Direct and inverse spectral continuity for Dirac operators

Pavel Gubkin; Roman Bessonov

REVIEW 2 major objections 5 minor 1 cited by

Direct and inverse spectral continuity for Dirac operators

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves the first explicit two-sided uniform stability estimate for the Dirac spectral correspondence with $L^2$ potentials.

desk verdict Solid Kronig-Penney result and a plausible main claim, but the proof of Theorem 1.3 has a gap in Lemma 3.10 that is more than a typo; the paper deserves serious refereeing but needs repair. read the letter →

arxiv 2505.00485 v1 pith:TQ6BRJMA submitted 2025-05-01 math.SP

Roman Bessonov , Pavel Gubkin This is my paper · ORCID

classification math.SP MSC 34L40

keywords DiracoperatorsspectralcontinuitySchur'salgorithmKronig–PenneymodelinverseproblemnonlinearFouriertransformWieneralgebradelta-interactions

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 establishes that the half-line Dirac spectral correspondence—mapping a square-integrable potential $q$ to its Schur function $f_q$—has explicit, uniform, two-sided continuity estimates. Concretely, for any two $L^2$ potentials $q,\tilde q$, the weighted $L^1$ distance between the potentials and the weighted Wiener-algebra distance between their spectral data are comparable up to universal constants, provided the weight parameter is large enough relative to the two norms. Previous knowledge of this Sylvester–Winebrenner correspondence was qualitative: it is a homeomorphism, but not uniformly continuous. The paper supplies the first uniform bound with explicit constants, and it is new even when one potential is zero. The proof works by solving the inverse spectral problem exactly for Dirac operators with $\delta$-interactions on a lattice via Schur's algorithm, then approximating arbitrary $L^2$ potentials by such discrete measures.

What carries the argument

The engine is Schur's algorithm. For a Schur function $F$ in the unit disk, the iteration $(zF_{k+1})(z)=(F_k(z)-F_k(0))/(1-\overline{F_k(0)}F_k(z))$ produces recurrence coefficients $F_k(0)$; for a periodic Schur function in the upper half-plane the same iteration reduces to a sequence of disk recurrence coefficients. The paper proves (Theorem 1.4) that for a Dirac potential supported on the lattice $\ell\mathbb Z_+$, the point mass at $\ell k$ is exactly $\kappa(F_k(0))$, where $\kappa(w)=\frac{w}{2|w|}\log\frac{1+|w|}{1-|w|}$. This exact dictionary turns the spectral correspondence for Kronig–Penney-type potentials into a two-sided estimate for Schur's algorithm in weighted Wiener algebras (Theorem 3.1). General $L^2$ potentials are then approximated by lattice measures with $\ell\to 0$, with convergence of potentials and spectral data controlled by lemmas proved in Sections 3.2–3.3.

What would settle it

Take two compactly supported piecewise-constant potentials $q,\tilde q$ with known matrix-exponential solutions, compute $f_q,f_{\tilde q}$ by solving the Dirac system, choose $A=12\max(\|q\|_{L^2}^2,\|\tilde q\|_{L^2}^2)$, and check whether the ratio $\|f_q-f_{\tilde q}\|_{W^1_A}/\|q-\tilde q\|_{L^1_{2A}}$ lies between $\sqrt{\pi/2}$ and $2\sqrt{2\pi}$. A single pair with ratio outside this interval, computed to numerical tolerance, would refute Theorem 1.3's constants; a search over random piecewise-constant pairs would also test whether the stated threshold on $A$ is needed.

Watch

Extended reading notes

Core claim

Theorem 1.3 states that for $q,\tilde q\in L^2(\mathbb R_+)$ with Schur functions $f_q,f_{\tilde q}$, one has $$ c_1\|q-\tilde q\|_{$L^{1}$_{2A}(\mathbb R_+)} \le \|f_q-f_{\tilde q}\|_{$W^{1}$_A(\mathbb R_+)} \le c_2\|q-\tilde q\|_{$L^{1}$_{2A}(\mathbb R_+)}, $$ with $c_1=\sqrt{\pi/2}$, $c_2=2\sqrt{2\pi}$, and $A\ge 12\max(\|q\|^2_{L^2},\|\tilde q\|^2_{L^2})$. Here the weighted $L^1$ norm is $\int_{\mathbb R_+}|q(\xi)|e^{-A\xi}d\xi$ and the spectral norm is the $L^1$ norm of the Fourier transform of $f_q-f_{\tilde q}$ with weight $e^{-A\xi}$. The paper also proves that on bounded sets this homeomorphism is not uniformly continuous in either direction (Theorem 1.2), so the two-sided weighted estimate is the sharp type of uniform control that holds.

Load-bearing premise

The argument relies on the previously established theorem that the map $q\mapsto f_q$ is a bijection from $L^2(\mathbb R_+)$ onto the class of Schur functions with integrable $\log(1-|f|^2)$, together with the sum rule $\|q\|_{L^2}^2=\frac1\pi\int -\log(1-|f_q|^2)$; if either fails, the limiting step that transfers the lattice estimates to general $L^2$ potentials would not go through.

Editorial extensions

If this is right

The direct problem is stable with rate: if one potential changes slightly in weighted $L^1$, the spectral Schur function changes proportionally in weighted Wiener norm, independent of the pair.
The inverse problem inherits the same bound: spectral data close in the weighted Wiener metric force potentials close in the weighted $L^1$ metric.
Approximation by lattice potentials converges quantitatively: the discrete measures $q_\ell$ in (3.17) give spectral data whose loss is controlled as $\ell\to0$, so the exactly solvable model is a usable numerical and discretization scheme.
The result supplies explicit constants for the nonlinear Fourier transform associated with the massless Dirac operator, so it bears on quantitative stability of inverse scattering for nonlinear Schr\"odinger-type problems.
Uniform continuity in the original unweighted metrics fails (Theorem 1.2); the weighted norms in Theorem 1.3 identify the correct quantitative framework.

Reading between the lines

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

My inference: the same Schur-algorithm route could yield explicit stability estimates for other exactly solvable one-dimensional models, such as Schr\"odinger operators with point interactions, by following the same two-step approximation; the paper does not claim this.
My inference: the constants $\sqrt{\pi/2}$ and $2\sqrt{2\pi}$ are unlikely to be sharp; the proof passes through an $\varepsilon$-approximation and a general Schur-theorem estimate with lossy factors, so I would expect the optimal ratio to be smaller and possibly expressible in terms of the $\kappa$ map alone.
My inference: a natural testable extension is to replace the exponential weights by polynomial weights and check whether a two-sided estimate of the same form holds with constants depending on the polynomial degree; the structure of the proof does not immediately preclude or imply this.

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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

Main take: this paper has two genuinely new pieces—an explicit two-sided stability estimate for the half-line Dirac spectral correspondence in the L2 class, and an exact description of the massless Dirac Kronig-Penney model via Schur's algorithm. The second is solid and elegant. The first is plausible and important, but the printed proof has a gap that is not just a typo.

The good parts: Theorem 1.4 and its corollaries are well proved. The reduction of measure-valued Dirac operators on a lattice to Schur's algorithm works, and the applications (Rakhmanov, Baxter, Szegő-Golinskii-Ibragimov analogues) are nice. Section 4 on non-uniform continuity of the Sylvester-Winebrenner map contains explicit constructions and reads correctly. The historical remarks are accurate and useful.

The problem is in the proof of Theorem 1.3, specifically Lemma 3.10. Two issues. First, the scaling: with r=e^{-Aℓ}, the conclusion has e^{-Aξ/2}, not e^{-Aξ}, so it does not match (1.9). That can be fixed by taking r=e^{-2Aℓ}. Second and more serious, the proof of Lemma 3.10 claims an ℓ-uniform Lipschitz bound for h=√(2π)(ĝ_qℓ−ĝ_q̃ℓ). But h is the Fourier transform of a function supported on an interval of length π/ℓ, so its derivative is bounded by (π/ℓ) times the L1 norm, not by an absolute constant times that L1 norm. The factor 1/ℓ changes the Riemann-sum error from O(ℓ^{1/2}) to O(1/√ℓ), which diverges. As printed, the passage from discrete Fourier coefficients to the continuous Fourier norm is unjustified. This is the load-bearing step for Theorem 1.3. I do not see a way to save it with a small edit; it needs a different argument, possibly using the special structure of Schur functions or a different approximation scheme.

The theorem may well be true—the strategy is natural and the estimate has the right scaling—but the current manuscript does not prove it. The weakest point is not the reliance on Denisov's bijection and sum rule; those are external and legitimate. It is the internal limiting argument in Lemma 3.10.

Who is this for? Spectral theorists working on Dirac operators, inverse scattering, and OPUC. The Kronig-Penney part alone is worth a serious referee. I would send it to referees, with instructions that the proof of Theorem 1.3 needs significant repair before publication.

Referee Report

2 major / 5 minor

Summary. The paper studies the spectral correspondence q ↦ f_q between L2 potentials on the half-line and Schur functions in S2(C+) for Dirac operators. The main new result, Theorem 1.3, claims an explicit two-sided uniform estimate with constants c1 = √(π/2), c2 = 2√(2π) between the weighted L1 norm of the potential difference and the Wiener-type W1_A norm of the difference of Schur functions, for A ≥ 12 max(||q||², ||q̃||²). The proof strategy is to approximate general L2 potentials by discrete measures supported on ℓZ+, to solve the direct and inverse spectral problems for the resulting Kronig-Penney models exactly via Schur's algorithm, to establish quantitative continuity estimates for Schur iterates (Theorem 3.1), and then to pass to the limit ℓ → 0 using approximation lemmas. The paper also proves Theorem 1.2 on the failure of uniform continuity in both directions of the Sylvester-Winebrenner homeomorphism, and derives several corollaries for the relativistic Kronig-Penney model. The central announced result therefore depends on the discrete approximation machinery in Section 3 and on the limiting passage in Lemma 3.10 and Section 3.4.

Significance. If Theorem 1.3 is correct, it would be the first explicit two-sided uniform quantitative stability estimate for the Sylvester-Winebrenner spectral correspondence for half-line Dirac operators with general L2 potentials. The paper combines a substantial exact-solvable model (Dirac operators with δ-interactions on a half-lattice) with Schur's algorithm and gives explicit constants. It also proves a non-uniformity theorem in both directions, which is a valuable cautionary complement to the positive estimate. The authors provide detailed proofs of many auxiliary statements, and the use of the external bijection theorem and sum rule from Denisov's work is legitimate rather than circular. However, the proof of the main theorem as printed contains two distinct technical gaps: a weight mismatch in the limiting step and a false uniform Lipschitz estimate in Lemma 3.10. These gaps occur precisely at the load-bearing passage from the discrete ℓ-dependent estimates to the continuous Fourier norms in (1.9).

major comments (2)

[Section 3.4, Eq. (3.39)-(3.43)] The choice r = e^{-Aℓ} is incompatible with Lemma 3.10. Lemma 3.10 asserts a limit with weight e^{-Aξ/2} on the right-hand side, whereas Theorem 1.3 and (3.39) require the weight e^{-Aξ}. With r = e^{-Aℓ}, equation (3.36) shows that the sum converges to (1/√(2π))∫_{R+} e^{-Aξ/2}|f̂_q - f̂_q̃| dξ, not to the W1_A norm. Consequently the limiting inequality obtained from (3.43) bounds W1_{A/2}, not W1_A, and (1.9) is not established as printed. Replacing r by e^{-2Aℓ} would align the weights: the Riemann sum then tends to (1/√(2π))∫ e^{-Aξ}|f̂_q - f̂_q̃| dξ, and combining (3.43) with Lemma 3.7 with weight 2A yields exactly the constants c1 = √(π/2) and c2 = 2√(2π) after ε → 0. This repair is local, but as it stands the printed proof does not prove the stated theorem.
[Lemma 3.10, proof of (3.37)-(3.38)] The claimed uniform Lipschitz bound |h(ξ) - h(2ℓk)| ≤ C|ξ - 2ℓk| · ||g_{qℓ} - g_{q̃ℓ}||_{L1} is false. Since h is the Fourier transform of g_{qℓ} - g_{q̃ℓ}, one has |h'(ξ)| ≤ (1/√(2π)) ∫ |x| |g_{qℓ}(x) - g_{q̃ℓ}(x)| dx ≤ (π/(2ℓ)) ||g_{qℓ} - g_{q̃ℓ}||_{L1}, because the supports of g_{qℓ} and g_{q̃ℓ} are contained in an interval of length π/ℓ. Using this correct bound, each term in the sum in (3.38) is of size O(ℓ^{1/2} e^{-Aℓk}) rather than O(ℓ^{3/2} e^{-Aℓk}), and the total error becomes O(ℓ^{-1/2}) instead of O(ℓ^{1/2}). The convergence claimed in (3.37) is therefore not justified by the argument given. This is a load-bearing step in the discrete-to-continuous limit, so the proof of Lemma 3.10 needs a substantially different argument or an additional structural property of the functions g_{qℓ}.

minor comments (5)

[Section 3.3, Eq. (3.36)] In the displayed line preceding (3.37), the expression "2r|ĝ_{qℓ}(2ℓk) - ...|" should read "2ℓ|...|"; the variable r is not defined at that point and the intended Riemann sum step is 2ℓ.
[Lemma 3.8, proof] In the sentence "where the factor 2π appears in the last inequality", the reference should be to the last equality; the normalization of m_T accounts for a factor 2π, not an inequality.
[Section 3.4, opening paragraph] The theorem statement and (3.39) assume A ≥ 12 max(||q||², ||q̃||²), which forces A > 0, but (3.39) is introduced with "A ∈ R". This should be corrected to A > 0 for clarity.
[Throughout] There are several typographical issues in the displayed formulas, e.g., stray OCR artifacts such as "/greaterorequalslant" in place of ≥ and "OPERA TORS" in the title header. These do not affect the mathematics but should be cleaned up in the final version.
[Section 4.4, Proposition 4.12] The proof of Proposition 4.14 relies on Example 6.1 from [9] for the existence of non-unique reflection coefficients; this is an acceptable external input, but the dependence should be stated explicitly in the main text so that the reader understands that Theorem 1.2 inherits the non-injectivity result from [9].

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.3 is derived from the Schur algorithm, the exact lattice solution, and approximation lemmas; self-citations affect only the secondary non-uniformity statement.

full rationale

The main estimate (1.9) is not an input or a renamed version of an input. The proof in Section 3 approximates q and q-tilde by lattice measures q_l and q-tilde_l (Lemmas 3.5-3.7), connects lattice potentials to Schur recurrence coefficients via Theorem 1.4 (derived in Section 2 from the Schur algorithm and standard OPUC facts), proves the two-sided Schur-algorithm estimate Theorem 3.1 from scratch, and passes to the limit with Lemmas 3.9 and 3.10. The external inputs from [15] (bijection and sum rule) and [51] (Schur/OPUC parametrization) are stated assumptions external to the target inequality; they are not fitted to the conclusion. No parameter is tuned to the data being predicted; the constants c1 and c2 are absolute. Self-citations [9] and [7] occur in Section 4.4, where Theorem 1.2's inverse non-uniformity is obtained by importing the non-injectivity example from the first author's prior work with Denisov. That is load-bearing for the secondary negative result but does not feed into Theorem 1.3, and the cited constructions are prior published external evidence rather than circular reuse of the present theorem. Possible weight mismatches in the limit passage (e.g., e^{-A*xi/2} versus e^{-A*xi}) are correctness concerns, not circularity, and are not counted here. Overall, no step of the derivation is equivalent by construction to its input.

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

The paper contains no fitted parameters; all constants are explicit and derived. It relies on standard spectral theory and on the earlier bijection/sum-rule theorem from [15]. The new map kappa is a defined function, not a postulated entity.

assumptions (4)

standard math Krein-de Branges theorem: the map from singular canonical Hamiltonians H to Weyl functions m_H is a homeomorphism of compact metric spaces.
Used in Lemma 3.9 and Section 4 to transfer convergence of Hamiltonians to convergence of Schur functions; stated as Theorem 2.1 in the paper.
domain assumption Bijection theorem and sum rule for Dirac operators with L2 potentials from Denisov [15]: q maps to f_q is onto S2(C+) and ||q||^2 = (1/pi) integral of -log(1-|f_q|^2).
Not reproved in the paper; used in Lemma 3.8 and Section 3.4 to identify limits needed for Theorem 1.3.
standard math Geronimus' theorem equating recurrence coefficients of a Schur function and of the associated measure's orthogonal polynomials.
Used in proofs of Corollaries 1.5-1.7 to translate Rakhmanov, Baxter, and Szegő-Golinskii-Ibragimov theorems.
domain assumption Theory of Dirac operators with measure potentials due to Zeng [61], including unitary equivalence to canonical systems.
Used in Section 2.2 to define operators and justify Proposition 2.5.

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Pith. "Pith review of Direct and inverse spectral continuity for Dirac operators." pith.science (2026). https://pith.science/paper/TQ6BRJMA

@misc{pith2026250500485,
  author       = {Pith},
  title        = {Pith review of: Direct and inverse spectral continuity for Dirac operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQ6BRJMA}},
  note         = {Machine review of arXiv:2505.00485}
}

abstract

The half-line Dirac operators with $L^2$-potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general $L^2$-case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with $\delta$-interactions on a half-lattice in terms of the Schur's algorithm for analytic functions.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gelfand-Levitan condition for Dirac operators
math.SP 2025-08 unverdicted novelty 5.0 of 10

A generalization of Dirac operators is introduced whose solutions are only of bounded variation, with a spectral theory and a Gelfand-Levitan recovery condition.

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Works this paper leans on

62 extracted references · 59 canonical work pages · cited by 1 Pith paper

[1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden . Solvable models in quantum mechanics . Texts and Monographs in Physics. Springer-Verlag, New York, 1988. 4

work page 1988
[2]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden . Solvable models in quantum mechanics . AMS Chelsea Publishing, Providence, RI, second edition, 2005. With an a ppendix by P. Exner. 4

work page 2005
[3]

ﬁnite-section

G. Baxter. A norm inequality for a “ﬁnite-section” Wiene r-Hopf equation. Illinois J. Math. , 7(1):97–103, 1963. 16, 19

work page 1963
[4]

M. I. Belishev and V. S. Mikhaylov. Uniﬁed approach to cla ssical equations of inverse problem theory. J. Inverse Ill-Posed Probl. , 20(4):461–488, 2012. 4

work page 2012
[5]

Bessonov and S

R. Bessonov and S. Denisov. A spectral Szegő theorem on th e real line. Adv. Math. , 359:106851, 41, 2020. 2, 8

work page 2020
[6]

Bessonov and S

R. Bessonov and S. Denisov. De Branges canonical systems with ﬁnite logarithmic integral. Anal. PDE, 14(5):1509– 1556, 2021. 2

work page 2021
[7]

Bessonov and P

R. Bessonov and P. Gubkin. Stability of Schur’s iterates and fast solution of the discrete integrable NLS. J. Spectr. Theory, 2024. Published online ﬁrst. 33

work page 2024
[8]

R. V. Bessonov. Szegő condition and scattering for one-d imensional Dirac operators. Constr. Approx. , 51(2):273– 302, 2020. 9, 31

work page 2020

Show all 62 references

[9]

R. V. Bessonov and S. A. Denisov. Sobolev norms of L2-solutions to the nonlinear Schrödinger equation. Paciﬁc J. Math. , 331(2):217–258, 2024. 3, 33, 34

2024
[10]

Carlone, M

R. Carlone, M. Malamud, and A. Posilicano. On the spectr al theory of Gesztesy–Seba realizations of 1-D Dirac operators with point interactions on a discrete set. Journal of Diﬀerential Equations , 254(9):3835–3902, 2013. 5

2013
[11]

Chelkak, P

D. Chelkak, P. Kargaev, and E. Korotyaev. Inverse probl em for harmonic oscillator perturbed by potential, char- acterization. Comm. Math. Phys. , 249(1):133–196, 2004. 1

2004
[12]

Damanik, B

D. Damanik, B. Eichinger, and P. Yuditskii. Szegő’s the orem for canonical systems: the Arov gauge and a sum rule. J. Spectr. Theory , 11(3):1255–1277, 2021. 2

2021
[13]

Damanik, R

D. Damanik, R. Killip, and B. Simon. Perturbations of or thogonal polynomials with periodic recursion coeﬃcients. Ann. of Math. (2) , 171(3):1931–2010, 2010. 2

1931
[14]

de Branges

L. de Branges. Hilbert spaces of entire functions . Prentice-Hall, Inc., Englewood Cliﬀs, NJ, 1968. 7

1968
[15]

S. A. Denisov. Continuous analogs of polynomials ortho gonal on the unit circle and Kre ˘ ın systems. IMRS Int. Math. Res. Surv. , pages 1–148, 2006, Art. ID 54517. 2, 4, 5, 6, 12, 28, 30, 34

2006
[16]

Dym and H

H. Dym and H. P. McKean. Gaussian processes, function theory, and the inverse spect ral problem, volume Vol. 31 of Probability and Mathematical Statistics . Academic Press, New York-London, 1976. 7

1976
[17]

Eckhardt, A

J. Eckhardt, A. Kostenko, and G. Teschl. Spectral asymp totics for canonical systems. J. Reine Angew. Math. , 736:285–315, 2018. 7

2018
[18]

Eichinger, M

B. Eichinger, M. Lukic, and B. Simanek. An approach to un iversality using weyl m-functions. arXiv preprint arXiv:2108.01629, 2021. 7

2021 arXiv
[19]

Faddeev and L

L. Faddeev and L. Takhtajan. Hamiltonian methods in the theory of solitons . Classics in Mathematics. Springer, Berlin, english edition, 2007. Translated from the 1986 Rus sian original by Alexey G. Reyman. 5

2007
[20]

J. B. Garnett. Bounded analytic functions , volume 96 of Pure and Applied Mathematics . Academic Press Inc., New York, 1981. 2, 15, 16, 33, 34

1981
[21]

J. S. Geronimo and A. Martínez-Finkelshtein. On extens ions of a theorem of baxter. J. Approx. Theory , 139(1- 2):214–222, 2006. 19

2006
[22]

Geronimus

J. Geronimus. On polynomials orthogonal on the circle, on trigonometric moment-problem and on allied Carathéodory and Schur functions. Rec. Math. [Mat. Sbornik] N.S. , 15/57:99–130, 1944. 15

1944
[23]

Gesztesy and P

F. Gesztesy and P. Šeba. New analytically solvable mode ls of relativistic point interactions. 13:345–358, 1987. 5

1987
[24]

B. L. Golinskii and I. A. Ibragimov. A limit theorm of G. S zegő. Izv. Akad. Nauk SSSR Ser. Mat. , 35:408–427,
[25]

Golinskii

L. Golinskii. On Schur functions and Szegő orthogonal p olynomials. In Topics in interpolation theory (Leipzig, 1994), volume 95 of Oper. Theory Adv. Appl. , pages 195–203. Birkhäuser, Basel, 1997. 16

1994
[26]

Hassi, H

S. Hassi, H. De Snoo, and H. Winkler. Boundary-value pro blems for two-dimensional canonical systems. Integral Equations Operator Theory , 36(4):445–479, 2000. 6

2000
[27]

Real and abstract analysis

Edwin Hewitt and Karl Stromberg. Real and abstract analysis. A modern treatment of the theory of functions of a real variable . Springer-Verlag, New York-Berlin, 1969. 11 36 R. V. BESSONOV AND P. V. GUBKIN

1969
[28]

Horváth and M

M. Horváth and M. Kiss. Stability of direct and inverse e igenvalue problems for Schrödinger operators on ﬁnite intervals. Int. Math. Res. Not. IMRN , (11):2022–2063, 2010. 1

2022
[29]

R. O. Hryniv. Analyticity and uniform stability in the i nverse singular Sturm-Liouville spectral problem. Inverse Problems, 27(6):065011, 25, 2011. 1

2011
[30]

R. O. Hryniv. Analyticity and uniform stability in the i nverse spectral problem for Dirac operators. J. Math. Phys. , 52(6):063513, 17, 2011. 1

2011
[31]

R. J. Hughes. Relativistic Kronig-Penney-type Hamilt onians. Integral Equations Operator Theory , 31(4):436–448,
[32]

I. S. Kac and M. G. Krein. On the spectral functions of the string. Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems , 1968. English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 19–102. 7

1974
[33]

Killip and B

R. Killip and B. Simon. Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math. (2) , 158(1):253–321, 2003. 2

2003
[34]

Killip and B

R. Killip and B. Simon. Sum rules and spectral measures o f Schrödinger operators with L2 potentials. Ann. of Math. (2) , 170(2):739–782, 2009. 2

2009
[35]

E. L. Korotyaev and D. S. Chelkak. The inverse Sturm-Lio uville problem with mixed boundary conditions. Algebra i Analiz , 21(5):114–137, 2009. 1

2009
[36]

Kostenko and M

A. Kostenko and M. Malamud. 1-D Schrödinger operators w ith local point interactions: a review. In Spectral analysis, diﬀerential equations and mathematical physics : a festschrift in honor of Fritz Gesztesy’s 60th birthday , volume 87 of Proc. Sympos. Pure Math. , pages 235–2...

2013
[37]

R. De L. Kronig, W. G. Penney, and R. H. Fowler. Quantum me chanics of electrons in crystal lattices. Proceedings of the Royal Society of London. Series A, Containing Papers of a M athematical and Physical Character, 130(814):499– 513, 1931. 4, 5

1931
[38]

B. M. Levitan. Sturm-Liouville and Dirac operators, 19 91. 2
[39]

B. M. Levitan and I. S. Sargsjan. Sturm-Liouville and Dirac operators , volume 59 of Mathematics and its Ap- plications (Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1991. Trans lated from the Russian. 5

1991
[40]

Lotoreichik and S

V. Lotoreichik and S. Simonov. Spectral analysis of the half-line Kronig-Penney model with Wigner–Von Neumann perturbations. Rep. Math. Phys. , 74(1):45–72, 2014. 5

2014
[41]

Makarov and A

N. Makarov and A. Poltoratski. Etudes for the inverse sp ectral problem. J. Lond. Math. Soc. (2) , 108(3):916–977,
[42]

Marchenko

V. Marchenko. The generalized shift, transformation o perators, and inverse problems. In Mathematical events of the twentieth century , pages 145–162. Springer, Berlin, 2006. 4

2006
[43]

Nevai and V

P. Nevai and V. Totik. Orthogonal polynomials and their zeros. Acta Sci. Math.(Szeged) , 53(1-2):99–104, 1989. 19

1989
[44]

J. Persson. Fundamental theorems for linear measure di ﬀerential equations. Mathematica scandinavica, pages 19–43,
[45]

Poltoratski

A. Poltoratski. Pointwise convergence of the non-line ar Fourier transform. Ann. of Math. (2) , 199(2):741–793, 2024. 6

2024
[46]

E. A. Rakhmanov. The asymptotic behavior of the ratio of orthogonal polynomials. Mat. Sb. (N.S.) , 103(145)(2):237–252, 319, 1977. 16

1977
[47]

Ch. Remling. Spectral theory of canonical systems , volume 70 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, 2018. 3, 6, 7, 10, 12

2018
[48]

R. Romanov. Canonical systems and de Branges spaces. preprint arXiv:1408.6022, 2014. 6, 7, 9, 10, 12

2014 arXiv
[49]

A. M. Savchuk and A. A. Shkalikov. Inverse problems for t he Sturm-Liouville operator with potentials in Sobolev spaces: uniform stability. Funktsional. Anal. i Prilozhen. , 44(4):34–53, 2010. 1

2010
[50]

J. Schur. Über potenzreihen, die im innern des einheits kreises beschränkt sind. J. Reine Angew. Math. , 147:205–232,
[51]

B. Simon. Orthogonal polynomials on the unit circle. Part 1 , volume 54, Part 1 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2005. Cla ssical theory. 4, 5, 15, 16, 18, 19, 22

2005
[52]

B. Simon. Orthogonal polynomials with exponentially d ecaying recursion coeﬃcients. CRM Proceedings and lecture notes, 42, 2007. 19

2007
[53]

B. Simon. Szegő’s theorem and its descendants . M. B. Porter Lectures. Princeton University Press, Prince ton, NJ,
[54]

Sylvester and D

J. Sylvester and D. P. Winebrenner. Linear and nonlinea r inverse scattering. SIAM J. Appl. Math. , 59(2):669–699,
[55]

G. Szegő. Orthogonal polynomials, volume Vol. XXIII of Amer. Math. Soc. Colloquium Publications . Amer. Math. Soc., Providence, RI, fourth edition, 1975. 5

1975
[56]

Tao and Ch

T. Tao and Ch. Thiele. Nonlinear Fourier analysis. Prep rint ArXiv 2003, 2012, https://arxiv.org/abs/1201.5129, 2003, 2012. 3

2003 arXiv
[57]

B. Thaller. The Dirac equation . Texts and Monographs in Physics. Springer-Verlag, Berlin , 1992. 5

1992
[58]

Volberg and P

A. Volberg and P. Yuditskii. On the inverse scattering p roblem for Jacobi matrices with the spectrum on an interval, a ﬁnite system of intervals or a Cantor set of positive length . Comm. Math. Phys. , 226(3):567–605, 2002. 3, 34

2002
[59]

Williams

D. Williams. Probability with martingales . Cambridge Mathematical Textbooks. Cambridge University Press, Cam- bridge, 1991. 34

1991
[60]

Yuditskii

P. Yuditskii. Killip-Simon problem and Jacobi ﬂow on GM P matrices. Adv. Math. , 323:811–865, 2018. 2

2018
[61]

J. Zeng. Spectral theory of Dirac operators with measur es. PhD thesis, Unirversity of Oklahoma, Norman , 2023. 8, 11 SPECTRAL CONTINUITY 37 R.V.Bessonov: roman.bessonov@fmf.uni-lj.si University of Ljubljana Jadranska 19, 1000 Ljubljana, SLOVENIA St. Petersburg Department of S...

2023
[2011]

Spectral theory for L2 perturbations of orthogonal polynomials. 2

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden . Solvable models in quantum mechanics . Texts and Monographs in Physics. Springer-Verlag, New York, 1988. 4

work page 1988

[2] [2]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden . Solvable models in quantum mechanics . AMS Chelsea Publishing, Providence, RI, second edition, 2005. With an a ppendix by P. Exner. 4

work page 2005

[3] [3]

ﬁnite-section

G. Baxter. A norm inequality for a “ﬁnite-section” Wiene r-Hopf equation. Illinois J. Math. , 7(1):97–103, 1963. 16, 19

work page 1963

[4] [4]

M. I. Belishev and V. S. Mikhaylov. Uniﬁed approach to cla ssical equations of inverse problem theory. J. Inverse Ill-Posed Probl. , 20(4):461–488, 2012. 4

work page 2012

[5] [5]

Bessonov and S

R. Bessonov and S. Denisov. A spectral Szegő theorem on th e real line. Adv. Math. , 359:106851, 41, 2020. 2, 8

work page 2020

[6] [6]

Bessonov and S

R. Bessonov and S. Denisov. De Branges canonical systems with ﬁnite logarithmic integral. Anal. PDE, 14(5):1509– 1556, 2021. 2

work page 2021

[7] [7]

Bessonov and P

R. Bessonov and P. Gubkin. Stability of Schur’s iterates and fast solution of the discrete integrable NLS. J. Spectr. Theory, 2024. Published online ﬁrst. 33

work page 2024

[8] [8]

R. V. Bessonov. Szegő condition and scattering for one-d imensional Dirac operators. Constr. Approx. , 51(2):273– 302, 2020. 9, 31

work page 2020