How does a damped pole arise from the limit of a discrete system?

Question

In Fetter and Walecka Many Particle Physics we see a sketch for a quantum system how the limit of a quantum system embedded in a bath of harmonic oscillators can give rise to a pole off of the real frequency axis. My understanding is that a finite system will have all its poles on the real frequency axis, yet as the system becomes large the poles on the real axis merge giving rise to a broadened peak on the real frequency axis and a pole off the real frequency axis. In mechanical engineering people typically jump straight to the continuum limit for friction by assuming a damping term in the transfer function which gives rise to a similar complex frequency pole. Can anyone point me to a somewhat rigorous derivation, ideally common to classical and quantum systems, that shows how complex frequency poles can arise from taking the continuum limit of a finite discrete system? My apologies for the vaguely formed question!