A family $\mathbf{A}_\alpha$ 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 $\sigma_{a.c.}$ of the operator $\mathbf{A}_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $\mathbf{A}_0$ is purely absolutely continuous and admits an explicit description. It turns out that for $\alpha < \sqrt 2$ one has $\sigma_{a.c.}(\mathbf{A}_\alpha) = \sigma_{a.c.}(\mathbf{A}_0)$, including the multiplicity. For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary Jacobi matrix which is related to the operator $\mathbf{A}_\alpha$. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.