Characterization of the time-dependent free Schr\"odinger operator by the Galilei invariance

Abstract

The time-dependent free Schr\"odinger operator is shown to be characterized as the only linear partial differential operator of the second order that is invariant under the Galilei group in the Euclidean space-time $\mathbb R\times\mathbb R^n$. The method of proof depends on the analysis of the invariance of polynomials given by the application of the linear partial differential operators to monochromatic plane waves under space rotations and pure Galilei transformations.