The purpose of this article is to define the radiation fields on asymptotically hyperbolic manifolds and to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was

Abstract. In the present paper we consider Riemannian coverings (X, g) → (M, g) with residually finite covering group Γ and compact base space (M, g). In particular, we give two general procedures resulting in a family of deformed coverings (X, gε) → (M, gε) such that the spectrum of the Laplacian ∆ (Xε,gε) has at least a prescribed finite number of spectral gaps provided ε is small enough. If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec ∆ (X,gε) has, in addition, band-structure and there is an asymptotic estimate for the number N(λ) of components of spec ∆ (X,gε) that intersect the interval [0, λ]. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results. 1.

Let NN(λ) and ND(λ) be the counting functions of the Dirichlet and Neumann Laplacian on a domain Ω ⊂ R n. If λ is not a Dirichlet or Neumann eigenvalue then (*) NN(λ) = ND(λ) + g − (λ),

Introduction The operator of linear elasticity is a good example of an non scalar operator : its principal symbol is not an homothety. In the theory of Elasticity one studies the deformation due to displacement of solid bodies regarded as continuous media. Let (M; g) be a riemannian manifold with boundary of dimension m 2, and r the Levi-Civita covariant derivative. A deformation of the "shape" of M under an infinitesimal displacement given by the vector field X is then given by the Lie derivative : LX (g); the so called strain-tensor. Recall that the metric defines a scalar product on every tensor bundle, we shall note it (:; :) g . Its defines also a natural isomorphism between the tangent bundle and the cotangent bundle. Recall the musical symbols [BGM

Abstract Let Ω0 be a domain in the cube (0, 2π) n, and let χτ (x) be a function that equals 1 inside Ω0, equals τ in (0, 2π) n \ Ω0, and that is extended periodically to R n. It is known that, in the limit τ → ∞, the spectrum of the operator −∇χτ (x) ∇ exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands. 1.

Dedicated with great pleasure to Sergio Albeverio on the occasion of his 70th birthday Abstract. The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems. 1.

Abstract. In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case. 1.