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

我的建议致力于数学物理中产生的一些相互关联的问题，这些问题在现代分析的几个领域中起着核心作用，这些问题的解决将导致加拿大数学家在这些领域取得重大进展和领导地位。受奇异矢量场（漂移）扰动的布朗运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（SDE）的解。寻找漂移的最大可容许奇点，即使相应的SDE具有唯一解，吸引了许多数学家的兴趣，但仍远未完成。我打算实质性地推进这一搜索，达到临界阶奇点，通过应用新的算子理论技术，最近允许我第一次将漂移的临界点和临界超表面奇点结合起来（在这个问题的一个较弱的变体中，即构建一个相关的Feller过程）。接下来，我打算开发研究此类SDEs解所需的工具，包括相应Kolmogorov倒算符基本解的（非高斯）双面界。2。我继续致力于解决长期存在的在d=3或更高维R^d上薛定谔算子不存在正特征值的问题，以及薛定谔算子特征函数的唯一延拓（UC）的相关问题。我打算通过利用与UC的链接的算子理论技术（扩展了我与L. Shartser的早期工作）和不依赖于UC的技术（一种新方法），在缺乏正特征值的问题上获得新的、接近最优的结果。项目一和项目二的目标是将现代算子理论技术引入扩散过程和独特延续领域。3。最近，我（与A. Brudnyi）通过将Cartan定理A和B （Oka-Cartan理论）推广到这些代数的谱上的相干型束，建立了Stein流形覆盖上全纯函数的某些fr<s:1>代数内的复函子理论的基本结果（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，如Anderson局部化）。这项工作表明，Oka-Cartan理论作为研究d-bar方程的复函数理论的一种替代方法，在复流形的经典设置之外是有效的。我打算将已开发的技术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的代数，例如多盘上Hardy代数的某些子代数（获得了这些代数的一个冕定理），旨在确定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.