Istituto di Matematica Applicata e Tecnologie Informatiche     
Markowich P. A., Pietra P., Carsten P. Weak limits of finite difference schemes for Schroedinger-type equations. Preprint ercim.cnr.ian//1997-1035, 1997.
Many problems of solid state physics require the solution of the Schroedinger equation in the case of a small (scaled) Planck constant. In classical quantum physics the wave function is an auxiliary quantity, used to compute the primary physical quantities, which are quadratic functions of the wave function, e.g. the position density and the current density. It is well known that the highly oscillatory nature of the wave function inhibits strong convergence as the Planck constant goes to zero. Clearly, the weak convergence of the wave function is not sufficient for passing to the limit in the macroscopic densities, and the analysis of the so-called semi-classical limit is a mathematically rather complex issue. Exactly the same problem, i.e. the highly oscillatory nature of the solutions, has to be coped with when the Schroedinger equation with small Planck constant is solved numerically. Even for stable discretization schemes (or under mesh size restrictions which guarantee stability) the oscillations may very well pollute the solution in such a way that the quadratic macroscopic quantities and other physical observables come out completely wrong when the scaled Planck constant is small. Here we adapt the Wigner-transform techniques used to analyse the semi-classical limit for the continuous Schroedinger equation to the analysis of finite difference discretizations. The spatial discretizations are general arbitrary-order symmetric schemes. Several time discretizations are considered: the Crank-Nicolson scheme, the leap frog scheme (both belonging to the set of mostly used methods for the Schroedinger equation) and one step (asymmetric) implicit and explicit time discretizations. For all these methods we identify the semiclassical Wigner measure for all (sensible) combinations of the Planck constant and of the time and space mesh sizes. The macroscopic quantities (e.g. position and current density) can be retrieved as moments (zeroth and first order) of the Wigner measure. We clearly have uniform convergence for the average values of the macroscopic quantities in those cases, for which the Wigner measure of the numerical scheme is identical to the Wigner measure of the Schroedinger equation itself. From this theory we obtain sharp (i.e. necessary and sufficient) conditions on the mesh sizes which guarantee a good approximation quality of the macroscopic quantities uniformly as the Planck constant goes to zero. The numerical tests confirm the predicted behaviour: in cases where the Wigner measure of the discrete scheme does not coincide with the Wigner measure of the Schroedinger equation we get wrong results for the macroscopic quantities for small Planck constant. We remark that the theory does not depend on whether caustics develop or not. Even for nonoscillating macroscopic quantities the results are wrong for badly chosen mesh parameters.
Subject 65-XX

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