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We consider the three-dimensional Laplacian with a magnetic field created byan infinite rectilinear current bearing a constant current. The spectrum of theassociated hamiltonian is the positive half-axis as the range of an infinity ofband functions all decreasing toward 0. We make a precise asymptotics of theband function near the ground energy and we exhibit a semi-classical behavior.We perturb the hamiltonian by an electric potential. Helped by the analysis ofthe band functions, we show that for slow decaying potential, an infinitenumber of negative eigenvalues are created whereas only finite number ofeigenvalues appears for fast decaying potential. Our results show differentborderline type conditions that in the case where there is no magnetic field.

On the ground state of the Laplacian in presence of a magnetic field created by a rectilinear current