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

TL;DR

This paper characterizes the free Schrödinger operator as the unique second-order linear PDE invariant under Galilei transformations, using polynomial invariance analysis of plane waves.

Contribution

It provides a novel characterization of the Schrödinger operator based on Galilei invariance, expanding understanding of symmetry properties in quantum mechanics.

Findings

Schrödinger operator uniquely invariant under Galilei group

Method based on polynomial invariance of plane waves

Clarifies symmetry constraints in quantum PDEs

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.