On the global approximate controllability in small time of semiclassical 1-D Schrödinger equations between two states with positive quantum densities

In this paper, we study, in the semiclassical sense, global approximate controllability small time of quantum density and momentum 1-D cubic Schrödinger equation with two controls between states positive densities. We first control asymptotic expansions zeroth order physical observables via Agrachev–Sarychev method. Then conclude proof through techniques approximation nonlinear equation.