The Schrödinger 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, including 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
1.
The first author was supported in part by the Faculty of Mathematics of the Vienna University, DFG grant 436 RUS 113/929/0-1,
FWF grant P19138-N13, and by Alexander von Humboldt Research Award. The second author was supported in part by Max-Planck
Institute for Mathematics in the Sciences (Leipzig) and by the NSF Grants DMS-0434698 and DMS-0600863.