On a selection principle for multivalued semiclassical flows

Athanassoulis, Agissilaos and Katsaounis, Theodoros and Kyza, Irene (2014) On a selection principle for multivalued semiclassical flows. (Submitted)

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Abstract

We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture.

Item Type: Article
Subjects: Q Science > QA Mathematics
Depositing User: prof Theodoros Katsaounis
Date Deposited: 07 Apr 2014 08:22
Last Modified: 03 Nov 2017 00:32
URI: http://preprints.acmac.uoc.gr/id/eprint/303

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