Spectral theory of dynamical systems

The paper (a survey) states Theorem 4: for an irrational rotation Tx=x+α, if ξ has nonzero degree, is absolutely continuous, and ξ' has bounded variation, then the weighted operator V_{ξ,T} has Lebesgue spectrum, citing Iwanik–Lemańczyk–Rudolph (1993). The statement appears explicitly and without proof in Section 4.1 of the uploaded survey (Theorem 4) . The candidate model provides a complete proof via a strict positive-commutator (Mourre/Putnam) method. This proof is sound modulo a minor fix: when the degree n is negative, one should replace the averaged conjugate operator A_N by sign(n)·A_N to obtain a positive lower bound. The use of Denjoy–Koksma for BV functions is consistent with the survey’s surrounding context (cf. its discussion of DK in the degree-zero case) . Therefore, both the paper’s claim and the model’s proof are correct, and the approaches are different (survey citation vs. commutator proof).