Control of a quantum particle in a moving box

This 1-D Schrödinger equation describes the non relativistic motion of a single charged particle (mass m = 1, = 1) with a potential V in a uniform electric field t → u(t). With v̈ = −u, (1) represents also the dynamics of a particle in a non Galilean frame of absolute position v (see, e.g., [9]). A change of independent variables (t, q) → (t, z) and dependent variable ψ → φ, transform (1) into (2) where the control appears as a shift on the space variable. These classical transformations are as follows (see, e.g., [2]). Instead of considering u as control, take v defined by v̈ = −u as control. Then