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

Abstract

A family $A_α$ of differential operators depending on a real parameter $α\ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $σ_{a.c.}$ of the operator $A_α$ 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 $α<\sqrt 2$ one has $σ_{a.c.}(A_α)= σ_{a.c.}(A_0)$, including the multiplicity. For $α\ge\sqrt2$ an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator $A_α$. 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.