We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.

Scattering on compact manifolds with infinitely thin hornsJochen BRÜNING and Vladimir GEYLER IntroductionIn the paper [25] L. D. Faddeev initiated the investigation of the quantum mechanical scattering on manifolds of constant negative curvature with cusps (sometimes also called "horns" [38]); further developments of this theory are presented e.g. in [20], [26], [53], [68]. It is interesting to note that an explicit expression for the reflection coefficient in the case of one horn was obtained earlier by R. Godement [34]. Note also that M. G. Gutzwiller has revealed a relation between the scattering theory on manifolds with horns and the description of chaotic behavior of quantum systems [38], [39], [40].If we imagine the width of the horns tending to zero, then we obtain a so-called hedgehog-shaped topological space (or "horned manifold"). Strictly speaking, we consider the limit of a family of horned spaces in the sense of the Hausdorff-Gromov distance [36]. The simplest specimen of such a manifold is the Euclidean plane with an attached half-line. The quantum mechanical scattering in this system has been investigated for the first time by P. Exner and P.Šeba [21]; in [23] these authors consider a compact plane domain with a half-line glued to it. A series of significant physical applications of the corresponding results as well as an intensive bibliography related to the subject in question may be found in [24]; we may add that the considered problem is also connected with the scattering on graphs In this paper we consider the quantum mechanical scattering in a hedgehog-shaped space which is constructed by gluing a finite number of half-lines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum

The effect of the surface curvature on the magnetic moment and persistent currents in twodimensional (2D) quantum rings and dots is investigated. It is shown that the surface curvature decreases the spacing between neighboring maxima of de Haas -van Alphen (dHvA) type oscillations of the magnetic moment of a ring and decreases the amplitude and period of AharonovBohm (AB) type oscillations. In the case of a quantum dot, the surface curvature reduces the level degeneracy at zero magnetic fields. This leads to a suppression of the magnetic moment at low magnetic fields. The relation between the persistent current and the magnetic moment is studied. We show that the surface curvature decreases the amplitude and the period of persistent current oscillations.

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.