Jacobi and CMV Matrices with Coefficients of Generalized Bounded Variation

Abstract

We consider Jacobi and CMV matrices with coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner–von Neumann type potentials cos$(n\varnothing\,+\,\alpha)/n^\gamma$ with $\gamma\,> \,0$.

Our results show preservation of the absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

Mathematics Subject Classification (2010). Primary 42C05,47B36.