Perturbations of orthogonal polynomials with periodic recursion coefficients

Abstract

We extend the results of Denisov-Rakhmanov, Szegő-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.Underlying the association of measures and recursion coefficients are matrix representations. For OPRL, we take the matrix for multiplication by x in the

The absolutely continuous spectrum of Jacobi matrices

“…This is in fact obvious because cutting into two half lines (at n = 0) amounts to replacing a(0) by 0, which is a rank two perturbation. In this generality, the theme of studying (1.4) was introduced and investigated by Damanik, Killip, and Simon in [7]; see especially [7,Theorem 1.2]. By using Theorem 1.4, we can go beyond the results of [7].…”

“…6. In [7], Damanik, Killip, and Simon prove Theorem 1.8 in the special case where the coefficients (a, b) ∈ T N (E) are periodic. This imposes restrictions on the spectrum E and is not the generic case; in general, the coefficients are only quasiperiodic.…”

Spectral theory of a class of block Jacobi matrices and applications

“…More generally, eigenvectors of h(p), p ∈ [0, π] are nothing but the so-called Bloch solutions studied in the Floquet theory developed for periodic Jacobi operators. In this sense, Theorem 10.2 is well known with different proofs, see [11,20,23,28]. We will give however a proof, mainly for completeness, but also because it is elementary and based on standard linear algebra.…”

“…Of course, the case where N = 1 corresponds to the usual scalar Jacobi operators that are well studied by different approaches, see e.g. [2,6,7,8,10,11,13,15,16,17,18,20,23,25,27,28] and their references. (This list is very restrictive and contains only some papers related directly to the present one).…”

Lieb–Thirring Inequalities for Finite and Infinite Gap Jacobi Matrices

“…As in the continuous case, the inequality is false for p < 1/2. More recently, the p = 1/2 case of (1.4) was extended to finite gap Jacobi matrices in [5,8,11]. In the setting of periodic and almost periodic parameters, the role of J 0 as a natural limiting point is taken over by the so-called isospectral torus, denoted T E .…”