We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {H} has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer \mathsf {H} at infinity.

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph.

Various qualitative methods for analyzing the discrete spectrum of SchrOdinger operators have been elaborated during the last three decades-for a review see Eef. 1, Section XIII. 3. They are of a great physical interest, especially because they can provide us with an information about -)*the few-body and many-body Hamlltonians, for which the quantitative methods are usually difficult to be applied. lt is not strange, therefore, that new re3ulta of this type continue to appear.

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.