Spectral analysis of time changes for horocycle flows

We prove (under the condition of A. G. Kushnirenko) that all time changes for the horocycle flow have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes for horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators.