Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Cantero, Maŕıa José; Ituarte, Leandro del Moral; Velázquez, L.

doi:10.1016/s0024-3795(02)00457-3

Cited by 258 publications

(394 citation statements)

References 49 publications

(50 reference statements)

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“…It is known that any unitary matrix can be presented in a form with two diagonals and 2 antidiagonals [28]; this is related to CMV theory [29]. Assume that e −iT J is in that form in the register basis ℓ⟩, where ℓ = 0, 1, .…”

Section: Resultsmentioning

confidence: 99%

Quantum spin chains with fractional revival

2016

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Abstract. A systematic study of fractional revival at two sites in XX quantum spin chains is presented and analytic models with this phenomenon are exhibited. The generic models have two essential parameters and a revival time that does not depend on the length of the chain. They are obtained by combining two basic ways of realizing fractional revival in a spin chain each bringing one parameter. The first proceeds through isospectral deformations of spin chains with perfect state transfer. The second arises from the recurrence coefficients of the para-Krawtchouk polynomials with a bi-lattice orthogonality grid. It corresponds to an analytic model previously identified that can possess perfect state transfer in addition to fractional revival.

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Section: Resultsmentioning

confidence: 99%

Quantum spin chains with fractional revival

2016

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“…Cantero, Moral, and Velazquez [8] proved that the Verblunsky coefficients associated with (LM, e 1 ) are precisely (α 0 , . .…”

Section: Circular Jacobi Ensembles and Deformed Verblunsky Coefficienmentioning

confidence: 99%

“…In particular, they obtained a sparse matrix model which is five-diagonal, called CMV (after the names of the authors Cantero, Moral, and Velásquez [8]). In this framework, there is not a natural underlying measure such as the Haar measure; the matrix ensemble is characterized by the laws of its elements.…”

Section: The Circular Jacobi Ensemblementioning

confidence: 99%

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Bourgade¹,

Nikeghbali²,

Rouault³

2009

International Mathematics Research Notices

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Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analog of the Jacobi ensemble:with Reδ > −1/2. If e is a cyclic vector for a unitary n × n matrix U , the spectral measure of the pair (U , e) is well parameterized by its Verblunsky coefficients (α 0 , . . . , α n−1 ). We introduce here a deformation (γ 0 , . . . , γ n−1 ) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ 0 ) · · · r(γ n−1 ) of elementary reflections parameterized by these coefficients. If γ 0 , . . . , γ n−1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above.These deformed Verblunsky coefficients also allow us to prove that, in the regime δ = δ(n) with δ(n)/n → βd/2, the spectral measure and the empirical spectral distribution

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“…and p(n) = E n 2 respectively, where as usual, E[x] denotes the integer part of x (see [3], [9] and [10] for other properties for these particular orderings). In the sequel we will denote by {φ n (z)} ∞ n=0 the sequence of monic orthogonal Laurent polynomials for the measure µ and the generating sequence {p(n)} ∞ n=0 , that is, when the leading coefficients are equal to 1 for all n ≥ 0 (coefficients of z q(n) or z −p(n) when s(n) = 0 or s(n) = 1 respectively).…”

Section: Orthogonal Laurent Polynomials On the Unit Circle Preliminamentioning

confidence: 99%

“…On the other hand, the rapidly growing interest on problems on the unit circle, like quadratures, Szegő polynomials and the trigonometric moment problem has suggested to develop a theory of orthogonal Laurent polynomials on the unit circle introduced by Thron in [36], continued in [26], [21], [11] and where the recent contributions of Cantero, Moral and Velázquez in [4], [3] and [6] has meant an important and definitive impulse for the spectral analysis of certain problems on the unit circle. Here, it should be remarked that the theory of orthogonal Laurent polynomials on the unit circle establishes features totally different to the theory on the real line because of the close relation between orthogonal Laurent polynomials and the orthogonal polynomials on the unit circle (see [9]).…”

Section: Introductionmentioning

confidence: 99%

A matrix approach to the computation of quadrature formulas on the unit circle

Cantero¹,

Cruz-Barroso²,

González-Vera³

2008

Applied Numerical Mathematics

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In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Cited by 258 publications

References 49 publications

Quantum spin chains with fractional revival

Quantum spin chains with fractional revival

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

A matrix approach to the computation of quadrature formulas on the unit circle

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