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An Accurate Approximation of Exponential Integrators for the Schrödinger Equation

Abstract

Numerical time propagation of semi-linear equations of the Schrödinger type can be performed by the use of exponential integrators. The main difficulty for efficient implementation of this type of schemes lies in the evaluation of \(\varphi \)-functions of a matrix argument. We develop a Chebyshev series approximation for these functions and propose a simple algorithm for the evaluation of the series coefficients. The domain of convergence of the series is consistent with the spectrum of Schrödinger type operators. This approximation is shown to be accurate and performs favorably in comparison to other state of the art methods for approximation of \(\varphi \)-functions.

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Acknowledgements

The author wishes to thank Adi Ditkowski for valuable discussions of the method presented in this paper.

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Correspondence to A. Y. Meltzer.

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Meltzer, A.Y. An Accurate Approximation of Exponential Integrators for the Schrödinger Equation. J Sci Comput 81, 1493–1508 (2019). https://doi.org/10.1007/s10915-019-01075-1

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  • DOI: https://doi.org/10.1007/s10915-019-01075-1

Keywords

  • Chebyshev expansion
  • Schrödinger equation
  • Exponential integrators
  • Explicit scheme
  • Bessel series