On global attractors and radiation damping for nonrelativistic particle coupled to scalar field

The Hamiltonian system of a scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner is considered. The particle is also subject to a confining external potential. The stationary solutions of the system are Coulomb type wave fields centered at those particle positions for which the external force vanishes. It is proved that the solutions of finite energy converge, in suitable local energy seminorms, to the set S of all stationary states in the long time limit t → ± ∞. Next it is shown that the rate of relaxation to a stable stationary state is determined by the spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave that is a solution of the free wave equation. Similar relaxation has been proved previously for the case of a relativistic particle when the speed of the particle is less than the wave speed. Now these results are extended to a nonrelativistic particle with velocity, including that greater than the wave speed. However, the research is restricted to the plane particle trajectories in ℝ³. Extension to the general case remains an open problem.