Bogolyubov transformation in the problem of capture of a massive particle by a quantum field

A Bogolyubov transformation to collective coordinates is used to study the problem of the capture of a heavy particle by a quantum field. It is shown that for an appropriate choice of the parameters in the problem the large mass of the particle is almost completely “eaten up” by the interaction. The spectrum of excited states of the particle – field system near the ground state is studied. The equations for the Green's function in the first nonvanishing approximation in the reciprocal coupling constant are investigated. It is shown that the elements of the $S$ matrix associated with the free part of the excitation spectrum satisfy the condition of elastic unitarity. The effective mass of the system is determined.