Kalnins, E.G., Miller, W., Jr. & Pogosyan, G.S.(2004). Infinite order symmetries for two-dimensional separable Schrödinger equations. In Proceedings of Institute of Mathematics of NAS of Ukraine, 50(1), 184-195.
Permanent Research Commons link: https://hdl.handle.net/10289/1763
Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables, there is a calculus to describe the (in general) infinite-order differential operator symmetries of the Schrödinger equation. The calculus is formal but can be made rigorous when all functions in the eigenvaue equation are analytic. The infinite-order calculus exhibits structure that is not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. We go further and extend the calculus to the situation where the Schrödinger equation admits a second-order symmetry operator, not necessarily associated with orthogonal separable coordinates.
Institute of Mathematics of the National Academy of Sciences of Ukraine
This article has been published in Proceedings of Institute of Mathematics of NAS of Ukraine. ©2004 Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv(Kiev), Ukraine.