Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square--summable variation

Details Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square--summable variation Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square--summable variation

2009-12-06

arxiv.org/abs/0912.1142v2

Mathematics

Spectral Theory

18pp; revised version

Scientific paper

We study self-adjoint bounded Jacobi operators of the form: (J \psi)(n) = a_n \psi(n + 1) + b_n \psi(n) +a_{n-1} \psi(n - 1) on $\ell^2(\N)$. We assume that for some fixed q, the q-variation of $\{a_n\}$ and $\{b_n\}$ is square-summable and $\{a_n\}$ and $\{b_n\}$ converge to q-periodic sequences. Our main result is that under these assumptions the essential support of the absolutely continuous part of the spectrum of J is equal to that of the asymptotic periodic Jacobi operator. This work is an extension of a recent result of S.A.Denisov.

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