Piecewise absolutely continuous cocycles over irrational rotations

For an irrational rotation cc of the circle group T = R/Z and a piecewise a bsolutely continuous function f:T --> R, the unitary operator Vh(x) = e(2 p i if(x))h(x + alpha) on L-2(T) is studied. It is shown that iff has a singl e discontinuity with non-integer jump then V is kappa-weakly mixing for som e kappa with 0 < \kappa\ < 1. In particular V has continuous singular spect rum. The property of kappa-weak mixing (with possible change of the value o f kappa, 0 < \kappa\ < 1) holds for all irrational rotations and, given a, is stable under perturbations off by functions with sufficiently small O(1/ n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular f or one value of a and Lebesgue for another.