Asymptotic Analysis

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.

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)=Ve^u 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.

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(ε^α) with 0<α<1/2 in the L²-norm is shown as well as weak convergence in the W^1,p spaces for 1<p<2.