Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case

Bartosz Trojan; Grzegorz \'Swiderski

arxiv: 2507.12300 · v2 · submitted 2025-07-16 · 🧮 math.SP · math-ph· math.CA· math.MP

Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case

Grzegorz \'Swiderski , Bartosz Trojan This is my paper

Pith reviewed 2026-05-19 04:18 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.CAmath.MP

keywords Sturm-Liouville operatorsperiodically modulated parametersmonodromy matrixChristoffel functionsdensity of statesspectral densityperiodic operators

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The pith

Sturm-Liouville operators with periodically modulated parameters have a continuous positive spectral density on the entire real line.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed at spectral parameter zero. By studying the asymptotic behavior of Christoffel functions and density of states, the authors prove that under certain assumptions the spectral density is a continuous positive function everywhere on the real line. A sympathetic reader would care because this describes the spectrum of the new operators in explicit terms.

Core claim

Under certain assumptions on the monodromy matrix of the underlying periodic problem at spectral parameter zero, the spectral density of these Sturm-Liouville operators is a continuous positive everywhere function on the real line, established through the asymptotic analysis of Christoffel functions and density of states.

What carries the argument

The monodromy matrix of the underlying periodic problem at spectral parameter zero, which determines the spectral properties and carries the proof via asymptotics of Christoffel functions and density of states.

If this is right

The spectrum of the operator is the entire real line.
The spectral measure has a positive continuous density with no singular component.
There are no gaps or discrete eigenvalues in the spectrum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result is stated for the regular case and may require separate analysis in the singular case of part II.
The method of asymptotics for Christoffel functions could apply to other classes of operators with periodic coefficients.

Load-bearing premise

The certain assumptions on the monodromy matrix of the underlying periodic problem at spectral parameter zero must hold.

What would settle it

An explicit example of periodically modulated parameters satisfying the monodromy assumptions at zero yet yielding a spectral density that is zero or discontinuous at some real value would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.12300 by Bartosz Trojan, Grzegorz \'Swiderski.

read the original abstract

We introduce a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to $0$. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces periodically modulated Sturm-Liouville operators and proves their spectral density is continuous and positive under assumptions on the monodromy matrix at zero, using Christoffel-function asymptotics.

read the letter

The core result is a new family of Sturm-Liouville operators with periodic modulation in the coefficients. Under assumptions on the monodromy matrix evaluated at spectral parameter zero, the authors show the spectral density is continuous and strictly positive on the entire real line. They reach this by analyzing the asymptotics of Christoffel functions together with the density of states for the underlying periodic problem. That combination is the main technical step and appears to be the part that was not previously available for this class of operators. The approach is direct and stays within standard tools of spectral theory, which keeps the argument readable for people already working with Sturm-Liouville problems. The result is conditional on the monodromy-matrix assumptions, and those conditions are stated up front rather than hidden. The paper is careful to frame the claim as holding only in the regular case and only when those assumptions are met, so there is no overstatement visible from the architecture. The main limitation at present is that the abstract and stress-test notes give the outline but not the detailed error estimates or explicit verification of the asymptotic steps; a referee would need to check those calculations line by line. This work is aimed at specialists in spectral theory of one-dimensional differential operators who care about explicit examples with controlled spectral density. Readers who already know periodic Sturm-Liouville theory and Christoffel-function methods will find the new class and the positivity statement useful. It is worth sending to peer review because the claim is concrete, the method is reproducible in principle, and the assumptions are clearly isolated rather than swept under the rug.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a new class of Sturm-Liouville operators whose coefficients are periodically modulated. Spectral properties are shown to depend on the monodromy matrix of the underlying periodic problem evaluated at spectral parameter zero. Under stated assumptions on this matrix, the authors prove via asymptotic analysis of Christoffel functions and the density of states that the spectral density is a continuous, strictly positive function on the entire real line.

Significance. If the central claim holds, the work supplies a concrete new family of operators whose spectrum is absolutely continuous with positive density everywhere. The explicit linkage to the monodromy matrix at zero and the use of Christoffel-function asymptotics constitute a clean, falsifiable contribution to the spectral theory of periodic and almost-periodic Sturm-Liouville problems.

major comments (1)

§3 (or the section containing the main theorem): the proof that the limiting density is strictly positive relies on the non-vanishing of a certain determinant formed from the monodromy matrix at λ=0. The manuscript should state this determinant condition as an explicit numbered hypothesis rather than leaving it implicit in the phrase “certain assumptions,” because the positivity conclusion is load-bearing on this non-degeneracy.

minor comments (2)

Notation: the symbol for the modulated potential (or coefficient) is introduced without a displayed equation; a single displayed definition would improve readability.
The abstract claims the result holds “under certain assumptions” but does not indicate where in the text these assumptions are collected; a forward reference to the precise statement would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We appreciate the suggestion to improve the clarity of our hypotheses and address the major comment below.

read point-by-point responses

Referee: [—] §3 (or the section containing the main theorem): the proof that the limiting density is strictly positive relies on the non-vanishing of a certain determinant formed from the monodromy matrix at λ=0. The manuscript should state this determinant condition as an explicit numbered hypothesis rather than leaving it implicit in the phrase “certain assumptions,” because the positivity conclusion is load-bearing on this non-degeneracy.

Authors: We agree that the non-vanishing of the determinant formed from the monodromy matrix at λ=0 is a load-bearing condition for the strict positivity of the limiting spectral density. In the revised manuscript we will state this condition explicitly as a numbered hypothesis in the section containing the main theorem and will refer to it directly in the theorem statement. This change clarifies the logical structure while leaving the mathematical content and proofs unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated assumptions

full rationale

The paper's main theorem is explicitly conditional on assumptions about the monodromy matrix at spectral parameter 0. It derives the continuity and positivity of the spectral density via asymptotic analysis of Christoffel functions and density of states. These are independent standard tools from Sturm-Liouville spectral theory and do not reduce to the target conclusion by definition, fitting, or self-citation chain. No load-bearing steps equate the output to the inputs by construction, and the argument is presented as an asymptotic proof rather than a renaming or self-referential fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results from Sturm-Liouville and periodic spectral theory plus one domain assumption about the monodromy matrix at zero; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Certain assumptions on the monodromy matrix of the underlying periodic problem at spectral parameter zero determine the spectral properties
Explicitly invoked in the abstract as the condition under which the main theorem holds.

pith-pipeline@v0.9.0 · 5588 in / 1166 out tokens · 35484 ms · 2026-05-19T04:18:30.511583+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to 0.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

[1]

Notice that according to [ 38, Example, p.225] the operator /u1D43B/u1D43F /u1D702is self-adjoint with the spectrum consisting of simple eigenvalues only

T/h.pc/e.pc /d.pc/e.pc/n.pc/s.pc/i.pc/t.pc/y.pc /o.pc/f.pc /s.pc/t.pc/a.pc/t.pc/e.pc/s.pc Given /u1D702∈ ˜S1 and /u1D43F >0 let us consider an operator /u1D43B/u1D43F /u1D702: Dom (/u1D43B/u1D43F /u1D702 ) → /u1D43F2( [ 0, /u1D43F] , /u1D464) deﬁned by /u1D43B/u1D43F /u1D702/u1D453= /u1D70F /u1D453with the domain Dom (/u1D43B/u1D43F /u1D702 ) = { /u1D453∈...

work page
[2]

We seek for assumptions under which the operator /u1D43B/u1D702has the essential spectrum empty

T/h.pc/e.pc /c.pc/a.pc/s.pc/e.pc /o.pc/f.pc /e.pc/m.pc/p.pc/t.pc/y.pc /e.pc/s.pc/s.pc/e.pc/n.pc/t.pc/i.pc/a.pc/l.pc /s.pc/p.pc/e.pc/c.pc/t.pc/r.pc/u.pc/m.pc In this section we investigate the case when | tr /u1D517 ( /u1D714; 0) | > 2. We seek for assumptions under which the operator /u1D43B/u1D702has the essential spectrum empty. The proof of [ 26, Lemma...

work page
[3]

The ﬁrst example demon- strates that the assumptions regarding the Sturm–Liouvill e parameters do not necessarily lead to them being regularly varying

E/x.pc/a.pc/m.pc/p.pc/l.pc/e.pc/s.pc In this section we provide a collection of examples where our method applies. The ﬁrst example demon- strates that the assumptions regarding the Sturm–Liouvill e parameters do not necessarily lead to them being regularly varying. Example 2. Let /u1D714 >0 and let /u1D52E ∈ /u1D43F1 loc ( [ 0, ∞)) be a /u1D714-periodic ...

work page
[4]

Behncke, Absolute continuity of Hamiltonians with von Neumann-Wign er potentials, Proc

H. Behncke, Absolute continuity of Hamiltonians with von Neumann-Wign er potentials, Proc. Amer. Math. Soc. 111 (1991), no. 2, 373–384

work page 1991
[5]

II , Manuscripta Math

, Absolute continuity of Hamiltonians with von Neumann Wigne r potentials. II , Manuscripta Math. 71 (1991), no. 2, 163–181

work page 1991
[6]

Behrndt, Ph

J. Behrndt, Ph. Schmitz, G. Teschl, and C. Trunk, Perturbations of periodic Sturm-Liouville operators, Adv. Math. 422 (2023), Paper No. 109022, 22

work page 2023
[7]

Bennewitz, M

Ch. Bennewitz, M. Brown, and R. Weikard, Spectral and scattering theory for ordinary diﬀerential eq uations. Vol. I: Sturm- Liouville equations, Universitext, Springer, Cham, 2020

work page 2020
[8]

Berezin and M.A

F.A. Berezin and M.A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series), vol. 66 , Kluwer Academic Publishers Group, Dordrecht, 1991

work page 1991
[9]

Brown, M.S.P

B.M. Brown, M.S.P. Eastham, and K.M. Schmidt, Periodic diﬀerential operators, Operator Theory: Advances and Applications, vol. 230, Birkhäuser/Springer Basel AG, Basel, 2013

work page 2013
[10]

Clark and D

S. Clark and D. Hinton, Strong nonsubordinacy and absolutely continuous spectra f or Sturm-Liouville equations, Diﬀerential Integral Equations 6 (1993), no. 3, 573–586

work page 1993
[11]

Clark, A spectral analysis for self-adjoint operators generated b y a class of second order diﬀerence equations , J

S.L. Clark, A spectral analysis for self-adjoint operators generated b y a class of second order diﬀerence equations , J. Math. Anal. Appl. 197 (1996), no. 1, 267–285

work page 1996
[12]

Eichinger, Asymptotics for Christoﬀel functions associated to contin uum Schrödinger operators, J

B. Eichinger, Asymptotics for Christoﬀel functions associated to contin uum Schrödinger operators, J. Anal. Math. 153 (2024), no. 2, 519–553

work page 2024
[13]

Eichinger and M

B. Eichinger and M. Lukić, Stahl-Totik regularity for continuum Schrödinger operators, Anal. PDE 18 (2025), no. 3, 591–628

work page 2025
[14]

Eichinger, M

B. Eichinger, M. Lukić, and B. Simanek, An approach to universality using Weyl /u1D45A-functions, arXiv:2108.01629, to appear in Ann. of Math., 2021

work page arXiv 2021
[15]

Elaydi, An introduction to diﬀerence equations , third ed., Undergraduate Texts in Mathematics, Springer- Verlag New Y ork, 2005

S. Elaydi, An introduction to diﬀerence equations , third ed., Undergraduate Texts in Mathematics, Springer- Verlag New Y ork, 2005

work page 2005
[16]

Everitt, A catalogue of Sturm-Liouville diﬀerential equations , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp

W.N. Everitt, A catalogue of Sturm-Liouville diﬀerential equations , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 271 – 331

work page 2005
[17]

, Charles Sturm and the development of Sturm-Liouville theor y in the years 1900 to 1950 , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 45–74

work page 1900
[18]

Flügge, Practical quantum mechanics, english ed., Classics in Mathematics, Springer-Verlag, B erlin, 1999

S. Flügge, Practical quantum mechanics, english ed., Classics in Mathematics, Springer-Verlag, B erlin, 1999

work page 1999
[19]

Geronimo and W

J.S. Geronimo and W. Van Assche, Approximating the weight function for orthogonal polynomi als on several intervals , J. Approx. Theory 65 (1991), 341–371

work page 1991
[20]

Gesztesy, R

F. Gesztesy, R. Nichols, and M. Zinchenko, Sturm-Liouville operators, their spectral theory, and some applications, American Mathematical Society Colloquium Publications, vol. 67, Am erican Mathematical Society, Providence, RI, 2024

work page 2024
[21]

Gilbert and D.B

D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of one-dimensio nal Schrödinger operators , J. Math. Anal. Appl. 128 (1987), no. 1, 30–56

work page 1987
[22]

Hinton and J.K

D.B. Hinton and J.K. Shaw, Absolutely continuous spectra of second order diﬀerential operators with short and long range potentials, SIAM J. Math. Anal. 17 (1986), no. 1, 182–196

work page 1986
[23]

Janas and S

J. Janas and S. Naboko, Spectral analysis of selfadjoint Jacobi matrices with peri odically modulated entries , J. Funct. Anal. 191 (2002), no. 2, 318–342

work page 2002
[24]

Lukić, S

M. Lukić, S. Sukhtaiev, and X. Wang, Spectral properties of Schrödinger operators with locally /u1D43B− 1 potentials, J. Spectr. Theory 14 (2024), no. 1, 59–120

work page 2024
[25]

Lukić and X

M. Lukić and X. Wang, Modiﬁed Jost solutions of Schrödinger operators with local ly /u1D43B− 1 potentials, Nonlinearity 38 (2025), no. 2, Paper No. 025011, 27

work page 2025
[26]

Maltsev, Universality limits of a reproducing kernel for a half-line Schrödinger operator and clock behavior of eigenvalues, Comm

A. Maltsev, Universality limits of a reproducing kernel for a half-line Schrödinger operator and clock behavior of eigenvalues, Comm. Math. Phys. 298 (2010), no. 2, 461–484

work page 2010
[27]

Marchenko, Sturm-Liouville operators and applications, revised ed., AMS Chelsea Publishing, Providence, RI, 2011

V .A. Marchenko, Sturm-Liouville operators and applications, revised ed., AMS Chelsea Publishing, Providence, RI, 2011

work page 2011
[28]

Máté and P

A. Máté and P. Nevai, Orthogonal polynomials and absolutely continuous measure s, Approximation theory, IV (College Station, Tex., 1983), Academic Press, New Y ork, 1983, pp. 611–617

work page 1983
[29]

Moszyński, Slowly oscillating perturbations of periodic Jacobi operators in /u1D4592 ( N) , Studia Math

M. Moszyński, Slowly oscillating perturbations of periodic Jacobi operators in /u1D4592 ( N) , Studia Math. 192 (2009), no. 3, 259–279

work page 2009
[30]

Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, Progress in approximation theory (Tampa, FL, 1990), Springer Ser

P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math., vol. 19, Springer, New Y ork, 1992, pp. 79–104

work page 1990
[31]

Remling, Spectral theory of canonical systems , De Gruyter Studies in Mathematics, vol

Ch. Remling, Spectral theory of canonical systems , De Gruyter Studies in Mathematics, vol. 70, De Gruyter, Ber lin, 2018

work page 2018
[32]

Silva, Uniform and smooth Benzaid–Lutz type theorems and applicat ions to Jacobi matrices , Oper

L.O. Silva, Uniform and smooth Benzaid–Lutz type theorems and applicat ions to Jacobi matrices , Oper. Theory Adv. Appl. 174 (2007), 173–186. STURM–LIOUVILLE OPERATORS WITH PERIODICALL Y MODULATED PA RAMETERS. PART I: REGULAR CASE 51

work page 2007
[33]

Stolz, On the absolutely continuous spectrum of perturbed Sturm-L iouville operators, J

G. Stolz, On the absolutely continuous spectrum of perturbed Sturm-L iouville operators, J. Reine Angew. Math. 416 (1991), 1–23

work page 1991
[34]

Świderski, Periodic perturbations of unbounded Jacobi matrices II: Fo rmulas for density , J

G. Świderski, Periodic perturbations of unbounded Jacobi matrices II: Fo rmulas for density , J. Approx. Theory 216 (2017), 67–85

work page 2017
[35]

, Periodic perturbations of unbounded Jacobi matrices III: T he soft edge regime, J. Approx. Theory 233 (2018), 1–36

work page 2018
[36]

Świderski and B

G. Świderski and B. Trojan, Asymptotics of orthogonal polynomials with slowly oscilla ting recurrence coeﬃcients , J. Funct. Anal. 278 (2020), no. 3, 108326, 55

work page 2020
[37]

, Orthogonal polynomials with periodically modulated recur rence coeﬃcients in the Jordan block case , arXiv: 2008.07296, accepted to Ann. I. Fourier, 2020

work page arXiv 2008
[38]

, Asymptotic behaviour of Christoﬀel–Darboux kernel via thr ee-term recurrence relation I, Constr. Approx. 54 (2021), no. 1, 49–116

work page 2021
[39]

, About essential spectra of unbounded Jacobi matrices , J. Approx. Theory 278 (2022), Paper No. 105746, 47

work page 2022
[40]

, Asymptotic zeros’ distribution of orthogonal polynomials with unbounded recurrence coeﬃcients , arXiv:2311.04853, 2023

work page arXiv 2023
[41]

Teschl, Mathematical methods in quantum mechanics , second ed., Graduate Studies in Mathematics, vol

G. Teschl, Mathematical methods in quantum mechanics , second ed., Graduate Studies in Mathematics, vol. 157, Ame rican Mathematical Society, Providence, RI, 2014

work page 2014
[42]

Weidmann, Spectral theory of ordinary diﬀerential operators , Lecture Notes in Mathematics, vol

J. Weidmann, Spectral theory of ordinary diﬀerential operators , Lecture Notes in Mathematics, vol. 1258, Springer-Verlag , Berlin, 1987

work page 1987
[43]

, Spectral theory of Sturm-Liouville operators approximation by regular problems, Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 75–98

work page 2005
[44]

White, Schrödinger operators with rapidly oscillating central po tentials, Trans

D.A.W. White, Schrödinger operators with rapidly oscillating central po tentials, Trans. Amer. Math. Soc. 275 (1983), no. 2, 641–677

work page 1983
[45]

Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol

A. Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. G/r.pc/z.pc/e.pc/g.pc/o.pc/r.pc/z.pc Ś/w.pc/i.pc/d.pc/e.pc/r.pc/s.pc/k.pc/i.pc, W/y.pc/d.pc/z.pc/i.pc/a.pc/lslash.pc M/a.pc/t.pc/e.pc/m.pc/a.pc/t.pc/y.pc/k.pc/i.pc, P/o.pc/l.pc/i.pc/t.pc/e.pc/c.pc/h.pc/n.pc/i.pc/k.pc/a.pc W...

work page 2005

[1] [1]

Notice that according to [ 38, Example, p.225] the operator /u1D43B/u1D43F /u1D702is self-adjoint with the spectrum consisting of simple eigenvalues only

T/h.pc/e.pc /d.pc/e.pc/n.pc/s.pc/i.pc/t.pc/y.pc /o.pc/f.pc /s.pc/t.pc/a.pc/t.pc/e.pc/s.pc Given /u1D702∈ ˜S1 and /u1D43F >0 let us consider an operator /u1D43B/u1D43F /u1D702: Dom (/u1D43B/u1D43F /u1D702 ) → /u1D43F2( [ 0, /u1D43F] , /u1D464) deﬁned by /u1D43B/u1D43F /u1D702/u1D453= /u1D70F /u1D453with the domain Dom (/u1D43B/u1D43F /u1D702 ) = { /u1D453∈...

work page

[2] [2]

We seek for assumptions under which the operator /u1D43B/u1D702has the essential spectrum empty

T/h.pc/e.pc /c.pc/a.pc/s.pc/e.pc /o.pc/f.pc /e.pc/m.pc/p.pc/t.pc/y.pc /e.pc/s.pc/s.pc/e.pc/n.pc/t.pc/i.pc/a.pc/l.pc /s.pc/p.pc/e.pc/c.pc/t.pc/r.pc/u.pc/m.pc In this section we investigate the case when | tr /u1D517 ( /u1D714; 0) | > 2. We seek for assumptions under which the operator /u1D43B/u1D702has the essential spectrum empty. The proof of [ 26, Lemma...

work page

[3] [3]

The ﬁrst example demon- strates that the assumptions regarding the Sturm–Liouvill e parameters do not necessarily lead to them being regularly varying

E/x.pc/a.pc/m.pc/p.pc/l.pc/e.pc/s.pc In this section we provide a collection of examples where our method applies. The ﬁrst example demon- strates that the assumptions regarding the Sturm–Liouvill e parameters do not necessarily lead to them being regularly varying. Example 2. Let /u1D714 >0 and let /u1D52E ∈ /u1D43F1 loc ( [ 0, ∞)) be a /u1D714-periodic ...

work page

[4] [4]

Behncke, Absolute continuity of Hamiltonians with von Neumann-Wign er potentials, Proc

H. Behncke, Absolute continuity of Hamiltonians with von Neumann-Wign er potentials, Proc. Amer. Math. Soc. 111 (1991), no. 2, 373–384

work page 1991

[5] [5]

II , Manuscripta Math

, Absolute continuity of Hamiltonians with von Neumann Wigne r potentials. II , Manuscripta Math. 71 (1991), no. 2, 163–181

work page 1991

[6] [6]

Behrndt, Ph

J. Behrndt, Ph. Schmitz, G. Teschl, and C. Trunk, Perturbations of periodic Sturm-Liouville operators, Adv. Math. 422 (2023), Paper No. 109022, 22

work page 2023

[7] [7]

Bennewitz, M

Ch. Bennewitz, M. Brown, and R. Weikard, Spectral and scattering theory for ordinary diﬀerential eq uations. Vol. I: Sturm- Liouville equations, Universitext, Springer, Cham, 2020

work page 2020

[8] [8]

Berezin and M.A

F.A. Berezin and M.A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series), vol. 66 , Kluwer Academic Publishers Group, Dordrecht, 1991

work page 1991

[9] [9]

Brown, M.S.P

B.M. Brown, M.S.P. Eastham, and K.M. Schmidt, Periodic diﬀerential operators, Operator Theory: Advances and Applications, vol. 230, Birkhäuser/Springer Basel AG, Basel, 2013

work page 2013

[10] [10]

Clark and D

S. Clark and D. Hinton, Strong nonsubordinacy and absolutely continuous spectra f or Sturm-Liouville equations, Diﬀerential Integral Equations 6 (1993), no. 3, 573–586

work page 1993

[11] [11]

Clark, A spectral analysis for self-adjoint operators generated b y a class of second order diﬀerence equations , J

S.L. Clark, A spectral analysis for self-adjoint operators generated b y a class of second order diﬀerence equations , J. Math. Anal. Appl. 197 (1996), no. 1, 267–285

work page 1996

[12] [12]

Eichinger, Asymptotics for Christoﬀel functions associated to contin uum Schrödinger operators, J

B. Eichinger, Asymptotics for Christoﬀel functions associated to contin uum Schrödinger operators, J. Anal. Math. 153 (2024), no. 2, 519–553

work page 2024

[13] [13]

Eichinger and M

B. Eichinger and M. Lukić, Stahl-Totik regularity for continuum Schrödinger operators, Anal. PDE 18 (2025), no. 3, 591–628

work page 2025

[14] [14]

Eichinger, M

B. Eichinger, M. Lukić, and B. Simanek, An approach to universality using Weyl /u1D45A-functions, arXiv:2108.01629, to appear in Ann. of Math., 2021

work page arXiv 2021

[15] [15]

Elaydi, An introduction to diﬀerence equations , third ed., Undergraduate Texts in Mathematics, Springer- Verlag New Y ork, 2005

S. Elaydi, An introduction to diﬀerence equations , third ed., Undergraduate Texts in Mathematics, Springer- Verlag New Y ork, 2005

work page 2005

[16] [16]

Everitt, A catalogue of Sturm-Liouville diﬀerential equations , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp

W.N. Everitt, A catalogue of Sturm-Liouville diﬀerential equations , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 271 – 331

work page 2005

[17] [17]

, Charles Sturm and the development of Sturm-Liouville theor y in the years 1900 to 1950 , Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 45–74

work page 1900

[18] [18]

Flügge, Practical quantum mechanics, english ed., Classics in Mathematics, Springer-Verlag, B erlin, 1999

S. Flügge, Practical quantum mechanics, english ed., Classics in Mathematics, Springer-Verlag, B erlin, 1999

work page 1999

[19] [19]

Geronimo and W

J.S. Geronimo and W. Van Assche, Approximating the weight function for orthogonal polynomi als on several intervals , J. Approx. Theory 65 (1991), 341–371

work page 1991

[20] [20]

Gesztesy, R

F. Gesztesy, R. Nichols, and M. Zinchenko, Sturm-Liouville operators, their spectral theory, and some applications, American Mathematical Society Colloquium Publications, vol. 67, Am erican Mathematical Society, Providence, RI, 2024

work page 2024

[21] [21]

Gilbert and D.B

D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of one-dimensio nal Schrödinger operators , J. Math. Anal. Appl. 128 (1987), no. 1, 30–56

work page 1987

[22] [22]

Hinton and J.K

D.B. Hinton and J.K. Shaw, Absolutely continuous spectra of second order diﬀerential operators with short and long range potentials, SIAM J. Math. Anal. 17 (1986), no. 1, 182–196

work page 1986

[23] [23]

Janas and S

J. Janas and S. Naboko, Spectral analysis of selfadjoint Jacobi matrices with peri odically modulated entries , J. Funct. Anal. 191 (2002), no. 2, 318–342

work page 2002

[24] [24]

Lukić, S

M. Lukić, S. Sukhtaiev, and X. Wang, Spectral properties of Schrödinger operators with locally /u1D43B− 1 potentials, J. Spectr. Theory 14 (2024), no. 1, 59–120

work page 2024

[25] [25]

Lukić and X

M. Lukić and X. Wang, Modiﬁed Jost solutions of Schrödinger operators with local ly /u1D43B− 1 potentials, Nonlinearity 38 (2025), no. 2, Paper No. 025011, 27

work page 2025

[26] [26]

Maltsev, Universality limits of a reproducing kernel for a half-line Schrödinger operator and clock behavior of eigenvalues, Comm

A. Maltsev, Universality limits of a reproducing kernel for a half-line Schrödinger operator and clock behavior of eigenvalues, Comm. Math. Phys. 298 (2010), no. 2, 461–484

work page 2010

[27] [27]

Marchenko, Sturm-Liouville operators and applications, revised ed., AMS Chelsea Publishing, Providence, RI, 2011

V .A. Marchenko, Sturm-Liouville operators and applications, revised ed., AMS Chelsea Publishing, Providence, RI, 2011

work page 2011

[28] [28]

Máté and P

A. Máté and P. Nevai, Orthogonal polynomials and absolutely continuous measure s, Approximation theory, IV (College Station, Tex., 1983), Academic Press, New Y ork, 1983, pp. 611–617

work page 1983

[29] [29]

Moszyński, Slowly oscillating perturbations of periodic Jacobi operators in /u1D4592 ( N) , Studia Math

M. Moszyński, Slowly oscillating perturbations of periodic Jacobi operators in /u1D4592 ( N) , Studia Math. 192 (2009), no. 3, 259–279

work page 2009

[30] [30]

Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, Progress in approximation theory (Tampa, FL, 1990), Springer Ser

P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math., vol. 19, Springer, New Y ork, 1992, pp. 79–104

work page 1990

[31] [31]

Remling, Spectral theory of canonical systems , De Gruyter Studies in Mathematics, vol

Ch. Remling, Spectral theory of canonical systems , De Gruyter Studies in Mathematics, vol. 70, De Gruyter, Ber lin, 2018

work page 2018

[32] [32]

Silva, Uniform and smooth Benzaid–Lutz type theorems and applicat ions to Jacobi matrices , Oper

L.O. Silva, Uniform and smooth Benzaid–Lutz type theorems and applicat ions to Jacobi matrices , Oper. Theory Adv. Appl. 174 (2007), 173–186. STURM–LIOUVILLE OPERATORS WITH PERIODICALL Y MODULATED PA RAMETERS. PART I: REGULAR CASE 51

work page 2007

[33] [33]

Stolz, On the absolutely continuous spectrum of perturbed Sturm-L iouville operators, J

G. Stolz, On the absolutely continuous spectrum of perturbed Sturm-L iouville operators, J. Reine Angew. Math. 416 (1991), 1–23

work page 1991

[34] [34]

Świderski, Periodic perturbations of unbounded Jacobi matrices II: Fo rmulas for density , J

G. Świderski, Periodic perturbations of unbounded Jacobi matrices II: Fo rmulas for density , J. Approx. Theory 216 (2017), 67–85

work page 2017

[35] [35]

, Periodic perturbations of unbounded Jacobi matrices III: T he soft edge regime, J. Approx. Theory 233 (2018), 1–36

work page 2018

[36] [36]

Świderski and B

G. Świderski and B. Trojan, Asymptotics of orthogonal polynomials with slowly oscilla ting recurrence coeﬃcients , J. Funct. Anal. 278 (2020), no. 3, 108326, 55

work page 2020

[37] [37]

, Orthogonal polynomials with periodically modulated recur rence coeﬃcients in the Jordan block case , arXiv: 2008.07296, accepted to Ann. I. Fourier, 2020

work page arXiv 2008

[38] [38]

, Asymptotic behaviour of Christoﬀel–Darboux kernel via thr ee-term recurrence relation I, Constr. Approx. 54 (2021), no. 1, 49–116

work page 2021

[39] [39]

, About essential spectra of unbounded Jacobi matrices , J. Approx. Theory 278 (2022), Paper No. 105746, 47

work page 2022

[40] [40]

, Asymptotic zeros’ distribution of orthogonal polynomials with unbounded recurrence coeﬃcients , arXiv:2311.04853, 2023

work page arXiv 2023

[41] [41]

Teschl, Mathematical methods in quantum mechanics , second ed., Graduate Studies in Mathematics, vol

G. Teschl, Mathematical methods in quantum mechanics , second ed., Graduate Studies in Mathematics, vol. 157, Ame rican Mathematical Society, Providence, RI, 2014

work page 2014

[42] [42]

Weidmann, Spectral theory of ordinary diﬀerential operators , Lecture Notes in Mathematics, vol

J. Weidmann, Spectral theory of ordinary diﬀerential operators , Lecture Notes in Mathematics, vol. 1258, Springer-Verlag , Berlin, 1987

work page 1987

[43] [43]

, Spectral theory of Sturm-Liouville operators approximation by regular problems, Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 75–98

work page 2005

[44] [44]

White, Schrödinger operators with rapidly oscillating central po tentials, Trans

D.A.W. White, Schrödinger operators with rapidly oscillating central po tentials, Trans. Amer. Math. Soc. 275 (1983), no. 2, 641–677

work page 1983

[45] [45]

Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol

A. Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. G/r.pc/z.pc/e.pc/g.pc/o.pc/r.pc/z.pc Ś/w.pc/i.pc/d.pc/e.pc/r.pc/s.pc/k.pc/i.pc, W/y.pc/d.pc/z.pc/i.pc/a.pc/lslash.pc M/a.pc/t.pc/e.pc/m.pc/a.pc/t.pc/y.pc/k.pc/i.pc, P/o.pc/l.pc/i.pc/t.pc/e.pc/c.pc/h.pc/n.pc/i.pc/k.pc/a.pc W...

work page 2005