On New Relations Between Spectral Properties of Jacobi Matrices and Their Coefficients

“…Recently the area of sum rules, for orthogonal polynomials as well as Schrödinger operators, saw a rapid development starting with papers by Deift-Killip [2] and Killip-Simon [9], which were followed by many others (e.g., [3,10,11,12,14,16,20,23,24]). …”

Section: Introductionmentioning

confidence: 99%

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Golinskiĭ

Zlatoš

2007

Constr Approx

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Abstract. Let µ be a non-trivial probability measure on the unit circle ∂D, w the density of its absolutely continuous part, α n its Verblunsky coefficients, and Φ n its monic orthogonal polynomials. In this paper we compute the coefficients of Φ n in terms of the α n . If the function log w is in L 1 (dθ), we do the same for its Fourier coefficients. As an application we prove that if α n ∈ ℓ 4 and Q(z) ≡ N m=0 q m z m is a polynomial, then withQ(z) ≡ N m=0q m z m and S the left shift operator on sequences we haveWe also study relative ratio asymptotics of the reversed polynomials Φ *and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures µ and ν for this difference to converge to zero uniformly on compact subsets of D.

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“…Kupin's result is among the strongest of several recent results (see [12,14,24]) in which an ℓ p requirement for V itself and an ℓ 2 requirement for some type of variation of V combine to ensure that Σ ac (H V ) = Σ ac (∆) = (−2, 2). In most other results of this genre, p is a concrete number, such as 3 or 4.…”

Section: Introductionmentioning

confidence: 84%

Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

Last

2007

Commun. Math. Phys.

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On New Relations Between Spectral Properties of Jacobi Matrices and Their Coefficients

Cited by 37 publications

References 16 publications

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

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