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A. Arnold, C. Klein, B. Ujvari:
"WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment";
in: "ASC Report 19/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 17.

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis for the method is presented for a not explicitly known WKB phase, and secondly the phase and its derivatives will be computed with spectral methods. The efficiency of the approach is illustrated for several examples

Uniformly accurate scheme, Schrödinger equation, highly oscillating wave func-tions, higher order WKB-approximation, asymptotically correct finite difference scheme, spectral methods

http://www.asc.tuwien.ac.at/preprint/2018/asc19x2018.pdf