Global well-posedness for the Schroedinger equation coupled to a   nonlinear oscillator
  The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$. 
Alexander Komech and Andrew Komech