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

TL;DR

This paper proves that solutions of a coupled scalar wave and nonrelativistic particle system converge to stationary states over time, with the relaxation rate depending on initial data decay, extending previous relativistic results to nonrelativistic particles.

Contribution

It extends the analysis of relaxation to stationary states from relativistic to nonrelativistic particles with arbitrary superlight velocities, under the restriction of plane trajectories.

Findings

Solutions converge to stationary states as time approaches infinity.

Relaxation rate depends on the spatial decay of initial data.

Radiation of dispersion waves occurs during relaxation.

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.