Asymptotic Analysis - Volume 58, issue 3

Authors: Briet, Philippe | Raikov, Georgi | Soccorsi, Eric

Article Type: Research Article

Abstract: We consider a 2D Schrödinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H=H0 +V. First, we establish a Mourre estimate, and as a corollary prove that the singular continuous spectrum of H is empty, and any compact subset of the complement of the threshold set may contain at most a finite set of …eigenvalues of H, each of them having a finite multiplicity. Next, we introduce the Krein spectral shift function (SSF) for the operator pair (H, H0 ). We show that this SSF is bounded on any compact subset of the complement of the threshold set, and is continuous away from the threshold set and the eigenvalues of H. The main results of the article concern the asymptotic behaviour of the SSF at the thresholds, which is described in terms of the SSF for a pair of effective Hamiltonians. Show more

Keywords: Schrödinger operators, magnetic field, Mourre estimates, spectral shift function, effective Hamiltonians

Citation: Asymptotic Analysis, vol. 58, no. 3, pp. 127-155, 2008

Authors: Bartolucci, Daniele | Orsina, Luigi

Article Type: Research Article

Abstract: We extend the Harnack type inequality proved in C. R. Acad. Sci. Paris 315(2) (1992), 159–164, to the solutions of -div(A∇u)=Veu in Ω, with no boundary conditions. Here A is a symmetric, uniformly elliptic matrix and Ω⊂R2 is open and bounded. As an application we are able to generalize the quantization results of Ind. Univ. Math. J. 43(4) (1994), 1255–1270, to the uniformly elliptic case.

Keywords: Harnack type inequalities, Liouville equations, uniformly elliptic equations, renormalized solutions

Citation: Asymptotic Analysis, vol. 58, no. 3, pp. 157-169, 2008

Authors: Amirat, Youcef | Touzani, Rachid

Article Type: Research Article

Abstract: We derive a mathematical model for eddy currents in two-dimensional geometries where the conductors are thin domains. We assume that the current flows in the x3 -direction and the inductors are domains with small diameters of order O(ε). The model is derived by taking the limit ε→0. A convergence rate of O(εα ) with 0<α<1/2 in the L2 -norm is shown as well as weak convergence in the W1,p spaces for 1<p<2.

Citation: Asymptotic Analysis, vol. 58, no. 3, pp. 171-188, 2008

Authors: Lu, Songsong

Article Type: Correction

Citation: Asymptotic Analysis, vol. 58, no. 3, pp. 189-190, 2008