2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

This paper follows on from recent work of Evans, Levitin and Vassiliev on trapped modes for an infinitely long acoustic waveguide with a smooth obstacle [4]. In that paper the authors proved the existence of localised (L

Introduction There has been some interest recently to Laplacians on strips or layers. Such a system is trivial when the manifold is straight and the boundary conditions are translationally invariant, so there is a natural separation of variables. On the other hand, the spectral properties become nontrivial if the transverse modes are coupled, which can be achieved, e.g., if the manifold is bent, locally deformed, or coupled to another one [E S, DE, BGRS, E STV, EV1, EV2]. The interest stems from two sources. On the physical side, such operators with Dirichlet boundary conditions are used as models of various mesoscopic semiconductor structures. The corresponding solid--state literature is rather rich --- see [DE, E STV] for some references. On the other hand, bound states in systems with open geometries pose also mathematical questions such as the weak--coupling limit, validity of the semiclassical approximation, resonance scattering in such structures,<F55.

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Abstract not found

We study the behavior of a quantum particle confined to a hard–wall strip of a constant width in which there is a finite number N of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein’s formula, we analyze its spectral and scattering properties. The bound state–problem is analogous to that of point interactions in the plane: since a two–dimensional point interaction is never repulsive, there are m discrete eigenvalues, 1 ≤ m ≤ N, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and infinite heigth, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point–perturbation array in the plane, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations.

. We study the elasticity operator in a semi-strip subject to free boundary conditions. In the case of zero Poisson ratio we prove the existence of a positive eigenvalue embedded in the essential spectrum. Physically, the eigenvalue corresponds to a "trapped mode", that is, a harmonic oscillation of the semi-strip localized near the edge. This effect, known in mechanics as the "edge resonance", has been extensively studied numerically and experimentally. Our result provides a mathematical justification. 1. Introduction In this paper we study a particular self-adjoint operator A with spectrum oe(A) = [0; +1) and prove the existence of a positive eigenvalue e . Our proof is based on the use of certain symmetries of A which allow us to identify suitable invariant subspaces. The crucial step is to choose an invariant subspace in such a way that the restriction of A to this subspace has essential spectrum separated from zero. Then the existence of an eigenvalue below this essential spectr...

. We consider a two-dimensional infinitely long acoustic waveguide formed by two parallel lines containing an arbitrarily shaped obstacle. The existence of trapped modes that are the eigenfunctions of the Laplace operator in the corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin & Vassiliev (1994) for obstacles symmetric about the centreline of the waveguide. In our paper we deal with the situation when the obstacle is shifted with respect to the centreline and study the resulting complex resonances. We are particularly interested in those resonances which are perturbations of (real) eigenvalues. We study how an eigenvalue becomes a complex resonance moving from the real axis into the upper half--plane as the obstacle is shifted from its original position. The shift of the eigenvalue along the imaginary axis is predicted theoretically and the result is compared with numerical computations. The total number of resonances lying inside a sequence of e...

The problem of finding necessary and sufficient conditions for the existence of trapped modes in waveguides has been known since 1943, [8]. The problem is the following: consider an infinite strip M in R 2 (or an infinite cylinder with the smooth boundary in R n). The spectrum of

A simple method of calculating eigenvalues and resonances in domains with infinite regular ends ∗