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

A family $A_{\alpha }$ of differential operators depending on a real parameter $\alpha \ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum ${\sigma }_{a\mathrm{.}c\mathrm{.}}$ of the operator $A_{\alpha }$ and its multiplicity for all values of the parameter. The spectrum of $A_0$ is purely a.c. and admits an explicit description. It turns out that for $\alpha <\sqrt{2}$ one has ${\sigma }_{a\mathrm{.}c\mathrm{.}}(A_{\alpha })={\sigma }_{a\mathrm{.}c\mathrm{.}}(A_0)$, including the multiplicity. For $\alpha \ge \sqrt{2}$ an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator $A_{\alpha }$. 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.