Equality of the Spectral and Dynamical Definitions of Reflection

Abstract

For full-line Jacobi matrices, Schrödinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of m-functions, is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x = −∞ as t → −∞ goes entirely to x = ∞ as t → ∞. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.

Right Limits and Reflectionless Measures for CMV Matrices

“…Assume E is reflectionless in the sense of Definition 1. 3. Then it follows from the discussion at the beginning of this section that µ n is reflectionless for all n. Now assume µ n is reflectionless in the sense of Definition 1.6 for all n. Since two α's are not zero, it follows from Lemma 3.1 that there are infinitely many nonzero α's.…”

“…Definition 1. 3. Let {α n } n∈Z be a doubly-infinite sequence of Verblunsky coefficients and let E be the associated whole-line CMV matrix.…”

Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences

“…Thus, if q = p j=1 q j , we can write λ(w) as Hence, from Theorem 2.5, the weight function w(θ) associated to µ (2) is such that…”

“…Finally, we give a complete characterization about the singular part of the measure µ (2) in terms of the parameters b 1 , b 2 and c.…”

Polinômios Ortogonais no Círculo Unitário com Relação a Certas Medidas Associadas a Coecientes de Verblunsky Periódicos

“…Além disso, os coeficientes de Verblunsky {α n } ∞ n=0 , associados aos polinômios ortogonais com respeito a µ, são dados pela relação α n−1 = τ n− 1 1 − 2m n − ic n 1 − ic n , n ≥ 1, onde τ 0 = 1, τ n = n k=1 (1−ic k )/(1+ic k ), n ≥ 1 e {m n } ∞ n=0é a sequência de parâmetros minimal para {d n } ∞ n=1 . Neste trabalho, impondo algumas restrições sobre o par de sequências reais {{c n } ∞ n=1 , {m n } ∞ n=1 }, mostramos queé possível obter medidas de probabilidade não triviais no círculo unitário cujos respectivos coeficientes de Verblunsky são periódicos.…”