Asymptotic Analysis - Volume 68, issue 1-2

Authors: Raymond, Nicolas

Article Type: Research Article

Abstract: In this paper we are interested in the semiclassical estimates of the spectrum of the Neumann Laplacian in dimension 3. This work aims to present a complementary case to the one presented in the paper of Helffer and Morame in the case of constant magnetic field. More precisely, in the case when the magnetic field is variable and under the most generic condition for which boundary localizations can be observed, we prove a three terms upper bound for the lowest eigenvalue and establish some semiclassical behaviour of the spectrum.

Keywords: spectral theory, semiclassical analysis, Neumann Laplacian, magnetic field

DOI: 10.3233/ASY-2010-0978

Citation: Asymptotic Analysis, vol. 68, no. 1-2, pp. 1-40, 2010

Authors: Kian, Yavar

Article Type: Research Article

Abstract: We obtain global Strichartz estimates for the solution u of the wave equation ∂t 2 u−divx (a(t, x)∇x u)=0 with time-periodic metric a(t, x) equal to 1 outside a compact set with respect to x. We assume a(t, x) is a non-trapping perturbation and, moreover, we suppose that there are no resonances zj ∈C with |zj |≥1.

Keywords: time-dependent perturbation, non-trapping metric, Strichartz estimates

DOI: 10.3233/ASY-2010-0982

Citation: Asymptotic Analysis, vol. 68, no. 1-2, pp. 41-76, 2010

Authors: Elbaz, Claude

Article Type: Research Article

Abstract: For a scalar field propagating at light velocity c, we show that kinematic and dynamic properties of almost monochromatic standing waves, with frequency Ω(x, t)=ω±δΩ(x, t), where ω is constant, and δΩ(x, t)�ω, are formally identical with mechanical properties of matter. They are both described by equations with the same mathematical structure. The energy conservation stems from stability in time, while the variational principle stems from stability in space. In classical mechanics of a particle, the relativist equations correspond to the geometrical optics approximation as ω→∞. The quantum mechanical equations correspond to the wave optics approximation, in which wave homogeneous Fourier relations are …replaced by the material Heisenberg relations. Show more

Keywords: relativist mechanics, quantum mechanics, progressive and standing waves, almost progressive waves, almost standing waves, duality, variational principles

DOI: 10.3233/ASY-2010-0985

Citation: Asymptotic Analysis, vol. 68, no. 1-2, pp. 77-88, 2010

Authors: Shubov, Marianna A. | Rojas-Arenaza, Miriam

Article Type: Research Article

Abstract: The paper is devoted to rigorous spectral analysis of recently developed mathematical model of a double-walled carbon nanotube. The model is governed by a system of four partial differential equations describing vibrations of two Timoshenko beams coupled through distributed van der Waals forces. The system is equipped with a four-parameter family of nonconservative boundary conditions. The corresponding initial boundary-value problem has been reduced to an evolution equation in the state space. The dynamics generator is a nonselfadjoint matrix differential operator with purely discrete spectrum. It is shown that the entire spectrum asymptotically splits up into four spectral branches. Asymptotical representation …has been derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity. To prove the results, a two-step procedure involving construction of the left and right reflection matrices has been used. Show more

Keywords: carbon nanotubes, Timoshenko system, dynamics generator, matrix differential operator, reflection matrices, asymptotics of the spectrum

DOI: 10.3233/ASY-2010-0991

Citation: Asymptotic Analysis, vol. 68, no. 1-2, pp. 89-123, 2010