_M + 1/2. It appears on the Kepler side after the logarithmic radial Liouville transformation r = e^{-X} and a genuine coupling-constant metamorphosis, and on the Landau side after horocyclic reduction at fixed momentum p_y = C together with a half-density normalization that produces the universal 1/4 shift. The two parameter dictionaries—C^2 = -2E_K and λ = γ/C on one side, C = p_y and λ = B on the other—are what make the bridge a single spectral equation rather than a formal analogy. The Darboux shape-invariance intertwiners provide the complementary parameter-shifting spectrum-generat

Load-bearing premise

The bridge equates the continuous horocyclic momentum C = p_y on the Landau side with the discrete, bound-state-normalized values C = γ/(n_r + |ℓ| + 1/2) on the Kepler side, assuming these two parametrizations can be joined within a single Morse spectral problem without an additional physical constraint fixing C or B.

What would settle it

A direct spectral calculation of the hyperbolic Landau problem at fixed horocyclic momentum p_y = C, for a non-half-integer magnetic field B, should show no bounded threshold resonance at E_M = 0 and a continuum edge at E_L = B^2 + 1/4; if a bounded edge state appears for generic B, the Morse-dictionary claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The Kepler bound-state spectrum is reorganized as Morse chains: each shell n_r + |ℓ| = N maps to a finite Morse chain n = 0, ..., N-1 plus a threshold resonance at n = N, so the Morse radial quantum number equals the Kepler radial quantum number.
  • Half-integer magnetic fields B = N + 1/2 correspond to integer Morse families, where the highest would-be Landau level reaches the continuum edge and the reflection amplitude factorizes into a finite Blaschke-type product.
  • Landau time evolution in the attractive branch takes the Kepler-conic form, so bound, threshold, and scattering trajectories of the Morse system correspond respectively to closed magnetic circles, horocycles, and open hypercycles in the hyperbolic plane.
  • Classically, the magnetic SL(2,R) Casimir reduces to the Morse Hamiltonian, while quantum mechanically the Darboux intertwiners shift the shape parameter (equivalently B) by integers, giving a parameter-shifting spectrum-generating structure.
  • The relation γ = B p_y expresses the Coulomb coupling as a product of magnetic field and conserved horocyclic momentum, suggesting an electric–magnetic metamorphosis rather than a strict duality.

Where Pith is reading between the lines

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

  • A natural next step, not taken in the paper, is to oxidize the bridge by promoting C back to a dynamical momentum and restoring the suppressed angular and time variables; a successful oxidation would turn the differential-equation identity into a true canonical correspondence between the two parent systems.
  • The half-density 1/4 shift on the Landau side mirrors the radial centrifugal shift on the Kepler side; one could test whether this shift appears in horocyclic reductions of magnetic systems on other constant-curvature surfaces, which would generalize the mechanism.
  • The paper notes the threshold resonance at E_M = 0 for integer Morse families; a concrete physical probe would be to look for such a bounded-but-not-normalizable edge state in scattering data of the hyperbolic Landau problem, where it would appear as a zero-width feature in the reflection amplitude.
  • The paper's closing remarks connect the Morse system to Whittaker and boundary Liouville physics; a speculative extension, beyond the paper's claims, would be a concrete AdS2/CFT1 dictionary built on this bridge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 3 minor

Summary. This paper constructs a classical and quantum correspondence between the planar Kepler–Coulomb problem and the hyperbolic Landau problem, mediated by the one-dimensional Morse Hamiltonian. On the Kepler side, the radial equation after the logarithmic substitution r=e^{-X} and the Liouville transformation ψ=r^{-1/2}u becomes the Morse spectral equation with E_M=-ℓ², C²=-2E_K, λ=γ/C. On the Landau side, reduction at fixed horocyclic momentum p_y=C produces the same Morse Hamiltonian with λ=B and E_M=E_L-B²-1/4. The paper derives a classical orbit dictionary, a Kepler-conic form of Landau time evolution, integer Morse families corresponding to Kepler shells, half-integer magnetic fields at which the highest Morse level becomes a threshold resonance, and complementary algebraic structures from the magnetic SL(2,R) Casimir and from Darboux shape invariance.

Significance. If accepted, the paper provides a concrete, explicit bridge between a flat electric central-potential problem and a curved magnetic problem through a solvable one-dimensional system. The main transformations are standard and the spectral dictionaries are derived in detail rather than fitted; the Landau and Morse spectra used are independently known, and the paper is candid about the limitations of the correspondence (formal endpoint, distinct Hilbert-space quantizations, parameter-level identification of p_y and C). The claimed novelty — the combination of the two reductions into one dictionary — is credible and likely to be of interest to researchers working on integrable systems, supersymmetric quantum mechanics, and Landau problems in curved spaces.

major comments (1)
  1. [Abstract and Sec. 6] The claim that the ℓ=0 member of each Kepler shell is 'represented by' the Morse threshold resonance is stronger than what the map actually provides. Under ψ=r^{-1/2}u, r=e^{-X}, the ℓ=0 Kepler bound state satisfies u∼r^{1/2} (up to logarithms) near r=0, hence ψ→const as X→∞, so ψ is not in L²(R,dX) and is not a normalizable Morse eigenstate. The paper correctly calls this a 'formal endpoint' in Sec. 6 and notes the distinct Hilbert-space quantizations in Sec. 7, but the abstract and the phrase 'At the quantum level' in Sec. 6 can be misread. Please make explicit in the abstract and in Sec. 6 that the ℓ=0 correspondence holds at the level of the reduced differential equation / generalized eigenfunction, or specify a weighted inner product or self-adjoint extension that makes the threshold resonance a genuine image of the Kepler state.
minor comments (3)
  1. [Sec. 7] The paragraph on the different spectral status of C is helpful. Consider adding an explicit sentence in Sec. 1 or the abstract that the relation γ=B p_y is a relation between parameters of reduced systems, not an operator identity in a joint Hilbert space, to preempt the natural reading of a physical equality.
  2. [Sec. 6, Eq. (28)] The special values B=N+1/2 are discrete points in the continuous Landau parameter space; it may be worth saying explicitly that A_M=N is a discrete subset selected by the integer Kepler-shell condition, not a quantization of B in the Landau problem itself.
  3. [Sec. 7] There is a typographical spacing issue in 'de Alfaro–Fubini– Furlan' (extra space before Furlan). Also, the phrase 'the state is bounded but not square-integrable' in Sec. 6 could be expanded by one sentence explaining that this is the generalized eigenfunction at the continuum edge, to clarify its physical status.

Circularity Check

0 steps flagged

No significant circularity: the Kepler–Morse and Landau–Morse correspondences are explicit transformations of known Hamiltonians, and the spectral dictionary is derived algebraically.

full rationale

The derivation chain is self-contained. On the Kepler side, Eq. (3) with r=e^{-X} and ψ=r^{-1/2}u gives Eq. (5), and with C^2=-2E_K, λ=γ/C this is exactly the Morse equation (6). On the Landau side, the horocyclic reduction of H_B at p_y=C gives Eq. (13), and the quantum half-density normalization gives Eq. (14). These are explicit transformations of the known Kepler and hyperbolic-Landau Hamiltonians, not fitted inputs. The spectral identifications use standard closed-form spectra: the Morse spectrum (22), the Kepler spectrum (25), and the Landau levels (23). The claim that physical Kepler bound states select integer Morse families A_M=n_r+|ℓ| is an algebraic consequence of the known Kepler spectrum together with the dictionary C=γ/(n_r+|ℓ|+1/2), λ=γ/C; it is not a fitted parameter renamed as a prediction. The ℓ=0 threshold-resonance identification is explicitly qualified in the paper: 'the state is bounded but not square-integrable' (Sec. 6), and Sec. 7 notes that 'the distinct Hilbert-space quantizations of the parent systems then select different subsets or interpretations of the same Morse spectral data.' That is a mathematical/completeness caveat, not a circular reduction. Self-citations appear only as contextual references to prior conformal-bridge work and are not load-bearing for the central equivalence. No circular step is identifiable in the paper's own equations.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles, forces, or entities. The only 'parameters' are the standard physical parameters (Coulomb coupling γ, magnetic field B, horocyclic momentum p_y, angular momentum ℓ) and the Morse parameters C and λ, all of which are defined by the systems themselves. The central claim rests on standard mathematical transformations (Liouville, CCM, horocyclic reduction) and the standard spectra of known solvable models.

free parameters (3)
  • Morse scale C = C = sqrt(-2E_K) = γ/(n_r+|ℓ|+1/2) on Kepler side; C = p_y (continuous) on Landau side
    C is not fitted to data; it is a free parameter of the Morse Hamiltonian, identified with either the Kepler energy or the horocyclic momentum. It is a free scale parameter whose value depends on the sector, but no data fitting is involved.
  • Morse shape parameter λ = B = λ = B = A_M + 1/2
    λ is the shape parameter of the Morse well; on the Kepler side it equals γ/C (rescaled Coulomb strength) and on the Landau side it equals the magnetic field B. It is a parameter of the theory, not fitted.
  • Kepler angular momentum ℓ = integer ℓ
    ℓ labels the Kepler angular modes and becomes the Morse spectral parameter E_M = -ℓ^2. It is a quantum number, not fitted.
axioms (6)
  • domain assumption The Kepler problem separates in polar coordinates with angular momentum ℓ conserved.
    Standard separation of variables for the planar Kepler problem, used in §2.
  • standard math The logarithmic Liouville transformation r = e^{-X} with ψ = r^{-1/2}u and Schwarzian correction {r,X} = -1/2 gives the transformed radial equation.
    Standard Liouville transformation of the radial equation, used in §2 (Eqs. (3)-(5)).
  • domain assumption The hyperbolic Landau Hamiltonian in horocyclic coordinates has the form H_B = p_X^2 + (e^{-X}p_y - B)^2 with mass m=1/2.
    This is the definition of the Landau problem on H^2 in the chosen coordinates, used in §3 (Eq. (12)). It fixes the normalization convention.
  • standard math Quantum half-density normalization in the horocyclic reduction introduces the +1/4 shift.
    The transformation Φ = e^{-X/2}ψ gives the shifted operator with +1/4, used in §3 (Eq. (14)).
  • standard math The Morse bound-state spectrum is E_{M,n} = -(A_M - n)^2 with n < A_M.
    Textbook spectrum of the Morse potential, quoted in §6 (Eq. (22)) from the cited literature [13–15].
  • domain assumption The classical Casimir of the magnetic sl(2,R) algebra satisfies C_L = J_0^2 - J_- J_+ = H_B - B^2.
    Algebraic statement about the Noether charges of the Landau system, used in §4 (Eq. (16)).

pith-pipeline@v1.3.0-alltime-deepseek · 10923 in / 8543 out tokens · 64953 ms · 2026-08-02T09:00:41.039967+00:00 · methodology

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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 radial Liouville transformation and genuine coupling-constant metamorphosis produce the Morse spectral problem; classically, the Kepler polar angle becomes proportional to the Morse evolution parameter. On the Landau side, horocyclic reduction at fixed momentum gives the same Morse Hamiltonian, while quantum half-density normalization produces the universal $1/4$ spectral shift. Consequently, the Kepler bound-state problem and the attractive fixed-horocyclic-momentum sectors of the hyperbolic Landau problem are encoded in a common Morse spectral equation, which organizes the Kepler shell structure together with the threshold, resonance and reflection data of the Morse and reduced Landau systems. At the quantum level, each bound-state Kepler shell selects a Morse system from a distinguished integer-parameter family: the Morse level number coincides with the Kepler radial quantum number, while the zero-angular-momentum member of the shell is represented by the Morse threshold resonance. 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. Algebraically, the classical magnetic $SL(2,\mathbb R)$ Casimir reduces to the classical Morse Hamiltonian, whereas at the quantum level Darboux shape invariance provides a complementary parameter-shifting spectrum-generating structure.

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 ↗

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