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REVIEW 2 major objections 4 minor 24 references

On the exact solutions of a two-dimensional hydrogen atom in a constant magnetic field

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The exact polynomial solutions for the two-dimensional hydrogen atom in a constant magnetic field are not eigenvalues of the radial equation; each matches a true eigenstate only at isolated field strengths where its curve crosses a…

desk verdict A useful clarification of quasi-exact solutions in 2D hydrogen in a magnetic field, but the paper overstates its main claim and leans on a shaky Hellmann-Feynman argument. read the letter →

arxiv 2506.07765 v2 pith:O77LRYBZ submitted 2025-06-09 quant-ph

Francisco M. Fernández This is my paper
classification quant-ph
keywords quasi-exactsolvablemodelstwo-dimensionalhydrogenatomconstantmagneticfieldFrobeniusmethodRayleigh-Ritzthree-termrecurrencerelationpolynomialsolutionsspectralinterpretation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is about the meaning of exact-looking polynomial solutions for the two-dimensional hydrogen atom in a constant magnetic field. It argues that the energies $W_s^{(n)}=\gamma(n+s+1)/2$ produced by truncating the Frobenius series are not the spectrum of the radial Schrödinger equation, because they are positive for all parameters and the polynomial index $n$ is not the radial quantum number. Comparing quasi-exact curves with Rayleigh-Ritz variational eigenvalues computed for $Z=1$, $s=0$, the paper shows that each polynomial solution becomes a true eigenstate only at special field values $\gamma_s^{(n,i)}$ where its curve crosses a variational eigenvalue curve. The point matters because such solutions have sometimes been read as exact spectra of quasi-solvable models; the paper gives a criterion for when they are physically meaningful.

What carries the argument

The central object is the truncated Frobenius ansatz $R_s^{(n)}(r)=r^s e^{-\gamma r^2/4}\sum_{j=0}^n c_j r^j$, whose coefficients obey the three-term recurrence relation (7). The truncation conditions $c_{n+1}=c_{n+2}=0$ force $W_s^{(n)}=\gamma(n+s+1)/2$ and leave a polynomial equation whose roots are the special field values $\gamma_s^{(n,i)}$. The comparison machinery is the Rayleigh-Ritz method with non-orthogonal basis sets $u_{is}=r^{i+s}e^{-\gamma r^2/4}$ for large $\gamma$ and $v_{is}=r^{i+s}e^{-\alpha r}$ for small $\gamma$; the secular determinants factor out the QS energies at the special field values, displaying the crossings as exact.

What would settle it

Pick a value of $\gamma$ where a quasi-exact curve $W_s^{(n)}(\gamma)$ crosses a converged Rayleigh-Ritz eigenvalue curve but $\gamma$ is not one of the special roots $\gamma_s^{(n,i)}$ (for instance, an unmarked crossing visible in the paper's Figure 1), and solve the radial equation (1) to high precision by an independent method such as numerical integration. If the resulting eigenenergy equals $W_s^{(n)}(\gamma)$, the central claim is wrong; if it differs, the claim is supported.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the quasi-exactly-solvable (QS) energies $W_s^{(n)}(\gamma)$ obtained from the exact polynomial solutions are not eigenvalues of the radial equation (1). They satisfy the wrong monotonicity in $Z$, they are always positive, and $n$ does not count nodes. What is true is that for each polynomial degree $n$ there are special field strengths $\gamma_s^{(n,i)}$, roots of a polynomial equation coming from $c_{n+1}=0$, such that the pair $(\gamma_s^{(n,i)}, W_s^{(n,i)})$ coincides with a genuine eigenpair of the Schrödinger equation; the paper exhibits this coincidence by showing that the Rayleigh-Ritz secular determinant factors out the QS energy at those points. Thus the exact polynomial solutions are isolated exact solutions for special parameters, not a description of the spectrum.

Load-bearing premise

The whole comparison assumes the Rayleigh-Ritz eigenvalues with the chosen finite basis sets converge to the true spectrum, so that a crossing between a quasi-exact curve and a variational curve really marks an eigenpair.

Editorial extensions

If this is right

  • For $Z>0$ the truncation method never yields the ground state of the model, since it cannot produce a nodeless solution; the paper's figures and tables show this state missing from the QS curves.
  • Away from the discrete crossing points $\gamma_s^{(n,i)}$, the quasi-exact energy $W_s^{(n)}(\gamma)$ carries no spectral information, so plots that treat those curves as eigenvalues are misleading.
  • Because $W_s^{(n)}$ is always positive while true bound-state eigenvalues are negative at $\gamma=0$ and become positive only above critical fields $\gamma^c_{\nu s}$, the QS solutions necessarily miss the entire negative-energy part of the spectrum.
  • The scaling $\gamma_s^{(n)}(Z)=\gamma_s^{(n)}(1)Z^2$ and $c_j(Z)=c_j(1)Z^j$ reduces the two-parameter problem to $Z=1$ without loss of generality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same 'crossing, not spectrum' interpretation should apply to any quasi-exactly-solvable model where a truncated power series is compared with a variational spectrum; the secular-determinant factorization used here is a general diagnostic for locating the crossings.
  • A stricter test would optimize the small-$\gamma$ basis exponent $\alpha$ variationally and increase the basis dimension until the eigenvalues stop moving; that would confirm that the unmarked crossings in the figures are not numerical artifacts.
  • The special parameter pairs $(\gamma_s^{(n,i)}, W_s^{(n,i)})$ are exact solutions of the differential equation and could be used as benchmark tests for numerical methods, even though they do not represent the spectrum globally.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper revisits the exact polynomial (quasi-exact) solutions of the two-dimensional hydrogen atom in a constant magnetic field. The author uses the Frobenius method to rederive polynomial solutions of degree n with energy W_s^(n) = γ(n+s+1)/2, where the termination condition fixes γ as a function of Z (or vice versa). He then compares these quasi-exact results with Rayleigh-Ritz numerical eigenvalues for Z=1, s=0, showing that the quasi-exact energies appear as exact RRM eigenvalues only at the specific field values γ_s^(n,i) predicted by the truncation condition, while the straight lines W_s^(n)(γ) also cross the true eigencurves at other points. The conclusions are that the polynomial solutions are isolated quasi-exact states, that they do not form a complete spectral family, and that the degree n is not the radial quantum number in general.

Significance. The intended message is sound and valuable: the polynomial solutions are eigenfunctions only at special parameter values, and the energy formula W_s^(n)=γ(n+s+1)/2 should not be read as a spectral curve at fixed Z. The paper's strengths are its clean Frobenius derivation, the explicit RRM tables (Tables 1 and 2) that reproduce the quasi-exact energies at the expected γ values, and the clear graphical demonstration (Figures 1 and 2) that some intersections of the quasi-exact lines with true eigencurves are not quasi-exact points. The novelty is limited, since Taut (1995) and Le et al. (2017) already established the quasi-exact nature of these solutions; the present contribution is mainly a pedagogical re-presentation and numerical confirmation. With the central statements made precise, the paper would be a useful cautionary note.

major comments (2)
  1. [Section 3, after Eq. (8)] The sentence 'W_s^(n)(Z) are not the eigenvalues of the radial equation (1)' is too strong and, as written, false. For every Z the termination condition fixes γ_s^(n)(Z), and at that field value the polynomial R_s^(n)(r) is a square-integrable solution of (1) with eigenvalue W_s^(n)(Z); hence W_s^(n)(Z) is an eigenvalue of the Hamiltonian with that specific γ. The accompanying claim that the positivity of W_s^(n) shows the solutions cannot describe the whole spectrum is also not a sufficient reason, because true eigenvalues become positive for sufficiently large γ. The correct statement, which the paper itself uses in Section 4, is that W_s^(n)(γ) is not a spectral curve for fixed Z except at the roots γ_s^(n,i), and that at non-QS intersections the energy equals a true eigenvalue whose eigenfunction is not polynomial. Please revise this paragraph with explicit quantifiers and give the actual reason for the isolated nature of the QS states.
  2. [Section 3, paragraph citing Eq. (3)] The appeal to the Hellmann-Feynman theorem is not valid as an explanation of the Z-dependence. Equation (3) states ∂W_νs/∂Z < 0 at fixed γ, but along a quasi-exact curve γ = γ_s^(n)(Z) depends on Z; the total derivative of W_s^(n)(Z) includes the positive term (∂W_s^(n)/∂γ)(dγ_s^(n)/dZ), so no contradiction with the HFT arises. The reason the quasi-exact energies are not a spectral family is that the polynomial ansatz is a solution only when γ coincides with a root of the termination condition. Please remove or rewrite the sentence 'which is the reason why they do not exhibit the correct behaviour with respect to Z (see equation (3))'.
minor comments (4)
  1. [Section 4, Eq. (22)] The secular determinant expressions in Eq. (22) are asserted without derivation or a description of how they were obtained; including the matrix elements or a short derivation would make the benchmark more transparent.
  2. [Section 4, basis sets] The RRM is presented as providing upper bounds, but the completeness of the bases (21) and (23) is not stated; a sentence noting their completeness (e.g., via generalized Laguerre polynomials) would make the convergence claim rigorous, and the text should clarify that α=1 in (23) is a fixed, non-optimized choice.
  3. [Introduction, line after Ref. [10]] The phrase 'see allso [10]' contains a typo and should read 'see also [10]'.
  4. [Figure 1] The horizontal axis label appears to contain a duplicate tick value ('3 3'); please check the axis labeling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quasi-exact energies are derived from the Frobenius recurrence, and the Rayleigh-Ritz comparison is an independent numerical benchmark.

full rationale

The paper's derivation chain is self-contained. The ansatz (6) and the recurrence relation (7) follow directly from the differential equation, and the truncation conditions cn+1 = cn+2 = 0 with Bn = 0 yield Eq. (8), W_s^(n) = gamma(n+s+1)/2, while the condition cn+1 = 0 determines the discrete gamma_s^(n,i) values. These QS energies are not fitted to the RRM data; they are algebraic outputs of the recurrence. The RRM in Section 4 is used only as an independent upper-bound benchmark, with convergence checked in Tables 1 and 2 and crossings identified in Figures 1 and 2. The paper does cite the author's earlier papers [9,10] for the general warning that QS solutions should not be mistaken for the full spectrum, but that claim is independently supported here by the explicit polynomial construction and by comparison with Taut [14], Le et al [12], and the RRM. The Hellmann-Feynman argument leading to the blanket statement that W_s^(n)(Z) are not eigenvalues is mathematically imprecise, since the polynomial is an eigenfunction at the companion field gamma_s^(n,i), but this is a correctness concern rather than a circular reduction. No parameter is fitted and then renamed a prediction, and no load-bearing result is imported solely from a self-citation. Therefore no significant circularity is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard spectral theory of the radial Schrodinger equation, on the termination condition for a three-term recurrence, and on the Rayleigh-Ritz convergence theorem. The only hand-chosen numerical parameter is alpha=1 in the small-gamma basis; it affects convergence speed but not the converged eigenvalues. No invented entities are introduced.

free parameters (1)
  • RRM basis exponent alpha = 1
    In the small-gamma basis v_is(r)=r^(i+s)e^(-alpha r), the exponent is set to alpha=1 by hand for all calculations (Section 4). It affects convergence speed, not the converged eigenvalues, so it is not used to fit the central claim.
assumptions (4)
  • domain assumption The radial eigenvalue equation (1) with bound-state condition (2) describes the 2D hydrogen atom in a constant magnetic field.
    Taken from prior literature [12,14]; all analytical and numerical results are about this operator.
  • domain assumption The asymptotic ansatz R(r) ~ r^s e^(-gamma r^2/4) sum c_j r^j (Eq. 6) captures all square-integrable solutions for gamma>0.
    Standard Frobenius asymptotic analysis; not proved in this paper but standard for this potential.
  • standard math A polynomial solution of degree n is obtained exactly when c_n != 0 and c_(n+1)=c_(n+2)=0 in the three-term recurrence (7).
    Termination of the recurrence gives a finite sum that solves the differential equation; this is the standard truncation condition.
  • domain assumption Rayleigh-Ritz eigenvalues with increasing basis dimension converge from above to the true eigenvalues (MacDonald's theorem).
    Invoked via refs [16,17]; the comparison between QS energies and RRM curves relies on this convergence at the displayed dimensions.

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Cite this review

Pith. "Pith review of On the exact solutions of a two-dimensional hydrogen atom in a constant magnetic field." pith.science (2026). https://pith.science/paper/O77LRYBZ

@misc{pith2026250607765,
  author       = {Pith},
  title        = {Pith review of: On the exact solutions of a two-dimensional hydrogen atom in a constant magnetic field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O77LRYBZ}},
  note         = {Machine review of arXiv:2506.07765}
}
read the original abstract

We discuss the exact polynomial solutions for the two-dimensional hydrogen atom in a constant magnetic field already studied earlier by other authors. In order to provide a suitable meaning for such solutions we compare them with numerical results provided by the Rayleigh-Ritz method.

Figures

Figures reproduced from arXiv: 2506.07765 by the authors.

Figure 1
Figure 1. RRM eigenvalues Wν0, ν = 0, 1, 2, 3 (blue continuous lines), TTRR eigenvalues W (n) 0 (γ), n = 0, 1, . . . , 6 (red dashed lines) and W (n) 0  γ (n) 0  (red points) 0.0 0.1 0.2 0.3 0.4 0.5 -0.1 0.2 0.5 0.8 1.1 1.4 1.7 γ W(n) 0 , Wν 0 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. RRM eigenvalues Wν0, ν = 1, 2, . . . , 5 (blue continuous lines), TTRR eigenvalues W (n) 0 (γ), n = 0, 1, . . . , 8 (red dashed lines) and W (n) 0  γ (n) 0  (red points) 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

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Works this paper leans on

24 extracted references · 22 canonical work pages

  1. [1]

    Cohen-Tannoudji, B

    C. Cohen-Tannoudji, B. Diu, and F. Lalo¨ e, Quantum Mechanics ( John Wiley & Sons, New York, 1977)

    work page 1977

  2. [2]

    F. L. Pilar, Elementary Quantum Chemistry (McGraw-Hill, New York , 1968)

    work page 1968

  3. [3]

    P Flessas, Phys

    G. P Flessas, Phys. Lett. A 72, 289 (1979)

    work page 1979

  4. [4]

    P Flessas and K

    G. P Flessas and K. P. Das, Phys. Lett. A 78, 19 (1980)

    work page 1980

  5. [5]

    P Flessas, J

    G. P Flessas, J. Phys. A 14, L209 (1981)

    work page 1981

  6. [6]

    P Flessas, Phys

    G. P Flessas, Phys. Lett. A 81, 17 (1981)

    work page 1981

  7. [7]

    P Flessas and A

    G. P Flessas and A. Watt, J. Phys. A 14, L315 (1981)

    work page 1981

  8. [8]

    A. V. Turbiner, Phys. Rep. 642, 1 (2016). arXiv:1603.02992 [quant-ph]

    work page Pith review arXiv 2016

Show all 24 references
  1. [9]

    F. M. Fern´ andez, Ann. Phys.434, 168645 (2021). arXiv:2109.11545 [quant- ph]

    2021 arXiv

  2. [10]

    Amore and F

    P. Amore and F. M. Fern´ andez, J. Math. Phys. 62, 032106 (2021). arXiv:2110.14526 [quant-ph]

    2021 arXiv

  3. [11]

    M. S. Child, S-H. Dong, and X-G. Wang, J. Phys. A 33, 5653 (2000). 9

    2000

  4. [12]

    D-N. Le, N-T. D. Ngoc-Tram D. Hoang, and V-H. Le, J. Math. P hys. 58, 042102 (2017)

    2017

  5. [13]

    Bildstein and M

    S. Bildstein and M. Grabowski, J. Math. Phys. 66, 012103 (2025)

    2025

  6. [14]

    M. Taut, J. Phys. A 28, 2081 (1995)

    1995

  7. [15]

    A. V. Turbiner and Escobar-Ruiz M. A., J. Phys. A 46, 295204 (2013)

    2013

  8. [16]

    J. K. L. MacDonald, Phys Rev. 43, 830 (1933)

    1933

  9. [17]

    F. M. Fern´ andez, J. Math. Chem. 63, 911 (2025). arXiv:2206.05122 [quant- ph]

    2025 arXiv

  10. [18]

    G¨ uttinger, Z

    P. G¨ uttinger, Z. Phys. 73, 169 (1932)

    1932

  11. [19]

    R. P. Feynman, Phys. Rev. 56, 340 (1939)

    1939

  12. [20]

    F. M. Fern´ andez, Dimensionless equations in non-relativistic qu antum me- chanics, arXiv:2005.05377 [quant-ph]

    2005 arXiv

  13. [21]

    Hoang-Do, D-L

    N-T. Hoang-Do, D-L. Pham, and V-H. Le, Physica B 423, 31 (2013)

    2013

  14. [22]

    F. M. Fern´ andez, A. M. Mes´ on, and E. A. Castro, J. Phys. A 18, 1389 (1985)

    1985

  15. [23]

    F. M. Fern´ andez, J. Math. Chem. 62, 2083 (2024). arXiv:2405.10340 [quant-ph]

    2024 arXiv

  16. [24]

    F. M. Fern´ andez, Q. Ma, and R. H. Tipping, Phys. Rev. A 39, 1605 (1989). 10 Table 1: Lowest RRM eigenvalues for Z = 1, s = 0 and γ = 4 N ν = 0 ν = 1 ν = 2 ν = 3 4 -1.449885589 4 8.34525977 17.66452696 5 -1.458156835 4 8.344361267 12.69095166 6 -1.459389343 4 8.344349784 12.5...

    1989

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