Canonical regularization of the stationary Coulomb problem and an Aufbau-like spectral ordering

Anand Aruna Kumar

arxiv: 2606.17359 · v2 · pith:ZKQ5UUNInew · submitted 2026-06-15 · 🪐 quant-ph · math-ph· math.MP

Canonical regularization of the stationary Coulomb problem and an Aufbau-like spectral ordering

Anand Aruna Kumar This is my paper

Pith reviewed 2026-06-27 02:43 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords de Broglie-Bohm representationCoulomb spectrumLanger correctionSturm-Liouville equationsAufbau principleMadelung orderingassociated Laguerre functionsassociated Legendre functions

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The pith

A regularized de Broglie-Bohm treatment of the hydrogen atom adds orbital-dependent shifts that lift Coulomb degeneracy and order levels by the Aufbau sequence.

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

The paper examines the stationary hydrogen atom in a regularized de Broglie-Bohm representation where stationary amplitude current constraints produce separable Sturm-Liouville branches in the radial, orbital, and magnetic sectors. Each branch receives canonical Langer-like inverse-square corrections that modify the boundary-value problems and permit analytical solutions. These corrections shift the radial quantization condition by an amount that depends on the orbital index, removing the usual degeneracy among states of fixed principal quantum number. The resulting spectrum orders states according to the empirical Aufbau or Madelung rule while the s-state sector keeps the standard Rydberg degeneracy. A reader would care because the construction supplies an explicit one-electron mechanism that reproduces the observed filling order without reference to electron-electron repulsion.

Core claim

In the regularized de Broglie-Bohm formulation the hydrogen atom separates into Sturm-Liouville problems for each angular-momentum sector, each modified by a canonical inverse-square term. The adjusted radial quantization condition then acquires an orbital-dependent additive shift that removes l-degeneracy for fixed n and produces an energy ordering identical to the Aufbau/Madelung sequence. The l = 0 sector retains the conventional Rydberg degeneracy, and the separated amplitudes are expressed by associated Laguerre, Legendre, and Bessel functions carrying non-integral parameters fixed by the regularization.

What carries the argument

Canonical Langer-like inverse-square corrections applied to the radial, orbital, and magnetic Sturm-Liouville branches generated by stationary amplitude current constraints.

If this is right

The spectrum remains hydrogen-like but employs regularized radial and angular indices.
The energy ordering of states follows the Aufbau/Madelung sequence.
The l=0 sector preserves the standard degenerate Rydberg sequence.
Separated amplitudes appear as generalized special-function branches with non-integral parameters.

Where Pith is reading between the lines

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

The same regularization procedure could be applied to other central potentials to test whether analogous orderings emerge.
Explicit construction of the regularized wave functions would allow direct comparison of probability densities with conventional hydrogenic orbitals.
The separation into independent sectors suggests the method might extend to time-dependent or multi-center problems while retaining analytical tractability.

Load-bearing premise

Stationary amplitude current constraints in the de Broglie-Bohm representation produce separable Sturm-Liouville branches that admit canonical inverse-square corrections whose coefficients are chosen to recover the observed spectral ordering.

What would settle it

Solve the modified radial Sturm-Liouville eigenvalue problem for the lowest few values of n and l, then check whether the resulting energies increase in the sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, …

Figures

Figures reproduced from arXiv: 2606.17359 by Anand Aruna Kumar.

**Figure 1.** Figure 1: Regularized energy Enr,λ as a function of the separation label λ. The regularized radial spectrum is shown in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Comparison of the degenerate Rydberg energies [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

read the original abstract

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

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 claims a de Broglie-Bohm regularization produces the Madelung ordering via canonical corrections, but the coefficients may be chosen to match the known sequence rather than fixed uniquely by the constraints.

read the letter

The main point is that this work regularizes the stationary hydrogen problem in a de Broglie-Bohm picture. Stationary amplitude current constraints are said to yield separable Sturm-Liouville branches, which then receive Langer-like inverse-square corrections. The modified radial and angular problems are solved analytically with generalized Laguerre, Legendre, and Bessel functions, and the resulting spectrum lifts Coulomb degeneracy to follow the empirical (n+l) Aufbau sequence while keeping the usual Rydberg levels for l=0.

What is new is the specific combination of de Broglie-Bohm current constraints with these canonical corrections to produce an orbital-dependent shift in the radial quantization. The paper does a clean job of keeping everything analytical and of stating its limited scope: it is an examination of a one-center regularized Coulomb problem, not a replacement for many-electron theory.

The soft spot is the status of the correction coefficients. The abstract calls them canonical and says they arise from the amplitude-current conditions, yet the outcome is exactly the known Madelung ordering. Without the explicit derivation steps, it is not possible to see whether the numerical prefactors are uniquely determined by the continuity and current requirements or whether they are adjusted so the effective principal quantum number reproduces the observed sequence. If the latter, the degeneracy lifting is an input rather than a consequence. The abstract also gives no numerical benchmarks or comparison against exact solutions.

The work is aimed at readers who want an analytical handle on spectral ordering inside a modified one-particle Coulomb problem, such as those teaching atomic physics or quantum chemistry. It will not interest people looking for new many-electron mechanisms.

The paper shows clear engagement with the relevant literature on Langer corrections and Sturm-Liouville treatments, so the thinking is serious on its own terms. I would not send it to peer review until the derivation of the correction coefficients is written out in full and shown to be independent of the target ordering.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that a regularized de Broglie-Bohm representation of the stationary hydrogen atom, based on amplitude current constraints, yields separable Sturm-Liouville problems whose radial, orbital, and magnetic sectors acquire canonical Langer-like inverse-square corrections. These modified boundary-value problems admit analytical solutions in terms of generalized associated Laguerre, Legendre, and Bessel functions with non-integral parameters. The resulting orbital-dependent shift in the radial quantization condition lifts the Coulomb l-degeneracy and produces a spectral ordering that follows the empirical Aufbau/Madelung (n+l) sequence, while the l=0 sector retains the standard Rydberg degeneracy.

Significance. If the inverse-square prefactors are shown to be uniquely fixed by the de Broglie-Bohm current constraints without reference to the observed ordering, the work would supply an analytical, single-particle mechanism linking a regularized Coulomb problem to the Madelung rule. The explicit construction of generalized special-function solutions and the retention of the Rydberg series for s-states constitute technical strengths that could be of interest to foundational quantum mechanics and atomic spectroscopy.

major comments (1)

[Abstract] Abstract (paragraph on modified boundary value problems): The claim that the Langer-like corrections are 'canonical' and that the resulting spectrum 'follows the Aufbau/Madelung sequence' requires an explicit derivation demonstrating that the numerical coefficients of the 1/r² terms are uniquely determined by the stationary amplitude-current continuity conditions alone. Without this step, it remains possible that the prefactors are selected to enforce the (n+l) ordering, rendering the degeneracy lifting an input rather than a derived consequence of the regularization.

minor comments (1)

The abstract mentions 'generalized special function branches' but does not specify the precise non-integral parameter values or the explicit form of the modified radial equation; adding these in the main text would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the derivation of the inverse-square corrections. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph on modified boundary value problems): The claim that the Langer-like corrections are 'canonical' and that the resulting spectrum 'follows the Aufbau/Madelung sequence' requires an explicit derivation demonstrating that the numerical coefficients of the 1/r² terms are uniquely determined by the stationary amplitude-current continuity conditions alone. Without this step, it remains possible that the prefactors are selected to enforce the (n+l) ordering, rendering the degeneracy lifting an input rather than a derived consequence of the regularization.

Authors: The coefficients of the 1/r² terms are fixed in Sections 3 and 4 by imposing the stationary amplitude-current continuity conditions on the de Broglie-Bohm representation in spherical coordinates. These conditions require the radial probability current to vanish at the origin and at infinity while preserving separability, which uniquely determines the effective inverse-square strengths in the radial, orbital, and magnetic Sturm-Liouville operators without any reference to empirical spectral ordering. The (n+l) sequence then follows as a derived output of the modified radial quantization condition applied to the resulting generalized Laguerre equation. We agree that the abstract paragraph could more explicitly signal the location of this derivation and will revise it to include a short clause referencing the current-constraint origin of the coefficients. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation framed as independent consequence of amplitude-current constraints

full rationale

The provided abstract states that stationary amplitude current constraints generate separable Sturm-Liouville branches that admit canonical Langer-like inverse-square corrections, after which the modified boundary-value problems produce an orbital-dependent shift and the Aufbau/Madelung ordering as a consequence. No equation, definition, or statement in the text equates the correction coefficients to the target ordering, fits them to data, or imports the result via self-citation. The chain is presented as self-contained from the de Broglie-Bohm representation onward, with the ordering emerging rather than presupposed.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on the de Broglie-Bohm stationary-current constraint and the choice of Langer-like corrections whose precise coefficients are not derived from external data or theorems in the abstract. No new particles or forces are introduced.

free parameters (1)

Langer correction coefficients
Inverse-square correction strengths in radial and angular sectors; their values are required to produce the Aufbau ordering but are not shown to be fixed by independent principles.

axioms (2)

domain assumption Stationary amplitude current constraints generate separable Sturm-Liouville branches
Invoked in the opening paragraph of the abstract to justify separation into radial, orbital, and magnetic sectors.
standard math Generalized associated Laguerre, Legendre, and Bessel functions with non-integral parameters remain valid eigenfunctions
Standard special-function theory is assumed to extend without additional proof.

pith-pipeline@v0.9.1-grok · 5791 in / 1636 out tokens · 50785 ms · 2026-06-27T02:43:58.098453+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages

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

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Second–order differential equations. Conditions of complete integrability,

V. P. Ermakov, “Second–order differential equations. Conditions of complete integrability,” Univ. Izv. Kiev Series III,9, 1–25 (1880)

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Hamiltonian formulation of quantum mechanics with Ermakov invariants,

G. Reinisch, “Hamiltonian formulation of quantum mechanics with Ermakov invariants,” Physica A206, 229–252 (1994)

1994

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

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

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A regularization method to obtain analytical solutions to the de Broglie–Bohm wave equation,

Anand Aruna Kumar, S. K. Srivatsa, and R. Tengli, “A regularization method to obtain analytical solutions to the de Broglie–Bohm wave equation,” International Journal of Quantum Foundations12(2), 313–334 (2026). https://ijqf.org/archives/8075

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Ermakov–Lewis invariants in stationary Bohm–Madelung quantum mechanics,

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On the connection formulas and the solutions of the wave equation,

R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev.51, 669–676 (1937)

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Langer modification, quantization condition and barrier penetration in quantum mechanics,

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E. Madelung, Die Mathematischen Hilfsmittel des Physikers, 3rd ed., Springer, Berlin, 1936

1936

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On the first appearance of atomic electrons withl,n,n r, andn+lgiven,

V. M. Klechkovskii, “On the first appearance of atomic electrons withl,n,n r, andn+lgiven,” Sov. Phys. JETP3(1), 125–127 (1956)

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Justification of the rule for successive filling of (n+l) groups,

V. M. Klechkovskii, “Justification of the rule for successive filling of (n+l) groups,” Sov. Phys. JETP 14(2), 334–335 (1962)

1962

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n+lfilling rule in the periodic system and focusing potentials,

Yu. N. Demkov and V. N. Ostrovskii, “n+lfilling rule in the periodic system and focusing potentials,” Sov. Phys. JETP35(1), 66–69 (1972)

1972

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Second quantization for the Kepler problem,

J. C. Baez, “Second quantization for the Kepler problem,” Journal of Mathematical Physics67(6), 062301 (2026)

2026

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

2013

[15] [15]

Legendre functions of fractional degree: transformations and evaluations,

R. S. Maier, “Legendre functions of fractional degree: transformations and evaluations,” Proc. R. Soc. A 472, 20160097 (2016). https://doi.org/10.1098/rspa.2016.0097

work page doi:10.1098/rspa.2016.0097 2016

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L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non–Relativistic Theory, Course of Theoretical Physics, Vol. 3, 3rd ed., Butterworth–Heinemann, 1998

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F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, Release 1.2.4 of 2025-03-15. https://dlmf.nist.gov/. 11

2025