Mathematical Sciences: Eigenvalues of Elliptic Operators in Gaps of the Essential Spectrum

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9401417 Hempel The research will focus on magnetic phenomena, starting with situations where a thin layer of solid matter is locally penetrated by a strong magnetic field. The mathematical problem is to analyze the eigenvalues of the perturbed Schrodinger operator in the gaps of the essential spectrum. The perturbation occurs as a first order differential operator. A portion of the project involves numerical computations of the eigenvalues. A second project is to obtain a model for crystals with impurities. In an attempt to be realistic, these models will consider the case of a random distribution of impurities. This project involves research in mathematical physics. In mathematical physics one has to frequently analyze the effect of local perturbations in a homogeneous or periodic reference material. An important example is provided by solid state physics where impurities in insulators or semi-conductors lead to new energy levels, so-called impurity levels. These impurity levels are basic for the quantum mechanical theory of the color of crystals; in particular, lasers operate on such impurity levels. Similarly these impurity levels affect the conductivity properties of doped semi-conductors. These impurity levels are studied mathematically by analyzing eigenvalues of perturbed differential equations. ***