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An Optimized Perfectly Matched Layer for the Schrödinger Equation

Published online by Cambridge University Press:  20 August 2015

Anna Nissen  and
Gunilla Kreiss
Anna Nissen*
Affiliation:
Department of Information Technology, Uppsala University, P.O. Box 337, SE-75105 Uppsala, Sweden
Gunilla Kreiss*
Affiliation:
Department of Information Technology, Uppsala University, P.O. Box 337, SE-75105 Uppsala, Sweden
*
Corresponding author.Email:anna.nissen@it.uu.se
Email:gunilla.kreiss@it.uu.se
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Abstract

We derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.

Keywords

35B6535P2535Q4165M0665M15Schrödinger equationperfectly matched layer

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 1 , January 2011 , pp. 147 - 179
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Hagstrom, T., Radiation boundary condition for the numerical simulation of waves, Acta. Numerica., 8 (1999), 47–106.CrossRefGoogle Scholar
[2]Hagstrom, T., New results on absorbing layers and radiation boundary conditions, Topics in computational wave propagation, Lecture Notes in Comput. Sci. Engrg., 31 (1999), 1–42.Google Scholar
[3]Givoli, D., Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 1–29.Google Scholar
[4]Serban, I., Werschnik, J. and Gross, E. K. U., Optimal control of time-dependent targets, Phys. Rev. A., 71 (2005), 053810.Google Scholar
[5]Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185–200.Google Scholar
[6]Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729–796.Google Scholar
[7]Jiang, S. and Greengard, L., Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions, Comm. Pure. Appl. Math., 61 (2008), 261–288.CrossRefGoogle Scholar
[8]Sjoögreen, B. and Petersson, N. A., Perfectly matched layers for Maxwell’s equations in second order formulation, J. Comput. Phys., 209 (2005), 19–46.Google Scholar
[9]Kormann, K., Holmgren, S. and Karlsson, H. O., Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian, J. Chem. Phys., 128 (2008), 184101-1–184101-11.Google Scholar
[10]Moiseyev, N., Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Phys. Rep., 302 (1998), 211–293.Google Scholar
[11]Karlsson, H. O., Accurate resonances and effective absorption of flux using smooth exterior scaling, J. Chem. Phys., 109 (1998), 9366–9371.Google Scholar
[12]Chew, W. C. and Weedon, W. H., A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Micro. Opt. Tech. Lett., 7 (1994), 599–604.Google Scholar
[13]Sommerfeld, T. and Tarantelli, F., Subspace iteration techniques for the calculation of resonances using complex symmetric Hamiltonians, J. Chem. Phys., 112 (1999), 2106–2110.Google Scholar
[14]Kreiss, H. O. and Lorenz, J., Initial-Boundary Value Problems and the Navier-Stokes Equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004.Google Scholar
[15]Kormann, K. and Nissen, A., Error control for simulations of a dissociative quantum system, Proceedings of ENUMATH 2009, the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsala, Sweden, June 29–July 3, 2009.Google Scholar
[16]Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004.Google Scholar
[17]Gustafsson, B., Kreiss, H-O. and Oliger, J., Time-Dependent Problems and Difference Methods, John Wiley & Sons, Inc., 1995.Google Scholar