Semiclassical analysis of discretizations of Schrodinger-type equations

We apply Wigner-transform techniques to the analysis of difference methods for Schrodinger-type equations in the case of a small Planck constant. In t his way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planc k constant tends to zero. The theory developed in this paper is not based o n local and global error estimates and does not depend on whether caustics develop or not. Numerical examples are presented to help interpret the theory.