2013/12/13 by Andronikos Paliathanasis, Michael Tsamparlis · 1 citation
Mathematics · Physics and Astronomy · #Adjoint representation #Euclidean space #Geodesic #Homogeneous space #Homothetic transformation #Homotopy and Cohomology in Algebraic Topology #Lie group #Lie theory #Nonlinear Waves and Solitons #Point (geometry) #Quantum Mechanics and Non-Hermitian Physics #Spacetime symmetries #math-ph #math.AP #math.MP
paper · pdf · doi:10.1142/s0219887814500376
published as Int. J. Geom. Methods Mod. Phys. 11, 1450037 (2014) · to be published in IJGMMP, 18 pages
arxiv created 2013/12/13 · openalex publication_date 2014/01/29 · arxiv updated 2014/09/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.