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Smilansky's model of irreversible quantum graphs: I. The absolutely continuous spectrum

In the model suggested by Smilansky (2004 Waves Random Media 14 143–53) one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at several different points in the graph. The present paper is the first one in which the case K > 1 is investigated. For the sake of simplicity, we consider K = 2, but our argument is of a general character. In this first of two papers on the problem, we describe the absolutely continuous spectrum. Our approach is based upon scattering theory.