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Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

Published online by Cambridge University Press:  20 August 2015

Pauline Klein ,
Xavier Antoine ,
Christophe Besse  and
Matthias Ehrhardt
Pauline Klein*
Affiliation:
Institut Elie Cartan Nancy, Nancy-Université, CNRS UMR 7502, INRIA CORIDA Team, Boulevard des Aiguillettes B.P. 239, 54506 Vandœuvre-lès-Nancy, France
Xavier Antoine*
Affiliation:
Institut Elie Cartan Nancy, Nancy-Université, CNRS UMR 7502, INRIA CORIDA Team, Boulevard des Aiguillettes B.P. 239, 54506 Vandœuvre-lès-Nancy, France
Christophe Besse*
Affiliation:
Laboratoire Paul Painlevé, CNRS UMR 8524, Simpaf Project Team-Inria CR Lille Nord Europe, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Matthias Ehrhardt*
Affiliation:
Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaufistr. 20, 42119 Wuppertal, Germany
*
Email:pauline.klein@iecn.u-nancy.fr
Corresponding author.Email:xavier.antoine@iecn.u-nancy.fr
Email:christophe.besse@math.univ-lille1.fr
Email:ehrhardt@math.uni-wuppertal.de
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Abstract

We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

Keywords

35J1065M6065N30Absorbing boundary conditionsstationary Schrödinger equationsunbounded domainspatially dependent potentialground states computation

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Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1280 - 1304
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]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(4) (2008), 729–796.Google Scholar
[2]Antoine, X., Besse, C. and Klein, P., Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential, J. Comput. Phys., 228(2) (2009), 312–335.Google Scholar
[3]Antoine, X., Besse, C., Ehrhardt, M. and Klein, P., Modeling boundary conditions for solving stationary Schrödinger equations, Report Preprint 10/04 (http://www.math.uni-wupperta l.de), Lehrstuhl für Angewandte Mathematik und Numerische Mathematik.Google Scholar
[4]Bao, W., Chern, I.-L. and Lim, F. Y., Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates, J. Comput. Phys., 219(2) (2006), 836–854.CrossRefGoogle Scholar
[5]Bao, W., Cai, Y. and Wang, H., Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229(20) (2010), 7874–7892.CrossRefGoogle Scholar
[6]Bao, W., Li, H. and Shen, J., A Generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput., 31(5) (2009), 3685–3711.Google Scholar
[7]Bao, W. and Shen, J., A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates, J. Comput. Phys., 227(23) (2008), 9778–9793.Google Scholar
[8]Ledoux, V., Van, M. Daele and Vanden Berghe, G., MATSLISE: A MATLAB package for the numerical solution of Sturm-Liouville and Schrödinger equations, ACM Trans. Math. Softw., 31(4) (2005), 532–554.Google Scholar
[9]Ledoux, V., Ixaru, L. G., Rizea, M., Van Daele, M. and Vanden Berghe, G., Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited, Comput. Phys. Commun., 175(9) (2006), 612–619.CrossRefGoogle Scholar
[10]Ledoux, V., Study of Special Algorithms for Solving Sturm-Liouville and Schrödinger Equations, Ph.D. Thesis, Dept. of Applied Mathematics and Computer Science, Ghent University, 2007.Google Scholar
[11]Ben Abdallah, N., Degond, P. and Markowich, P. A., On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys., 48(1) (1997), 135–155.Google Scholar
[12]Arnold, A., Mathematical concepts of open quantum boundary conditions, Trans. Theory Stat. Phys., 30 (2001), 561–584.Google Scholar
[13]Lent, C. and Kirkner, D., The quantum transmitting boundary method, J. Appl. Phys., 67(10) (1990), 6353–6359.Google Scholar
[14]Kirkner, D., Lent, C. and Sivaprakasam, S., The numerical simulation of electron transmission through a two-dimensional quantum device by the finite element method, Int. J. Numer. Meth. Eng., 29 (1990), 1527–1537.Google Scholar
[15]Heinen, M. and Kull, H.-J., Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction, Phys. Rev. E, 79(5) (2009), 056709.Google Scholar
[16]Heinen, M. and Kull, H.-J., Numerical calculation of strong-field laser-atom interaction: an approach with perfect reflection-free radiation boundary conditions, Laser Phys., 20(3) (2010), 581–590.CrossRefGoogle Scholar
[17]Ehrhardt, M. and Mickens, R. E., Solutions to the discrete Airy equation: application to parabolic equation calculations, J. Comput. Appl. Math., 172(1) (2004), 183–206.Google Scholar
[18]Ehrhardt, M. and Zisowsky, A., Fast calculation of energy and mass preserving solutions of Schrödinger-Poisson systems on unbounded domains, J. Comput. Appl. Math., 187(1) (2006), 1–28.CrossRefGoogle Scholar
[19]Moyer, C., Numerov extension of transparent boundary conditions for the Schrödinger equation in one dimension, Phys, Am. J.., 72(3) (2004), 351–358.Google Scholar
[20]Moyer, C., Numerical solution of the stationary state Schrödinger equation using transparent boundary conditions, Comput. Sci. Eng., 8(4) (2006), 32–40.CrossRefGoogle Scholar
[21]Ehrhardt, M. and Zheng, C., Exact artificial boundary conditions for problems with periodic structures, J. Comput. Phys., 227(14) (2008), 6877–6894.Google Scholar
[22]Burgnies, L., Mècanismes de Conduction en Règime Ballistique dans les Dispositifs Electroniques Quantiques, Ph.D. thesis, Universitè des Sciences et Technologies de Lille, 1997.Google Scholar
[23]Frensley, W., Boundary conditions for open quantum driven far from equilibrium, Rev. Mod. Phys., 62 (1990), 745–791.Google Scholar
[24]Markowich, P., Ringhofer, C. and Schmeiser, C., Semiconductor Equations, Springer, New-York, 1990.Google Scholar
[25]Schmeiser, C., Computational methods for semiclassical and quantum transport in semiconductor devices, Acta. Numer., 3 (1997), 485–521.Google Scholar
[26]Taylor, M., Pseudodifferential Operators, Vol. 34 of Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1981.Google Scholar
[27]Chazarain, J. and Piriou, A., Introduction to the Theory of Linear Partial Differential Equations, North-Holland, Amsterdam/New-York, 1982.Google Scholar
[28]Shao, H. and Wang, Z., Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation, Comput. Phys. Commun., 180(1) (2009), 1–7.CrossRefGoogle Scholar
[29]Klein, P., Construction et Analyse de Conditions aux Limites Artificielles Pour des Equations de Schrödinger avec Potentiels et non Linèaritès, Ph.D. thesis, Nancy Universitè, France (http://www.iecn.u-nancy.fr/~klein/These_PK_hr.pdf), 2010.Google Scholar
[30]Gross, E., Structure of a quantized vortex in boson systems, Nuovo Cimento, 20(3) (1961), 454–477.Google Scholar
[31]Pitaevskii, L., Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13(2) (1961), 451–454.Google Scholar
[32]Bao, W., The nonlinear Schrödinger equation and applications in Bose-Einstein condensation and plasma physics, in: Dynamics in Models of Coarsening, Condensation and Quantization, Vol. 9 of IMS Lecture Notes Series, World Scientific, 2007, 215–255.Google Scholar
[33]Bao, W. and Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25(5) (2004), 1674–1697.Google Scholar
[34]Bao, W. and Tang, W., Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187(1) (2003), 230–254.Google Scholar
[35]Dehghan, M. and Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Commun., 181 (2010), 43–51.Google Scholar
[36]Thelwell, R., Carter, J. and Deconinck, B., Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrödinger equation, J. Phys. A, 39 (2006), 73–84.Google Scholar
[37]Carr, L., Clark, C. and Reinhardt, W., Stationary solutions of the one-dimensional nonlinear Schrödinger equation, I: case of repulsive nonlinearity, Phys. Rev. A, 62 (2000), 063610.CrossRefGoogle Scholar
[38]Carr, L., Clark, C. and Reinhardt, W., Stationary solutions of the one-dimensional nonlinear Schrödinger equation, II: case of attractive nonlinearity, Phys. Rev. A, 62 (2000), 063611.Google Scholar
[39]Bransden, B. and Joachain, C., Physics of Atoms and Molecules, Prentice Hall (Pearson Education Ltd, Harlow, England), 2003.Google Scholar
[40]Betcke, M. and Voss, H., Analysis and efficient solution of stationary Schrödinger equation governing electronic states of quantum dots and rings in magnetic feld, Report 143, TU Hamburg-Harburg (2010) submitted to Journal of Computational Physics.Google Scholar
[41]Zisowsky, A., Arnold, A., Ehrhardt, M. and Koprucki, T., Discrete transparent boundary conditions for transient k• p-Schrödinger equations with application to quantum-heterostructures, J. Appl. Math. Mech., 85(11) (2005), 793–805.Google Scholar
[42]Klindworth, D., Discrete Transparent Boundary Conditions for Multiband Effective Mass Approximations, Master’s thesis, Technische Universität Berlin, 2009.Google Scholar
[43]Odermatt, S., Luisier, M. and Witzigmann, B., Bandstructure calculation using the k•p method for arbitrary potentials with open boundary conditions, J. Appl. Phys., 97 (2005), 046104.Google Scholar
[44]Antoine, X., Besse, C. and Mouysset, V., Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math. Comput., 73(248) (2004), 1779–1799.Google Scholar