{"id":"27191af9-bd1d-4cf3-965d-58292afa5544","arxiv_id":"1908.03457","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims uniqueness of the potential from Weyl function, two spectra, spectral data, and a Hochstadt-Lieberman half-inverse theorem for a conformable fractional Sturm-Liouville operator, but the half-inverse proof has a fatal asymptotic gap for fractional orders.","lead":"This paper derives uniqueness theorems for inverse problems of a Sturm-Liouville operator built with conformable fractional derivatives. The central half-inverse theorem is not established for fractional orders because the proof's asymptotic bound fails.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4 is not established for 0<α<1: the claimed decay of χ(λ)=H(λ)/Δ(λ) in §3.3 is false, since H grows like e^{2τ(π/2)^α/α} while Δ grows only like e^{τπ^α/α}; after t=x^α/α the fixed potential covers less than half the interval, the non-unique regime.","rationale":"After applying the substitution t=x^α/α, the conformable fractional operator is exactly (−d²/dt²)+Q(t) with the same boundary form; this is standard and exposes that Theorem 4 is a Hochstadt–Lieberman statement with the potential fixed on [T/2^α,T]. For α<1 this interval is shorter than half of [0,T], precisely the regime where classical uniqueness is false. The reader's criticism is correct and is the load-bearing flaw: the entire-function argument in §3.3 depends on a decay bound for χ that fails by an exponential factor for every α<1. I verified the growth rates from (5)–(8) and (9): H has growth exponent 2(π/2)^α/α in the upper half-plane, while Δ has lower growth exponent π^α/α, so χ cannot tend to zero. No later argument supplies the missing decay. Theorems 1–3 are essentially translations of well-known Borg/Marchenko results, but even if accepted, they do not make the advertised half-inverse theorem valid. The reader's REJECT verdict is appropriate.","tokens_in":7008,"tokens_out":11349,"duration_ms":125394,"concrete_test":"Set α=1/2, take q=~q≡0 on (π/2,π] and q−~q≡c≠0 on [0,π/2] with h=~h=H=~H=0. Solve the two problems numerically in the t-coordinate t=2√x on [0,2√π]. For λ=−R^2, R=10^2,10^3,10^4, compute χ(λ)=H(λ)/Δ(λ) from the boundary Wronskians. If |χ| grows like R^{-2} exp(R(2√π)(√2−1)) instead of decaying like O(1/R), the estimate in §3.3 is false and the proof of Theorem 4 collapses.","verdict_should_be":"REJECT","load_bearing_attack":"Under t=x^α/α, D_α=d/dt, so L_α is an ordinary Sturm–Liouville operator on [0,T], T=π^α/α, with potential Q(t)=q((αt)^{1/α}) and boundary data y_t(0)−hy(0)=0, y_t(T)+Hy(T)=0. Theorem 4 therefore prescribes Q only on [T/2^α,T]; for α<1, 2^α<2, so this is strictly less than half of [0,T]. The classical Hochstadt–Lieberman uniqueness theorem requires at least half of the potential, and less-than-half data is the known non-unique regime. The proof's specific error is the claimed bound on χ. For large λ with τ=Im√λ>0, (9) gives |Δ(λ)|≥C|√λ|e^{τT}, while H(λ)=h−~h+∫_0^{π/2}(q−~q)φ~φ d_αt is, for generic q−~q with support reaching x=π/2, of order e^{2τ(π/2)^α/α}. The quotient therefore grows like exp(τπ^α(2^{1−α}−1)/α) along λ=−R^2, not like 1/|√λ|. The Liouville step forcing H≡0 in §3.3 is unsupported; the theorem's advertised conclusion does not follow. Theorems 1–3 are translations of classical results and do not repair this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the conformable-fractional Sturm-Liouville boundary value problem -D_x^\\alpha D_x^\\alpha y + q(x)y = \\lambda y on (0,\\pi), where D_x^\\alpha is the conformable derivative of order 0<\\alpha\\le 1, with boundary conditions D_x^\\alpha y(0)-hy(0)=0 and D_x^\\alpha y(\\pi)+Hy(\\pi)=0. It states four uniqueness theorems: recovery of q, h, H from the Weyl function (Theorem 1); recovery from two spectra (Theorem 2); recovery from eigenvalues and norming constants (Theorem 3); and a half-inverse Hochstadt-Lieberman-type theorem (Theorem 4) in which the spectrum, the right boundary parameter H, and q on (\\pi/2,\\pi) determine q on [0,\\pi/2] and h. The proofs of Theorems 1-3 follow classical Marchenko/Gelfand-Levitan/Borg patterns, using asymptotic formulas and a product representation imported from [28]. Theorem 4 is attacked with an entire-function argument on the characteristic function \\Delta(\\lambda).","tokens_in":7284,"tokens_out":17235,"duration_ms":178575,"significance":"If established, the results would extend standard inverse spectral uniqueness results to the conformable-fractional setting. A useful implicit fact is that the change of variables t=x^\\alpha/\\alpha converts the operator to an ordinary Sturm-Liouville operator, so much of the content is inherited from classical theory. Theorems 1-3 appear plausible provided the asymptotic and product formulas from [28] are accepted. The advertised Hochstadt-Lieberman theorem, however, is not established for 0<\\alpha<1: the proof's key growth estimate is false in that range, and the underlying interval geometry is outside the classical half-interval uniqueness regime. The paper therefore cannot be accepted in its present form.","major_comments":[{"comment":"The assertion |\\chi(\\lambda)| \\le C/|\\sqrt{\\lambda}| immediately after the definition of \\chi is unsupported and, for 0<\\alpha<1, is false in general. From (5), |\\phi(x,\\lambda)| and |\\tilde\\phi(x,\\lambda)| are O(\\exp(|\\tau|x^\\alpha/\\alpha)), so the integral defining H(\\lambda) is O(\\exp(2|\\tau|(\\pi/2)^\\alpha/\\alpha)). From (9), |\\Delta(\\lambda)| \\ge C|\\sqrt{\\lambda}|\\exp(|\\tau|\\pi^\\alpha/\\alpha) on G_\\delta. Hence the quotient has upper bound C|\\sqrt{\\lambda}|^{-1}\\exp(|\\tau|\\pi^\\alpha(2^{1-\\alpha}-1)/\\alpha), whose exponent is positive precisely when \\alpha<1. Taking \\lambda=-R^2 (so \\sqrt{\\lambda}=iR and |\\tau|=R) and choosing q-\\tilde q that does not vanish up to x=\\pi/2 shows that the exponential growth is real, not merely an artifact of crude estimates. Thus \\chi need not be bounded or decaying, Liouville's theorem cannot be applied, and the conclusion H\\equiv 0 is not obtained. This is load-bearing because the rest of Theorem 4, including (16)-(17), depends on H\\equiv 0.","section":"Section 3.3, after the definition of chi(lambda)"},{"comment":"Even if H\\equiv 0 were established, the last step of Theorem 4 is not justified. The proof defines \\psi from \\phi, obtains (17), and then says 'Taking into account Theorem 1' to conclude q=\\tilde q on [0,\\pi/2] and h=\\tilde h. Theorem 1 requires equality of the Weyl functions of two boundary value problems, whereas (17) is an equality of Wronskians at one endpoint. No argument is given that (17) implies equality of the relevant Weyl functions, and the auxiliary boundary value problem on (0,\\pi/2) is not formulated with the hypotheses needed to invoke Theorem 1. This step therefore requires a separate proof.","section":"Section 3.3, final paragraph"},{"comment":"Theorem 2 relies on the product representation (14) to conclude \\Delta\\equiv\\tilde\\Delta from equality of the two spectra. As displayed, however, the product runs over n=0 and contains n^2 in the denominator, so the expression is not well-defined for n=0. The authors should state the correct product formula, with the right normalization and starting index, and either prove it or cite the precise statement from [28]. This is a necessary correction because the proof of Theorem 2 depends directly on this representation.","section":"Section 3.2, Eq. (14)"}],"minor_comments":[{"comment":"There are several typographical errors, including 'OPERA TOR' in the title, 'derivates' in the abstract, and 'equiped' in the introduction; these should be corrected.","section":"Title and abstract"},{"comment":"Reference [28] is dated 2009, although conformable fractional calculus was introduced in 2014; the publication year and bibliographic details should be checked and corrected.","section":"Reference [28]"},{"comment":"The proof states that P_1 and P_2 are O(1) and O(1/\\sqrt{\\lambda}) from (5)-(9), but (9) is stated only on G_\\delta; the authors should briefly explain why the estimates extend to the excluded small disks around the zeros of \\Delta so that Liouville's theorem can be applied.","section":"Section 3.1, proof of Theorem 1"}],"recommendation":"reject","confidential_remarks":"The paper is a straightforward adaptation of classical inverse Sturm-Liouville arguments to the conformable-fractional setting, with most of the technical input coming from [28]. The main advertised theorem has a genuine gap for 0<\\alpha<1, and the underlying geometry suggests the statement is outside the classical uniqueness regime, so I do not see a local fix within the current scope. Theorems 1-3 may be salvageable, but the paper as a whole is not suitable for publication in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper is mostly a translation exercise, and the one new-looking result, the Hochstadt-Lieberman-type half-inverse theorem, has a genuine proof gap. I would not send it to peer review as is.\n\nWhat the paper does well: the authors carefully set up the conformable fractional Sturm-Liouville problem and prove uniqueness from the Weyl function, two spectra, and eigenvalues plus norming constants. The proofs follow the standard classical pattern—defining entire functions, applying Liouville's theorem, and using the known asymptotics from [28]. There is no circularity and no self-citation; the reliance on [28] is explicit and appropriate. If you work inside the conformable calculus formalism, the statements at least look like the right analogues of the classical theorems.\n\nBut the mathematical content is not new. The conformable derivative D_alpha is just d/dt after the change of variables t = x^alpha/alpha. Under that map, the operator becomes an ordinary regular Sturm-Liouville operator on [0, pi^alpha/alpha] with potential Q(t) = q((alpha t)^{1/alpha}). Theorems 1-3 are therefore the classical uniqueness theorems of Marchenko, Borg, and the spectrum-plus-norming-constants theorem, rewritten in the x-coordinate. The paper never mentions this equivalence, which is a serious omission in the novelty framing.\n\nTheorem 4 is the real problem. The proof defines chi(lambda) = H(lambda)/Delta(lambda) and asserts |chi(lambda)| <= C/|sqrt(lambda)| for large lambda. That bound is false for 0 < alpha < 1. From the asymptotics in (5)-(8), H(lambda) can grow like exp(2 tau (pi/2)^alpha/alpha), while the lower bound on Delta in (9) is only exp(tau pi^alpha/alpha). For alpha < 1, the quotient grows exponentially along lambda = -R^2; it does not decay. So the Liouville step forcing H identically zero is unsupported. The classical coordinate confirms the intuition: prescribing q on (pi/2, pi) corresponds to prescribing Q only on [T/2^alpha, T], which for alpha < 1 is strictly less than half the interval. That is exactly the known non-unique regime for the classical Hochstadt-Lieberman theorem. The gap is load-bearing, not cosmetic.\n\nMinor point: even if H identically zero were established, the final step invoking Theorem 1 on a half-interval with a transformed potential is terse and would need more justification. But the chi decay issue already sinks the theorem.\n\nWho is this for? Possibly someone collecting examples of how conformable fractional problems reduce to classical ones, but the paper does not state that reduction. For inverse spectral theory, it offers nothing new. The citation pattern is clean, and the writing is honest, but the central claim fails.\n\nRecommendation: desk reject. If the authors remove Theorem 4 and explicitly state the change-of-variables equivalence, they might have a modest pedagogical note, but the current manuscript should not consume referee time.","headline":"The paper's advertised half-inverse theorem is unproven for alpha<1 due to a false decay bound, and the rest of the results are classical Sturm-Liouville theorems in conformable disguise.","tokens_in":7902,"tokens_out":2458,"would_cite":false,"duration_ms":27634,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31B20","34A55","34B24","26A33"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims four inverse-spectral uniqueness theorems for a conformable fractional Sturm–Liouville operator, including a half-inverse theorem that uses one spectrum plus the potential on one half.","keywords":["inverse spectral problem","conformable fractional derivative","Sturm-Liouville operator","Weyl function","norming constants","half-inverse problem","Hochstadt-Lieberman theorem","uniqueness"],"falsifier":"Using the paper's asymptotic formulas (5)-(8) and (9), evaluate $|\\chi(i\\tau)|$ for a fixed $\\alpha<1$: the numerator is controlled by $\\exp(2|\\tau|(\\pi/2)^\\alpha/\\alpha)$ and the denominator by $|\\tau|\\exp(|\\tau|\\pi^\\alpha/\\alpha)$, so the quotient can grow like $\\exp(|\\tau|(2^{1-\\alpha}-1)\\pi^\\alpha/\\alpha)$. Exhibiting such growth would refute the boundedness claim and the conclusion $\\chi\\equiv 0$; conversely, proving boundedness for all $\\alpha\\in(0,1]$ would complete the half-inverse theorem.","tokens_in":6733,"feed_emoji":"🧮","tokens_out":13089,"duration_ms":120933,"temperature":0.7,"pith_summary":"The paper studies the inverse problem for the conformable fractional Sturm–Liouville equation $-D^\\alpha_x D^\\alpha_x y+q(x)y=\\lambda y$ on $0<x<\\pi$, where the conformable fractional derivative acts on differentiable $f$ as $x^{1-\\alpha}f'(x)$, with real continuous potential $q$ and real boundary parameters $h,H$. It claims that the operator is uniquely determined by three classical data sets: the Weyl function, two full spectra, or eigenvalues together with norming constants. It also claims a half-inverse theorem in the spirit of Hochstadt and Lieberman: if two operators share one full spectrum, the same right boundary parameter $H$, and the same potential on $(\\pi/2,\\pi)$, then the potential on the left half and the left boundary parameter $h$ are forced. The proofs work by comparing entire and meromorphic solutions of the two operators and using boundedness to force equality. If the claims are correct, conformable fractional Sturm–Liouville problems inherit the same inverse-spectral uniqueness structure as the classical theory.","feed_headline":"One spectrum plus half the potential fixes the other half","feed_subtitle":"The paper proves a fractional generalization of the Hochstadt–Lieberman half-inverse theorem.","key_machinery":"The mechanism is the conformable fractional Wronskian $W_\\alpha[\\psi,\\phi]=\\psi D^\\alpha_x\\phi-\\phi D^\\alpha_x\\psi$, which is independent of $x$, together with the Weyl function $M(\\lambda)=-\\psi(0,\\lambda)/\\Delta(\\lambda)$. The comparison arguments form quotients such as $P_1,P_2$ and $\\chi=H(\\lambda)/\\Delta(\\lambda)$ from solutions of the two operators; the asymptotic formulas for $\\phi$ and $\\psi$ are used to show these quotients are entire and bounded, so Liouville's theorem forces them to be constant or zero. In the two-spectra argument, equality of spectra is converted into equality of characteristic functions through the canonical product representation $\\Delta(\\lambda)=\\frac{\\pi^{3\\alpha-2}}{\\alpha^3}(\\lambda_0-\\lambda)\\prod_{n=0}^\\infty \\frac{\\lambda_n-\\lambda}{n^2}$. For the half-inverse theorem, the key identity is $D^\\alpha_x[\\tilde\\phi D^\\alpha_x\\phi-\\phi D^\\alpha_x\\tilde\\phi]=(q-\\tilde q)\\phi\\tilde\\phi$, and the reflected solution $\\psi(x,\\lambda)=\\phi(((\\pi/2)^\\alpha-x^\\alpha)^{1/\\alpha},\\lambda)$ carries right-half information to the left half.","core_discovery":"The central claim is a set of uniqueness theorems for $L_\\alpha(q,h,H)$, the boundary value problem $-D^\\alpha_x D^\\alpha_x y+q(x)y=\\lambda y$ with $D^\\alpha_x y(0)-hy(0)=0$ and $D^\\alpha_x y(\\pi)+Hy(\\pi)=0$. Theorem 1 asserts that equality of the Weyl functions $M(\\lambda)=-\\psi(0,\\lambda)/\\Delta(\\lambda)$ forces $q=\\tilde q$ almost everywhere, $h=\\tilde h$, and $H=\\tilde H$. Theorem 2 asserts the same conclusion from equality of two spectra, the original spectrum and that of the Dirichlet-at-$0$ problem, and Theorem 3 from equality of eigenvalues plus norming constants. Theorem 4, the half-inverse or Hochstadt–Lieberman-type theorem, asserts that if the spectra agree, $H$ agrees, and $q=\\tilde q$ on $(\\pi/2,\\pi)$, then $q=\\tilde q$ almost everywhere on $[0,\\pi]$ and $h=\\tilde h$.","pith_inferences":["The split point $\\pi/2$ is likely a convenience: the same reflection construction should adapt to any $a\\in(0,\\pi)$ by using $((a^\\alpha-x^\\alpha)^{1/\\alpha})$, so the half-inverse statement may hold with arbitrary partial-interval data.","If the half-inverse theorem is valid, it suggests a layer-stripping reconstruction algorithm for fractional Sturm–Liouville problems: measure one spectrum, know the potential on one side, and recover the other side from the induced initial-value problem.","The proof's boundedness estimate for $\\chi(\\lambda)$ is a natural stress test for $\\alpha<1$; checking $|\\chi(i\\tau)|$ numerically could reveal whether the theorem needs a different argument below order one."],"forward_implications":["Equality of Weyl functions would be enough to fix $q$, $h$, and $H$, so a full spectral response measurement cannot be ambiguous.","Two spectra alone would determine the operator without norming constants, matching the classical Borg-type picture.","When only one spectrum is known, adding norming constants would restore uniqueness.","The half-inverse theorem would mean that knowing the right half of the potential plus one full spectrum leaves no freedom in the left half or in the left boundary parameter."],"supporting_citations":[{"why":"Supplies the asymptotic formulas for $\\phi$ and $\\psi$, the $x$-independence of the fractional Wronskian, the canonical product representation of $\\Delta$, and the norming-constant identity $\\beta_n\\alpha_n=-\\Delta'(\\lambda_n)$.","marker":"[28]"},{"why":"The classical Hochstadt–Lieberman half-inverse theorem that Theorem 4 generalises to the conformable fractional setting.","marker":"[15]"},{"why":"Introduces the conformable fractional derivative used to define the operator under study.","marker":"[17]"},{"why":"Provides the conformable calculus rules, including the chain rule used to construct the reflected solution in Theorem 4.","marker":"[1]"}],"fun_headline_variants":["Fractional Sturm–Liouville inverse problems solved","Half-inverse theorem goes fractional","Equality of spectra forces potential","Fractional Hochstadt–Lieberman theorem proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The half-inverse proof assumes the quotient $\\chi(\\lambda)=H(\\lambda)/\\Delta(\\lambda)$ decays as $C/|\\sqrt{\\lambda}|$ at infinity; the paper's estimates justify that bound only for $\\alpha\\ge 1$, so for $0<\\alpha<1$ the Liouville step is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Sturm–Liouville inverse problems solved","Half-inverse theorem goes fractional","Equality of spectra forces potential","Fractional Hochstadt–Lieberman theorem proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000878,"raw_usage":{"total_tokens":3727,"prompt_tokens":806,"completion_tokens":2921,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":2867}},"tokens_in":422,"tokens_out":2921,"duration_ms":20707,"temperature":1.0,"reasoning_tokens":2867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:16:30.680304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using the paper's asymptotic formulas (5)-(8) and (9), evaluate $|\\chi(i\\tau)|$ for a fixed $\\alpha<1$: the numerator is controlled by $\\exp(2|\\tau|(\\pi/2)^\\alpha/\\alpha)$ and the denominator by $|\\tau|\\exp(|\\tau|\\pi^\\alpha/\\alpha)$, so the quotient can grow like $\\exp(|\\tau|(2^{1-\\alpha}-1)\\pi^\\alpha/\\alpha)$. Exhibiting such growth would refute the boundedness claim and the conclusion $\\chi\\equiv 0$; conversely, proving boundedness for all $\\alpha\\in(0,1]$ would complete the half-inverse theorem.","supporting_citations":[{"cited_title":"Inverse Problems in S cience and Engineering, 1-32 (2009)","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic formulas for $\\phi$ and $\\psi$, the $x$-independence of the fractional Wronskian, the canonical product representation of $\\Delta$, and the norming-constant identity $\\beta_n\\alpha_n=-\\Delta'(\\lambda_n)$."},{"cited_title":"SIAM Journal on Applied Mathematics","cited_arxiv_id":null,"evidence_quote":"The classical Hochstadt–Lieberman half-inverse theorem that Theorem 4 generalises to the conformable fractional setting."},{"cited_title":"J Comput Appl Math","cited_arxiv_id":null,"evidence_quote":"Introduces the conformable fractional derivative used to define the operator under study."},{"cited_title":"J C omput Appl Math","cited_arxiv_id":null,"evidence_quote":"Provides the conformable calculus rules, including the chain rule used to construct the reflected solution in Theorem 4."}],"review_version":1}