Asymptotic Analysis - Volume 51, issue 1

Authors: Klein, M. | Rama, J. | Wüst, R.

Article Type: Research Article

Abstract: Let H0 be self-adjoint with E0 a possibly degenerate eigenvalue embedded in the continuous spectrum σc (H0 ) and ψ0 a normalized eigenfunction. For a certain class of perturbations W and H=H0 +W we investigate the asymptotics of the (naive) resonance state e−itH ψ0 in the limit W→0. This amplifies previous results of Merkli and Sigal.

Citation: Asymptotic Analysis, vol. 51, no. 1, pp. 1-16, 2007

Authors: Duyckaerts, Thomas

Article Type: Research Article

Abstract: This work is dedicated to the study of a linear model arising in fluid-structure interaction and introduced by Rauch, Zhang and Zuazua. The system is formed of a heat and a wave equation, taking place in two distinct domains, and coupled by transmission conditions at the interface of the domains. Two different transmission conditions are considered. In both cases, when the interface geometrically controls the wave domain, we show the quick polynomial decay of the energy for solutions with smooth initial data, improving the rate of decay obtained by the previous authors. The polynomial stability is deduced from an …optimal observability inequality conjectured in their work. The proof of this estimate mainly relies on a known generalized trace lemma for solutions of partial differential equations and the results of Bardos, Lebeau and Rauch on the control of the wave equation. Without the geometric condition, we show, using a Carleman inequality of Lebeau and Robbiano and an abstract theorem of N. Burq, a logarithmic decay for solution of the system with one of the two transmission conditions. This result improves the speed of decay obtained by Zhang and Zuazua, and is also optimal in some geometries. Show more

Keywords: asymptotic stability, non-uniform stability, observability, coupling, hyperbolic equations, parabolic equations, fluid-structure interaction

Citation: Asymptotic Analysis, vol. 51, no. 1, pp. 17-45, 2007

Authors: Gomes, Diogo A. | Valls, Claudia

Article Type: Research Article

Abstract: In this paper we investigate the quantum action problem using Wigner measures on the torus. We prove existence, study its main properties, and prove convergence to Mather measures in the semiclassical limit. We also indicate how to extend these techniques to the study of stochastic Mather measures.

Citation: Asymptotic Analysis, vol. 51, no. 1, pp. 47-61, 2007

Authors: Pushnitski, A. | Sloushch, V.

Article Type: Research Article

Abstract: Let H be the Stark operator in $\mathbb{R}^{3}$ and let V=V(x) be a non-negative potential which decays at infinity as |x|−l for a sufficiently large l>0. The spectral shift function for the pair of operators H+tV, H is studied in the asymptotic regime t→∞. The leading term of the asymptotics is obtained; the result is in agreement with the phase space volume considerations.

Keywords: Stark operator, spectral shift function, large coupling constant

Citation: Asymptotic Analysis, vol. 51, no. 1, pp. 63-89, 2007