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

Abstract

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.