Abstract
We apply Wigner-transform techniques to the analysis of difference methods for Schrödinger-type equations in the case of a small Planck constant. In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not.Numerical examples are presented to help interpret the theory.
Highlights
Many problems of solid-state physics require the solution of the Schr6dinger equation in the case of a small Planck constant c" euet C2 i--f Au + iV(x)u O, xm t 0) (x) x (1.1b)Here V is a given electrostatic potential, 0 < e
The consistency-stability concept of classical numerical analysis provides a framework for the convergence analysis of finite difference discretizations of linear partial differential equations
We shall see that the set of Wigner-measures of the difference schemes
Summary
Many problems of solid-state physics require the (numerical) solution of the Schr6dinger equation in the case of a small (scaled) Planck constant c". For the linear Schr6dinger equation classical numerical analysis methods (like the stabilityconsistency concept) are sufficient to derive meshing strategies for discretizations which guarantee (locally) strong convergence of the discrete wave functions to u when c > 0 is fixed (cf [21, 1,2,3]). We clearly have convergence for the average values of all (regular) observables in those cases, for which the Wigner measure of the numerical scheme is identical to the Wigner measure of the Schr6dinger equation itself From this theory we obtain sharp (i.e., necessary and sufficient) conditions on the mesh sizes which guarantee good approximation quality of all (smooth) observables for e small. In order to avoid taking subsequences we shall assume for the following u. w/ is the unique Wigner-measure of (A3)
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