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

我的建议致力于一些起源于数学物理的相互关联的问题，在现代分析的几个领域中发挥着核心作用，这些问题的解决将导致加拿大数学家在这些领域的重大进步和领导。*I.被奇异向量场扰动的布朗运动(漂移)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(SDE)的解。寻找漂移的最大允许奇点，即使相应的SDE有唯一解，引起了许多数学家的兴趣，但仍远未完成。我打算通过应用新的算子论技术，第一次将漂移的临界点和临界超曲面奇点结合起来(在这个问题的一个较弱的变体中，即构造一个相关的Feller过程)，大大推进这一探索，达到临界阶奇性。接下来，我打算发展研究这类随机微分方程解所需的工具，包括相应的Kolmogorov向后算子基本解的(非高斯)双边界。*II.我继续致力于解决R^d上的薛定谔算子在d=3或更高维时不存在正本征值的问题，以及与此相关的薛定谔算子特征函数的唯一连续性(UC)问题。我打算通过利用一种使用到UC的链接的算符理论技术(扩展了我与L.Shartser的早期工作)和一种不依赖UC的技术(一种新方法)，在缺乏正本征值的问题上获得新的、接近最优的结果。*项目I和II的目标是将现代算符理论技术引入扩散过程和唯一连续的领域。*III.最近，I(与A.Brudnyi)通过将Cartan定理A和B(Oka-Cartan理论)推广到这些代数(模型例子：全纯概周期函数，出现在各种分析和数学物理问题中，例如在Anderson局部化中)，在Stein流形的覆盖上的全纯函数的某些Fréchet代数上建立了复函数论的基本结果。这项工作表明，Oka-Cartan理论作为研究d-bar方程的复函数论的另一种方法，在复流形的经典设置之外是有效的。我打算将所发展的技术推广到在某种意义上具有相似的局部结构但具有不同的全局结构的全纯函数的代数，例如多圆盘上的Hardy代数的某些子代数(得到这些代数的一个冠状定理)，目的是确定Oka-Cartan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.*****