On the spectrum of periodic perturbations of certain unbounded Jacobi operators

Sahbani, Jaouad

doi:10.7494/opmath.2016.36.6.807

Cited by 7 publications

(3 citation statements)

References 12 publications

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“…(I) if |tr X 0 (0)| < 2, then, under some regularity assumptions on Jacobi parameters, the measure µ is purely absolutely continuous on R with positive continuous density, see [23,25,56,58,59]; (II) if |tr X 0 (0)| = 2, then we have two subcases: TOME 0 (0), FASCICULE 0 (a) if X 0 (0) is diagonalizable, then, under some regularity assumptions on Jacobi parameters, there is a compact interval I ⊂ R such that the measure µ is purely absolutely continuous on R\I with positive continuous density and it is purely discrete in the interior of I, see [11,12,13,15,16,21,22,27,49,56,57,63]; (b) if X 0 (0) is not diagonalizable, then only certain examples have been studied, see [9,14,24,26,40,41,41,42,43,44,47,55,71].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Świderski,

Trojan

2024

Annales De l'Institut Fourier

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We study orthogonal polynomials with periodically modulated recurrence coefficients when 0 lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.Résumé. -Nous étudions des polynômes orthogonaux avec coefficients de récurrence périodiquement modulés, lorsque 0 est une véritable extrémité du spectre de la matrice de Jacobi périodique correspondante. En particulier, nous montrons que leur mesure d'orthogonalité est purement absolument continue sur une demidroite réelle et purement discrète sur son complémentaire. De plus, nous fournissons une formule constructive pour la densité, en termes de déterminants de Turán. Nous déterminons aussi le comportement asymptotique exact des polynômes orthogonaux. Enfin, nous étudions les limites d'échelle du noyau de Christoffel-Darboux. the moments Rx n dµ(x) are finite.

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Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Świderski,

Trojan

2024

Annales De l'Institut Fourier

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“…More specifically, we distinguish four cases: I. if | tr X 0 (0)| < 2, then, under some regularity assumptions on Jacobi parameters, the measure µ is purely absolutely continuous on R with positive continuous density, see [19,21,45,47,51]; II. if | tr X 0 (0)| = 2, then we have two subcases: a) if X 0 (0) is diagonalizable, then, under some regularity assumptions on Jacobi parameters, there is a compact interval I ⊂ R such that the measure µ is purely absolutely continuous on R \ I with positive continuous density and it is purely discrete in the interior of I, see [7][8][9]11,12,17,18,23,40,45,46,49]; b) if X 0 (0) is not diagonalizable, then only certain examples have been studied, see [5, 10, 20, 22, 33, 34, 34-37, 39, 44, 58]. Then usually the measure µ is purely absolutely continuous on a real half-line and discrete on its complement; III.…”

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Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Świderski¹,

Trojan²

2020

Preprint

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GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN A. We study orthogonal polynomials with periodically modulated entries when 0 lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.

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“…Additive perturbations. Finally, let us consider the additive periodic perturbations of sequences satisfying the assumptions of Theorem C. The case when α n ≡ 1, β n ≡ 0 and ã being some regular sequence (usually ãn = (n + 1) α for α ∈ (0, 1]) has been examined extensively, see [6,7,8,10,11,15,17,24].…”

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confidence: 99%

Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime

Świderski

2017

Preprint

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We present pretty detailed spectral analysis of Jacobi matrices with periodically modulated entries in the case when 0 lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that the studied operators are always self-adjoint irrespective of the modulated sequence. Moreover, if the growth of the modulated sequence is superlinear, then the spectrum of the considered operators is always discrete. Finally, we study regular perturbations of this class in the linear and the sublinear cases. We impose conditions assuring that the spectrum is absolute continuous on some regions of the real line. A constructive formula for the density in terms of Turán determinants is also provided.

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On the spectrum of periodic perturbations of certain unbounded Jacobi operators

Abstract: Abstract.It is known that a purely off-diagonal Jacobi operator with coefficients an = n α , α ∈ (0, 1], has a purely absolutely continuous spectrum filling the whole real axis. We show that a 2-periodic perturbation of these operators creates a non trivial gap in the spectrum.

Cited by 7 publications

References 12 publications

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime

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