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Infinite order symmetries for two-dimensional separable Schrödinger equations

Infinite order symmetries for two-dimensional separable Schrödinger equations

##### Abstract

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.

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Conference Contribution

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##### Citation

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.

##### Date

2004

##### Publisher

Institute of Mathematics of the National Academy of Sciences of Ukraine

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##### Rights

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.