{"id":"03f9b3ec-091f-4b8c-af9f-edcdab780cf4","arxiv_id":"2504.18928","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Energy levels of a 2D hydrogen atom in a circular box with a Stark field are computed variationally, revealing an exact accidental degeneracy at beta=3/4 and electric-field-induced level splitting.","lead":"This paper computes the energy levels of a two-dimensional hydrogen atom trapped in a circular box with an electric field applied, using a standard variational calculation. It finds an exactly solvable special case where energy levels collide, and shows how the electric field splits degenerate levels and creates avoided crossings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The accidental-degeneracy claim at β=3/4 is supported only by N=12 RRM eigenvalues, with no convergence certificate and the degeneracy proof explicitly deferred; a truncation shift in the crossing would invalidate the paper's central result.","rationale":"The exact ground state (13) is real independent support: direct substitution verifies the equation and boundary condition, so part of the paper's analytical core is solid. The concern is not that the numerics are necessarily wrong but that the paper's main new phenomenon is asserted from a single unbenchmarked RRM run. The reader's weakest assumption—no convergence proof, N=12, figures at r0=10—is exactly the place where the degeneracy claim is least secure. I would keep the CONDITIONAL verdict: require a convergence study plus an independent check of the degeneracy, or a proof or clear separation of the conjecture, before the central claim is accepted as established. This is a routine numerical-support condition, not a rejection.","tokens_in":5045,"tokens_out":13995,"duration_ms":144647,"concrete_test":"Recompute the zero-field radial eigenvalues for (0,2) and (1,0) at β around 3/4 with an independent method such as a shooting solution of the radial ODE or a high-order FEM with fine mesh, resolving eigenvalues to at least 10 digits, and repeat with the RRM at N=16, 20, and 25. If the crossing parameter or the equality E02=E10 at β=3/4 shifts by more than plotting resolution, or if E_{20}=E_{12} fails for the first few conjectured pairs, the central degeneracy claim is unsupported; conversely, if it persists to 10 digits, the conjecture becomes credible and the conditional may be lifted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the accidental threefold degeneracy E02=E10 at β=3/4 and the field-induced splitting of that level. The exact state (13) proves only E00=-1/8; it does not fix the degenerate pair. The equality E02=E10 is established by Fig. 1 from the truncated Rayleigh-Ritz basis (9) with N=12, and Section 4 explicitly declines to prove the wider conjecture E_{n0}=E_{n-1,2}. No convergence test, truncation-error estimate, or independent computation for λ≠0 is supplied, and the authors say the polynomial basis is not expected to be suitable for too large values of r0 although Figures 2-4 go to r0=10. Since the RRM gives upper bounds, an apparent crossing can be an artifact of unequal convergence of two variational states: if the (0,2) eigenvalue has converged from above less tightly than (1,0), the crossing parameter and energy move with N. The paper's main narrative—appearance of accidental degeneracy and splitting of degenerate energy levels—therefore rests on a numerical coincidence that could be an artifact of the N=12 truncation. This is a correctness risk, not a stylistic issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper revisits the two-dimensional hydrogen atom confined to a circular impenetrable box, with the nucleus clamped at the origin, in the presence of a uniform electric field. Working with dimensionless Hamiltonians obtained by different length scalings, the authors derive scaling limits (r0→0 reduces to the particle in a circular box; r0→∞ recovers the free 2D hydrogen energies), prove that the eigenvalues are even functions of the field strength λ under a unitary transformation, and separate the even/odd parity sectors. The main numerical tool is a Rayleigh-Ritz method (RRM) with a truncated polynomial basis. At zero field, for the dimensionless box parameter β=3/4, the authors find a three-fold accidental degeneracy among the states (0,0), (0,2), and (1,0), and give an exact ground-state wavefunction R00(r) with energy -1/8 (Eq. 13). They conjecture the wider ladder E_{n0}=E_{n-1,2} at this β. For nonzero field, they present spectra showing the splitting of the degenerate levels and avoided crossings.","tokens_in":5303,"tokens_out":15548,"duration_ms":145282,"significance":"The paper contains several verifiable analytical results: the exact ground state in Eq. (13) checks by direct substitution; the small-box limit in Table 1 agrees with independent particle-in-a-box eigenvalues; and the symmetry E(-λ)=E(λ) is proven. If the accidental degeneracy is confirmed, the model is a useful benchmark for confined Stark systems. However, the central degeneracy claim currently rests on a single truncated RRM calculation with no convergence certification, which limits the paper's significance.","major_comments":[{"comment":"The three-fold degeneracy E00=E02=E10 at β=3/4 is the paper's central novel feature, but it is established only by the truncated N=12 RRM calculation. The exact state (13) proves only E00=-1/8; it does not constrain E02 or E10. Since RRM eigenvalues are upper bounds and the convergence rates differ between the ν=0 and ν=2 sectors, the apparent crossing in Fig. 1 could shift or disappear with increasing N. Please provide a convergence study (e.g., a table of E02(3/4) and E10(3/4) for increasing N, or an independent shooting-method calculation) before presenting the degeneracy as a fact in Section 5.","section":"Section 4, Figure 1"},{"comment":"No truncation-error estimate or convergence test is provided for any λ≠0 eigenvalue. The authors state that the polynomial basis (9) is 'not expected to be suitable for too large values of r0', yet Figs. 3 and 4 extend to r0=10. The assertion that the N=12 basis provides 'sufficiently accurate eigenvalues' is unsupported. Please provide convergence data for representative (r0,λ) points and either truncate the plotted range to the converged regime or quantify the error bars.","section":"Section 3, Figures 2-4"},{"comment":"The claimed discussion of the large-box-radius limit is incomplete for λ≠0. For λ>0, the potential -λ r cosφ is unbounded below as r→∞, so no bound-state limit r0→∞ exists for fixed λ; the large-r0 parts of Figs. 3-4 therefore require interpretation or removal. The paper discusses the r0→∞ limit only for f=0 (Eq. 8), which should be stated explicitly.","section":"Section 2/5"}],"minor_comments":[{"comment":"The paper should state explicitly that after scaling L=ℏ²/(m_e K) the box radius becomes r0 (in units of L) and that this quantity coincides with the parameter β introduced in Section 4.","section":"Section 2, Eq. (5)"},{"comment":"The normalization constant is printed as '√3e − 8', which is ambiguous; it should read √(3e−8). Also, the remark 'R(r→∞)=0' is confusing for a finite-box problem and should be rephrased as 'the function (13) decays as r→∞' or removed.","section":"Section 4, Eq. (13)"},{"comment":"Typos: 'stisfies' should be 'satisfies' (Section 4); 'higly' should be 'highly' (Introduction); 'r2_0' should be 'r_0^2' in the Table 1 caption and in the text.","section":"Section 4 and Table 1"},{"comment":"For reproducibility, please include a small table of the lowest eigenvalues for representative parameters (e.g., r0=3/4 with λ=0 and λ=1) instead of relying entirely on figures without numerical data.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is a plausible candidate for publication in a quantum-chemistry or mathematical-physics journal, but the central numerical claim needs verification. If the authors provide a convergence study or an independent numerical method confirming the β=3/4 crossing, I would support acceptance after minor revision. The exact ground state (13) is a noteworthy analytical result that should be highlighted more."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has one genuinely exact result that checks out, and everything else is a fairly ordinary Rayleigh-Ritz calculation whose numerical reliability is asserted rather than demonstrated. The exact ground state at β = 3/4 is the thing to take away; the accidental-degeneracy story is the thing to worry about.\n\nCredit where it is earned. Equation (13), R00(r) = (1−r)e^((1−r)/2)/√(3e−8) with E00 = −1/8, does verify by direct substitution, and the authors are careful to present it as an exact state, not a fit. The r0→0 limit against the particle-in-a-box spectrum is a sound independent check, and the E(λ)=E(−λ) symmetry via the φ→φ+π unitary transformation is clean and correct. The authors also deserve credit for explicitly labeling the degeneracy ladder En0 = En−1,2 as a conjecture and not trying to slide it in as a theorem.\n\nThe soft spots are real, though. The RRM eigenvalues come from a basis truncated at N=12, and the paper offers no convergence study, no truncation-error estimate, and no independent benchmark for λ≠0. The authors themselves say the polynomial basis is not suitable for large r0, yet the figures extend to r0=10. The stress-test concern is pointed: because RRM gives upper bounds, the E02/E10 crossing at β=0.75 could shift if the two states converge at different rates, and the crossing is exactly what the \"accidental degeneracy\" narrative leans on. That is a genuine risk, not a stylistic quibble. It is not fatal to the whole paper, because the exact ground state and the symmetry/limits stand independently, but the degeneracy claim as presented is a numerical observation in need of a convergence certificate or a proof.\n\nMinor gripes: no code or data are shipped, the comparison to Longo et al. is a bit brisk, and the discussion of the nucleus location is tangential to the actual calculation. None of these affect the core math.\n\nWho is this for? People working on confined Coulomb systems or using RRM on similar two-dimensional problems. It is a modest but honest contribution. A serious referee should engage with it, primarily to ask for convergence data and to separate the conjecture from the results. With that support, the exact state alone justifies publication.\n\nRecommendation: send it to peer review, but expect revision focused on numerical evidence.","headline":"A verifiable exact ground state and clean symmetry arguments sit inside a numerical RRM calculation whose convergence is never certified; the degeneracy claim needs support before it carries weight.","tokens_in":5839,"tokens_out":1935,"would_cite":false,"duration_ms":20577,"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":"A two-dimensional hydrogen atom in a circular box has an exact degenerate point that an electric field splits.","keywords":["two-dimensional hydrogen atom","circular box confinement","Stark effect","Rayleigh-Ritz method","accidental degeneracy","exact ground state","avoided crossings"],"falsifier":"Substitute $R_{00}(r)=(1-r)e^{(1-r)/2}/\\sqrt{3e-8}$ into the radial Schrödinger equation at $\\beta=3/4$ and confirm $E=-1/8$; then compute $E_{10}-E_{02}$ at $\\beta=3/4$ with an independent high-accuracy method, such as finite differences or a basis with exponential asymptotics. If the gap does not vanish to numerical precision, the three-fold accidental degeneracy is an artifact of the Rayleigh-Ritz truncation.","tokens_in":4835,"feed_emoji":"⚛️","tokens_out":10560,"duration_ms":94061,"temperature":0.7,"pith_summary":"This paper studies the quantum spectrum of a two-dimensional hydrogen atom in an impenetrable circular box, with the nucleus fixed at the centre and a uniform electric field applied. The main analytical result is that when the dimensionless box-strength parameter is $\\beta=3/4$, the zero-field ground state is exactly $R_{00}(r)=(1-r)e^{(1-r)/2}/\\sqrt{3e-8}$ with energy $E_{00}=-1/8$, and at the same parameter the levels $E_{02}$ and $E_{10}$ cross, producing a three-fold degeneracy that the reflection symmetry of the problem does not require. The paper presents Rayleigh-Ritz eigenvalues for nonzero field and shows that this accidental degeneracy splits when the field is turned on, with avoided crossings appearing only between states of the same parity. If the exact solution and the degeneracy are genuine, the model offers a concrete confined-Stark system where a closed-form state and a degeneracy-splitting mechanism can be studied quantitatively.","feed_headline":"Exact degeneracy found for the circular-box 2D hydrogen atom","feed_subtitle":"At box strength beta = 3/4, two levels cross into a three-fold degeneracy that an electric field splits.","key_machinery":"The load-bearing objects are the accidental degeneracy at $\\beta=3/4$ and the exact radial function $R_{00}(r)$. The degeneracy is produced by the crossing of the energy curves $E_{02}(\\beta)$ and $E_{10}(\\beta)$, and the exact ground state is a closed-form eigenfunction at that same point. The numerical engine is the Rayleigh-Ritz method with the polynomial basis sets $r^i(r_0-r)\\cos(j\\varphi)$ and $r^i(r_0-r)\\sin(j\\varphi)$; because the method yields eigenvalues that approach the exact ones from above, the apparent crossings and splittings have a variational guarantee of accuracy within the chosen truncation.","core_discovery":"For the dimensionless Hamiltonian $H=-\\tfrac12\\nabla^2-\\beta/r$ on the unit disk, the authors exhibit, at $\\beta=3/4$, the closed-form radial ground state $R_{00}(r)=(1-r)e^{(1-r)/2}/\\sqrt{3e-8}$ with eigenvalue $E_{00}=-1/8$; this function satisfies the wall condition $R_{00}(1)=0$ and also decays at infinity. Their Rayleigh-Ritz spectrum shows that this same coupling is where the states $E_{02}$ and $E_{10}$ cross, giving a three-fold accidental degeneracy: $E_{02}$ carries $|m|=2$ and is therefore two-fold, while $E_{10}$ carries $m=0$. The numerical results further suggest the stronger pattern $E_{n0}=E_{n-1,2}$ for all $n\\ge1$ at this value of $\\beta$. When the electric field with dimensionless strength $\\lambda$ is turned on, the degeneracy splits; because the eigenvalues are even functions of $\\lambda$, the spectrum is symmetric under reversal of the field, and avoided crossings occur only between states with the same parity under $\\phi\\to-\\phi$.","pith_inferences":["If the pattern $E_{n0}=E_{n-1,2}$ at $\\beta=3/4$ is exact, the model likely possesses a hidden dynamical symmetry beyond its geometric symmetries; the closed form of $R_{00}$ resembles the ground state of a shape-invariant potential and may be the first rung of an exact ladder of solutions.","A direct extension is to compute the small-$\\lambda$ Stark splitting of the degenerate manifold: a linear splitting would measure an effective dipole moment of the doublet, while a quadratic splitting would measure a polarizability, both quantities being checkable with the same Rayleigh-Ritz code at finer truncations.","Because the polynomial basis is designed for moderate box sizes, the large-$r_0$ portions of the spectrum would be a natural place to repeat with a basis carrying the correct exponential decay; the avoided crossings at large $r_0$ are the most likely features to be affected by truncation error."],"forward_implications":["At $\\beta=3/4$ the confined atom has at least one exactly known eigenstate, so the model provides a closed-form benchmark for testing numerical methods on confined Stark systems.","The three-fold accidental degeneracy means a uniaxial electric field splits the level into distinct components, giving a concrete prediction for the Stark pattern of a confined two-dimensional hydrogen atom.","If the conjectured pattern $E_{n0}=E_{n-1,2}$ at $\\beta=3/4$ holds, the spectrum at that box size has a hidden degeneracy structure absent at other sizes.","The symmetry $E(-\\lambda)=E(\\lambda)$ implies Stark shifts are invariant under reversing the field direction, a testable property of the confined model.","In the small-box limit $r_0^2 E_{n\\nu}$ approaches the eigenvalues of a particle in a unit circular box, so the box radius acts as a tunable parameter interpolating between hard-wall and free-atom behaviour."],"supporting_citations":[{"why":"Supplies the earlier treatment of the two-dimensional hydrogen atom in a circular box with an electric field, including a movable nucleus, which this paper revisits with the nucleus clamped.","marker":"[6]"},{"why":"Provides the scaling and dimensionless-Hamiltonian technique used to define the two dimensionless forms of the model and to derive the $r_0\\to 0$ and $r_0\\to\\infty$ limits.","marker":"[9]"},{"why":"States the variational theorem that Rayleigh-Ritz eigenvalues approach the exact eigenvalues from above, the property that justifies trusting the crossings and splittings.","marker":"[12]"},{"why":"Gives the analogous harmonic-oscillator-in-a-circular-box study whose polynomial basis sets and limiting behaviour the present model is compared with.","marker":"[15]"},{"why":"Supplies the general discussion of avoided crossings used to interpret the level repulsion seen in the field-on spectrum.","marker":"[16]"}],"fun_headline_variants":["Circular-box 2D hydrogen: accidental triple degeneracy at beta=3/4","At beta=3/4, the 2D hydrogen atom in a box shows triple degeneracy","Electric field splits accidental degeneracy of boxed 2D hydrogen atom","Closed-form solution reveals triple degeneracy in 2D hydrogen box","Accidental three-fold degeneracy in a circular-box 2D hydrogen atom"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical eigenvalues are treated as converged for every box radius and field strength shown, even though no truncation-error estimate or independent check is given for nonzero field and the polynomial basis is acknowledged to be unreliable for large boxes.","fun_headline_variants_meta":{"raw":{"variants":["Circular-box 2D hydrogen: accidental triple degeneracy at beta=3/4","At beta=3/4, the 2D hydrogen atom in a box shows triple degeneracy","Electric field splits accidental degeneracy of boxed 2D hydrogen atom","Closed-form solution reveals triple degeneracy in 2D hydrogen box","Accidental three-fold degeneracy in a circular-box 2D hydrogen atom"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000688,"raw_usage":{"total_tokens":3085,"prompt_tokens":878,"completion_tokens":2207,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2101}},"tokens_in":494,"tokens_out":2207,"duration_ms":14154,"temperature":1.0,"reasoning_tokens":2101,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:06:37.043278+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute $R_{00}(r)=(1-r)e^{(1-r)/2}/\\sqrt{3e-8}$ into the radial Schrödinger equation at $\\beta=3/4$ and confirm $E=-1/8$; then compute $E_{10}-E_{02}$ at $\\beta=3/4$ with an independent high-accuracy method, such as finite differences or a basis with exponential asymptotics. If the gap does not vanish to numerical precision, the three-fold accidental degeneracy is an artifact of the Rayleigh-Ritz truncation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the earlier treatment of the two-dimensional hydrogen atom in a circular box with an electric field, including a movable nucleus, which this paper revisits with the nucleus clamped."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the variational theorem that Rayleigh-Ritz eigenvalues approach the exact eigenvalues from above, the property that justifies trusting the crossings and splittings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the analogous harmonic-oscillator-in-a-circular-box study whose polynomial basis sets and limiting behaviour the present model is compared with."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the general discussion of avoided crossings used to interpret the level repulsion seen in the field-on spectrum."}],"review_version":1}