On quantum algebra symmetries of discrete Schr\"odinger equations

TL;DR

This paper explores two quantum deformations of the Schr"odinger algebra that serve as symmetry algebras for space or time discretized Schr"odinger equations, linking deformation parameters to lattice steps.

Contribution

It identifies specific quantum algebra symmetries for discretized Schr"odinger equations and discusses their relation to lattice steps and nonlinear maps.

Findings

Quantum deformations correspond to lattice discretization steps.

Symmetry algebras are linked to space or time discretizations.

Discussion on full space-time discretization symmetry.

Abstract

Two non-standard quantum deformations of the (1+1) Schr\"odinger algebra are identified with the symmetry algebras of either a space or time uniform lattice discretization of the Schr\"odinger equation. For both cases, the deformation parameter of the corresponding Hopf algebra can be interpreted as the step of the lattice. In this context, the introduction of nonlinear maps defining Schr\"odinger and $sl(2,\R)$ quantum algebras with classical commutation rules turns out to be relevant. The problem of finding a quantum algebra linked to the full space-time discretization is also discussed.