Symmetry breaking interactions for the time dependent Schrödinger equation

A systematic study of the symmetry porperties of the Schrodinger 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 Schrodinger 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 Schrodinger equation is in general a nonlinear one [the ordinary Schrodinger 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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