  We study the problem of convergence to equilibrium for evolution equations
associated to general quadratic operators. Quadratic operators are
non-selfadjoint differential operators with complex-valued quadratic symbols.
Under appropriate assumptions, a complete description of the spectrum of such
operators is given and the exponential return to equilibrium with sharp
estimates on the rate of convergence is proven. Some applications to the study
of chains of oscillators and the generalized Langevin equation are given.
