Given a complete noncompact surface embedded in R3, we consider the Dirichlet Laplacian in the layer that is defined as a tubular neighborhood of constant width about . Using an intrinsic approach to the geometry of , we generalize the spectral results of the original paper by Duclos et al. [Commun. Math. Phys. 223, 13 (2001)] to the situation when does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain .

We show that there is a family Schrdinger operators with scaled potentials which approximates the '-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.

Nous etudions Ie Laplacien sur une bande rectiligne plane, avec des conditions aux bords de Neuman sur un intervalle de largeur 2a Ie long de la frontiere et de Dirichlet ailleurs. Pour a suffisamment petit cet operateur a une valeur propre simple 6 (a); nous montrons quil existe deux nombres positifs c1, c2 tels que E (a)-(03C0/d) 2~-c2a4. La meme conclusion s applique pour une paire de bandes paralleles avec conditions aux bords de Dirichlet, de largeurs differentes et, sur la frontiere commune, une fenetre de largeur 2a. de lInstitut Henri Poincar-Physique theorique-0246-0211 Vol. 65/96/0!/$4.00/@ Gauthier-Villars

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or-type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.