Difference analogues of the free Schrodinger equation

We propose an infinite family of difference equations, which are derived fr om the first principle that they are invariant with respect to the Schrodin ger algebra. The first member of this family is a difference analogue of th e free Schrodinger equation. These equations are obtained via a purely alge braic construction from a corresponding family of singular vectors in Verma modules over the Schrodinger algebra. The crucial moment in the constructi on is the realization of the Schrodinger algebra through additive differenc e vector fields, i.e. vector fields with difference operators instead of di fferential operators. Our method produces also differential-difference equa tions in which only space- or time-differentiation is replaced with the cor responding difference operators.