Jacobi Operators with Singular Continuous Spectrum

Abstract

We show that for two classes of pairs of sequences (c(n), v(n)), the operator H acting on \(\ell^{2} (\mathbb{Z})\) by $$(Hu)(n) = c(n)u(n + 1) + \overline{c(n - 1)}u(n - 1) + v(n)u(n)$$ has no eigenvalues. Applying these results to a class of quasiperiodic operators of magnetic origin, we show, that in a region where purely pure point spectrum has been established for almost every phase and frequency [5], the operator has purely singular continuous spectrum for a dense Gδ in phase and a dense Gδ in frequency. This generalizes similar results obtained for the Schrodinger case.