Title Scattering of Solitons for Schr\"odinger Equation Coupled to a   Particle
  We establish soliton-like asymptotics for finite energy solutions to the Schr\"odinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold in the Hilbert phase space. 
Alexander Komech, Elena Kopylova