  We investigate trend to equilibrium for the damped wave equation with a
confining potential in the Euclidean space. We provide with necessary and
sufficient geometric conditions for the energy to decay exponentially
uniformly. The proofs rely on tools from semiclassical analysis together with
the construction of quasimodes of the damped wave operator. In addition to the
Geometric Control Condition, which is familiar in the context of compact
Riemannian manifolds, our work involves a new geometric condition due to the
presence of turning points in the underlying classical dynamics which rules the
propagation of waves in the high-energy asymptotics.
