Asymptotic Analysis - Volume 58, issue 1-2

Authors: Khenissi, Moez | Vodev, Georgi

Article Type: Research Article

Abstract: We obtain free of resonances regions for the elasticity system in the exterior of a strictly convex body in R3 with dissipative boundary conditions under some natural assumptions on the behaviour of the geodesics on the boundary. To do so, we use the properties of the parametrix of the Neumann operator constructed in Duke Math. J. 78 (1995), 677–714. As a consequence, we obtain time decay estimates for the local energy of the solutions of the corresponding mixed boundary value problems.

Keywords: boundary stabilization, exterior domain, local energy decay, Rayleigh resonances

DOI: 10.3233/ASY-2008-0869

Citation: Asymptotic Analysis, vol. 58, no. 1-2, pp. 1-16, 2008

Authors: Buslaev, Vladimir S. | Sulem, Catherine

Article Type: Research Article

Abstract: We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε ∂t ψ=A(t)ψ, A(t)=A0 +V(t), ψ(0)=ψ0 , in the limit ε→0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ0 is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t), we derive a formal asymptotic solution of the problem whose leading order has an explicit representation. A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical …pseudo-differential operator ℳ=M(t, iε ∂t ) with compact operator-valued symbol M(t, E)=V1 (t)(A0 −EI)−1 V2 (t), V(t)=V2 (t)V1 (t), and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schrödinger equation on the axis with the δ-function potential: A(t)=−∂xx +α(t)δ(x). Show more

Keywords: adiabatic evolution, Schrödinger equation, semiclassical analysis

DOI: 10.3233/ASY-2008-0874

Citation: Asymptotic Analysis, vol. 58, no. 1-2, pp. 17-45, 2008

Authors: Hassi, E. Ait Ben | Bouslous, H. | Maniar, L.

Article Type: Research Article

Abstract: In this paper, we prove the compactness of the difference between the thermoelasticity semigroup and its decoupled one on some irregular domains. This will be achieved by proving the norm continuity of this difference and the compactness of the difference between the resolvent of their generators and using Theorem 2.3 in Semigroup Forum 65 (2002), 58–70. An application to a thermoelastic system on a concrete irregular domain is given.

Keywords: thermoelasticity, semigroup, compactness, norm continuity and fractional powers

DOI: 10.3233/ASY-2008-0878

Citation: Asymptotic Analysis, vol. 58, no. 1-2, pp. 47-56, 2008

Authors: Alexandrova, Ivana | Bony, Jean-François | Ramond, Thierry

Article Type: Research Article

Abstract: We study the scattering amplitude for Schrödinger operators at a critical energy level, which is a unique non-degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiié, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.

Keywords: scattering amplitude, critical energy, Schrödinger equation

DOI: 10.3233/ASY-2008-0877

Citation: Asymptotic Analysis, vol. 58, no. 1-2, pp. 57-125, 2008