Commun. Comput. Phys., 11 (2012), pp. 435-455.


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

Przemyslaw Klosiewicz 1*, Jan Broeckhove 1, Wim Vanroose 1

1 Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, B-2020 Antwerpen, Belgium.

Received 12 December 2009; Accepted (in revised version) 5 January 2011
Available online 24 October 2011
doi:10.4208/cicp.121209.050111s

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 Schrodinger equation. We extend previous work on the subject [1] to systems of coupled equations. Bound and resonant states of the Schrodinger 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 Schrodinger equation are varied. We have applied this approach to a number of model problems with satisfying results.


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PACS: 03.65.Ge, 03.65.Nk, 82.20.Xr, 47.20.Ky
Key words: Resonances, numerical continuation, coupled channels, Schrodinger equation.

*Corresponding author.
Email: przemyslaw.klosiewicz@ua.ac.be (P. Klosiewicz)
 

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