The spectral properties of hypoelliptic operators and the problem of return to equilibrium for the associated parabolic equations are topics of current interest. This paper deals with evolution equations associated with general quadratic operators \[ {\partial u\over\partial t}+ q^w(x, D_x)u= 0,\quad u_{t=0}= u_0\in L^2(\mathbb{R}^n), \] where \(q^w(x, D_x)\) is defined in the standard Weil quantization by the complex-valued quadratic form \(q(x,\xi)\), \(\text{Re\,}q\geq 0\) as a symbol. Quadratic operators are non-selfadjoint differential operators. Under appropriate assumptions (zero singular space \(S= \{0\}\) of \(q(x,\xi)\)), a complete description of the spectrum of \(q^w(x,D_x)\) is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proved. More precisely, the first eigenvalue in the bottom of the spectrum has algebraic multiplicity one with an eigenspace spanned by a ground state of exponential type.
At the end of the paper, several applications to the study of chains of oscillators and to the generalized Langevin equation are proposed.