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We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

We consider an electron with an anomalous magnetic moment, <i>g</i> &gt; &gt; 2, confined to a plane and interacting with a nonhomogeneous magnetic field <i>B</i>, and investigate the corresponding Pauli Hamiltonian. We prove a lower bound on the number of bound states for the case when <i>B</i> is of a compact support and the related flux is <i>N</i> + <i></i>, <i></i>(0, 1]. In particular, there are at least <i>N</i> + 1 bound states if <i>B</i> does not change sign. We also consider the situation where the magnetic field is due to a localized rotationally symmetric electric current vortex in the plane. In this case the flux is zero; there is a pair of bound states for a weak coupling, and higher orbital-momentum spin-down states appearing as the current strength increases.