Large time behavior in nonlinear Schrodinger equation with time dependent potential
arXiv:1305.4089
Abstract
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.
18 pages
References in corpus (5)
- KAM for the quantum harmonic oscillator
- The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
- A pseudoconformal compactification of the nonlinear Schrödinger equation and applications
- Global well-posedness and scattering for the defocusing cubic NLS in four dimensions
- Long time dynamics for the one dimensional non linear Schrödinger equation