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A High Order Numerical Method for Computing Physical Observables in the Semiclassical Limit of the One-Dimensional Linear Schrödinger Equation with Discontinuous Potentials

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Abstract

We develop a fourth order numerical method for the computation of multivalued physical observables (density, momentum, etc.) in the semiclassical limit of the one-dimensional linear Schrödinger equation in the case of discontinuous potentials. We adopt the level set framework developed in (Jin et al. in J. Comput. Phys. 210:497–518, 2005) which allows one to compute the multivalued physical observables via solving the classical Liouville equation with bounded initial data and approximating delta function integrals. We achieve high order accuracy for our method by studying two issues. The first is to highly accurately compute the solution and its derivatives of the Liouville equation with bounded initial data and discontinuous potentials. The second is to design high order numerical methods to evaluate one-dimensional delta function integrals with discontinuous kernel functions. Numerical examples are presented to verify that our method achieves the fourth order L 1-norm accuracy for computing multivalued physical observables of the one-dimensional linear Schrödinger equation with general discontinuous potentials.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, China

    Xin Wen

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  1. Xin Wen
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Correspondence to Xin Wen.

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Research supported in part by the Knowledge Innovation Project of the Chinese Academy of Sciences grant K3502012S8 and NSFC grant 10601062.

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Wen, X. A High Order Numerical Method for Computing Physical Observables in the Semiclassical Limit of the One-Dimensional Linear Schrödinger Equation with Discontinuous Potentials. J Sci Comput 42, 318–344 (2010). https://doi.org/10.1007/s10915-009-9326-9

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