Nonlinear adiabatic theorem for the Gross-Pitaevskii equation

Abstract

We prove a nonlinear adiabatic theorem for the repulsive Gross-Pitaevskii (GP) equation with a time-dependent trapping potential $V(x,t)$. If either (i) $V(\cdot,t)$ is smooth, subquadratic in space and with uniformly bounded spatial derivatives of order $\geq 2$, or (ii) $V(\cdot,t)$ is a uniformly bounded perturbation of a fixed non-negative potential which is asymptotically $|x|^m, m\geq 2$, then the GP equation has a global solution, which remains uniformly close to the instantaneous minimizer of the time-independent GP functional, provided the initial condition is a ground state. In fact, we show that the existence of a solution of the GP equation is the real bottleneck: If existence is assumed, then adiabaticity holds under very minimal assumptions.