Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

Lukić, Milivoje

doi:10.4171/jst/273

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…Translated into the language of random operators, the asymptotics of N (x) near E 0 corresponds to what is called Lifshitz tails [24]. For a random Schrödinger operator on Z d , Lifshitz predicted N (x) ∼ c 1 e −c 2 (x−E 0 ) −d/2 , which was later confirmed in some cases (for a related result, see [28]). From [21] it follows that for random operators corresponding to an operator of convolution given by 1 4 (a + a −1 + b + b −1 ), the function 1 − N (x) behaves as e −(x−1) −1 near E + .…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 87%

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

Grigorchuk¹,

Simanek²

2021

Trans. Amer. Math. Soc.

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We show that the lamplighter group L = Z/2Z Z has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family a + a −1 + b + b −1 − µc of convolution operators on L where µ is a real parameter.Our results show that the spectrum is a pure point spectrum for each value of µ, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when parameter µ passes the points 1 and −1. Namely if |µ| ≤ 1 the spectrum is the interval while when |µ| > 1 it is a union of the interval and a countable set of points accumulating to a point outside the interval.

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Section: Concluding Remarks and Open Problemsmentioning

confidence: 87%

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

Grigorchuk¹,

Simanek²

2021

Trans. Amer. Math. Soc.

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“…In fact, this set is not sufficiently well-understood, see e.g. [36]; 2] : The measure µ restricted to this set is always discrete.…”

Section: Annales De L'institut Fouriermentioning

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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Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

Cited by 2 publications

References 12 publications

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

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

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