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

小行星9401417 研究将集中在磁现象，从一个薄的固体物质层被强磁场局部穿透的情况开始。数学问题是分析扰动薛定谔算子在本质谱间隙中的本征值。扰动作为一阶微分算子发生。该项目的一部分涉及本征值的数值计算。第二个项目是获得杂质晶体的模型。 为了符合实际，这些模型将考虑杂质随机分布的情况。 这个项目涉及数学物理的研究。在数学物理中，人们必须经常分析均匀或周期性参考材料中局部扰动的影响。一个重要的例子是固态物理学，其中绝缘体或半导体中的杂质导致新的能级，即所谓的杂质能级。这些杂质水平是晶体颜色的量子力学理论的基础;特别是，激光器在这样的杂质水平上工作。 类似地，这些杂质水平影响掺杂半导体的导电性。通过分析扰动微分方程的本征值，对这些杂质能级进行了数学研究。 ***

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. ***