Nonlinear dynamics from linear quantum evolutions

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group phi(t) of transformations on a Hilbert space, H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrodinger equation to a submanifold invariant under the flow phi(t). The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation the flow phi(t). Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems. (C) 2019 Elsevier Inc. All rights reserved.