Volume 11, Issue 2
Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations

Przemyslaw Klosiewicz, Jan Broeckhove & Wim Vanroose

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

Published online: 2012-12

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  • 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. 

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@Article{CiCP-11-435, author = {}, title = {Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations}, journal = {Communications in Computational Physics}, year = {2012}, volume = {11}, number = {2}, pages = {435--455}, 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. 

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.121209.050111s}, url = {http://global-sci.org/intro/article_detail/cicp/7371.html} }
TY - JOUR T1 - Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations JO - Communications in Computational Physics VL - 2 SP - 435 EP - 455 PY - 2012 DA - 2012/12 SN - 11 DO - http://doi.org/10.4208/cicp.121209.050111s UR - https://global-sci.org/intro/article_detail/cicp/7371.html KW - AB -

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. 

Przemyslaw Klosiewicz, Jan Broeckhove & Wim Vanroose. (2020). Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations. Communications in Computational Physics. 11 (2). 435-455. doi:10.4208/cicp.121209.050111s
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