arxiv: 2605.03324 · v2 · submitted 2026-05-05 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Superposition of quasi coherent Bohm Madelung waves

Anand Aruna Kumar

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:53 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bohm-Madelungquasi-coherent statesErmakov-PinneyMathieu equationFourier-Besselnonlinear superpositionquantum interferenceJacobi-Anger expansion

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The pith

Nonlinear Bohm-Madelung superposition recovers linear Fourier-Bessel spectra in the quasi-coherent regime

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

The paper investigates superposition of stationary quantum states within the Bohm-Madelung hydrodynamic formulation, where nonlinear coupling between amplitude and phase generally invalidates linear superposition. In the quasi-coherent regime of near-degenerate states, the problem reduces to a mean amplitude satisfying an Ermakov-Pinney equation with a Wronskian invariant and a difference amplitude satisfying a parametrically driven Hill-Mathieu equation set by the energy splitting. A Jacobi-Anger expansion then restores a linear spectral structure as a Fourier-Bessel series whose coefficients are square-summable and whose weights are translation covariant. This structure accounts for how amplitude modulation and phase sidebands shape interference in concrete aperture and shifted-source geometries.

Core claim

Despite the intrinsic nonlinearity of the Bohm-Madelung equations, a linear spectral structure re-emerges in the quasi-coherent regime through a Jacobi-Anger expansion. This yields a Fourier-Bessel representation with square summable coefficients and translation covariant weights, after the dynamics separate into mean amplitude evolution governed by an Ermakov-Pinney equation and difference amplitude obeying a Hill-Mathieu equation determined by energy splitting.

What carries the argument

The quasi-coherent separation of amplitude dynamics into mean and difference components, enabling the Jacobi-Anger expansion to produce the Fourier-Bessel representation.

If this is right

Amplitude modulation and phase-induced sidebands organize interference patterns for aperture geometries in the nonlinear amplitude-phase framework.
One dimensional shifted sources exhibit organized interference through the same mechanism.
The resulting Fourier-Bessel representation has square summable coefficients.
The weights in the representation are translation covariant.

Where Pith is reading between the lines

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

The translation covariance may imply that the representation remains stable under small displacements of the sources.
Similar decompositions could apply to time-dependent cases if the Mathieu equation is solved for varying parameters.

Load-bearing premise

The near-degeneracy of stationary branches in the quasi-coherent regime permits the clean separation of the amplitude dynamics into a mean part and a difference part.

What would settle it

Measuring the intensity pattern from two slightly energy-split quantum sources in one dimension and comparing the observed sideband amplitudes against the predicted square-summable Fourier-Bessel coefficients would test the re-emergence of the linear spectral structure.

read the original abstract

The problem of superposition of stationary quantum states in the Bohm Madelung formulation is examined in a regime where amplitude and phase obey coupled nonlinear equations and linear superposition is not generally valid. In the quasi coherent regime of near degenerate stationary branches, the dynamics separates into a hierarchical structure: the mean amplitude evolves according to an Ermakov Pinney equation governed by a Wronskian invariant, while the difference amplitude obeys a parametrically driven Hill Mathieu equation determined by the energy splitting. Despite this intrinsic nonlinearity, a linear spectral structure re emerges through a Jacobi Anger expansion, yielding a Fourier Bessel representation with square summable coefficients and translation covariant weights. Applications to aperture geometries and one dimensional shifted sources demonstrate how amplitude modulation and phase induced sidebands organise interference patterns within a nonlinear amplitude phase framework. Keywords Bohm Madelung, Ermakov Pinney, nonlinear superposition, quasi coherent states, Mathieu Hill, Fourier Bessel, quantum interference.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a split of near-degenerate states into Ermakov-Pinney mean amplitude and Mathieu difference amplitude inside the nonlinear Bohm-Madelung equations, then recovers a Fourier-Bessel spectrum via Jacobi-Anger, but the decoupling step is asserted without visible algebra.

read the letter

The paper examines superposition for stationary states in the Bohm-Madelung formulation. Normally the nonlinear coupling between amplitude and phase blocks simple linear combinations. In the quasi-coherent regime where the states are near-degenerate, it claims the system separates hierarchically. The mean amplitude satisfies an Ermakov-Pinney equation with its Wronskian invariant, and the difference amplitude follows a parametrically driven Mathieu equation set by the energy difference. From there it applies the Jacobi-Anger expansion to recover a Fourier-Bessel representation. The coefficients are square summable and the weights are translation covariant. It then shows examples with aperture geometries and one-dimensional shifted sources, where amplitude modulation and phase sidebands shape the interference. What stands out is the concrete use of these classical special-function tools on the nonlinear quantum hydrodynamics problem. The applications give a sense of how the patterns organize without falling back to the linear Schrödinger picture. The main soft spot is the separation itself. The abstract states that the dynamics separates into those two equations, but the stress-test note is right to flag that no intermediate steps are given showing how the nonlinear terms cancel or get folded into the parametric drive. If the full manuscript walks through the algebra from the coupled amplitude-phase system and confirms no residual coupling, then the spectral structure follows cleanly. Without that, the later claims rest on an unverified reduction. This work is aimed at people interested in quantum foundations and hydrodynamical approaches to quantum mechanics. A reader working on nonlinear formulations or interference in such settings could find the examples useful. It is not a broad advance, but the specific combination is new enough to warrant checking. I would recommend sending it for peer review. A referee can verify the derivation steps and the summability in the expansion, which would clarify if the approach holds up.

Referee Report

1 major / 3 minor

Summary. The manuscript examines superposition of stationary quantum states within the Bohm-Madelung formulation, where amplitude and phase satisfy coupled nonlinear equations and linear superposition does not hold in general. In the quasi-coherent regime of near-degenerate stationary branches, it asserts that the dynamics decouples hierarchically: the mean amplitude obeys an Ermakov-Pinney equation with Wronskian invariant, while the difference amplitude satisfies a parametrically driven Hill-Mathieu equation set by the energy splitting. A Jacobi-Anger expansion is then applied to recover a linear spectral structure in the form of a Fourier-Bessel representation whose coefficients are square-summable and whose weights are translation-covariant. The framework is illustrated with applications to aperture geometries and one-dimensional shifted sources, showing how amplitude modulation and phase-induced sidebands organize interference patterns.

Significance. If the claimed exact decoupling from the nonlinear Madelung system and the subsequent emergence of the Fourier-Bessel structure can be established, the work would supply a concrete route to controlled nonlinear superposition in the hydrodynamic formulation, with direct relevance to interference phenomena that remain inaccessible to linear quantum mechanics. The reliance on well-studied special-function equations (Ermakov-Pinney, Mathieu, Jacobi-Anger) is a methodological strength that could facilitate analytic progress and falsifiable predictions for near-degenerate states.

major comments (1)

[Abstract and the quasi-coherent regime analysis (prior to the Jacobi-Anger expansion)] The central claim that the nonlinear amplitude-phase system exactly decouples into an Ermakov-Pinney equation for the mean amplitude and a parametrically driven Hill-Mathieu equation for the difference amplitude is asserted in the abstract and introduction but is not accompanied by the intermediate algebraic steps. No explicit reduction is shown demonstrating how the nonlinear convective term and the quantum-potential contributions cancel or are absorbed into the parametric drive arising solely from the energy splitting. This decoupling is load-bearing for the subsequent Jacobi-Anger expansion and the assertion of square-summable, translation-covariant coefficients; without it, residual nonlinear coupling would invalidate the linear spectral ansatz.

minor comments (3)

[Abstract] The phrase 're emerges' in the abstract should be hyphenated as 're-emerges' for standard English usage.
[Keywords] The keyword list contains 'Bohm Madelung' without the hyphen that appears in the title and abstract; consistency would improve readability.
[Abstract and applications section] The abstract refers to 'square summable coefficients' and 'translation covariant weights' without defining the precise function space or the translation operator; a brief clarification in the main text would aid readers unfamiliar with the Fourier-Bessel context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater transparency in the derivation of the decoupling. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and the quasi-coherent regime analysis (prior to the Jacobi-Anger expansion)] The central claim that the nonlinear amplitude-phase system exactly decouples into an Ermakov-Pinney equation for the mean amplitude and a parametrically driven Hill-Mathieu equation for the difference amplitude is asserted in the abstract and introduction but is not accompanied by the intermediate algebraic steps. No explicit reduction is shown demonstrating how the nonlinear convective term and the quantum-potential contributions cancel or are absorbed into the parametric drive arising solely from the energy splitting. This decoupling is load-bearing for the subsequent Jacobi-Anger expansion and the assertion of square-summable, translation-covariant coefficients; without it, residual nonlinear coupling would invalidate the linear spectral ansatz.

Authors: We agree that the submitted manuscript did not display the intermediate algebraic reduction in sufficient detail, which leaves the decoupling claim insufficiently supported for a reader. In the revised version we will insert a new subsection (immediately following the statement of the Madelung system) that carries out the explicit separation. Defining the mean and difference amplitudes as R_mean = (R1 + R2)/2 and R_diff = (R1 - R2)/2 together with the corresponding mean and difference velocities, the convective nonlinearity and the quantum-potential terms from the two stationary branches combine such that all cross terms proportional to the product R_mean R_diff vanish identically once the Wronskian invariant of the underlying linear Schrödinger equation is imposed. The mean equation then reduces exactly to the Ermakov-Pinney form whose invariant is fixed by the stationary phase condition, while the difference equation becomes a linear Hill-Mathieu equation whose only parametric coefficient is the energy splitting ΔE. No residual nonlinear coupling survives inside the quasi-coherent regime. The revised text will also note that this linear structure is what licenses the subsequent Jacobi-Anger expansion and the square-summability plus translation covariance of the Fourier-Bessel coefficients. We believe the added derivation removes the objection. revision: yes

Circularity Check

0 steps flagged

No circularity: separation into Ermakov-Pinney and Mathieu equations presented as derived result, not by construction or self-citation

full rationale

The paper states that in the quasi-coherent regime the nonlinear amplitude-phase system decouples such that the mean amplitude satisfies the Ermakov-Pinney equation while the difference amplitude obeys a parametrically driven Hill-Mathieu equation. This is offered as an emergent hierarchical structure from the Bohm-Madelung equations rather than a definitional assumption or a parameter fitted to the target spectral representation. The subsequent Jacobi-Anger expansion is then applied to the resulting linear spectral structure. No load-bearing step reduces to a self-citation whose content is unverified, nor does any prediction collapse to a fitted input by construction. The cited equations are standard and their invariants (Wronskian, parametric drive from energy splitting) are independent of the final Fourier-Bessel coefficients. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that near-degenerate branches permit clean separation into mean and difference amplitudes governed by the named classical nonlinear equations; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Near-degenerate stationary branches allow separation of amplitude dynamics into mean (Ermakov-Pinney) and difference (Mathieu) components
This regime assumption is required for the hierarchical structure and subsequent expansion to hold.

pith-pipeline@v0.9.0 · 5448 in / 1270 out tokens · 46810 ms · 2026-05-11T00:53:09.271976+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the mean amplitude evolves according to an Ermakov–Pinney equation governed by a Wronskian invariant, while the difference amplitude obeys a parametrically driven Hill/Mathieu equation
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

a linear spectral structure re-emerges through a Jacobi–Anger expansion, yielding a Fourier–Bessel representation with square-summable coefficients

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references

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Madelung, Quantentheorie in hydrodynamischer Form, Z

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys.40, 322–326 (1927)

1927
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H. R. Lewis, Classical and quantum systems with time-dependent harmonic-oscillator type Hamiltonians, Phys. Rev. Lett.18, 510–512 (1967)

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Pinney, The nonlinear differential equationy ′′ +p(x)y+cy −3 = 0, Proc

E. Pinney, The nonlinear differential equationy ′′ +p(x)y+cy −3 = 0, Proc. Am. Math. Soc.1, 681 (1950). 10

1950
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P. R. Holland, The Quantum Theory of Motion, Cambridge University Press, 1993

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Schuch, Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics, Fundamental Theories of Physics, 101, Springer, 2018

D. Schuch, Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics, Fundamental Theories of Physics, 101, Springer, 2018

2018
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A. B. Nassar and S. Miret-Art´ es, Bohmian Mechanics, Open Quantum Systems and Continuous Measure- ments, Springer, 2017

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V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989

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S. H. Strogatz, Nonlinear Dynamics and Chaos, Westview Press, 2015

2015
[12]

G. W. Hill, On the part of the motion of the lunar perigee, Acta Math.8, 1–36 (1886)
[13]

N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press, 1947

1947
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A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, 1979

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[15]

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013

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G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2019

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Born and E

M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 1999

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R. W. Boyd, Nonlinear Optics, 4th ed., Academic Press, 2020

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[19]

Gilberto Silva-Ortigoza, Ram´ on Silva-Ortigoza, and Isaac Morales-Lozano, J. Opt. Soc. Am. B43(5), 958–966 (2026). Appendix A Fourier–Bessel representation The quasi-coherent parent state may be written as ψ(x) =R 0(x) exp i ℏ S(x) ,(A.1) where, to first order inε, R0(x)≃ √ A h 1 + ε 2 sin2(k0x) i ,(A.2) and exp i ℏ S(x) = ∞X n=−∞ Jn ε 4 ei(2n+1)k0x.(A.3...

2026