Computation of the Semiclassical Limit of the Schrödinger Equation with Phase Shift by a Level Set Method

Authors:
Shi Jin;Xu Yang
Affiliations:
Department of Mathematics, University of Wisconsin, Madison, USA 53706;Department of Mathematics, University of Wisconsin, Madison, USA 53706
Venue:
Journal of Scientific Computing
Year:
2008

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Abstract

In this paper, we show how the level set method, developed in [Cheng, Liu and Osher, (2003). Comm. Math. Sci. 1(3), 593---621; Jin, Liu, Osher and Tsai, (2005). J. comp. Phys. 205, 222---241; Jin and Osher, (2003). Comm. Math. Sci. 1(3), 575---591] for the numerical computation of the semiclassical limit of the Schrödinger equation, can be amended to include the phase shift using the Keller-Maslov index. This gives a more accurate approximation of the physical observables for multivalued solutions in the semiclassical limit. Numerical examples in one and two spaces dimensions demonstrate the improved accuracy of our approach away from caustics.