Symmetries of linear difference equations via umbral calculus
We propose a general approach to the study of Lie symmetries of difference equations. This approach is based on the axiomatic theory of finite difference operators (introduced by G.C. Rota). In particular, our formalism is applied to the Schroedinger equation in order to obtain a realization of quantum mechanics on a lattice which preserves the properties of integrability, superintegrability and exact solvability.
This is joint work with Decio Levi and Pavel Winternitz