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

Abstract

We consider the Hamiltonian system of scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner. The particle is also 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 ${\cal S}$ of all stationary states in the long time limit $t\to\pm\infty$. Further we show that the rate of relaxation to a stable stationary state is determined by spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave which is a solution to the free wave equation. Similar relaxation has been proved previously for the case of relativistic particle when the speed of the particle is less than the speed of light. Now we extend these results to nonrelativistic particle with arbitrary superlight velocity. However, we restrict ourselves by the plane particle trajectories. The extension to general case remains an open problem.