Operator-theoretic approach to problems of Analysis and Partial Differential Equations

我的建议是致力于一些相互关联的问题，起源于数学物理，发挥了核心作用，在几个领域的现代分析，其解决方案将导致一个重大的进展和领导加拿大数学家在这些领域。 I.布朗运动受奇异向量场（漂移）的扰动是许多数学物理模型的主要组成部分。它被构造为相应的随机微分方程（ε）的解。寻找漂移的最大可容许奇点，即使得相应的漂移具有唯一解，吸引了许多数学家的兴趣，但仍然远远没有完成。我打算通过应用新的算子理论技术来大大推进这一搜索，达到临界阶奇点，这些技术最近使我能够首次将漂移的临界点和临界超曲面奇点（在这个问题的一个较弱的变体中，即构建一个相关的费勒过程）联合收割机。接下来，我打算开发研究此类偏微分方程解所需的工具，包括相应Kolmogorov后向算子基本解的（非高斯）双侧界。 二.我继续致力于解决长期存在的问题的缺乏正特征值的薛定谔算子在R^d，在d=3或更高的维度，和相关的问题的唯一延续（UC）的本征函数的薛定谔算子。我打算利用算子理论的技术，使用UC的链接（扩展我以前的工作与L。Shartser），以及不依赖于UC的技术（新方法）。* 项目I和II的目标是将现代算子理论技术引入扩散过程和独特连续性领域。 三.最近，我（与A。Brudnyi）通过将Cartan定理A和B（Oka-Cartan理论）扩展到这些代数的谱上的相干型层（模型例子：全纯概周期函数，出现在分析和数学物理的各种问题中，例如在安德森局部化中），建立了Stein流形覆盖上的全纯函数的某些Fréchet代数内的复变函数理论的基本结果。这项工作表明，奥卡-嘉当理论，作为一种方法来研究的d-杆方程的替代复函数理论，是有效的，超越了经典的设置复杂的流形。我打算把已发展的技巧推广到在某种意义上具有相似局部结构但具有不同整体结构的全纯函数代数，例如多圆盘上的哈代代数的某些子代数（得到这些代数的冠定理），目的是确定奥卡-嘉当理论的“自然域”。

My proposal is devoted to a number of interrelated problems originating in Mathematical Physics, playing a central role in several areas of modern Analysis, whose solution would lead to a significant progress and leadership of Canadian mathematicians in these areas.*** I. A Brownian motion perturbed by a singular vector field (drift) is the principal component of many models of Mathematical Physics. It is constructed as a solution of the corresponding stochastic differential equation (SDE). The search for the maximal admissible singularities of the drift, i.e. such that the corresponding SDE has a unique solution, attracted the interest of many mathematicians, but is still far from being complete. I intend to substantially advance this search, reaching critical-order singularities, by applying new operator-theoretic techniques that recently allowed me to combine, for the first time, critical point and critical hypersurface singularities of the drift (in a weaker variant of this problem, i.e. constructing an associated Feller process). Next, I intend to develop the instruments needed to study solutions of such SDEs, including (non-Gaussian) two-sided bounds on the fundamental solution of the corresponding Kolmogorov backward operator.*** II. I continue to work towards solving the long-standing problem of absence of positive eigenvalues of Schroedinger operators on R^d, in dimension d=3 or higher, and related problem of unique continuation (UC) for eigenfunctions of Schroedinger operators. I intend to obtain new, close-to-optimal results on the problem of absence of positive eigenvalues by exploiting an operator-theoretic technique that uses the link to the UC (extending my earlier work with L. Shartser), and a technique that does not rely on the UC (a new approach).*** The goal of Projects I and II is to bring modern operator-theoretic techniques to the areas of diffusion processes and unique continuation.*** III. Recently, I (jointly with A. Brudnyi) established the basic results of complex function theory within certain Fréchet algebras of holomorphic functions on coverings of Stein manifolds by extending Cartan theorems A and B (Oka-Cartan theory) to coherent-type sheaves on the spectra of these algebras (model example: holomorphic almost periodic functions, arising in various problems of Analysis and Mathematical Physics, e.g. in Anderson localization). This work suggests that the Oka-Cartan theory, as an approach to complex function theory alternative to studying the d-bar equation, is valid beyond the classical setup of complex manifolds. I intend to extend the developed techniques to the algebras of holomorphic functions that have, in a sense, a similar local structure, but a different global structure, e.g. certain subalgebras of Hardy algebra on polydisk (obtaining a corona theorem for these algebras), aiming at determining the "natural domain" of Oka-Cartan theory.*****