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 Euclidian 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.

to be published in IJGMMP, 18 pages

References in corpus (3)

• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Lie point symmetries of a general class of PDEs: The heat equation
• Generalizing the autonomous Kepler Ermakov system in a Riemannian space

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