Purely absolutely continuous spectrum for some random Jacobi matrices

Abstract

We consider random Jacobi matrices of the form (J ω u)(n) = a n (ω)u(n + 1) + b n (ω)u(n) + a n−1 (ω)u(n − 1) on ℓ 2 (N), where a n (ω) =ã n + α n (ω), b n (ω) =b n + β n (ω), {ã n } and {b n } are sequences of bounded variation obeyingã n → 1 and b n → 0, and {α n (ω)} and {β n (ω)} are sequences of independent random variables on a probability space (Ω, dP (ω)) obeying ∞ n=1 Ω (α 2 n (ω) + β 2 n (ω)) dP (ω) < ∞ and Ω α n (ω) dP (ω) = Ω β n (ω) dP (ω) = 0 for each n. We further assume that there exists C 0 > 0 such that 1/C 0 < a n (ω) < C 0 and −C 0 < b n (ω) < C 0 for every n and P a.e. ω. We prove that, for P a.e. ω, J ω has purely absolutely continuous spectrum on (−2, 2).

Generalized Prüfer variables for perturbations of Jacobi and CMV matrices

“…In those cases, it is known that the essential spectrum of J consists of finitely many bands, and that (1.14) holds on each band. In this way, our result generalizes a result of Kaluzhny-Last [KL07], who prove Theorem 2 in the case that (1.17) holds for q = 1. Their argument is more elaborate, working directly with transfer matrices and using the bounded variation condition.…”

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

“…Breuer and Last [1] have recently shown that, for any Jacobi matrix, the a.c. spectrum which is associated with bounded generalized eigenfunctions (like the spectrum of a periodic Jacobi matrix) is stable under squaresummable random perturbations. In [12] we have shown that if both a square-summable random perturbation and a decaying perturbation of bounded variation are added to the free Laplacian, the a.c. spectrum is still preserved. But, to the best of our knowledge, there has been no significant progress made so far with the general deterministic case.…”

Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

“…A related result concerning general V 0 has been recently obtained by Breuer-Last [2], who show stability of absolutely continuous spectrum associated with bounded generalized eigenfunctions under random decaying ℓ 2 perturbation potentials (also see a related result by Kaluzhny-Last [9]). …”