Mathematical Sciences: Some Problems in Partial DifferentialEquations

将针对以下几个领域出现的问题开展工作： 偏微分方程 将努力 描述椭圆型微分算子的谱， 数学和物理学的基本问题。 特别 将注意力放在多粒子薛定谔算子上。 在这方面，将重点放在预测 N体原子束缚态的有限性与无限性 薛定谔算子的几何方法。 这些运营商 用无限重的原子核和N - 1来模拟原子 围绕原子核运动的电子。 工作也将完成调查的光谱和 散射理论的某些约化波方程， 通过分离时间变量从声学方程获得。 重点将放在波动方程与奇异 系数是强不连续的或不表现 在无穷远。 这项研究的应用可能是 波传播分析，例如发生在声学中的波传播分析 显像 这项工作的第三个方向是处理 经典Muntz逼近定理的建立 积极的措施。 解决这个问题的方法是 长期的猜想将使用最近的工作， 相关不连续系数波动方程边界 价值问题。 这个基本问题的解决方案将 证明了理论中所谓的“砰砰原理” 热方程的时间最优控制也是有效的。

Work will be done on problems arising in several areas of partial differential equations. Efforts will be made to characterize the spectrum of an elliptic differential operator, a fundamental question in both mathematics and physics. Special attention will be given to multiparticle Schrodinger operators. In this context, focus will be placed on predicting the finiteness or infiniteness of bound states for N-body atomic Schrodinger operators using geometric methods. These operators model an atom with an infinitely heavy nucleus and N - 1 electrons moving about the nucleus. Work will also be done investigating the spectral and scattering theory for certain reduced wave equations which are obtained from acoustic equations by separation the time variable. Emphasis will be placed on wave equations with singular coefficients which are strongly discontinuous or do not behave well at infinity. Applications of this research may be made to the analysis of wave propagation such as occurs in acoustic imaging. A third direction this work will take deals with the establishment of the classical Muntz approximation theorem for sets of positive measure. The approach to resolving this longstanding conjecture will use recent work relating properties of an associated discontinuos coefficient wave equation boundary value problem. A solution to this basic question will demonstrate that the so-called bang-bang principle in the theory of time optimal control for the heat equation is also valid.