Partial Differential Equations and Complex Variables

提案DMS-9801555 Francois Treves研究项目摘要 F.特里夫斯计划在未来几年内进行他的研究， 以下三个不同的领域：1。时间依赖的全局参数 非欧空间上的Schroedinger方程 黎曼度量这是与S的合作。Chanillo（Rutgers University）.的 第一步是在没有势能的情况下构造一个参数。下一 第一步是对一个势做同样的事情，它的导数在 无穷大在一个合适的速度（与类似的contions的度量）。他们 提出用参数化描述临界集和传播 奇点（在Lbesgue和Sobolev空间），特别注意 被困测地线的存在。2. Dolbeault上同调的边值问题 班特里夫斯正在完成一项研究计划，科达罗 （巴西圣保罗大学）和S. Gindikin（Rutgers University）.目标 有几个：获得二阶解的积分公式 双曲和超双曲偏微分方程;定义 广义函数空间（实际上是超函数）非常适合于 保留这些方程的经典群的表示;奠定 “解析可拓学”研究更复杂领域的基础 而不是楔子。3.二阶线性偏微分方程解的解析性 形式为“（解析的，真实的）平方和”的微分方程 向量场最近特里夫斯制定了一个必要的和充分的 这样的属性有效的条件;他计划使用微局部分析 来证明这个猜想 薛定谔方程的解是量子力学中的基本元素。 量子物理学中粒子状态的“波描述”。的 Chanillo和Treves试图构建的目的是提供明确的 解决薛定谔方程的公式-公式，使之成为可能 为了非常精确地描述粒子在大范围内的行为（以便 考虑到遥远的“辐射”）和“弯曲”空间（以便 考虑对粒子轨迹的特殊约束）。 Treves关于边界值的工作旨在打造新的计算工具， 再次明确的公式，这可以在理论中使用的氡和 X射线变换（位于层析成像的理论基础）。 最后，给出了向量平方和类微分方程 场”通过它们的解描述了各种布朗运动和随机过程（它们已经被广泛研究 概率理论家）。他们的解的解析性是一个精确的 数学公式的巨大平滑，以及表明的可能性， 计算它们的一些性质的方法。

Proposal DMS-9801555 Abstract of the Research Project of Francois Treves F. Treves plans to carry out his research in the coming years, in the following three distinct areas: 1. Global parametrices for the time-dependent Schroedinger equation on Euclidean space equipped with a non-Euclidean Riemannian metric. This is joint work with S. Chanillo (Rutgers Univ.). The first step is to construct a parametrix when there is no potential. The next step is to do the same with a potential, whose derivatives are decaying at infinity at a suitable rate (with similar contions on the metric). They propose to use the parametrix to describe the critical set and the propagation of singularities (in Lbesgue and Sobolev spaces), with special attention to the presence of trapped geodesics. 2. Boundary values of Dolbeault cohomology classes. Treves is completing a research program begun with P. D. Cordaro (Univ. of Sao Paulo, Brazil) and S. Gindikin (Rutgers Univ.). The objectives are several: to obtain integral formulas for the solutions of second-order hyperbolic and ultra-hyperbolic partial differential equations; to define spaces of generalized functions (actually hyperfunctions) well suited for the representations of the classical groups which preserve those equations; to lay the ground for the study of ``analytic extension" to domains more complicated than wedges. 3. Analyticity of the solutions of second-order linear partial differential equations of the form ``sums of squares of (analytic, real) vector fields". Recently Treves has formulated a necessary and sufficient condition for such a property to be valid; he plans to use microlocal analysis to initiate a proof of the conjecture. The solutions of the Schroedinger equation are the fundamental elements in the ``wave description" of the states of particles in Quantum Physics. The construction attempted by Chanillo and Treves is aimed at providing explicit formulas for solving the Schroedinger equation - formulas that make it possible to describe very precisely the behaviour of the particles in the large (so as to take into account distant ``radiation") and in ``curved" space (so as to take into account special constraints on the trajectories of the particles). The work of Treves on boundary values aims at forging new mathematicatools, and again explicit formulas, that could be used in the theory of the Radon and X-ray transforms (which lie at the theoretical foundations of tomography). Finally, the differential equations of the kind ``sums of squares of vector fields" describe, through their solutions, various kinds of Brownian motion and of random processes (they have been extensively studied by probability theorists). The analyticity of their solutions is a precise mathematical formulation of their great smoothness, as well as indicating the possibility of computing some of their properties in finitely many steps.