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

Abstract

We consider the Schrödinger operator $\mathbf{H}=(i\nabla +A{)}^2$ 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 $\mathbf{H}$ almost explicitly. In particular, we show that the operator $\mathbf{H}$ is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions $\exp (-i\mathbf{H}t)f$ of the time dependent Schrödinger 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.

6 pages