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A. IWANIK, M. LEMAN ! CZYK  C. MAUDUIT A For an irrational rotation α of the circle group T l R\Z and a piecewise absolutely continuous function f : T ,-R, the unitary operator Vh(x) l e# π if(x) h(xjα) on L#(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0 QκQ 1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0 QκQ 1) holds for all irrational rotations and, given α, is stable under perturbations of f 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 for one value of α and Lebesgue for another.