Partial Differential Equations and Complex Variables

弗朗索瓦·特里夫斯的研究项目摘要F.特里夫斯计划在未来几年开展他的研究，在以下三个不同的领域：1。欧几里得空间上具有非欧几里得黎曼度量的时变薛定谔方程的全局参数。这是与S. Chanillo（罗格斯大学）的合作。第一步是在没有势的情况下构造一个参数。下一步是对一个势做同样的事情，它的导数以一个合适的速率在无穷处衰减（在度规上有类似的条件）。他们建议使用参数来描述临界集和奇点的传播（在Lbesgue和Sobolev空间中），特别注意被困测地线的存在。2. Dolbeault上同类的边值。特里夫斯正在完成一个由Cordaro博士（巴西圣保罗大学）和S. Gindikin博士（罗格斯大学）共同发起的研究项目。目标有几个：得到二阶双曲型和超双曲型偏微分方程解的积分公式；定义广义函数（实际上是超函数）的空间，这些空间适合于保留这些方程的经典群的表示；为研究比楔形更复杂的域的“解析扩展”奠定基础。3. “（解析实）向量场平方和”形式的二阶线性偏微分方程解的解析性。最近，特里夫斯提出了这一性质成立的充分必要条件；他计划用微局部分析来证明这一猜想。薛定谔方程的解是量子物理学中粒子状态“波动描述”的基本要素。Chanillo和Treves所尝试的构造旨在为求解薛定谔方程提供明确的公式——这些公式可以非常精确地描述粒子在大空间（以便考虑到遥远的“辐射”）和“弯曲”空间（以便考虑到粒子轨迹上的特殊约束）中的行为。特里夫斯在边界值方面的工作旨在建立新的数学工具，以及可以用于氡和x射线变换理论的显式公式（这是断层扫描的理论基础）。最后，“向量场平方和”一类的微分方程通过其解描述了各种布朗运动和随机过程（它们已被概率理论家广泛研究）。它们的解的可解析性是它们巨大平滑性的精确数学表述，也表明了在有限多步中计算它们的某些性质的可能性。

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.