Long-time asymptotics for a classical particle interacting with a scalar wave field

We consider the Hamiltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t → ±∞. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data.