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

  • Anton Arnold
  • Matthias Ehrhardt
  • Ivan Sofronov
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


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].


Transformation Rule Fast Calculation Discretization Error Convolution Kernel Absorb Boundary Condition 
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  1. A. Arnold. Numerically absorbing boundary conditions for quantum evolution equations. VLSI Design, 6:313–319, 1998.Google Scholar
  2. 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.MathSciNetGoogle Scholar
  3. M. Ehrhardt and A. Arnold. Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma, 6:57–108, 2001.MathSciNetGoogle Scholar
  4. I. Sofronov. Artificial boundary conditions of absolute transparency for twoand threedimensional external time-dependent scattering problems. Euro. J. Appl. Math., 9:561–588, 1998.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anton Arnold
    • 1
  • Matthias Ehrhardt
    • 2
  • Ivan Sofronov
    • 3
  1. 1.Institut für Numerische MathematikUniversität MünsterMünsterGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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