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.

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The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrdinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.

We consider a two-dimensional electron with an anomalous magnetic moment, <i>g</i>&gt;2, interacting with a nonzero magnetic field <i>B</i> perpendicular to the plane which gives rise to a flux <i>F</i>. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the \mathcal{O}\left( {r^{ - 2 - \delta } } ight) decay at infinity: we show that if \mathcal{O}\left( {r^{ - 2 - \delta } } ight) exceeds an integer <i>N</i>, there is at least \mathcal{O}\left( {r^{ - 2 - \delta } } ight) bound states. Furthermore, we prove that weakly coupled bound states exist under mild regularity assumptions also in the zero flux case.

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a one-particle BirmanSchwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.