On a selection principle for multivalued semiclassical flows

Agissilaos Athanassoulis (1) , Theodoros Katsaounis (2, 3) , Irene Kyza (4)

1 Department of Mathematics [Leicester]
2 Department of Applied Mathematics [Heraklion]
3 ICS-FORTH - Institute of Computer Science [FORTH, Heraklion]
4 EPM - School of Engineering, Physics and Mathematics

We study the semiclassical behaviour of solutions of a Schrödinger equation with a scalar potential 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 corresponds 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 Schrödinger 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.

Mots clés: semiclassical limit for rough potential; smoothed Wigner transform; selection principle; a posteriori error control

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Agissilaos Athanassoulis, Theodoros Katsaounis, Irene Kyza. On a selection principle for multivalued semiclassical flows. 2014. ⟨hal-00973519v1⟩