arXiv 1012.01312010-12-22math-phmath.DSmath.MPphysics.comp-phquant-ph

耦合沟道薛定谔方程中共振态和束缚态的数值延拓

Numerical Continuation of resonances and bound states in coupled channel Schrödinger equations

Przemysław Kłosiewicz, Jan Broeckhove, Wim Vanroose

备注：: Accepted for publication in: Communications in Computational Physics (CICP)
发表信息：: Comm. Comput. Phys. 11 p 435 (2012)

摘要：

在这篇文章中，我们介绍了数值延拓方法和分歧理论，技术，找到他们的根源在动力系统的研究，跟踪量子力学薛定谔方程的束缚态和共振态的参数依赖性的问题。我们扩展以前的工作对这个问题的耦合方程组。薛定谔方程的束缚态和共振态可以通过S矩阵的极点来确定，S矩阵是一个可以从波函数的渐近形式导出的量。我们引入了一个正则化过程，本质上是将S-矩阵转化为它的逆，并提高其光滑性，从而使其适合数值延拓。当薛定谔方程中的参数变化时，这使我们能够自动跟踪束缚态和共振态。我们已经将这种方法应用于一些模型问题，取得了令人满意的结果。

英文摘要：

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 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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