On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current

Abstract

We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated Hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band functions near the ground state energy and we exhibit a semi-classical behavior. We perturb the Hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appear for fast decaying potential. The criterion about finiteness depends essentially on the decay rate of the potential with respect to the distance to the wire.