Correctness of the definition of the Laplace operator with delta-like potentials

B.E. Kanguzhin; K.S. Tulenov

REVIEW 3 major objections 3 minor 23 references

Correctness of the definition of the Laplace operator with delta-like potentials

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 that the formal Laplace operator with a delta-like potential, $-\Delta+\delta_s$, can be made rigorous by restricting the maximal Laplacian to functions with prescribed singular behavior at one point, yielding a unique…

desk verdict A novel restriction-of-maximal-operator scheme for delta-like potentials in bounded domains, but the sign convention in Lemma 1 makes the central construction internally inconsistent as printed. read the letter →

arxiv 1908.10277 v1 pith:PFKNHXK5 submitted 2019-08-26 math.FA

B.E. Kanguzhin , K.S. Tulenov This is my paper · ORCID

classification math.FA MSC 58J3235J0535J5635J08

keywords LaplaceoperatorDirichletproblemmaximaldelta-likepotentialresolventcorrectnesspointwiseperturbationKreinformula

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

Formally, $-\Delta+\delta_s$ is the Laplacian with a delta-like potential concentrated at an interior point $x_0$ of a bounded domain. This paper claims a rigorous definition of that operator by restriction rather than extension: the maximal Laplacian $B_M$ is restricted to functions whose singular part at the puncture is a prescribed linear combination of the Green function $G(x,x_0)$ and its spatial derivatives, with coefficients extracted by boundary functionals $\gamma_0,\dots,\gamma_d$. The central theorem states that the Dirichlet problem $B_Mu=f$ in the punctured ball, with $u|_{\partial\Omega}=0$ and internal conditions $\gamma_i(u)=\gamma_i(h)$, has a unique solution in $W^2_{2,\gamma}(\Omega_0)$ for every $f\in L^2(\Omega)$ and admissible $h$, given explicitly by $\int_\Omega G(x,\xi)f(\xi)\,d\xi+\gamma_0(h)G(x,x_0)+\sum_{i=1}^d \gamma_i(h)\partial G(x,x_0)/\partial\xi_i$. From this, the paper derives correctly solvable pointwise perturbations $B_K$, explicit resolvents, and a Krein-type trace formula, giving spectral information about the perturbed operator. If correct, this supplies a direct, quadratic-form-free route to delta potentials in several dimensions.

What carries the argument

The machinery is the pair consisting of the maximal operator and the singular Green function. The domain $W^2_{2,\gamma}(\Omega_0)$ consists of functions that are $W^2_{2,\mathrm{loc}}$ away from the puncture and whose singularity at $x_0$ is exactly $\gamma_0(h)G(x,x_0)+\sum_i\gamma_i(h)\partial G(x,x_0)/\partial\xi_i$ plus a regular part; the functionals $\gamma_0,\dots,\gamma_d$ are flux-type limits over shrinking spheres that read off those coefficients. The maximal operator $B_M$ acts by applying $\Delta$ to the regular remainder, so the singular coefficients turn into internal boundary conditions rather than source terms. The boundary form $\langle B_Mw,v\rangle-\langle w,B_Mv\rangle$ equals $\sum_i\gamma_i(w)\beta_i(v)-\sum_i\beta_i(w)\gamma_i(v)$, where $(\beta_0(v),\dots,\beta_d(v))$ are the values and derivatives at $x_0$ of the regular part; this makes $(\mathbb{C}^{d+1},\Gamma_1,\Gamma_2)$ a boundary triple, a description of boundary values as a pair of surjective maps into a finite-dimensional space, and drives the resolvent and trace computations.

What would settle it

Set $f=0$ and $h=G(x,x_0)$ in Theorem 2. The claimed solution is $u(x)=\gamma_0(h)G(x,x_0)=-G(x,x_0)$, but computing the flux integral defining $\gamma_0$ gives $\gamma_0(u)=+1$, not the required $\gamma_0(h)=-1$; equivalently, $B_MG=\Delta(2G)=2\delta(x-x_0)$, which is not in $L^2(\Omega)$. Evaluating either expression directly settles whether the internal boundary condition holds.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2: the restriction $B_M$ of the distributional Laplacian to the domain $W^2_{2,\gamma}(\Omega_0)$ is a well-defined maximal operator, and the boundary-value problem (2.7)--(2.9) has a unique solution given by the displayed Green formula. The proof rests on the decomposition of every element $h$ of the domain as $h_0+\gamma_0(h)G+\sum_i\gamma_i(h)\partial G/\partial\xi_i$ with $h_0\in W^2_2(\Omega)$, and on the removable singularity theorem for harmonic functions to get uniqueness. The paper then shows that choosing an operator $K:L^2(\Omega)\to W^2_{2,\gamma}(\Omega_0)$ and imposing $\gamma_j(u)=\gamma_j(KB_Mu)$ produces an invertible restriction $B_K$, which is the correctly defined pointwise perturbation; in particular, the choice $C_0(u)=ku(x_0)$, $C_1=\cdots=C_d=0$ yields the operator $\Delta+k\delta(x-x_0)$. A boundary triple $(\mathbb{C}^{d+1},\Gamma_1,\Gamma_2)$ with $\Gamma_1=(\gamma_i)$ and $\Gamma_2=(\beta_i)$ is exhibited, and the resolvent $(B_K-\lambda I)^{-1}$ is shown to be a finite-rank perturbation of the free resolvent satisfying Krein's formula with perturbation determinant $\Delta(\lambda)$.

Load-bearing premise

Everything rests on the sign convention that the singular functional $\gamma_0$ evaluates the Green function $G$ to $-1$ while the domain rule subtracts $\gamma_0(h)G$ before applying the Laplacian; those two choices contradict each other when $h=G$, so the claimed solution formula would violate the internal boundary condition it is meant to enforce.

Editorial extensions

If this is right

For every $f\in L^2(\Omega)$ and every admissible $h$, the punctured-domain Dirichlet problem has exactly one solution, so $B_M$ is a well-defined maximal operator and the displayed Green formula can be used to solve it.
Each finite-rank operator $K$ continuous in the sense of the paper selects a correctly solvable pointwise perturbation $B_K$; choosing $Kf = k\langle f,G(\cdot,x_0)\rangle G(\cdot,x_0)$ gives a rigorous meaning to $\Delta + k\delta(x-x_0)$.
The difference of resolvents $(B_K-\lambda I)^{-1}-(B_0-\lambda I)^{-1}$ is finite-rank, so its trace is well defined and equals $-d/d\lambda\,\ln\Delta(\lambda)$, locating the poles of the perturbed resolvent at the zeros of the perturbation determinant.
Eigenvalues of the free Dirichlet Laplacian that are invisible to the perturbation at the puncture, in the sense that $\omega_N(x_0)=0$ or the corresponding functional vanishes, remain eigenvalues of the perturbed operator; only a finite part of the spectrum changes.

Reading between the lines

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

The paper does not pursue it, but the same restriction scheme should extend to a finite set of punctures by taking a product of the functionals at each point; the boundary form and Krein formula would remain finite-rank with dimension multiplied by the number of points.
In the one-dimensional case the two functionals reduce to jumps of the function and of its derivative at the puncture, so the construction is the multi-dimensional analogue of the standard singular Sturm--Liouville operator; the boundary-triple formalism suggests versions on manifolds where an explicit Green function is available.
A direct numerical check of Theorem 2 for $h=G$, $f=0$ in a low-dimensional ball would settle whether the printed sign convention is consistent: evaluating the flux integral defining $\gamma_0$ on the claimed solution should give $-1$ if the internal boundary condition holds.

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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

Let me save you time: the reader's stress test is right. Lemma 1 sets γ0(G) = −1, and the domain definition then forces BMG = Δ(G − γ0(G)G) = Δ(2G) = 2δ, which is not in L2 for d > 2. So G is not in D(BM), Lemma 1 contradicts itself, and Theorem 2 fails for h = G: the formula gives u = G, which satisfies the external and internal boundary conditions but makes BMG = 2δ, not 0. This is not a technical gap; it is a sign error in the load-bearing convention. A consistent version would need γ0(G) = +1, or a compensating sign change in the domain definition, and the paper states neither.

What is genuinely new is the restriction-of-maximal-operator framework for the multidimensional ball: the functionals γ0 through γd, the basis {G, ∂G/∂ξ_i}, the resolvent formulas, and the Krein-type formula. As a program it is worthwhile—it aims at a direct, resolvent-friendly definition of −Δ + δ_s without quadratic forms, and the formal boundary-form computation in Theorem 3 is coherent once you accept the representation. The paper also engages honestly with prior work, including Savchuk–Shkalikov in one dimension and the quadratic-form approaches.

Other soft spots are secondary but real. Lemma 2's proof is circular: it uses uniqueness of the Dirichlet solution in W^2_2 to conclude w ∈ W^2_2, but w is not known to be in that class. The self-adjointness terminology in Theorem 5 is nonstandard, and Remark 11 later says B1 is not self-adjoint, which needs reconciling. Still, the sign inconsistency is the one that matters.

Who is this for: readers working on point interactions or singular perturbations might find the framework worth examining if the sign is repaired. As printed, the main theorems are unsupported. I would not cite it. I would give it a serious referee rather than desk-reject it outright, because the construction is novel and the error is localized; a referee could help determine whether the sign repair goes through cleanly. But I would not accept it anywhere near its current form.

Referee Report

3 major / 3 minor

Summary. The paper proposes a definition, in L2(Ω) for a ball Ω in Rd (d>2), of a maximal Laplace operator BM acting on functions in a punctured domain Ω0 with prescribed singular behaviour at an interior point x0. The singular part is expanded in a basis consisting of the Green's function G(·,x0) and its first derivatives, with coefficients recorded by functionals γ0,...,γd. The paper then constructs 'correctly solvable' restrictions BK representing −Δ plus delta-like potentials, gives resolvent formulas, a Krein-type trace formula, and spectral consequences. The central claim is Theorem 2, which asserts unique solvability of the Dirichlet problem BMu=f with internal conditions γi(u)=γi(h) and represents the solution explicitly. The proof of this theorem and the whole restriction scheme rest on Lemma 1, which states γ0(G)=−1 and γs(G)=0.

Significance. If the construction were correct, the paper would offer a direct, quadratic-form-free approach to point interactions in bounded domains, with explicit resolvents and a Krein formula. That would be a useful contribution to the literature on singular perturbations. The paper also tries to connect the construction to boundary triples and to the Savchuk–Shkalikov one-dimensional results. However, the central construction is internally inconsistent: the sign convention for γ0 in Lemma 1 contradicts the definition of the domain of BM, and Theorem 2's solution formula fails for the admissible input h=G. Because the defect occurs at the foundation of the paper and propagates through Lemma 2, Theorem 3, and all resolvent formulas, the claimed results are not established. I see no machine-checked proofs, reproducible code, or numerical verification that could compensate; the claims are purely analytical and they fail already on a concrete, admissible test case.

major comments (3)

[Section 2, definition of D(BM) and Lemma 1] Lemma 1 asserts γ0(G)=−1, but the definition of W^2_{2,γ}(Ω0) (display before Lemma 1) requires Δ_x(h−γ0(h)G−Σ_{s=1}^d γ_s(h)∂G/∂ξ_s) ∈ L2(Ω). Taking h=G and using γ0(G)=−1, γs(G)=0 gives BMG = Δ(G−(−1)G) = Δ(2G) = 2δ_{x0}, which is a distribution, not an L2 function, for d>2. Hence G∉D(BM), contradicting Lemma 1. This is not a technical gap in a proof; it is an internal inconsistency between the sign convention in Lemma 1 and the definition of the maximal operator.
[Theorem 2, Eq. (2.7)–(2.9)] For f=0 and h=G, the solution formula in Theorem 2 gives u(x)=γ0(h)G(x,x0)=−G(x,x0). Then γ0(u)=+1, whereas the internal boundary condition (2.9) requires γ0(u)=γ0(h)=−1. Equivalently, applying the domain operator with γ0(u)=+1 gives BMu=−2δ_{x0}≠0, so the claimed solution does not satisfy (2.7) even as an identity in distributions. Thus the central solvability theorem fails on a concrete admissible input, and the subsequent restriction construction based on Theorem 2 is unsupported.
[Lemma 2] Lemma 2 claims that every h∈W^2_{2,γ}(Ω0) has a unique representation h=h0+γ0(h)G+Σ_{i=1}^d γ_i(h)∂G/∂ξ_i with h0∈W^2_2(Ω). For h=G, using γ0(G)=−1, this representation gives h0=G+γ0(G)G = G−(−1)G = 2G (up to the derivative terms, which vanish). The Green function G behaves like |x−x0|^{2−d} near x0 and is not in L2(Ω) for d≥3, so h0 cannot belong to W^2_2(Ω). Lemma 2 is therefore also false under the stated sign convention. Since Lemma 2 is used to justify the boundary form and the restriction theory, this is another load-bearing failure.

minor comments (3)

[Section 4, definition of Δ(λ)] The determinant Δ(λ) is labelled as d×d, but it is a (d+1)×(d+1) matrix with indices 0,...,d; the dimension label should be corrected.
[Throughout] The manuscript contains numerous typographical errors and nonstandard spellings ('non-standart', 'Acknowlegment', 'delta-fuction', 'opertaor'), and several formulas, for example (4.9), are written as informal recursions without a precise statement of their domains of validity; a careful editorial revision would be needed.
[Section 5, Proposition 1] The integration by parts in Proposition 1 relies on the limiting flux identity ∫_{∂Π_δ^0} ∂ϕ0/∂ν dS → 1, but the proof does not state this value explicitly before using it; stating the dimension-dependent constants explicitly would make the computation easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the operator, restrictions, and resolvent/Krein formulas are explicitly constructed from stated definitions, with no fitted inputs or load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained rather than circular. In Section 2 the maximal operator B_M is defined explicitly by B_M u = Delta(u - gamma0(u)G - sum_s gamma_s(u) dG/dxi_s) with domain W^2_{2,gamma}(Omega0), and Lemma 2 gives the unique decomposition h = h0 + gamma0(h)G + sum_i gamma_i(h)dG/dxi_i with h0 in W^2_2(Omega). Theorem 2's solution formula is then a direct representation of this decomposition for the given boundary-value problem, not a quantity fitted from data or derived from the theorem itself. Theorem 4 defines restrictions B_K through an arbitrary continuous operator K and boundary conditions gamma_j(u)=gamma_j(K B_M u); the resolvent identities in Theorems 6-8 are obtained by substituting the fixed elements phi_i = G and dG/dxi_i, checking the equation and boundary traces, and solving the resulting linear algebraic system. These are algebraic consequences of the definitions, so no 'prediction' is equivalent to an input by construction. The self-citations [13]-[17] are used only as background for related regularized-trace questions and are not load-bearing for the main construction; the uniqueness argument in Theorem 2 cites the external removable-singularity result [19]. The sign inconsistency between Lemma 1 (gamma0(G)=-1) and the identity B_M G=0 used in Theorem 6 is a genuine mathematical correctness concern, but it is not a circularity: it does not reduce any claimed result to its own input. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities and fits no free parameters. Its assumptions are standard functional-analytic tools plus an inconsistent set of conventions for the singular functionals and the Green function. The inconsistency is counted as a domain-assumption axiom rather than as a fitted parameter.

assumptions (6)

standard math Green's function formula for the Dirichlet Laplacian in the unit ball (Theorem 1)
Invoked to define the singular basis functions and the solution formula in Theorem 2.
standard math Uniqueness and elliptic regularity for the Dirichlet problem Delta w = f, w|partial Omega = 0 in W^2_2(Omega)
Used in Lemma 2 to decompose W^2_{2,gamma} functions; regularity of w is not established from the stated hypotheses.
standard math Removable singularity theorem for harmonic functions, cited as [19, Theorem III.39]
Used in the proof of Theorem 2 for uniqueness of the homogeneous problem.
domain assumption The functionals gamma_j are well-defined and finite on W^2_{2,gamma}, and Lemma 1's stated values are correct
This is the load-bearing premise; as printed it is inconsistent with the domain definition because B_M G = 2 delta.
standard math Riesz representation theorem for the functionals gamma_j(K f)
Used in Remark 4 to represent K through L2 kernels c_j.
domain assumption The Dirichlet Laplacian B_0 in the bounded domain has discrete spectrum with an orthonormal eigenbasis
Used in Section 5 for the perturbation determinant and spectral calculations.

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Pith. "Pith review of Correctness of the definition of the Laplace operator with delta-like potentials." pith.science (2026). https://pith.science/paper/PFKNHXK5

@misc{pith2026190810277,
  author       = {Pith},
  title        = {Pith review of: Correctness of the definition of the Laplace operator with delta-like potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFKNHXK5}},
  note         = {Machine review of arXiv:1908.10277}
}

read the original abstract

In this paper, we give a correct definition of the Laplace operator with delta-like potentials. Correctly solvable pointwise perturbation is investigated and formulas of resolvent are described. We study some properties of the resolvent. In particular, we obtain Krein formula for these resolvents.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Pith tools

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

[1] [1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models in quantum mechanics. Second edition. With an appendix by Pavel Exner. AMS Chelsea Publishing, Providence, RI,

work page

[2] [2]

Albeverio, P

S. Albeverio, P. Kurasov, Singular perturbations of diﬀerential operators. Solvabl e Schrdinger type operators. London Mathematical Society Lecture Note Series, 271. Cambridge University Press, Cambridge, 2000. xiv+429 pp

work page 2000

[3] [3]

F. A. Berezin, L. D. Faddeev, Remark on the Schr¨ odinger equation with singular potentia l, Soviet Phys. Dokl., 2 (1961), 372–375

work page 1961

[4] [4]

Blanchard, R

Ph. Blanchard, R. Figari, and A. Mantile, Point interaction Hamiltonians in bounded domains , J. Math. Phys. 48, 082108 (2007); 10.1063/1.2770672

work page doi:10.1063/1.2770672 2007

[5] [5]

M. Sh. Birman, D. R. Yafaev, The spectral shift function. The papers of M. G. Kre ˇin and their further development . St. Petersburg Math. J., 4:5 (1993), 833–870

work page 1993

[6] [6]

M. Sh. Birman, T. A. Suslina, R. G. Shterenberg, Absolute continuity of the two-dimensional Schr¨ odinger operator with delta potential concentrated o n a periodic system of curves . St. Petersburg Math. J., 12:6 (2001), 983–1012

work page 2001

[7] [7]

M. Sh. Birman, T. A. Suslina, Second order periodic diﬀerential operators. Threshold pr oper- ties and homogenization , St. Petersburg Math. J., 15:5 (2004), 639–714

work page 2004

[8] [8]

M. M. Crum, Associated SturmLiouville systems , Quart. S. Math., 6 (1955), 121– 127

work page 1955