On the absolutely continuous spectrum in a model of irreversible quantum graph

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A family Aα of differential operators depending on a real parameter α≥ 0 is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum σₐ.c. of the operator Aα and its multiplicity for all values of the parameter. The spectrum of A₀ is purely a.c. and admits an explicit description. It turns out that for α<√ 2 one has σₐ.c.(Aα)= σₐ.c.(A₀), including the multiplicity. For α≥√2 an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator Aα. This birth of an extra-branch of a.c. spectrum is the exact mathematical expression of the effect which was interpreted by Smilansky as irreversibility.