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WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment

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Abstract

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.

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Authors and Affiliations

  1. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8-10, 1040, Wien, Austria

    Anton Arnold & Bernhard Ujvari

  2. Institut de Mathématiques de Bourgogne, Université de Bourgogne-Franche-Comté, 9 Avenue Alain Savary, Dijon, France

    Christian Klein

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  1. Anton Arnold
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  2. Christian Klein
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  3. Bernhard Ujvari
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Correspondence to Christian Klein.

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Communicated by Mechthild Thalhammer.

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The first author (AA) was supported by the FWF-doctoral school “Dissipation and dispersion in non-linear partial differential equations”, the bi-national FWF-project I3538-N32, and a sponsorship by Clear Sky Ventures. This work was supported by the ANR-FWF project ANuI. CK thanks for support by the isite BFC project NAANoD, the ANR-17-EURE-0002 EIPHI and by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE 2017 Grant Agreement No. 778010 IPaDEGAN.

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Arnold, A., Klein, C. & Ujvari, B. WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment. Bit Numer Math 62, 1–22 (2022). https://doi.org/10.1007/s10543-021-00868-x

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