We consider the Dirichlet Laplacian for astrip in \mathbb{R}^2 with one straight boundary and a width \mathbb{R}^2 , where \mathbb{R}^2 is a smooth function of acompact support with a length 2<i>b</i>. We show that in the criticalcase, \mathbb{R}^2 , the operator has nobound statesfor small \mathbb{R}^2 .On the otherhand, a weakly bound state existsprovided \mathbb{R}^2 . In thatcase, there are positive <i>c</i> ₁,<i>c</i> ₂ suchthat the corresponding eigenvalue satisfies \mathbb{R}^2 for all \mathbb{R}^2 sufficiently small.

We consider the Laplacian in a strip R(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a&gt;0. There is a sequence 0&lt;a1&lt;a2&lt; of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (aan)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.

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