Inverse problems for a conformable fractional Sturm-Liouville operator

A. Sinan Ozkan; \.Ibrahim Adalar

REVIEW 3 major objections 3 minor 33 references

Inverse problems for a conformable fractional Sturm-Liouville operator

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.03457 v1 pith:EOSLUNN4 submitted 2019-08-09 math.CA math.SP

A. Sinan Ozkan , İbrahim Adalar This is my paper · ORCID

classification math.CAmath.SP MSC 31B2034A5534B2426A33

keywords inversespectralproblemconformablefractionalderivativeSturm-LiouvilleoperatorWeylfunctionnormingconstantshalf-inverseHochstadt-Liebermantheoremuniqueness

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 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

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $h=\tilde h$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

But 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.

Theorem 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.

Minor 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.

Who 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.

Recommendation: 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.

Referee Report

3 major / 3 minor

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).

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 (3)

[Section 3.3, after the definition of chi(lambda)] 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 3.3, final paragraph] 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 3.2, Eq. (14)] 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.

minor comments (3)

[Title and abstract] There are several typographical errors, including 'OPERA TOR' in the title, 'derivates' in the abstract, and 'equiped' in the introduction; these should be corrected.
[Reference [28]] 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 3.1, proof of Theorem 1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation chain: the theorems use independent prior work [28] for asymptotics and do not assume their conclusions; the defect in Theorem 4 is a proof gap, not circularity.

full rationale

The paper's claimed derivations do not reduce to their inputs. Theorems 1–3 assume equality of Weyl functions, two spectra, or norming constants and use the resulting entire-function identities plus Liouville's theorem to force equality of the coefficients. The required asymptotic formulas and the product representation of Delta are quoted from [28], a work by different authors, so the citations are independent support rather than a self-citation chain. The target conclusions—q(x) = ~q(x), h = ~h, H = ~H—are not hidden in the hypotheses or in [28]. The only self-citation, [30], is one entry in a standard list of classical inverse Sturm-Liouville references and carries no load-bearing weight. The genuine weakness of the paper is located in Section 3.3, in the proof of Theorem 4: the asserted estimate |chi(lambda)| <= C/|sqrt(lambda)| is not justified for 0 < alpha < 1, because H(lambda) can grow like exp(2|tau|(pi/2)^alpha/alpha) while Delta(lambda) only grows like exp(|tau| pi^alpha/alpha), so the quotient is not shown to decay. That is a mathematical correctness gap, not a circularity: it does not amount to assuming the theorem or renaming fitted data as a prediction. Under the change of variables t = x^alpha/alpha the problem becomes an ordinary Sturm-Liouville operator, and the prescribed half-interval potential covers less than half of the new interval when alpha < 1; this explains why the classical Hochstadt-Lieberman conclusion may fail, but again this is an independent mathematical objection rather than an input-output identity. Accordingly, no circular step is found, and the score is low despite the serious proof defect.

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

No parameter is fitted and no new entity is introduced. The proofs lean on the prior paper [28] for all asymptotic and product formulas. Under the substitution t = x^alpha/alpha the whole problem is a classical Sturm-Liouville problem, which is why Theorems 1-3 carry over and why Theorem 4's half-interval claim is not justified.

assumptions (4)

domain assumption Asymptotic formulas (5)-(8) and the lower bound (9) for Delta(lambda) are taken from [28].
Used in every theorem to control entire functions, to identify eigenvalue zeros, and to estimate chi in Theorem 4.
domain assumption Product representation (14): Delta(lambda) = pi^(3alpha-2)/alpha^3 (lambda_0 - lambda) prod_{n=0}^infty (lambda_n - lambda)/n^2, cited from [28, Theorem 3.11].
This is the basis for Theorem 2 and is not derived in the paper.
domain assumption Norming constant identity beta_n alpha_n = -Delta'(lambda_n), cited from [28, Lemma 3.5].
This is the basis for part of Theorem 3 and is not derived in the paper.
domain assumption Eigenvalues are real, simple, and satisfy the asymptotic estimate (10), cited from [28].
Used to justify product representations and the quotient arguments in Theorems 2 and 3.

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Cite this review

Pith. "Pith review of Inverse problems for a conformable fractional Sturm-Liouville operator." pith.science (2026). https://pith.science/paper/EOSLUNN4

@misc{pith2026190803457,
  author       = {Pith},
  title        = {Pith review of: Inverse problems for a conformable fractional Sturm-Liouville operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOSLUNN4}},
  note         = {Machine review of arXiv:1908.03457}
}

read the original abstract

In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study on half-inverse problem and prove a Hochstadt and Lieberman-type theorem.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [28]

Inverse Problems in S cience and Engineering, 1-32 (2009)

Mortazaasl, H., Akbarfam, A.J.: Trace formula and inve rse nodal problem for a conformable fractional Sturm-Liouville problem. Inverse Problems in S cience and Engineering, 1-32 (2009)

work page 2009

[2] [1]

J C omput Appl Math

Abdeljawad, T.: On conformable fractional calculus. J C omput Appl Math. 279, 57–66 (2015)

work page 2015

[3] [2]

Ambarzumyan, V.A.: ¨Uber eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 ( 1929)

work page 1929

[4] [3]

Mathematical Methods in the Applied Sciences

Allahverdiev, Bilender P., Tuna H., Yal¸ cinkaya Y.: Con formable fractional Sturm-Liouville equation. Mathematical Methods in the Applied Sciences. 42 (10), 3508-3526 (2019)

work page 2019

[5] [4]

Electron J Diﬀer Equ

Al-Towailb, M.A: A q-fractional approach to the regular Sturm-Liouville problems. Electron J Diﬀer Equ. 88, 1–13 2017

work page 2017

[6] [5]

Open Math

Atangana A, Baleanu D, Alsaedi A.: New properties of conf ormable derivative. Open Math. 13, 889–898 (2015) 6 A. SINAN OZKAN AND ˙IBRAHIM ADALAR

work page 2015

[7] [6]

Springer, New York (2010)

Baleanu, D., Guvenc, Z.B., Machado, J.T.: New trends in n anotechnology and fractional calculus applications. Springer, New York (2010)

work page 2010

[8] [7]

Bestimmung der Diﬀerentialgleichung durch die Eigenwerte

Borg, G.: Eine Umkehrung der Sturm–Liouvilleschen Eige nwertaufgabe. Bestimmung der Diﬀerentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1946)

work page 1946