Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

Abstract

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an ℓ^p condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences β^(l), each of which has rotated bounded variation, i.e., ∑^∞_(n=0) │e^(iΦl)β^(l)_(n+1)−β_n^(l)│ < ∞ for some Φ_l. This includes a large class of discrete Schrödinger operators with almost periodic potentials modulated by ℓ^p decay, i.e. linear combinations of λ_ncos(2παn+Φ) with λ Є ℓ^p of bounded variation and any α. In all cases, we prove absence of singular continuous spectrum, preservation of absolutely continuous spectrum from the corresponding free case, and that pure points embedded in the continuous spectrum can only occur in an explicit finite set.

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