  We consider the large time behavior of solutions to defocusing nonlinear
Schrodinger equation in the presence of a time dependent external potential.
The main assumption on the potential is that it grows at most quadratically in
space, uniformly with respect to the time variable. We show a general
exponential control of first order derivatives and momenta, which yields a
double exponential bound for higher Sobolev norms and momenta. On the other
hand, we show that if the potential is an isotropic harmonic potential with a
time dependent frequency which decays sufficiently fast, then Sobolev norms are
bounded, and momenta grow at most polynomially in time, because the potential
becomes negligible for large time: there is scattering, even though the
potential is unbounded in space for fixed time.
