Asymptotic stability of stationary states in wave equation coupled to nonrelativistic particle

TL;DR

This paper proves that in a coupled wave-particle Hamiltonian system, solutions near stable stationary states asymptotically decompose into the stationary state plus a dispersive wave, under certain conditions.

Contribution

It establishes the asymptotic stability of stationary states in a wave equation coupled to a nonrelativistic particle with a Wiener condition on the charge density.

Findings

Solutions near stable states decompose into stationary and dispersive parts over time

The stationary solutions are Coulomb-like wave fields centered at equilibrium points

Dispersive waves evolve according to the free wave equation

Abstract

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 an 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. It is assumed that the charge density satisfies the Wiener condition which is a version of the "Fermi Golden Rule". We prove that in the large time approximation any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.