We attack the specific time-dependent Hamiltonian problem H = − 2 ( to t a ∂xx+ 1 2ω 2 ( t to )b x2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to a different time-dependent quadratic Schrödinger equations (TQ) and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-t...

We state an approximate controllability result for the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. This result applies both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. In addition we get some controllability properties for the density matrix. Finally we show, by me...

We propose an exact controllability result for Schrödinger equations in bounded domains under the Bardos–Lebeau–Rauch geometric control condition with an estimate of the control which is explicit with respect to the time of controllability. Also, we prove an explicit in time logarithmic observability estimate for the Schrödinger equation, where no geometrical conditions are supposed on the domain.