# Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems - Abstract

Arnold, Anton

Asymptotically correct finite difference schemes for highly oscillatory ODEs

We are concerned with the numerical integration of ODEs of the form $\epsilon2 \psi_{xx} + a(x)\psi=0$ for given $a(x)\ge\alpha>0$ in the highly oscillatory regime $0<\epsilon\ll 1$ (appearing as a stationary Schr\"odinger equation, e.g.). In two steps we derive an accurate finite difference scheme that does not need to resolve each oscillation: \begin{enumerate} \item With a WKB-ansatz the dominant oscillations are transformed out'', yielding a much smoother ODE. \item For the resulting oscillatory integrals we devise an asymptotic expansion both in $\epsilon$ and $h$. \end{enumerate} In contrast to existing strategies, the presented method has (even for a large spatial step size $h$) the same weak limit (in the classical limit $\epsilon\to 0$) as the continuous solution. Moreover, it has an error bound of the order $O(\epsilon3 h2)$. We shall give applications to $\bm{k\cdot p}$-Schr\"odinger systems and to the simulation of semiconductor-nanostructures.\\[2mm] Ref: A. ARNOLD, N. BEN ABDALLAH and C. NEGULESCU: WKB-based schemes for the Schr\"odinger equation in the semi-classical limit, preprint 2010.