On an optimal potential of Schr\"odinger operator with prescribed $m$ eigenvalue

Nurmukhamet Valeev; Yavdat Ilyasov

REVIEW 3 major objections 4 minor 16 references

On an optimal potential of Schr\"odinger operator with prescribed $m$ eigenvalue

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that the constrained minimization of ||V0−V||² subject to fixing the first m Dirichlet eigenvalues is attained, and that every minimizer takes the explicit form V0 minus a signed sum of squared eigenfunctions solving a…

desk verdict New m≥2 inverse-optimal spectral result with a solid variational core, but Theorem 1 as stated is false without a strict-ordering/feasibility hypothesis. read the letter →

arxiv 1908.07876 v1 pith:PI4OQ52S submitted 2019-08-21 math.AP math-phmath.MPmath.SP

Yavdat Ilyasov , Nurmukhamet Valeev This is my paper · ORCID

classification math.APmath-phmath.MPmath.SP MSC 34L0534A5547E0549J20

keywords inversespectralproblemSchrödingeroperatoroptimalpotentialprescribedeigenvaluesnonlinearsystemLagrangemultiplierDirichletvariationalmethod

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

This paper introduces an inverse optimal spectral problem for the one-dimensional Schrödinger operator: given a starting potential V0 and m target values E_1,...,E_m, find the potential V̂ closest to V0 in L² norm whose first m Dirichlet eigenvalues equal the targets. The main theorem states that a minimizer always exists, provided the targets are realizable, and that every minimizer has the explicit form V̂ = V0 − Σ σ_j û_j², where (û_1,...,û_m) solves the nonlinear boundary-value system (E) with σ_j ∈ {0, ±1}. This is new because it ties an inverse problem to a system of nonlinear differential equations that can be solved numerically. The result extends the authors' earlier one-parameter case (a single prescribed eigenvalue) to any finite number m, and the paper conjectures uniqueness for all m.

What carries the argument

The load-bearing identity is the Fréchet derivative of the k-th eigenvalue map, D_V E_k(V)(h) = ⟨φ_k(V)², h⟩/||φ_k(V)||², which makes E_k continuously differentiable on L² and turns the constrained minimization into a Lagrange multiplier equation. The other engines are Banach–Alaoglu compactness on the bounded minimizing sequence; the uniform $W^{{2,2}}$ bound for eigenfunctions of potentials with bounded L² norm; Sturm's nodal theorem, which fixes the indices of the limiting eigenpairs and rules out eigenvalue skipping; and the linear independence of the squares of the first m eigenfunctions, proved in the appendix, which forces the multiplier µ0 to be nonzero. Combining these yields the coupled nonlinear system (E), whose solution directly produces V̂ by (2.4).

What would settle it

Take m = 2, E_1 = E_2 = 0, and V0 ≡ 0 on (0,L): because any such operator has simple spectrum with E_1(V) < E_2(V), no potential can satisfy both constraints, so (P) has an empty feasible set and no minimizer, showing that Theorem 1's universal statement over arbitrary real E_i fails without the implicit strict-ordering hypothesis.

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Extended reading notes

Core claim

The central discovery is the equivalence between a constrained distance-minimization problem over potentials and a coupled nonlinear system. For fixed V0 and strictly ordered targets E_1<...<E_m, Theorem 1 asserts that the infimum of ||V0−V||² over all V∈L² with E_k(V)=E_k for k=1,...,m is attained, and that at any minimizer V̂ the gradient identity forces 2µ0(V0−V̂) = Σ µ_j φ_j(V̂)². Subtracting V̂ from the eigenvalue equations for φ_i(V̂) yields the system (E): −u_i''+V0 u_i = E_i u_i + Σ_{j=1}^m σ_j u_j² u_i, with Dirichlet boundary conditions; setting û_i = √|µ_i| φ_i(V̂) and σ_i = sign(µ_i) gives (2.4). Thus the optimal potential is V0 minus a signed sum of squared bound-state eigenfunctions. The proof obtains the minimizer by compactness of a minimizing sequence, identifies the weak limit's eigenpairs by nodal counts via Sturm's theorem, and applies the Lagrange multiplier rule to get the representation.

Load-bearing premise

The feasible set S = {V ∈ L² : E_k(V) = E_k, k = 1,...,m} must be nonempty and the prescribed numbers must be strictly ordered E_1 < ... < E_m; the paper states neither condition explicitly, but the proof needs S nonempty to start with a minimizing sequence.

Editorial extensions

If this is right

For any data (V0, E_1,...,E_m) for which the feasible set is nonempty, the infimum in (P) is attained, so the notion of a closest potential with prescribed spectral data is well defined.
Every minimizer V̂ satisfies V̂ = V0 − Σ_{j=1}^m σ_j û_j², where (û_j) is a weak solution of the nonlinear system (E); solving (E) gives a constructive route to the optimal potential.
When V0 already has the prescribed first m eigenvalues, the minimizer is V0 itself and all σ_j vanish, so the nonlinear system admits the zero solution.
The one-parameter results from the authors' earlier work become a special case of Theorem 1, and the paper states that the argument should extend to the whole line and to other boundary conditions.
The representation shows that the optimal correction to V0 is always a rank-m term built from squared eigenfunctions of the corrected operator, so the adjustment inherits the nodal structure of the target eigenfunctions.

Reading between the lines

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

If the conjecture of uniqueness for (E) holds, then for fixed V0 the map from prescribed strictly increasing m-tuples to optimal potentials is single-valued, making the multi-parameter inverse problem a well-posed parameterization of a codimension-m family of potentials.
A natural testable extension is to prescribe eigenvalue gaps (E_{k+1}−E_k) or weighted eigenvalue functionals instead of individual eigenvalues; the derivative formula still supplies the Lagrange multiplier structure, but the feasible sets become nonlinear and compactness would need to be rechecked.
Numerically, one could compute the same minimizer two ways—direct constrained optimization over V and solution of (E) followed by (2.4)—on a grid; any systematic discrepancy would indicate that the weak-solution class or the restricted sign values σ_j exclude some attainable corrections.
The theorem gives a minimal fine-tuning interpretation: the cheapest L² adjustment of V0 that moves m eigenvalues to target values is always a sum of squared eigenfunctions of the corrected operator, which may guide iterative algorithms for spectral design.

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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

Colleague,

This paper extends the authors' m=1 inverse optimal spectral work to m≥2: given V0 and m target eigenvalues, minimize the L2 distance to V0. The main theorem claims existence of a minimizer and characterizes it via system (E). That characterization is the genuinely new piece, and the variational strategy (weak compactness, Lagrange multipliers, Sturm count) is coherent. The derivation of (E) is earned: no fitted parameters, no circularity. The appendix's linear independence argument is a bit rough, but the statement is standard. So there is real substance here.

Now the soft spots, in order of seriousness.

First, Theorem 1 as stated is false. No ordering assumption is made on the E_i. For every potential in L^2, the Dirichlet eigenvalues are strictly increasing. If the prescribed tuple is not strictly increasing, the constraint set S is empty, so no minimizer exists. The proof in Section 3 picks a minimizing sequence from S; that sequence does not exist. This is not a mere typo: the theorem quantifies over arbitrary E_i. The fix is easy: add E_1 < ... < E_m and justify nonemptiness of S (cite Pöschel–Trubowitz or give a construction). Until then, the result is conditional.

Second, the index bound r_i ≤ i−1 is off by one; it should be r_i ≤ i, then orthogonality forces equality. This is a typo, but it sits in a key step.

Third, after (3.11) the formula for V̂ drops a factor of 1/(2µ0) and has the opposite sign from the preceding equation. It is repairable: one can rescale the multipliers to normalize µ0, and the σ_j ∈ {+1,-1,0} then come out correctly. But the displayed identity is not correct as written.

None of these kill the intended argument; the repaired theorem is plausible and the system (E) is a useful necessary condition. The paper would benefit from a careful rewrite of the statement and the multiplier step. This is for spectral theorists and people doing quantum design / inverse optimization. A serious referee should see it. I would send it to peer review with a request for revision, flagging the feasibility issue first.

Best.

Referee Report

3 major / 4 minor

Summary. The paper studies an inverse optimal spectral problem for the one-dimensional Dirichlet Schrödinger operator on (0,L): given a reference potential V0 and prescribed numbers E1,...,Em, find the L2 potential V̂ closest to V0 such that the first m eigenvalues of H_V̂ are exactly E1,...,Em. The main result, Theorem 1, asserts that a minimizer always exists and that every minimizer has the form V̂ = V0 - Σ σ_j û_j², where (û_1,...,û_m) solves the coupled nonlinear system (E) with σ_j ∈ {0,+1,-1}. The proof proceeds by taking a minimizing sequence, using weak compactness to produce a limit potential, showing that the limit has the prescribed eigenvalues by a nodal-count and orthogonality argument, and then applying a Lagrange multiplier rule to derive the nonlinear system. The paper also contains a short appendix intended to prove linear independence of the squares of the first m eigenfunctions.

Significance. If the hypotheses are repaired, the result is a meaningful contribution: it gives an existence theorem for a multi-parameter optimal inverse spectral problem and, more importantly, characterizes the optimal potential through the explicit nonlinear system (E). The Euler-Lagrange derivation of (E) is the paper's main novelty and is conceptually clean. The proof is largely self-contained and the statement of the characterization in (2.4) is concrete enough to be used in numerical computations. The paper does not address uniqueness or numerical implementation, but those are not promised. The main value is the existence-plus-characterization statement, which is a natural extension of the authors' earlier one-eigenvalue work to the multi-eigenvalue setting.

major comments (3)

[Section 2, Problem (P) and Theorem 1] The statement of Problem (P) and Theorem 1 omits a necessary hypothesis: the prescribed numbers E_1,...,E_m must be strictly increasing. For every V in L^2(0,L), the Dirichlet eigenvalues of H_V satisfy E_1(V) < E_2(V) < ..., so the feasible set S = {V : E_k(V)=E_k, k=1,...,m} is empty unless E_1 < ... < E_m. For example, with m=2, E_1=E_2=0 and V_0=0, no potential satisfies both constraints, so the minimum in (2.3) is not attained and Theorem 1 as literally stated is false. The proof in Section 3 begins by choosing a minimizing sequence in S, which is impossible if S is empty. The theorem needs the explicit strict-ordering hypothesis and, unless it is cited as standard, a proof or reference that every strictly increasing m-tuple is realizable as the first m Dirichlet eigenvalues of some L^2 potential.
[Section 3, paragraph following Eq. (3.9)] The inequality 'r_i ≤ i-1' is incorrect and should be 'r_i ≤ i'. The nodal count gives that φ_i* has at most i-1 zeros; since the r_i-th eigenfunction of H_V̂ has exactly r_i-1 zeros, the correct conclusion is r_i-1 ≤ i-1, i.e., r_i ≤ i. With the printed inequality, r_1 ≤ 0, which contradicts φ_1* ≠ 0, and the subsequent conclusion r_i = i cannot be derived. This is a fixable typo, but it appears at a load-bearing point in the proof of admissibility of V̂.
[Section 3, Eq. (3.11) and Section 5, Appendix] The proof that μ_0 ≠ 0 depends on the claim that the squares φ_j²(V̂), j=1,...,m, are linearly independent, but the appendix proof is not correct as written. In Eq. (5.12) the coefficients are inconsistently denoted α_k and α_i; after differentiating twice the derivation uses what appears to be a single energy E_k even though each eigenfunction has its own eigenvalue E_i; and the step leading from Σ α_i E_k φ_i φ_i' = 0 to Σ α_i E_k φ_i² = 0 requires an integration argument that is not stated. The final reduction to m-1 functions is also indexed incorrectly. The linear-independence statement may be true and provable by a different argument, but as it stands the proof does not justify μ_0 ≠ 0, which is essential for deriving system (E) from (3.10)-(3.11).

minor comments (4)

[Section 3, first paragraph] The text states that by Banach-Alaoglu one obtains a subsequence such that ||V0-V_j|| → ||V0-V̂|| and V_j ⇀ V̂ weakly. The norm convergence to ||V0-V̂|| is not a consequence of weak convergence alone; the subsequent argument only needs the lower semicontinuity inequality ||V0-V̂|| ≤ liminf ||V0-V_j||. The wording should be corrected.
[Section 3, after Eq. (3.10)] The proof says that Σ |μ_j| ≠ 0 follows from an assumption, but no such assumption is stated in Theorem 1. The theorem later mentions the case E_i' ≠ E_i'(V0) for some i' only as a conditional statement. The dichotomy between the trivial case E_k = E_k(V0) for all k and the nontrivial case should be stated explicitly before deriving the sign property of σ_j.
[Section 5, Eq. (5.12)] The summation index in Eq. (5.12) is inconsistent: α_k is used in the first sum and α_i in later expressions. Please reindex uniformly. The last displayed sum over j=1,...,m-1 with coefficients γ_k also needs correct indexing.
[References] The W^{2,2} bound for eigenfunctions is cited to the authors' own paper [3]; a standard elliptic regularity reference would be more appropriate. Several other self-citations for the m=1 case are fine, but the dependence of the key estimate on [3] should be reduced or justified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal-potential representation is derived from the variational principle, and the cited self-results are not load-bearing.

full rationale

The derivation chain in Sections 2 and 3 is self-contained in the relevant sense. Theorem 1 is proved by a direct variational argument: a minimizing sequence in the feasible set (when it is nonempty) is weakly compact, the weak limit is shown admissible by passing to the limit in the resolvent equation and using orthogonality plus nodal counts, and the representation (2.4) is then obtained from the Lagrange multiplier rule and the Fréchet derivative formula (3.5). The nonlinear system (E) is not assumed: it is derived by setting u_i = |\mu_i|^{1/2} \varphi_i(\hat V) in the identity 2\mu_0(V_0-\hat V)=\sum_j \mu_j \varphi_j^2(\hat V). Thus (2.4) and (E) are consequences of the minimization, not restatements of the constraints. The only self-citations ([2], [3], [6]) concern the m=1 case and a routine W^{2,2} bound for eigenfunctions; the latter is an externally verifiable standard estimate and neither supplies the central existence claim nor is equivalent to it. There is, however, a genuine hypothesis gap of a non-circular kind: the statement omits the necessary strict ordering E_1<...<E_m and the nonemptiness of the feasible set S, so if S is empty the minimizing sequence in Section 3 does not exist and Theorem 1 as literally stated is false. This is a correctness/missing-hypothesis problem, not a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; V0, L, and the target E_i are problem inputs, and the σ_i are signs of Lagrange multipliers whose existence is established, though not specified in advance. The main unstated input is feasibility: the target tuple must be strictly increasing and realizable, which the paper neither assumes explicitly nor proves. All other axioms are standard spectral theory or standard optimization results. No new physical entities are introduced.

assumptions (5)

standard math Sturm-Liouville spectral theory: H_V with V ∈ L²(0,L) and Dirichlet boundary conditions is self-adjoint with discrete simple spectrum; the k-th eigenfunction has exactly k−1 zeros in (0,L).
Invoked throughout Section 2 and in the index-matching argument after Eq. (3.8); standard reference cited is [8].
standard math Differentiability of eigenvalues and continuity/analyticity of eigenfunctions: E_k(V) is C¹ with D E_k(V)(h) = ⟨φ_k(V)², h⟩, and V ↦ φ_k(V) is analytic.
Lemma 1, Eq. (3.5); cited to [4] and [5].
domain assumption Feasibility: the set of potentials whose first m Dirichlet eigenvalues equal given strictly ordered numbers E_1 < ... < E_m is nonempty.
Used to select the minimizing sequence in Section 3. Not stated in Theorem 1 and not proven or cited; needed also because the eigenvalues of any H_V are strictly increasing, so non-increasing tuples are infeasible.
standard math Uniform W^{2,2} bound for normalized eigenfunctions when V is bounded in L² and E_k(V) is fixed.
Used before Eq. (3.6); the paper cites its own [3], though a standard elliptic bootstrap (via the Green's function G0) suffices.
standard math Lagrange multiplier rule with constraint qualification: at a minimizer, if the gradients of the constraints are linearly independent, then the Euler-Lagrange equation holds with a nonzero multiplier on the objective.
Applied after (3.9); the needed linear independence of {φ_j(V̂)²} is proven in the Appendix, but the text does not state the qualification condition or cite the theorem.

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Pith. "Pith review of On an optimal potential of Schr\"odinger operator with prescribed $m$ eigenvalue." pith.science (2026). https://pith.science/paper/PI4OQ52S

@misc{pith2026190807876,
  author       = {Pith},
  title        = {Pith review of: On an optimal potential of Schr\"odinger operator with prescribed $m$ eigenvalue},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PI4OQ52S}},
  note         = {Machine review of arXiv:1908.07876}
}

abstract

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

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

[1] [1]

Uber eine frage der eigenwerttheorie. Zeitschrift f\

V. Ambarzumian, \"Uber eine frage der eigenwerttheorie. Zeitschrift f\"ur Physik A Hadrons and Nuclei 53 (9) (1929), 690-695

work page 1929

[2] [2]

Borg, Eine umkehrung der Sturm0iouvilleschen eigenwertaufgabe

G. Borg, Eine umkehrung der Sturm0iouvilleschen eigenwertaufgabe. Acta Mathematica, 78 (1) (1946), 1-96

work page 1946

[3] [3]

D. E. Edmunds, W. D. Evans, Spectral theory and differential operators. Vol. 15. Oxford: Clarendon Press, 1987

work page 1987

[4] [4]

I. M. Gel'fand, B. M. Levitan, On the determination of a differential equation from its spectral function. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 15 (4) (1951), 309-360

work page 1951

[5] [5]

Y. Sh. Ilyasov, N. F. Valeev, On nonlinear boundary value problem corresponding to N -dimensional inverse spectral problem, J. Diff. Eq., 266.8 (2019) 4533-4543

work page 2019

[6] [6]

Il'yasov, N

Ya. Il'yasov, N. Valeev, On an inverse spectral problem and a generalized Sturm's nodal theorem for nonlinear boundary value problems, Ufa Math. J., 10:4 (2018) 122-128

work page 2018

[7] [7]

B. M. Levitan, I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Vol. 39. American Mathematical Soc., 1975

work page 1975

[8] [8]

Lord Rayleigh J. W. S. (1877): The Theory of Sound, Dover Publications, New York, 1945 [Tom I, p. 216)

work page 1945