Representations of the Schrodinger algebra and Appell systems

We investigate the structure of the Schrodinger algebra. Two constructions are given that yield the physical realization via general methods starting from the abstract Lie algebra. Representations are found on a Fock space with basis given by a canonical Appell system. Generalized coherent states are used in the construction of the Hilbert space of functions on which certain commuting elements act as self-adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the semidirect product structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space are found. Then certain evolution equations connected with canonical Appell systems on this algebra are shown.