{"id":"0707f174-7a21-4e21-8a17-24d691c792d7","arxiv_id":"2506.07765","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"The polynomial quasi-exact solutions for the 2D hydrogen atom in a magnetic field match true eigenstates only at special field strengths, so they are not the full spectrum.","lead":"This paper checks the so-called exact polynomial solutions of the two-dimensional hydrogen atom in a magnetic field against accurate numerical eigenvalues. It shows these solutions are not the full spectrum and only coincide with true eigenstates at special field strengths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The blanket claim that W_s^(n)(Z) are not eigenvalues is literally false and is not supported by the HFT argument; the defensible claim is that the polynomial wavefunctions are exact only at QS points, while the same energy can occur at non-QS crossings.","rationale":"The paper's core mathematical content is sound: the polynomial constructed by the truncation condition is an exact eigenfunction at the special field values gamma_s^(n,i), and those isolated solutions do not constitute the full spectrum. The RRM comparison in Section 4 is an appropriate interpretation, and at the large-gamma QS points the QS function lies exactly in the u_is basis, so the observed exact RRM eigenvalues for W_10 and W_20 are independent evidence rather than a fragile numerical accident. The reader's weakest_assumption about RRM completeness and convergence is therefore not the most load-bearing issue for the central claim: the exactness of the QS points is analytic, and the qualitative existence of extra intersections is robust. The real weakness is in the precise formulation of the central claim. Section 3's sentence that W_s^(n)(Z) are not eigenvalues is literally false under the natural reading because, at the gamma determined by the truncation condition, the polynomial energy is an eigenvalue. The paper's own Figure 1 shows that W_s^(n)(gamma) lines can cross true eigencurves at non-QS gamma values, so the energy value can be an eigenvalue even when the polynomial is not a solution. The supporting HFT argument is also misapplied, since Eq. (3) is a fixed-gamma derivative while gamma is varied along the QS family. The correct and sufficient claim is that the polynomial wavefunction is an eigenfunction only at the QS roots, and that the straight lines W_s^(n)(gamma) are not spectral curves. This does not overturn the paper's useful clarification, but it does require a careful rewriting of the headline assertion and a proof of the square-integrability of the non-terminating Frobenius solution to justify the 'not eigenvalues' wording. The reader's CONDITIONAL verdict already allows for such revision, so I keep the verdict unchanged.","tokens_in":6553,"tokens_out":31032,"duration_ms":372676,"concrete_test":"Fix Z=1, s=0 and compute the first-excited eigencurve W_1(gamma) with a high-accuracy spectral method (e.g., large-N RRM with the u_is basis and a Richardson extrapolation). Find the non-QS root gamma* of W_1(gamma) = gamma/2, the n=0 line, which Figure 1 indicates exists. At gamma*, construct the Frobenius solution of Eq. (7) with W = gamma*/2 and verify by asymptotic analysis or by direct numerical square-integrability that the series is non-terminating yet normalizable. If it is normalizable, the literal Section 3 statement that W_s^(n)(Z) are not eigenvalues is false, and the paper must be revised to say that the polynomial wavefunctions are exact only at QS points while the energy values can coincide with true eigenvalues at other field strengths.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 states that 'W_s^(n)(Z) are not the eigenvalues of the radial equation (1)' and attributes this to the wrong Z-dependence via the Hellmann-Feynman theorem (Eq. 3). As written, this is false: for each Z the truncation condition fixes gamma_s^(n)(Z), and at that companion field the polynomial R_s^(n)(r) is an exact square-integrable solution with energy W_s^(n)(Z), so the energy is an eigenvalue of that Hamiltonian. The HFT argument also does not apply directly because Eq. (3) is a derivative at fixed gamma, whereas gamma_s^(n)(Z) changes with Z along the QS curve; the derivative of W along that curve includes a gamma contribution. The defensible claim, which the paper itself uses in Section 4, is that the line W_s^(n)(gamma) = gamma(n+s+1)/2 is not a spectral curve for fixed Z and that the polynomial wavefunction is an eigenfunction only at the roots gamma_s^(n,i). The paper even shows intersections of W_s^(n)(gamma) with true eigencurves that are not QS points, meaning the same numerical energy can be a true eigenenergy realized by a non-polynomial eigenfunction. Without a precise quantifier, the central correction overstates its conclusion and could mislead readers into thinking these energies are never eigenvalues.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":6882,"tokens_out":18230,"duration_ms":204190,"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":[{"comment":"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.","section":"Section 3, after Eq. (8)"},{"comment":"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))'.","section":"Section 3, paragraph citing Eq. (3)"}],"minor_comments":[{"comment":"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.","section":"Section 4, Eq. (22)"},{"comment":"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.","section":"Section 4, basis sets"},{"comment":"The phrase 'see allso [10]' contains a typo and should read 'see also [10]'.","section":"Introduction, line after Ref. [10]"},{"comment":"The horizontal axis label appears to contain a duplicate tick value ('3 3'); please check the axis labeling.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely a re-presentation of results already in Taut (1995) and Le et al. (2017), and it relies heavily on the author's own earlier papers; if the journal values strong novelty this is a concern, but the clear numerical demonstration and the explicit correction of an overstatement in Bildstein and Grabowski may still make it acceptable as a pedagogical note."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Fernandez paper on the 2D hydrogen atom in a constant magnetic field. The useful part is the numerical interpretation: the RRM calculations show clearly that the QS polynomial energies coincide with true eigenvalues only at the special field values gamma_s^(n,i), and the paper adds a scaling relation gamma_s^(n)(Z)=gamma_s^(n)(1)Z^2 and a table of critical gamma values. That's a genuine, if modest, contribution.\n\nThe soft spot is the claim in Section 3 that the QS energies 'are not the eigenvalues of the radial equation (1)'. Taken at face value that is false: at each field value satisfying the truncation condition, the polynomial is an exact square-integrable solution of exactly that equation, so its energy is an eigenvalue of the Hamiltonian at that field. The HFT argument offered in support doesn't hold, because the derivative in Eq. (3) is at fixed gamma whereas the QS curve changes gamma with Z. The defensible conclusion, which the paper's own figures support, is that the line W = gamma(n+s+1)/2 is not a spectral curve for fixed Z and that true eigenvalue curves cross it at points that are not QS points. This is a fixable exposition problem, but it needs fixing: as written, the central claim overstates the result.\n\nThe rest is in better shape. The recurrence and truncation are clean, the RRM tables reproduce W10=4 and W20=1 as expected, and the comparison with Taut and Le et al. is honest. Minor annoyances: the secular determinants in Eq. (22) are quoted without derivation, and the small-gamma basis uses a hand-set alpha=1 with no completeness proof. These are minor because the printed RRM values already make the point.\n\nWho should read this? People who work on quasi-exact solvability in this model; it's a cautionary note with a couple of new numbers, not a paradigm shift. I'd send it to an expert referee rather than desk reject, but the author should be asked to correct the overbroad claim and reframe the HFT argument before publication.","headline":"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.","tokens_in":7378,"tokens_out":5636,"would_cite":true,"duration_ms":62346,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["quasi-exact solvable models","two-dimensional hydrogen atom","constant magnetic field","Frobenius method","Rayleigh-Ritz method","three-term recurrence relation","polynomial solutions","spectral interpretation"],"falsifier":"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.","tokens_in":6379,"feed_emoji":"🧲","tokens_out":10114,"duration_ms":104305,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact polynomial solutions are not the hydrogen-in-field spectrum","feed_subtitle":"Each polynomial energy matches a true eigenstate only where its curve crosses a Rayleigh-Ritz eigenvalue curve.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"presents the quasi-exact polynomial solutions for this model that the paper interprets.","marker":"[13]"},{"why":"earlier study of the same solutions; supplies the node theorem and a secular-equation comparison the paper builds on.","marker":"[12]"},{"why":"original Frobenius treatment of the model; notes that the truncation fails to give the nodeless ground state.","marker":"[14]"},{"why":"establishes the truncation rule and the general argument that QS solutions are often misread as spectra.","marker":"[9]"},{"why":"companion discussion of incorrect conclusions drawn from QS solutions.","marker":"[10]"},{"why":"textbook basis for the Rayleigh-Ritz method used for the numerical comparison.","marker":"[2]"},{"why":"theorem that Rayleigh-Ritz eigenvalues are upper bounds, justifying the comparison with true eigenvalues.","marker":"[16]"},{"why":"supports the convergence of the Rayleigh-Ritz eigenvalues used in the tables.","marker":"[17]"},{"why":"used to compute accurate critical field values where eigenvalues vanish, supporting the discussion of why QS solutions miss low-lying states.","marker":"[24]"}],"fun_headline_variants":["Polynomial solutions hit the spectrum only at crossings","Exact solutions exist, but only for special field strengths","Hydrogen in a field: polynomial answers are sparse","QS energies are only exact at isolated field strengths","Polynomial solutions: not a spectrum, just special points"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial solutions hit the spectrum only at crossings","Exact solutions exist, but only for special field strengths","Hydrogen in a field: polynomial answers are sparse","QS energies are only exact at isolated field strengths","Polynomial solutions: not a spectrum, just special points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000582,"raw_usage":{"total_tokens":2638,"prompt_tokens":746,"completion_tokens":1892,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":362,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":362,"tokens_out":1892,"duration_ms":13559,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:26:27.598327+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bildstein and M","cited_arxiv_id":null,"evidence_quote":"presents the quasi-exact polynomial solutions for this model that the paper interprets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"earlier study of the same solutions; supplies the node theorem and a secular-equation comparison the paper builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"original Frobenius treatment of the model; notes that the truncation fails to give the nodeless ground state."},{"cited_title":"A most misunderstood conditionally-solvable quantum-mechanical model","cited_arxiv_id":"2109.11545","evidence_quote":"establishes the truncation rule and the general argument that QS solutions are often misread as spectra."},{"cited_title":"An ubiquitous three-term recurrence relation","cited_arxiv_id":"2110.14526","evidence_quote":"companion discussion of incorrect conclusions drawn from QS solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"textbook basis for the Rayleigh-Ritz method used for the numerical comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"theorem that Rayleigh-Ritz eigenvalues are upper bounds, justifying the comparison with true eigenvalues."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"used to compute accurate critical field values where eigenvalues vanish, supporting the discussion of why QS solutions miss low-lying states."}],"review_version":1}