Adiabatic theorem for the Gross–Pitaevskii equation

Abstract

ABSTRACTWe prove an adiabatic theorem for the nonautonomous semilinear Gross–Pitaevskii equation. More precisely, we assume that the external potential decays suitably at infinity and the linear Schrödinger operator −Δ+Vs admits exactly one bound state, which is ground state, for any s∈[0,1]. In the nonlinear setting, the ground state bifurcates into a manifold of (small) ground state solutions. We show that, if the initial condition is at the ground state manifold, bifurcated from the ground state of −Δ+V0, then, for any fixed s∈[0,1], as 𝜀→0, the solution will converge to the ground state manifold bifurcated from the ground state of −Δ+Vs. Moreover, the limit is of the same mass to the initial condition.