Commutation relations for unitary operators II

Let ff be a regular non-constant symbol defined on the dd-dimensional torus TdTd with values on the unit circle. Denote respectively by κκ and LL, its set of critical points and the associated Laurent operator on l2(Zd)l2(Zd). Let UU be a suitable unitary local perturbation of LL. We show that the operator UU has finite point spectrum and no singular continuous component away from the set f(κ)f(κ). We apply these results and provide a new approach to analyze the spectral properties of GGT matrices with asymptotically constant Verblunsky coefficients. The proofs are based on positive commutator techniques. We also obtain some propagation estimates.