Asymptotic Analysis - Volume 29, issue 1

Authors: Gohberg, I. | Kaashoek, M.A. | Sakhnovich, A.L.

Article Type: Research Article

Abstract: Scattering problems are solved for a canonical differential system with a pseudo‐exponential potential v. The latter means that v is defined in terms of a triple consisting of an n×n matrix β and two n×m matrices γ1 and γ2 satisfying β* −β=iγ2 γ2  * , via the formula v(x)=−2iγ1  * eixα* Σ(x)−1 eixα γ, γ=−(γ1 +iγ2 ), where α=β−γ1 γ2 * and the matrix function Σ is given by Σ(x)=In +∫0 x Λ(t)Λ(t)*  dt, Λ(x)=[e−ixα γ1 eixα γ]. Such a potential may not be summable. Explicit formulas are presented for the scattering function and the …reflection coefficient of the system, and for other functions defined in terms of the asymptotic properties of the fundamental solution of the corresponding differential system. The corresponding inverse problems are also solved explicitly. The state space method from algebraic system theory is used as a basic tool. The results extend those in [11, 2, 5, 4]. Show more

Citation: Asymptotic Analysis, vol. 29, no. 1, pp. 1-38, 2002

Authors: Aramaki, Junichi

Article Type: Research Article

Abstract: We discuss semi‐classical analysis for the eigenvalues of the Schrödinger operator on ${\mathbb{R}}$ d with magnetic and electoric potentials. We consider the case where the electric potential vanishes precisely of even order at the wells. Our final aim is to clarify that when the operator has a non‐degenerate eigenvalue and then to get the complete asymptotics of the eigenvalue in the semi‐classical sense. Moreover, we give an example which seems to be new.

Citation: Asymptotic Analysis, vol. 29, no. 1, pp. 39-68, 2002

Authors: Motron, Mélissa

Article Type: Research Article

Abstract: In this article, we prove that the best constant for the Sobolev trace map from W1,1 ($\mathbb{R}$ N−1 ×$\mathbb{R}$ + ) into L1 ($\mathbb{R} $ N−1 ×{0}) is 1. We deduce from it that when Ω is a bounded open set whose boundary is piecewise C1 , the best first constant for the Sobolev trace map γ from W1,1 (Ω) into L1 (∂Ω) is also 1, in the sense that: for any ε>0, there exists Bε >0 such that for any u in W1,1 (Ω), ∫∂Ω |u|≤(1+ε)∫Ω |∇u|+Bε ∫Ω |u|. Independently, we prove that the best …second constant for Sobolev trace map from W1,1 (Ω) into L1 (∂Ω) is |∂Ω|/|{Ω}| (where |∂Ω| denotes the (N−1)‐dimensional measure of ∂Ω and |{Ω}| the N‐dimensional measure of Ω) when Ω is a connected open bounded set; in other words, there exists A>0 such that for any u in W1,1 (Ω): ∫∂Ω |u|≤A∫Ω |∇u|+ $\dfrac{|\partial{\Omega}|}{|{\Omega}|}$ ∫Ω |u|. Show more

Citation: Asymptotic Analysis, vol. 29, no. 1, pp. 69-90, 2002