{"id":"d585d458-58b4-4ebc-aa27-9fabd7b88b86","arxiv_id":"2608.23457","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a three-term asymptotic expansion for the eigenvalues and uniform L-infinity bounds for the eigenfunctions of the fractional Laplacian on an interval, settling conjectures from numerical experiments.","lead":"This paper derives a sharper formula for the energy levels of the fractional Laplacian in a one-dimensional interval, including the next correction term beyond the known two-term approximation. It also proves that the corresponding eigenfunctions stay uniformly bounded, confirming a numerical conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1(i) coefficient B_alpha is inconsistent with the proof: Proposition 5.4's own algebra yields tan(πα/2) in the numerator, while (1.7) defines B_alpha with tan(πα/2) in the denominator.","rationale":"The reader's weakest_assumption concerns the imported half-line representation from [23]. My concern is different and more direct: the paper contains an internal algebraic inconsistency in the proof of the central third-term formula. Theorem 1(i) is the main new eigenvalue asymptotics result, and its coefficient B_α is explicitly used to state the asymptotics. If the coefficient is wrong as printed, the central claim is not correct as stated. The issue is localized to the final simplification in Proposition 5.4 and the definition in (1.7): the displayed identity forces tan(πα/2) into the numerator, but the definition puts it in the denominator. The rest of the proof, including the half-line expansions, the residual estimates, and the uniform L∞ argument, appears coherent and I did not find a comparable flaw there. Because the fix is a single coefficient replacement and the proof contains the ingredients for it, I would not reject the paper outright; I would require the correction before acceptance. Hence CONDITIONAL rather than ACCEPT or REJECT.","tokens_in":27671,"tokens_out":46804,"duration_ms":423760,"concrete_test":"Take α = 1/4. Substitute (5.28), (5.9), and (5.10) into (5.3), expand D_L to order L^{−1−2α}, and pass to the Rayleigh quotient as in Proposition 5.4, using the exact identities for Γ(1−α)Γ(1+2α) and the integrals (5.21)–(5.22). Compare the resulting coefficient of μ_n^{−2−α} with (1.7). A simpler check: starting from the line immediately before the final 'Hence' in Proposition 5.4, multiply sin x(2 cos x − sec x) by c_α/Γ(1+α) = sin x/π; the product is cos(2x) tan x / π, so the denominator-tan formula in (1.7) cannot follow without an additional algebraic step.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equations (1.7) and the final simplification in Proposition 5.4 cannot both hold. In Proposition 5.4, the coefficient of −ε μ_n^{−2−α} is obtained from (5.29) as α c_α Γ(1+2α)/(Γ(1+α) 2^{1+2α}) (2 cos x − sec x), x = πα/2. Using c_α/Γ(1+α) = sin x/π and sin x(2 cos x − sec x) = cos(2x) tan x, this coefficient equals α Γ(1+2α) cos(πα) tan(πα/2)/(π 2^{1+2α}). The paper instead writes this expression with tan(πα/2) in the denominator and identifies it with B_α. Since Theorem 1(i) uses (1.7), the stated third term is off by a factor tan^2(πα/2) for every 0 < α < 1 except α = 1/2. This is an internal inconsistency, not a question about the imported half-line constants: the displayed algebra in the last paragraph of Section 5 is false as written. The proof itself appears to supply the numerator version, so the statement should be corrected.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fractional Laplacian (−Δ)^{α/2} with zero exterior condition on the interval I=(−1,1), 0<α<2. Theorem 1 claims a three-term asymptotic expansion for the n-th eigenvalue λ_n(α), with explicit constants κ_α, A_α, B_α, η_α, improving the two-term formula of Kwaśnicki and confirming the numerically predicted O_α(n^{−2}) remainder. Theorem 2 claims an absolute bound on the L^∞ norms of the normalized eigenfunctions, uniform in both α and n. The proofs construct quasimodes from half-line eigenfunctions imported from Kwaśnicki’s work, derive residual estimates in W^{1,1}, and then extract the third term in the eigenvalue expansion; the eigenfunction bound is obtained via heat-semigroup estimates, an antiderivative bound, and a truncated quadratic-form argument for small α.","tokens_in":27899,"tokens_out":29677,"duration_ms":239581,"significance":"If the main results are correct after the coefficient issue described below is fixed, this is a substantial contribution: it settles the conjectural third-order eigenvalue asymptotics for the fractional Laplacian in an interval and the Kwaśnicki conjecture on uniform L^∞ eigenfunction bounds. The paper is commendably explicit: all constants are given in closed form, the proofs are structured through verifiable propositions, and the argument does not fit parameters to the interval eigenvalues. The proof of Theorem 2 is especially strong, as it is uniform in both n and α. The use of the already-proved two-term formula to identify the auxiliary integral Q_α in (5.28) is a consistency check and is not circular, because the two-term formula is established before that point.","major_comments":[{"comment":"The final simplification in Proposition 5.4 is inconsistent with the definition of B_alpha in (1.7). Starting from the coefficient in (5.29), with x=πα/2, the coefficient of −ε μ_n^{−2−α} before the final identification is α c_α Γ(1+2α)/(Γ(1+α) 2^{1+2α}) (2 cos x − sec x). Using c_α/Γ(1+α)=sin x/π and sin x(2 cos x − sec x)=cos(2x) tan x, this equals α Γ(1+2α) cos(πα) tan(πα/2)/(π 2^{1+2α}). The paper instead writes this expression as equal to B_alpha, which has tan(πα/2) in the denominator. Consequently Theorem 1(i) and equation (1.9) are off by a factor tan^2(πα/2) for every 0<α<1 except α=1/2, where the coefficient vanishes. The proof itself supplies the numerator version, so the statement should be corrected by changing (1.7) to the numerator form (or, if the intended coefficient is the denominator form, by recomputing the preceding algebra in (5.29)). This is a load-bearing error in the headline three-term formula and must be fixed before acceptance.","section":"Section 5, Proposition 5.4 (final paragraph) and equation (1.7)"}],"minor_comments":[{"comment":"The notation √α/2 is ambiguous in the typeset text. In context it is consistently used to mean √(α/2), but this is not clear without parentheses. Please write √(α/2) explicitly in all displayed formulas.","section":"Throughout; especially (2.3), (2.13), (5.20), and the definitions of K_α and M_α"},{"comment":"The symbol P is used both for the pointwise quantity P_k = sup_x |∫_{−1}^x φ_k(t)dt| and for the scalar maximum P = max{...}. This is locally clear but could be confusing; renaming the scalar, for example P_max or M, would improve readability.","section":"Section 6, proof of Lemma 6.4"}],"recommendation":"major_revision","confidential_remarks":"The algebra slip in the coefficient B_alpha is localized and easily fixable, but because Theorem 1(i) is the central advertised result, I cannot recommend acceptance in the current form. Apart from this, the paper appears carefully executed: the residual estimates, the uniform eigenfunction argument, and the use of the imported half-line machinery are all consistent on my reading. I would be happy to recommend acceptance after the coefficient is corrected and the final algebra in Section 5 is re-checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know that the paper has a load-bearing algebraic error in the headline result. In Proposition 5.4 the coefficient computed for the −ε μ_n^{−2−α} term is α Γ(1+2α) cos(πα) tan(πα/2)/(π 2^{1+2α}), but Theorem 1(i) defines B_α with tan(πα/2) in the denominator. The paper's own identity sin x(2cos x−sec x)=cos(2x)tan x gives the numerator version; the displayed simplification to B_α as printed is false except at α=1/2. So the stated third term is off by tan^2(πα/2) for every 0<α<1, α≠1/2. This is not a subtle boundary issue.\n\nEverything else is in better shape. The three-term expansion for α≥1 (and the log term at α=1) appears consistent, the uniform L∞ bound is proved with a clever two-step argument and explicit constants, and the paper is honest about importing the half-line representation from Kwaśnicki. The proofs are long but structured, and I did not find a second error of similar magnitude.\n\nThe soft spots are exactly where you'd expect: the small-s expansion of I_α(s) and the pairing integrals are dense and easy to glitch past. But the B_α inconsistency is concrete and checkable, so the main theorem is not correct as stated. The fix is straightforward (change B_α to the numerator version) but the paper needs a real revision, not a copy edit.\n\nWho is this for? Spectral theorists working on fractional Laplacians and anyone tracking the open questions from Kwaśnicki's 2012 paper. Even with the error, it deserves a serious referee — the machinery is substantial and the uniform bound is genuinely new. But I would not cite Theorem 1(i) until the correction lands.\n\nRecommendation: send to peer review with an explicit request to check Section 5 and correct (1.7).","headline":"A substantial paper with a real algebraic error in the main three-term coefficient that needs fixing before it can be trusted.","tokens_in":28467,"tokens_out":4430,"would_cite":false,"duration_ms":34918,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["35P20","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The fractional Laplacian on the interval has a three-term eigenvalue expansion and eigenfunctions bounded uniformly in the fractional order and the index.","keywords":["fractional Laplacian","eigenvalue asymptotics","uniform eigenfunction bounds","interval","half-line eigenfunction","quasimode","Laplace transform","spectral theory"],"falsifier":"For a fixed $\\alpha\\in(1,2)$, evaluate $I_\\alpha(s)$ in (2.6) numerically near $s=0$ and check that $\\pi I_\\alpha(s)/s$ approaches $\\pi\\cot(\\pi/\\alpha)$; alternatively, compute $\\lambda_n(\\alpha)$ to high precision for a fixed $\\alpha$ and check that $\\lambda_n-\\mu_n^\\alpha-(-1)^n\\kappa_\\alpha\\mu_n^{-2}$ follows the claimed power of $\\mu_n$ with the claimed coefficient rather than a different exponent.","tokens_in":27427,"feed_emoji":"📈","tokens_out":7237,"duration_ms":65766,"temperature":0.7,"pith_summary":"This paper studies the eigenvalue problem for the fractional Laplacian on the interval $(-1,1)$ with zero exterior condition, for every fractional order $0<\\alpha<2$. It proves that the $n$th eigenvalue has a three-term asymptotic expansion in powers of the shifted index $\\mu_n = n\\pi/2 - (2-\\alpha)\\pi/8$, with explicit constants and a remainder that behaves differently in the three regimes $0<\\alpha<1$, $\\alpha=1$, and $1<\\alpha<2$. This improves earlier two-term asymptotics and confirms the numerically predicted $O(n^{-2})$ remainder. The paper also proves that the $L^2$-normalized eigenfunctions are uniformly bounded in the supremum norm by one constant independent of both $n$ and $\\alpha$, settling a conjecture suggested by numerical experiments. Sharp individual eigenvalue asymptotics and uniform eigenfunction bounds are basic structural facts about this widely used nonlocal operator, and they were previously known only partially.","feed_headline":"Fractional Laplacian eigenvalues pinned down to third order","feed_subtitle":"A new proof gives explicit remainder terms and uniform eigenfunction bounds for every fractional order between 0 and 2.","key_machinery":"The argument is carried by the generalized half-line eigenfunction $F_\\alpha(t)=\\sin(t+\\theta)-G_\\alpha(t)$, where $G_\\alpha$ is a nonnegative, completely monotone correction whose Laplace transform encodes the spectral information of the half-line operator. Gluing two reflected copies of $F_\\alpha(\\mu_n\\,\\cdot\\,)$ across the interval with a smooth partition of unity produces a quasimode $\\tilde\\phi_n$ whose Rayleigh quotient already yields the two-term formula; the third term is extracted by expanding the residual $(A_I-\\mu_n^\\alpha)\\tilde\\phi_n$ and pairing it with the quasimode. For $0<\\alpha<1$, where the cutoff error is the same size as the target term, the paper replaces the cut quasimode by an uncut combination $w_n$ and uses an exact quadratic-form identity to reach the sharper error. The uniform bound uses heat-semigroup estimates for orders bounded away from zero and, for small $\\alpha$, a residual estimate of size $o(\\alpha/n)$, an antiderivative bound for the eigenfunction, and a truncated quadratic-form test with $(\\varphi_n-\\Lambda)_+$.","core_discovery":"The central result is a complete three-term asymptotic formula for the eigenvalues of the fractional Laplacian in the interval. Writing $\\mu_n = n\\pi/2 - (2-\\alpha)\\pi/8$ and $\\kappa_\\alpha = \\alpha c_\\alpha/2^{\\alpha+1}$, the paper proves\n$$\\lambda_n(\\$\\alpha$) = \\mu_n^\\$\\alpha$ + (-1)^n \\kappa_\\$\\alpha$ \\$mu_n^{{-2}}$ + R_n(\\$\\alpha$),$$\nwhere for $0<\\alpha<1$ the remainder is $(A_\\alpha + (-1)^n B_\\alpha)\\mu_n^{-2-\\alpha} + o(\\mu_n^{-2-\\alpha})$; for $\\alpha=1$ it is $(-1)^{n+1}(2\\pi^2)^{-1}\\mu_n^{-3}\\log\\mu_n + O(\\mu_n^{-3})$; and for $1<\\alpha<2$ it is $(-1)^n \\eta_\\alpha \\mu_n^{-3} + o(\\mu_n^{-3})$, with explicit constants $A_\\alpha$, $B_\\alpha$, and $\\eta_\\alpha$. The same arguments show that the normalized eigenfunctions satisfy $\\|\\varphi_n\\|_{L^\\infty(I)}\\le C$ for an absolute constant $C$, uniformly in $n\\ge 1$ and $0<\\alpha<2$. Along the way the paper establishes that the $L^2$ distance between the true eigenfunction and its quasimode decays as $n^{-1-\\alpha}$, giving a quantitative spectral approximation.","pith_inferences":["A natural testable extension is to replace the fractional Laplacian by a general Bernstein function $\\psi(-\\Delta)$ on an interval; the same half-line representation should yield a three-term formula whose constants come from the tail of the corresponding correction term, provided an analogue of the Laplace-transform identity holds.","In higher dimensions the paper's methods do not directly apply; one could test numerically whether the uniform $L^\\infty$ bound persists for balls, where symmetry might allow a similar one-dimensional reduction, and whether it fails for domains with corners.","The explicit value $Q_\\alpha=\\alpha^2/(2K_\\alpha)$ obtained in the small-order analysis could be used to compute next-order heat-trace or partition-function corrections for the interval, connecting the eigenvalue asymptotics to the short-time heat-kernel expansion.","The proof suggests the crude bound $\\exp(100)$ is far from sharp; a numerical survey of $\\|\\varphi_n\\|_{L^\\infty}$ over $\\alpha$ and $n$ could locate the actual supremum and reveal whether any non-uniform effect appears as $\\alpha\\to 0$."],"forward_implications":["The eigenvalue expansion is now known through three terms for every fixed $0<\\alpha<2$, including the precise logarithmic term at $\\alpha=1$ and the parity-dependent signs $(-1)^n$.","The numerically conjectured $O_\\alpha(n^{-2})$ remainder in the two-term formula is confirmed, with explicit constants controlling the next order in each regime.","The uniform bound $\\|\\varphi_n\\|_{L^\\infty}\\le C$ holds with a single constant for all orders $\\alpha$ and all indices $n$, not only for $\\alpha\\ge 1/2$ as previously known.","The eigenfunction approximation result $\\|\\varphi_n - \\tilde\\phi_n/\\|\\tilde\\phi_n\\|\\|_{L^2} \\lesssim n^{-1-\\alpha}$ gives a constructive route to computing eigenfunctions and eigenvalues of the fractional Laplacian in one dimension."],"supporting_citations":[{"why":"Supplies the exact half-line eigenfunction representation $F_\\alpha(t)=\\sin(t+\\theta)-G_\\alpha(t)$ and the Laplace-transform identity (2.13), the foundation for the quasimode construction.","marker":"[23]"},{"why":"Provides the previous two-term eigenvalue asymptotics, the quasimode gluing construction, and the uniform eigenfunction bound for $\\alpha\\ge 1/2$ that this paper refines.","marker":"[24]"},{"why":"Extends the two-term asymptotics to more general operators $\\psi(-\\Delta)$ and calibrates the conjectured $O_\\alpha(n^{-2})$ remainder through numerical simulation.","marker":"[20]"},{"why":"Proves the $\\alpha=1$ case of both the eigenvalue asymptotics and the uniform eigenfunction bound, providing the baseline for the present three-term result.","marker":"[21]"},{"why":"Establishes the equivalence of the Fourier, singular-integral, and quadratic-form definitions of the fractional Laplacian used throughout the paper.","marker":"[22]"},{"why":"Supplies the heat-kernel comparison for the Dirichlet fractional Laplacian used in transferring semigroup bounds for the uniform eigenfunction estimate.","marker":"[4]"}],"fun_headline_variants":["Fractional Laplacian eigenvalues: third-order asymptotics","Uniform eigenfunction bounds for fractional Laplacian","Eigenvalue formula settles fractional Laplacian conjecture","Explicit remainders for fractional Laplacian eigenvalues","Three-term asymptotic for fractional Laplacian eigenvalues"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire calculation leans on imported exact formulas for the half-line eigenfunction and its Laplace transform; if any of those formulas were inaccurate, the explicit constants in the three-term expansion would come out wrong.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Laplacian eigenvalues: third-order asymptotics","Uniform eigenfunction bounds for fractional Laplacian","Eigenvalue formula settles fractional Laplacian conjecture","Explicit remainders for fractional Laplacian eigenvalues","Three-term asymptotic for fractional Laplacian eigenvalues"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1415,"prompt_tokens":967,"completion_tokens":448,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":583,"tokens_out":448,"duration_ms":4150,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-28T00:13:50.523714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $\\alpha\\in(1,2)$, evaluate $I_\\alpha(s)$ in (2.6) numerically near $s=0$ and check that $\\pi I_\\alpha(s)/s$ approaches $\\pi\\cot(\\pi/\\alpha)$; alternatively, compute $\\lambda_n(\\alpha)$ to high precision for a fixed $\\alpha$ and check that $\\lambda_n-\\mu_n^\\alpha-(-1)^n\\kappa_\\alpha\\mu_n^{-2}$ follows the claimed power of $\\mu_n$ with the claimed coefficient rather than a different exponent.","supporting_citations":[{"cited_title":"Kwaśnicki,Spectral analysis of subordinate Brownian motions on the half-line, Studia Math.206 (2011), no","cited_arxiv_id":null,"evidence_quote":"Supplies the exact half-line eigenfunction representation $F_\\alpha(t)=\\sin(t+\\theta)-G_\\alpha(t)$ and the Laplace-transform identity (2.13), the foundation for the quasimode construction."},{"cited_title":"Kwaśnicki,Eigenvalues of the fractional Laplace operator in the interval, J","cited_arxiv_id":null,"evidence_quote":"Provides the previous two-term eigenvalue asymptotics, the quasimode gluing construction, and the uniform eigenfunction bound for $\\alpha\\ge 1/2$ that this paper refines."},{"cited_title":"Kaleta, M","cited_arxiv_id":null,"evidence_quote":"Extends the two-term asymptotics to more general operators $\\psi(-\\Delta)$ and calibrates the conjectured $O_\\alpha(n^{-2})$ remainder through numerical simulation."},{"cited_title":"Kulczycki, M","cited_arxiv_id":null,"evidence_quote":"Proves the $\\alpha=1$ case of both the eigenvalue asymptotics and the uniform eigenfunction bound, providing the baseline for the present three-term result."},{"cited_title":"Kwaśnicki,Ten equivalent definitions of the fractional Laplace operator, Fract","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence of the Fourier, singular-integral, and quadratic-form definitions of the fractional Laplacian used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the heat-kernel comparison for the Dirichlet fractional Laplacian used in transferring semigroup bounds for the uniform eigenfunction estimate."}],"review_version":1}