Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T10:38:47.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations

Published online by Cambridge University Press:  20 August 2015

Przemysław Kłosiewicz ,
Jan Broeckhove  and
Wim Vanroose
Przemysław Kłosiewicz*
Affiliation:
Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, B-2020 Antwerpen, Belgium
Jan Broeckhove
Affiliation:
Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, B-2020 Antwerpen, Belgium
Wim Vanroose
Affiliation:
Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, B-2020 Antwerpen, Belgium
*
*Corresponding author.Email:[email protected]
Get access

Abstract

In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrödinger equation. We extend previous work on the subject [1] to systems of coupled equations.

Bound and resonant states of the Schrödinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schrödinger equation are varied. We have applied this approach to a number of model problems with satisfying results.

Keywords

03.65.Ge03.65.Nk82.20.Xr47.20.KyResonancesnumerical continuationcoupled channelsSchrödinger equation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 435 - 455
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Broeckhove, J., Klosiewicz, P., Vanroose, W., Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation, Journal of Computational and Applied Mathematics 234 (4) (2010) 1238–1248. doi:10.1016/j.cam.2009.07.054.Google Scholar
[2]Fahy, F., Gardonio, P., Sound and Structural Vibration: Radiation, Transmission and Response, Academic Press, 2007.Google Scholar
[3]Bonnet-Ben Dhia, A.-S., Mercier, J.-F., Resonances of an elastic plate in a compressible confined fluid, Quarterly Journal of Mechanics and Applied Mathematics 60 (4) (2007) 397.Google Scholar
[4]Taylor, J. R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications, Inc., Mineola, New York, 2006.Google Scholar
[5]Newton, R.G., Scattering Theory of Waves and Particles, 2nd Edition, Texts and Monographs in Physics, Springer-Verlag, 1982.Google Scholar
[6]Seydel, R., Practical Bifurcation and Stability Analysis, From Equilibrium to Chaos, Springer-Verlag, 1994.Google Scholar
[7]Doedel, E. J., Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer-Verlag, Dordrecht (2007) 51–75.Google Scholar
[8]Allgower, E.L., Georg, K., Numerical Continuation Methods - An Introduction, Vol. 13 of Springer Series in Computational Mathematics, Springer-Verlag, 1990.Google Scholar
[9]Keller, H. B., Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of Bifurcation Theory (ed. Rabinowitz, P. H.) (1977) 159–384.Google Scholar
[10]Keller, J. B., Antman, S. (Eds.), Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, New York, 1969.Google Scholar
[11]Doedel, E.J., AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congressus Numerantium 30 (1981) 265–284.Google Scholar
[12]Auto-software for continuation and bifurcation problems in ordinary differential equations, version AUTO-07p available online (August 2007). URL http://indy.es.concordia.ca/autoGoogle Scholar
[13]Salinger, A.G., Bou-Rabee, N.M., Pawlowski, R.P., Wilkes, E.D., LOCA 1.1 Library Of Continuation Algorithms: Theory and Implementation Manual, Sandia National Laboratories P.O. Box 5800 Albuquerque, NM 87185-1111,2002.Google Scholar
[14]Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S., An overview of the trilinos project, ACM Transactions On Mathematical Software 31 (3) (2005) 397–423. doi:10.1145/1089014.1089021.Google Scholar
[15]Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Mestrom, W., Riet, A.M., Sautois, B., MATCONT and CL_MATCONT: Continuation toolboxes in MATLAB, Universiteit Gent, Belgium and Utrecht University, The Netherlands (2006).Google Scholar
[16]Henderson, M.E., Multiple parameter continuation: Computing implicitly defined k-manifolds, International Journal of Bifurcation and Chaos 12 (3) (2002) 451–476.CrossRefGoogle Scholar
[17]Johnson, B.R., New numerical methods applied to solving the one-dimensional eigenvalue problem, The Journal of Chemical Physics 67 (1977) 4086–4093.Google Scholar
[18]Johnson, B.R., The renormalized numerov method applied to calculating bound states of the coupled-channel Schrödinger equation, The Journal of Chemical Physics 69 (1978) 4678–4688.Google Scholar
[19]Vanroose, W, McCurdy, C.W, Rescigno, T.N., Interpretation of low-energy electron-CO2 scattering, Physical Review A 66 (2002) 032720.CrossRefGoogle Scholar
[20]Ledoux, V., Daele, M. Van, Berghe, G. Vanden, Matslise: A matlab package for the numerical solution of Sturm-Liouville and Schrödinger equations, ACM Transactions on Mathematical Software 31 (2005) 532.Google Scholar
[21]Ledoux, V., Daele, M. Van, Berghe, G. Vanden, CPM{P,N} methods extended for the solution of coupled channel Schrödinger equations, Computer Physics Communications 174 (2006) 357–370.Google Scholar
[22]Ledoux, V., Daele, M. Van, Berghe, G. Vanden, A numerical procedure to solve the multichannel Schrödinger eigenvalue problem, Computer Physics Communications 176 (2007) 191–199.Google Scholar
[23]MATSCS: Matlab package implementing the CPM{P,N} methods for the numerical solution of the multichannel Schrödinger eigenvalue problems. URL http://www.nummath.ugent.be/SLsoftwareGoogle Scholar
[24]Abramowitz, M., Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1972.Google Scholar
[25]Newton, R. G., Analytic properties of radial wave functions, Journal of Mathematical Physics 1 (4) (1960) 319–347. doi:10.1063/1.1703665.Google Scholar