Large time behavior in nonlinear Schrodinger equation with time dependent potential

TL;DR

This paper studies the long-term behavior of solutions to the nonlinear Schrödinger equation with a time-dependent potential, establishing bounds on derivatives and momenta, and demonstrating scattering under certain decaying potential conditions.

Contribution

It provides new bounds on Sobolev norms and momenta for solutions with time-dependent potentials, including cases with decaying harmonic potentials, showing scattering phenomena.

Findings

Exponential control of first derivatives and momenta.

Double exponential bounds for higher Sobolev norms.

Scattering occurs when the potential decays sufficiently fast.

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.