Mathematical Sciences: Partial Differential Equations of Mathematical Physics

工作将在局部问题上进行。 微分方程产生于不同的模型 物理现象。 工作重点是线性 三种主要类型的微分算子。 第一个目标是在早期成功的基础上进行扩展 在研究等谱Schrodinger算子时， 周期势（如果你知道两个算子有 同样的光谱，它们是如何相关的？）。 技术 应适用于一般自伴椭圆 具有周期系数的算子 鉴于 已知作用于光滑算子谱 函数在环面上，这是一个基本域， 周期晶格，研究人员想确定 操作员的系数特性。 两 值得研究的重要案例将是 Laplace-Beltrami算子与非线性系统的哈密顿量 电子在磁场中通过晶格运动， 产生电势的离子。 第二个推力将集中在初始值上 定义在区域上的双曲型方程问题 有角的。 主要研究人员成功地 提供沿着“边缘”的条件， 椭圆方程的解存在。 在 双曲线设置奇点不仅发生在 超曲面相遇，但它们也在一些 复杂的方式。 特别是滑翔鳐 在边缘反射和透射， 光线锥 这项工作的长期目标之一是 将为一些错误提供理由 从衍射理论。 最后一个涉及的领域涉及近似， 固体物理学 这项工作的来源在于 Laplace-Beltrami算子 中 超小长度标度的设置 满足含时薛定谔 方程精确到一阶。 这些数据包随后 哈密顿量的轨迹 至少这是 习惯性假设。 这项工作的一部分将是 对整个模型进行仔细分析。

Work will be done on problems of partial differential equations arising from models of diverse physical phenomena. The work focuses on linear differential operators of three primary types. The first goal will to expand on earler successes in studying isospectral Schrodinger operators with periodic potentials (if you know that two operators have the same spectrum, how are they related?). Techniques should be applicable to general self-adjoint elliptic operators with periodic coefficients. Given that the spectrum is known for an operator acting on smooth functions on a torus which is a fundamental domain for the period lattice, the investigators want to determine characteristics of the operator's coefficients. Two important cases worthy of study will be that of the Laplace-Beltrami operator and the hamiltonian for an electron moving in a magnetic field through a lattice of ions which produce the potential. A second thrust will concentrate on initial-value problems for hyperbolic equations defined in domains with corners. The principal investigators succeeded in providing conditions along the "edge" for smooth solutions of elliptic equations to exist. In the hyperbolic setting singularities not only occur where hypersurfaces meet but they also propagate in some complex manner. In particular, gliding rays are reflected and transmitted at the edge giving rise to cones of rays. One of the long-term goals of this work will be to provide justification to some conjectures from the theory of diffraction. A final area involved concerns approximation in solid state physics. The source of this work lies in the study of the Laplace-Beltrami operator. In a setting of exceeding small length scale one arrives at wave packets satisfying the time-dependent Schrodinger equation up to a precise order. These packets follow trajectories of a hamiltonian. At least that is customarily assumed. Part of this effort will be directed to a careful analysis of the entire model.