This thesis is mainly concerned with the problem of exponential convergence to equilibrium
for open classical systems. We consider a model of a small Hamiltonian system coupled to
a heat reservoir, which is described by the Generalized Langevin Equation (GLE) and we
focus on a class of Markovian approximations to the GLE. The generator of these Markovian
dynamics is an hypoelliptic non-selfadjoint operator. We look at the problem of exponential
convergence to equilibrium by using and comparing three different approaches: classic ergodic
theory, hypocoercivity theory and semiclassical analysis (singular space theory). In particular,
we describe a technique to easily determine the spectrum of quadratic hypoelliptic operators
(which are in general non-selfadjoint) and hence obtain the exact rate of convergence to
equilibrium
