Wallis Products from the Four-Dimensional Singular Harmonic Oscillator

Bin Ye; Lei Yin; Ruitao Chen

arxiv: 2607.00340 · v1 · pith:HOOPQ7W7new · submitted 2026-07-01 · 🪐 quant-ph · math-ph· math.MP

Wallis Products from the Four-Dimensional Singular Harmonic Oscillator

Bin Ye , Ruitao Chen , Lei Yin This is my paper

Pith reviewed 2026-07-02 12:44 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords Wallis productvariational approximationsingular harmonic oscillatorGamma function ratiosKustaanheimo-Stiefel transformationinverse square potential

0 comments

The pith

The four-dimensional singular harmonic oscillator yields the Wallis product and its reciprocal from variational minimization of a quartic trial family.

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

The paper shows that the Wallis product can be obtained variationally from the four-dimensional singular harmonic oscillator. The inverse-square interaction is absorbed into an effective angular parameter ν with known exact energy ħω(ν+2). Using the trial radial function R_a(ρ)=Nρ^ν e^{-aρ^4}, the minimized energy produces an accuracy ratio from ratios of Gamma functions. Restricting ν to odd integers 2n−1 gives the standard Wallis product, while even integers 2n give the reciprocal. This demonstrates that Wallis products persist under inverse-square deformation and originate from the Gamma-function structure in the radial variational problem.

Core claim

The minimized variational energy yields an accuracy ratio governed by adjacent Gamma functions. Restricting ν to the odd sequence ν=2n−1 gives the standard Wallis product, whereas the even sequence ν=2n gives its reciprocal form. In the large-ν limit this ratio approaches unity.

What carries the argument

The quartic trial family R_a(ρ)=Nρ^ν e^{-aρ^4} whose variational minimum supplies a ratio of Gamma functions that recovers the Wallis product for integer sequences of the angular parameter ν.

If this is right

The accuracy ratio approaches unity in the large-ν semiclassical limit.
The two sequences correspond to integer and half-integer effective angular sectors in the dual Coulomb/MICZ description.
Wallis-type infinite products persist under an inverse-square deformation of the oscillator.

Where Pith is reading between the lines

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

The Kustaanheimo-Stiefel mapping suggests the same Gamma structure may appear in the dual three-dimensional Coulomb problem.
The approach could be tested by applying similar trial families to other deformed oscillators to generate additional product identities.

Load-bearing premise

The quartic trial family produces a variational energy whose accuracy ratio is governed by adjacent Gamma functions whose restriction to integer ν sequences exactly recovers the Wallis product.

What would settle it

Computing the variational minimum for a specific small odd ν such as ν=1 and checking whether the Gamma ratio exactly equals the corresponding partial product in the Wallis sequence would test the claim.

read the original abstract

We present a variational derivation of the Wallis product and its reciprocal from the four-dimensional singular harmonic oscillator. The inverse-square interaction is absorbed into an effective angular parameter $\nu$, so that the lowest exact energy in a fixed sector is $E_{4d,\mathrm{exact}}=\hbar\omega(\nu+2)$. Motivated by the radial Kustaanheimo--Stiefel relation $r=\rho^2$ between the four-dimensional oscillator and the three-dimensional Coulomb problem, we use the quartic trial family $R_a(\rho)=N\rho^\nu e^{-a\rho^4}$. The minimized variational energy yields an accuracy ratio governed by adjacent Gamma functions. In the large-$\nu$ semiclassical limit, this ratio approaches unity. Restricting $\nu$ to the odd sequence $\nu=2n-1$ gives the standard Wallis product, whereas the even sequence $\nu=2n$ gives its reciprocal form. The Coulomb-dual interpretation further relates the two branches to integer and half-integer effective angular sectors in the dual Coulomb/MICZ description. The result shows that Wallis-type infinite products persist under an inverse-square deformation of the oscillator and arise from a common Gamma-function structure in radial variational dynamics.

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 recovers the Wallis product as a Gamma ratio from a variational calculation, but only by picking a quartic trial function that is not the natural choice and appears tailored to manufacture the spacing needed for the identity.

read the letter

The new element is a variational calculation on the 4D singular oscillator that produces an accuracy ratio expressible as adjacent Gamma functions; restricting the effective angular momentum nu to odd or even integers then yields the Wallis product or its reciprocal. The authors absorb the inverse-square term into nu, take the exact energy as hbar omega (nu + 2), and minimize a trial energy with the family R_a(rho) = N rho^nu exp(-a rho^4). In the large-nu limit the ratio goes to one, and the dual Coulomb picture is mentioned in passing.

The calculation itself is straightforward once the ansatz is written down, and the link to Gamma functions is explicit. That is the part that works.

The soft spot is the trial function. The Kustaanheimo-Stiefel relation and the exact 4D oscillator ground state both point to a Gaussian in rho, which would give the exact energy for any nu and therefore no product at all. The quartic exponent is the minimal choice that turns the normalization and expectation integrals into Gamma functions whose arguments differ by 1/4; when nu is half-integer those arguments produce the Wallis form after minimization. The result is therefore an algebraic identity that holds for this specific family rather than a property forced by the oscillator dynamics or the duality. The abstract presents the construction as motivated by KS, yet nothing in the given description explains why quartic is preferred over the Gaussian that would make the variational energy exact. Without seeing the explicit integrals and the justification for the exponent, it is hard to tell whether the minimization step adds anything beyond the ansatz choice.

This is the sort of note that might interest people who collect variational derivations of classical identities, but the dependence on a non-obvious trial weakens the claim that the product arises from the quantum system. I would not send it to a serious referee in its current form; the central step needs a clearer defense of the ansatz before the result can be taken as more than a curiosity.

Referee Report

2 major / 2 minor

Summary. The paper claims a variational derivation of the Wallis product (and reciprocal) from the 4D singular harmonic oscillator. The inverse-square term is absorbed into effective angular parameter ν with exact ground energy E_exact = ħω(ν+2). Motivated by the KS map r=ρ², the trial radial function R_a(ρ)=N ρ^ν exp(-a ρ^4) is introduced; minimization of the variational energy produces an accuracy ratio E_var/E_exact expressible via adjacent Gamma functions. For the sequence ν=2n−1 this ratio recovers the partial Wallis product; for ν=2n it recovers the reciprocal form. The ratio approaches 1 for large ν, and a Coulomb/MICZ dual interpretation is sketched.

Significance. If the explicit reduction of the variational ratio to the Wallis product is shown to follow from the oscillator dynamics rather than solely from the algebraic properties of the chosen ansatz, the work would establish an interesting bridge between variational radial integrals and classical infinite products, with the inverse-square deformation and KS duality providing additional structure. The large-ν limit and dual-sector remarks are consistent with known semiclassical and mapping properties.

major comments (2)

[Abstract, trial family description] Abstract and trial-family paragraph: the quartic exponent −aρ⁴ is introduced as motivated by the KS relation r=ρ², yet the KS map maps the Coulomb exponential to a Gaussian e^{-bρ²} on the oscillator side and the exact 4D oscillator ground state (for any ν) is Gaussian in ρ. The chosen ansatz forces the substitution u=ρ⁴, producing Gamma arguments spaced by 1/4 that become half-integers precisely when ν=2n−1, allowing algebraic reduction to the Wallis product. The manuscript must demonstrate that the appearance of the product is not an identity true for any trial of this functional form but follows from the oscillator Hamiltonian itself.
[Abstract] Abstract and energy-ratio paragraph: the claim that the minimized variational energy 'yields an accuracy ratio governed by adjacent Gamma functions' that exactly equals the partial Wallis product for odd ν requires the explicit normalization integral ∫ R_a² ρ³ dρ and the expectation value of the radial Hamiltonian to be written out and reduced step-by-step. The abstract supplies only the final Gamma ratio; without these intermediate expressions it is impossible to verify that the ratio is not obtained by construction of the ansatz.

minor comments (2)

Notation: the effective angular parameter is called ν throughout, but the manuscript should explicitly state its relation to the usual 4D angular-momentum quantum number l (e.g., ν = l + 1 or similar) to avoid confusion with standard radial oscillator notation.
The large-ν semiclassical limit statement would benefit from a brief asymptotic expansion of the Gamma ratio to confirm it approaches unity independently of the specific sequence (odd or even).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific suggestions for improvement. The two major comments can be addressed by expanding the derivations in the revised manuscript.

read point-by-point responses

Referee: [Abstract, trial family description] Abstract and trial-family paragraph: the quartic exponent −aρ⁴ is introduced as motivated by the KS relation r=ρ², yet the KS map maps the Coulomb exponential to a Gaussian e^{-bρ²} on the oscillator side and the exact 4D oscillator ground state (for any ν) is Gaussian in ρ. The chosen ansatz forces the substitution u=ρ⁴, producing Gamma arguments spaced by 1/4 that become half-integers precisely when ν=2n−1, allowing algebraic reduction to the Wallis product. The manuscript must demonstrate that the appearance of the product is not an identity true for any trial of this functional form but follows from the oscillator Hamiltonian itself.

Authors: We agree that the connection to the dynamics must be made explicit. In the revision we will derive the variational energy from the radial Hamiltonian (kinetic term, harmonic potential, and effective centrifugal barrier ν(ν+2)/ρ²) applied to the trial function, showing that the stationarity condition for a is fixed by the relative coefficients of these terms. The resulting E_var/E_exact ratio therefore encodes the oscillator dynamics rather than being an arbitrary Gamma identity. A short comparison with a generic quartic trial (without the specific Hamiltonian coefficients) will be added to illustrate the distinction. revision: yes
Referee: [Abstract] Abstract and energy-ratio paragraph: the claim that the minimized variational energy 'yields an accuracy ratio governed by adjacent Gamma functions' that exactly equals the partial Wallis product for odd ν requires the explicit normalization integral ∫ R_a² ρ³ dρ and the expectation value of the radial Hamiltonian to be written out and reduced step-by-step. The abstract supplies only the final Gamma ratio; without these intermediate expressions it is impossible to verify that the ratio is not obtained by construction of the ansatz.

Authors: We accept the criticism. Although the body of the manuscript contains the integrals, the abstract and the energy-ratio section will be expanded to display the normalization ∫ R_a² ρ³ dρ, the separate expectation values of each Hamiltonian term, the explicit minimization with respect to a, and the subsequent algebraic reduction to the ratio of Gamma functions. This step-by-step presentation will confirm that the Wallis product emerges from the variational procedure applied to the oscillator rather than from the ansatz alone. revision: yes

Circularity Check

1 steps flagged

Quartically exponentiated trial ansatz algebraically forces Gamma ratio to equal Wallis product on odd ν

specific steps

self definitional [Abstract]
"Motivated by the radial Kustaanheimo--Stiefel relation r=ρ^2 between the four-dimensional oscillator and the three-dimensional Coulomb problem, we use the quartic trial family R_a(ρ)=Nρ^ν e^{-aρ^4}. The minimized variational energy yields an accuracy ratio governed by adjacent Gamma functions. ... Restricting ν to the odd sequence ν=2n−1 gives the standard Wallis product"

The quartic exponent is the minimal choice that spaces the Gamma arguments by 1/4 after the substitution u=ρ^4. For ν=2n−1 the resulting ratio of adjacent Gammas is algebraically identical to the partial Wallis product; the identity is true for any wave function of that functional form and does not depend on the oscillator dynamics, the variational minimization, or the KS duality. The paper therefore obtains the product by construction from the ansatz rather than deriving it from the physical model.

full rationale

The central claim is that a variational calculation on the 4D singular oscillator, using a KS-motivated trial, produces an accuracy ratio whose restriction to odd ν recovers the Wallis product. Inspection of the abstract shows the ratio is obtained solely from the normalization and expectation integrals of the specific quartic-exponent family; after the u=ρ^4 substitution these integrals are Gamma functions whose arguments differ by 1/4. Restricting ν=2n−1 then converts the ratio into the known partial Wallis product by the algebraic properties of the Gamma function. This identity holds for the chosen ansatz independently of the oscillator Hamiltonian or the KS map; a Gaussian trial (the natural image under r=ρ²) yields ratio exactly 1 for every ν and produces no product. The claimed derivation therefore reduces to re-expressing a pre-existing Gamma identity inside a variational wrapper whose functional form was selected to generate that identity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the exact energy formula E=ℏω(ν+2), the Kustaanheimo-Stiefel radial relation, and the specific quartic trial family whose integrals produce the Gamma ratio.

free parameters (1)

a
Variational parameter in the trial radial function R_a(ρ) that is minimized to obtain the energy expectation value.

axioms (2)

domain assumption The lowest exact energy in a fixed sector is E_{4d,exact}=ℏω(ν+2)
Stated directly in the abstract as the benchmark for the variational ratio.
domain assumption Radial Kustaanheimo-Stiefel relation r=ρ² between 4D oscillator and 3D Coulomb problem
Invoked in the abstract to motivate the dual Coulomb/MICZ interpretation.

pith-pipeline@v0.9.1-grok · 5748 in / 1559 out tokens · 40517 ms · 2026-07-02T12:44:35.523972+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 4 canonical work pages · 3 internal anchors

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J. Wallis,Arithmetica Infinitorum. Oxford, 1655
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work page internal anchor Pith review Pith/arXiv arXiv 2018
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Kustaanheimo and E

P. Kustaanheimo and E. L. Stiefel,Perturbation theory of Kepler motion based on spinor regularization,Journal f¨ ur die reine und angewandte Mathematik218(1965) 204–219

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Zwanziger,Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,Physical Review176(1968) 1480–1488

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Lavrenov,Generalized Kustaanheimo–Stiefel transformations,N-dimensional singular oscillator and generalized MICZ–Kepler system,arXiv preprint(2019) , [1908.03572]

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work page arXiv 2019
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NIST Digital Library of Mathematical Functions, “Gamma function: Asymptotic expansions, 5.11(iii) ratios.”https://dlmf.nist.gov/5.11, 2025. 11

2025

[1] [1]

Wallis,Arithmetica Infinitorum

J. Wallis,Arithmetica Infinitorum. Oxford, 1655

[2] [2]

Berggren, J

L. Berggren, J. M. Borwein and P. B. Borwein, eds.,Pi: A Source Book. Springer, New York, 1997

1997

[3] [3]

Quantum Mechanical Derivation of the Wallis Formula for $\pi$

T. Friedmann and C. R. Hagen,Quantum mechanical derivation of the wallis formula forπ, Journal of Mathematical Physics56(2015) 112101, [1510.07813]

work page internal anchor Pith review Pith/arXiv arXiv 2015

[4] [4]

O. I. Chashchina and Z. K. Silagadze,On the quantum mechanical derivation of the wallis formula forπ,Physics Letters A381(2017) 2593–2597, [1704.06153]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[5] [5]

Wallis formula from the harmonic oscillator

I. Cortese and J. A. Garc´ ıa,Wallis formula from the harmonic oscillator,Journal of Geometry and Physics124(2018) 249–254, [1709.01214]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[6] [6]

K. M. Case,Singular potentials,Physical Review80(1950) 797–806

1950

[7] [7]

de Alfaro, S

V. de Alfaro, S. Fubini and G. Furlan,Conformal invariance in quantum mechanics,Il Nuovo Cimento A34(1976) 569–612

1976

[8] [8]

Kustaanheimo and E

P. Kustaanheimo and E. L. Stiefel,Perturbation theory of Kepler motion based on spinor regularization,Journal f¨ ur die reine und angewandte Mathematik218(1965) 204–219

1965

[9] [9]

P. A. M. Dirac,Quantised singularities in the electromagnetic field,Proceedings of the Royal Society of London A133(1931) 60–72

1931

[10] [10]

H. V. McIntosh and A. Cisneros,Degeneracy in the presence of a magnetic monopole, Journal of Mathematical Physics11(1970) 896–916

1970

[11] [11]

Zwanziger,Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,Physical Review176(1968) 1480–1488

D. Zwanziger,Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,Physical Review176(1968) 1480–1488

1968

[12] [12]

Lavrenov,Generalized Kustaanheimo–Stiefel transformations,N-dimensional singular oscillator and generalized MICZ–Kepler system,arXiv preprint(2019) , [1908.03572]

A. Lavrenov,Generalized Kustaanheimo–Stiefel transformations,N-dimensional singular oscillator and generalized MICZ–Kepler system,arXiv preprint(2019) , [1908.03572]

work page arXiv 2019

[13] [13]

F. G. Tricomi and A. Erd´ elyi,The asymptotic expansion of a ratio of gamma functions, Pacific Journal of Mathematics1(1951) 133–142

1951

[14] [14]

Gamma function: Asymptotic expansions, 5.11(iii) ratios

NIST Digital Library of Mathematical Functions, “Gamma function: Asymptotic expansions, 5.11(iii) ratios.”https://dlmf.nist.gov/5.11, 2025. 11

2025