Quantum two-photon algebra from non-standard Uz(sl(2,R)) and a discrete time Schrodinger equation

The non-standard quantum deformation of the (trivially) extended sl(2,R) algebra is used to construct a new quantum deformation of the two-photon algebra he and its associated quantum universal R-matrix. A deformed one-boson representation for this algebra is deduced and applied to construct a first-order deformation of the differential equation that generates the two-photon algebra eigenstates in quantum optics. On the other hand, the isomorphism between h(6) and the (1+1) Schrodinger algebra leads to a new quantum deformation for the latter for which a differential-difference realization is presented. From it, a time discretization of the heat-Schrodinger equation is obtained and the quantum Schrodinger generators are shown to be symmetry operators.