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J. Geier, A. Arnold:
"WKB-based schemes for two-band Schrödinger equations in the highly oscillatory regime";
in: "ASC Report 25/2011", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

An efficient and accurate numerical method is presented for the solution of highly oscillatory differential
equations in one spatial dimension. While standard methods would require a very fine grid to resolve the
oscillations, the presented approach uses first an analytic WKB-type transformation, which filters out the
dominant oscillations. The resulting ODE-system is much smoother and can hence be discretized on a much
coarser grid, with significantly reduced numerical costs.
Here we are concerned with stationary two-band Schr¨odinger equations employed in quantum transport
applications. We focus on the Kane-model and the two band k·p-model. The accuracy of the presented
method is illustrated on a numerical example.

Schr¨odinger equation, Kane-model, two-band k · p-model, highly oscillating wave functions,

http://www.asc.tuwien.ac.at/preprint/2011/asc25x2011.pdf