  Integrable quantum many-body systems are considered to equilibrate to
generalized Gibbs ensembles (GGEs) characterized by the expectation values of
integrals of motion. We study the dynamics of exactly solvable quadratic
bosonic systems in the thermodynamic limit, and show a general mechanism for
the relaxation to GGEs, in terms of the diagonal singularity. We show
analytically and explicitly that a free bosonic system relaxes from a general
(not necessarily Gaussian) initial state under certain physical conditions to a
Gaussian GGE. We also show the relaxation to a Gaussian GGE in an exactly
solvable coupled system, a harmonic oscillator linearly interacting with
bosonic reservoirs.
