We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No–go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non– controllability for the linear Schrödinger equation and distributed additive control, a...

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 analyze the time–dependent solutions of the pseudo–differential Lévy–Schrödinger wave equation in the free (force-less) case, and we compare them with the associated Lévy processes. We first list the principal laws used to describe the time evolutions of both Lévy process densities and Lévy–Schrödinger wave packets. To have self–adjoint generators and unitary evolutions we will consider only...

We discuss aspects of gravitational modifications of Schrödinger dynamics proposed by Diósi and Penrose. We consider first the Diósi–Penrose criterion for gravitationally induced state vector reduction, and compute the reduction time expected for a superposition of a uniform density cubical solid in two positions displaced by a small fraction of the cube side. We show that the predicted effect ...

Essentially, the Darboux proposition is based on the covariance properties of ordinary and partial differential equations with respect to a gauge transformation in the special case of second order differential equations of the SturmLiouville type. In this work, the one-dimensional Schrödinger equation with a position-dependent mass (SEPDM) is transformed into a Schrödinger-like equation with a ...

Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both, the solution of the nonlinear Schrödinger equation as well as the solution of the semi-discretized equation, the actions of the linear Schrödinger equation are approximately conserved over long times. This also allows to show app...