Symmetry breaking interactions for the time dependent Schrödinger equation

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Abstract
A systematic study of the symmetry porperties of the Schrödinger equation uxx + iut = F (x,t,u,u*) is performed. The free particle equation (for F=0) is known to be invariant under the six-dimensional Schrödinger group 𝒮1. In this paper we find all continuous subgroups of 𝒮1 and for each subgroup we construct the most general interaction term F (x,t,u,u*), reducing the symmetry group of the equation from 𝒮1 to the considered subgroup. Since we allow for an arbitrary dependence of F on the wavefunction u (and its complex conjugate u*) the considered Schrödinger equation is in general a nonlinear one [the ordinary Schrödinger equation with a time dependent potential is recovered if F (x,t,u,u*) =uG (x,t)]. For each symmetry breaking interaction F the remaining symmetry group is used to obtain special solutions of the equations or at least to separate variables in the equation and to obtain some properties of the solutions.

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