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Approximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrödinger Equation

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Domain Decomposition Methods in Science and Engineering

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 40))

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Summary

We present a way to efficiently treat the well-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: firstly, to derive a discrete transparent boundary condition (DTBC) based on the Crank-Nicolson finite difference scheme for the governing equation. And, secondly, to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution. We illustrate the efficiency of the proposed method on several examples.

A much more detailed version of this article can be found in Arnold et al. [2003].

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References

  • A. Arnold. Numerically absorbing boundary conditions for quantum evolution equations. VLSI Design, 6:313–319, 1998.

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  • A. Arnold, M. Ehrhardt, and I. Sofronov. Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability. Comm. Math. Sci., 1:501–556, 2003.

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  • M. Ehrhardt and A. Arnold. Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma, 6:57–108, 2001.

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  • I. Sofronov. Artificial boundary conditions of absolute transparency for twoand threedimensional external time-dependent scattering problems. Euro. J. Appl. Math., 9:561–588, 1998.

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Author information

Authors and Affiliations

  1. Institut für Numerische Mathematik, Universität Münster, Einsteinstr. 62, D-48149, Münster, Germany

    Anton Arnold

  2. Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, D-10623, Berlin, Germany

    Matthias Ehrhardt

  3. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    Ivan Sofronov

Authors
  1. Anton Arnold
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  2. Matthias Ehrhardt
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  3. Ivan Sofronov
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Editor information

Editors and Affiliations

  1. Moffett Field, CA

    Timothy J. Barth

  2. Bonn

    Michael Griebel

  3. New York

    David E. Keyes  & Tamar Schlick  & 

  4. Espoo

    Risto M. Nieminen

  5. Leuven

    Dirk Roose

  6. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

    Ralf Kornhuber

  7. Department of Mathematics, University of Houston, Philip G. Hoffman Hall 651, 77204-3008, Houston, TX, USA

    Ronald Hoppe

  8. Dassault Aviation, 78 Quai Marcel Dassault Cedex 300, 92552, St. Cloud, France

    Jacques Périaux

  9. Laboratoire Jacques-Louis Lions, Université Paris VI, 175 Rue de Chevaleret, 75013, Paris, France

    Olivier Pironneau

  10. Courant Institute of Mathematical Science, New York University, Mercer Street 251, 10012, New York, USA

    Olof Widlund

  11. Department of Mathematics Eberly College of Science, Pennsylvania Sate University, McAllister Building 218, 16802, University Park, PA, USA

    Jinchao Xu

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© 2005 Springer-Verlag Berlin Heidelberg

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Arnold, A., Ehrhardt, M., Sofronov, I. (2005). Approximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrödinger Equation. In: Barth, T.J., et al. Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26825-1_10

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