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Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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ON THE COMPARISON OF THE DIRICHLET AND NEUMANN COUNTING FUNCTIONS
, 812
"... 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), ..."
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Cited by 3 (0 self)
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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),
Existence of spectral gaps, covering manifolds and residually finite groups
, 2007
"... 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) ..."
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Cited by 2 (2 self)
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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, bandstructure 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.
NONLOCAL ROBIN LAPLACIANS AND SOME REMARKS ON A PAPER BY FILONOV
, 2009
"... The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we e ..."
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Cited by 1 (1 self)
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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype 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.
A shift between Dirichlet and Neumann spectrum for generalized linear elasticity
"... 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 b ..."
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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 LeviCivita 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 straintensor. 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
ON THE DENSITY OF STATES OF PERIODIC MEDIA IN THE LARGE COUPLING LIMIT
, 2001
"... 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 bandgap structure. We establish t ..."
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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 bandgap structure. We establish the asymptotic behavior of the density of states function in the bands. 1.
SOME INEQUALITIES AND ASYMPTOTIC FORMULAS FOR EIGENVALUES ON RIEMANNIAN MANIFOLDS
, 906
"... 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 con ..."
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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 onedimensional case. 1.