REVIEW 3 major objections 3 minor 18 references
Singular perturbations of Laplace operator and their resolvents
T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper argues that cutting a closed lower-dimensional smooth manifold out of a ball still leaves the Laplace operator with invertible restrictions, a well-posed non-smooth Bitsadze-Samarskii problem, and a Krein trace formula for the…
desk verdict The manuscript under this arXiv ID is not the paper its abstract describes, so the announced Laplace/Bitsadze–Samarskii/Krein results have no support in the supplied text. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The object that carries the argument is the restricted Laplace operator on the cut ball: the Laplace operator whose domain is narrowed so that the removed lower-dimensional manifold functions as a boundary-like singular set. The identity that gives the paper its main output is M. G. Krein's trace formula for the difference of two resolvents. Krein's formula applies precisely when the difference of the resolvents is trace-class, and the paper's construction is arranged so that this trace-class condition holds for the studied restrictions. The non-smooth Bitsadze-Samarskii problem is the boundary-value counterpart of the same construction: it is the problem of solving $\Delta u = f$ with data that couple boundary values to values on the inner manifold. The removed manifold is what makes the perturbation singular, and the resolvent difference is the operator-theoretic record of that singularity.
What would settle it
Take the unit ball in $\mathbb{R}^3$ and remove a closed smooth curve such as a circle; form the two restrictions of the Laplace operator studied in the paper and compute the trace of $(\Delta_1-\lambda)^{-1}-(\Delta_2-\lambda)^{-1}$. If for some admissible pair of restrictions this difference is not trace-class, or if the trace disagrees with the Krein formula stated in the paper, the central claim fails.
Extended reading notes
Core claim
The central claim is that removing an inner closed smooth manifold of lower dimension from a multidimensional ball produces a region on which invertible restrictions of the Laplace operator are well defined. The restrictions differ from the ordinary Laplace operator by narrowing the domain of admissible functions, with the removed manifold acting as a singular set. With these restrictions in hand, the paper defines a well-posed non-smooth Bitsadze-Samarskii problem for the Laplace equation: a boundary-value problem whose data are tied to values on an inner manifold, now allowed to be non-smooth because of the cut. The paper further claims a trace formula of M. G. Krein for the difference of the resolvents of the studied operators, which presupposes that the resolvent difference is trace-class. Finally, it proves assertions on the spectrum of the non-smooth Bitsadze-Samarskii problem. Read sympathetically, the paper's claim is that the singularly perturbed domain still supports a complete, classical-looking spectral and resolvent theory.
Load-bearing premise
The construction rests on the assumption that cutting the lower-dimensional manifold out of the ball gives restrictions of the Laplace operator whose resolvent difference is trace-class; if that trace-class property fails, the Krein formula and the spectral assertions do not follow.
Editorial extensions
If this is right
- On a ball with an internal closed lower-dimensional cut, the Laplace operator has invertible restrictions, so singular domains of this kind admit a resolvent calculus rather than falling outside classical theory.
- The non-smooth Bitsadze-Samarskii problem for the Laplace equation is well-posed: it has a solution, and the solution is unique under the paper's conditions.
- The difference of the resolvents of the studied operators is trace-class, so M. G. Krein's formula can be written for its trace.
- The spectrum of the non-smooth Bitsadze-Samarskii problem has the structure asserted in the paper, extending spectral theory for the Laplace equation to domains with internal cuts.
Reading between the lines
- A note for the reader: the full text bundled with this record is a different manuscript about multidimensional singular integrals in Besov spaces; the Laplace-resolvent claims above are read from the title and abstract only.
- If the construction is as general as the abstract suggests, the same cut-and-restrict scheme should yield resolvent trace formulas for other elliptic operators on domains with internal slits or screens.
- Trace-class resolvent difference implies, though the paper does not state it, a spectral shift function for these singular restrictions, so the cut should have a scattering-theoretic reading.
- A numerical experiment on a specific case, such as a ball with a removed circle, could compute the trace of the resolvent difference and test the Krein formula directly.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper as submitted does not correspond to its abstract. The abstract announces results on singular perturbations of the Laplace operator in a ball with a lower-dimensional manifold removed: well-defined invertible restrictions, a well-posed non-smooth Bitsadze-Samarskii problem, Krein's formula for the trace of the difference of resolvents, and spectral assertions. The body of arXiv:1908.09537v1 is a separate manuscript, 'Multidimensional singular integrals and integral equations in fractional spaces I', which studies boundedness, invertibility, and solvability of multidimensional singular integral operators in Besov spaces. The body contains no Laplace operator, no resolvents, no Krein trace formula, no Bitsadze-Samarskii problem, and no spectral theorem. Thus the announced results cannot be verified from the supplied text; the manuscript under review does not contain the promised arguments.
Significance. If proved, the abstract claims would be a meaningful contribution to the spectral theory of singular perturbations of elliptic operators: a Krein trace formula for resolvent differences and well-posedness of a non-smooth Bitsadze-Samarskii problem are non-routine results. However, because the submitted body is an unrelated paper on singular integrals in Besov spaces, the significance is entirely hypothetical. The actual body may contain a serviceable treatment of multidimensional singular integrals in Besov spaces, but it does not address the announced topic. As submitted, the paper cannot be evaluated as a proof of its abstract, and the claimed results are unsupported by any inspectable argument.
major comments (3)
- [Abstract vs. full text] The abstract announces invertible restrictions of the Laplace operator after cutting out an inner smooth lower-dimensional manifold, a well-posed non-smooth Bitsadze-Samarskii problem, Krein's trace formula, and spectral assertions, but the body of the manuscript under this arXiv ID is titled 'Multidimensional singular integrals and integral equations in fractional spaces I' and contains no definition of such a cut domain, no Laplace operator, no resolvents, no Krein formula, and no spectral statement. This is not a presentation issue; the central claims of the abstract have no supporting argument anywhere in the supplied text.
- [First sentence of abstract or full text] The geometric operation 'an inner closed smooth manifold of a lower dimension is cut from a multidimensional ball' is never defined with mathematical hypotheses: the dimensions, codimension, regularity of the removed manifold, choice of boundary conditions on the cut, and the functional-analytic setting (self-adjointness, closedness, or essential self-adjointness of the restricted Laplace operator) are all absent. Without these hypotheses, the assertion that 'invertible restrictions of the Laplace operator are well defined' is not a theorem but an undefined claim, and this premise is load-bearing for all subsequent statements.
- [Section 3, Theorem 6] The only invertibility criterion proved in the body concerns translation-invariant singular integral operators in the ring Q_p and requires nonvanishing of the symbol on the unit sphere. This theorem concerns a completely different operator class and cannot be used to infer anything about restrictions of the Laplace operator on a punctured ball; no bridge between the two settings is supplied in the manuscript.
minor comments (3)
- [Title page] The title contains obvious spacing errors: 'MUL TIDIMENSIONAL SINGULAR INTEGRALS AND INTEGRAL EQUA TIONS IN FRACTIONAL SP ACES I' should read 'MULTIDIMENSIONAL SINGULAR INTEGRALS AND INTEGRAL EQUATIONS IN FRACTIONAL SPACES I'.
- [Abstract] The abstract has a diacritic error in 'Samarski\u0131', which should be 'Samarskii'.
- [Throughout the body] Equation numbering is inconsistent: the singular integral is introduced as (1.1), and later a different integral equation is also numbered (1.3) with variants (1.3') and (2.3); renumbering would improve readability, though this is secondary to the substantive mismatch.
Circularity Check
No circularity can be identified: the supplied full text is a different manuscript and contains no derivation of the abstract's Laplace-operator claims.
full rationale
The abstract announces results on invertible restrictions of the Laplace operator, a non-smooth Bitsadze-Samarskii problem, Krein's trace formula for resolvent differences, and spectral assertions. The full text supplied under arXiv:1908.09537v1 is a separate paper, 'Multidimensional singular integrals and integral equations in fractional spaces I' by N.K. Bliev and K.S. Tulenov, whose abstract states: 'In this paper, we investigate boundedness, invertibility and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations in Besov spaces.' The text contains singular integral operators in Besov spaces, Calderon-Zygmund estimates, symbols, and Fredholm-type equations, but no Laplace operator, no cut manifold, no resolvents, no Krein formula, and no Bitsadze-Samarskii problem. Because the claimed derivation chain for the abstract's results is absent from the supplied text, there is no equation or argument in which a prediction reduces to its own inputs. The mismatch is a verification/provenance failure rather than a circularity: no specific reduction can be quoted, and the hard rules require exhibiting a concrete circular step. Accordingly, the circularity score is 0, with no circular steps recorded.
Assumptions & free parameters
assumptions (3)
- domain assumption Cutting an inner closed smooth lower-dimensional manifold from a ball yields a well-defined domain and boundary conditions for restrictions of the Laplace operator.
- domain assumption The resolvent difference of the studied restrictions is trace-class, so Krein's trace formula applies.
- standard math Background spectral theory of self-adjoint extensions and resolvents of the Laplace operator is assumed.
Cite this review
Pith. "Pith review of Singular perturbations of Laplace operator and their resolvents." pith.science (2026). https://pith.science/paper/Z5HI2U76
@misc{pith2026190809537,
author = {Pith},
title = {Pith review of: Singular perturbations of Laplace operator and their resolvents},
year = {2026},
howpublished = {\url{https://pith.science/paper/Z5HI2U76}},
note = {Machine review of arXiv:1908.09537}
}
read the original abstract
An inner closed (without boundary) smooth manifold of a lower dimension is cut from a multidimensional ball. In this region, invertible restrictions of the Laplace operator are well defined. In particular, the well-posed non-smooth Bitsadze-Samarski\u{i} problem for the Laplace equation is defined. Moreover, we obtain M.G. Krein's formula for the trace of the difference of resolvents of the studied operators. We prove assertions on the spectrum of the non-smooth Bitsadze-Samarski\u{i} problem.
Reference graph
Works this paper leans on
-
[1]
Alexandrov, Combinatorial Topology, OGIZ, Moscow, 1947
P.S. Alexandrov, Combinatorial Topology, OGIZ, Moscow, 1947. [In Russian]
work page 1947
-
[2]
F. V. Atkinson, The normal solvability of linear equations in normed spaces , Mat. Sb. (N.S.), 28(70):1 (1951), 3–14
work page 1951
-
[3]
O.V. Besov, V.P. Il’in, and S.M. Nikol’ski˘i, Integral representations of function and embedding theorems, 1, 2. , John Willey, New York, 1978; 1979
work page 1978
-
[4]
Bliev N., Generalized analytic functions in fractional spaces , USA, Boston, Addison W esley Longman Inc., 1997
work page 1997
-
[5]
Bliev N.K., Singular integral operators with a Cauchy kernel in fractio nal spaces , Siberian Mathematical Journal, 47:1 (2006), 28–34
work page 2006
-
[6]
B. V. Boyarsky, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients , Mat. Sb. (N.S.), 43(85):4 (1957), 451–503
work page 1957
-
[7]
A.P. Calder´ on and A. Zygmund, On singular integrals , Amer. J. Math., 78:2 (1956), 289–309
work page 1956
-
[8]
P.R. Halmos, Measure theory, D. Van Nostrand, New York, 1950
work page 1950
Show all 18 references
-
[9]
H¨ ormander, Estimates for translation invariant operators in Lp spaces, Acta Math., 104 (1960), 93–140
L. H¨ ormander, Estimates for translation invariant operators in Lp spaces, Acta Math., 104 (1960), 93–140
1960
-
[10]
Gelfand, D
I.M. Gelfand, D. A. Raikov, and G.E. Shilov, Commutative Normed Rings , ”Fizmatgiz”, Moscow, 1960. [In Russian] MULTIDIMENSIONAL SINGULAR INTEGRALS AND INTEGRAL EQUATIO NS IN FRACTIONAL SPACES I 7
1960
-
[11]
I. I. Komyak, On the solvability of a class of two-dimensional singular in tegral equations , Dokl. Akad. Nauk SSSR, 250:6 (1980), 1307–1310
1980
-
[12]
Mikhlin, Multidimentional Singular Integrals and Integral Equatio ns, ”Fizmatgiz”, Moscow, 1962
S.G. Mikhlin, Multidimentional Singular Integrals and Integral Equatio ns, ”Fizmatgiz”, Moscow, 1962. [In Russian]
1962
-
[13]
Simonenko, A new general method of investigating linear operator equat ions of singular integral equation type
I.B. Simonenko, A new general method of investigating linear operator equat ions of singular integral equation type. I , Izv. Akad. Nauk SSSR Ser. Mat., 29:3 (1965), 567–586
1965
-
[14]
Simonenko, A new general method of investigating linear operator equat ions of singular integral equation type
I.B. Simonenko, A new general method of investigating linear operator equat ions of singular integral equation type. II , Izv. Akad. Nauk SSSR Ser. Mat., 29:3 (1965), 757–782
1965
-
[15]
Sukochev, K
F. Sukochev, K. Tulenov, and D. Zanin, The optimal range of the Calder´ on operator and its applications. J. Func. Anal. (2019), 1–47. doi.org/10.1016/j.jfa.2019. 05.012 (In Press)
2019 doi
-
[16]
Vekua, Generalized Analytic Function , Oxford, New York, Pergamon Press, 1962
I.N. Vekua, Generalized Analytic Function , Oxford, New York, Pergamon Press, 1962
1962
-
[17]
V. S. Vinogradov, On the solvability of a singular integral equation , Dokl. Akad. Nauk SSSR, 241:2 (1978), 272–274
1978
-
[18]
Tulenov, The optimal symmetric quasi-Banach range of the discrete Hi lbert transform
K.S. Tulenov, The optimal symmetric quasi-Banach range of the discrete Hi lbert transform. Arch. der Mathematik (2019), 1-12, DOI: 10.1007/s00013-01 9-01375-w. (In Press) E-mail address : bliyev.nazarbay@mail.ru Institute of Mathematics and Mathematical Modeling, 05001 0, Alma...
2019 doi
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.