SEMINAIRE MERCREDI 22 OCTOBRE 2003
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Dimitri Yafaev
Université de Rennes

Titre: A particle in the Bio-Savart-Laplace magnetic field: explicit solutions

Summary: We consider the Schr\"odinger operator ${\bf H}=(i\nabla+A)^2 $ in the
space $L_2({\mathbb R}^3)$ with a magnetic potential $A $ created by
an infinite straight current. We perform a spectral analysis of the
operator ${\bf H}$ almost explicitly. In particular, we show that the
operator ${\bf H}$ is absolutely continuous, its spectrum has infinite
multiplicity and coincides with the positive half-axis. Then we find
the large-time behavior of solutions $\exp(-i{\bf H}t)f$ of the time
dependent Schr\"odinger equation. Equations of classical mechanics are
also integrated. Our main observation is that both quantum and
classical particles have always a preferable (depending on its charge)
direction of propagation along the current and both of them are
confined in the plane orthogonal to the current.