The geometric origin of Lie point symmetries of the Schrödinger and the Klein–Gordon equations

Abstract

We determine the Lie point symmetries of the Schrödinger and the Klein–Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space, respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schrödinger equation and the Klein–Gordon equation are related with the Noether point symmetries of this Lagrangian. The general results are applied to two practical problems: (a) The classification of all two- and three-dimensional potentials in a Euclidean space for which the Schrödinger equation and the Klein–Gordon equation admit Lie point symmetries; and (b) The application of Lie point symmetries of the Klein–Gordon equation in the exterior Schwarzschild spacetime and the determination of the metric by means of conformally related Lagrangians.