Mathematical Sciences: Some Problems in Partial DifferentialEquations

将对偏微分方程的几个领域中出现的问题进行研究。我们将努力刻画椭圆微分算子的谱，这是数学和物理中的一个基本问题。我们将特别关注多粒子薛定谔算符。在这一背景下，重点将集中在用几何方法预测N体原子薛定谔算符束缚态的有限或无限。这些算符模拟了一个原子，该原子有一个无限重的原子核，N-1个电子在原子核周围运动。我们还将研究通过分离时间变量从声学方程中得到的某些约化波动方程的谱和散射理论。重点将放在具有奇异系数的波动方程上，这些波动系数在无穷远处是强烈不连续的或表现不佳的。这项研究可以应用于波传播的分析，例如在声成像中发生的波传播。第三个方向是建立经典的关于正测度集的Muntz逼近定理。解决这一长期猜想的方法将使用最近与相关间断系数波动方程边值问题的性质有关的工作。这一基本问题的解决将证明热方程时间最优控制理论中的所谓bang-bang原理也是有效的。

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.