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

分析和偏微分方程问题的算子理论方法

• 批准号: RGPIN-2017-05567
• 负责人: Kinzebulatov, Damir
• 金额: $ 3.06万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759464
• 关键词: Operator; theoretic; approach; problems; Analysis

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.

我的提案致力于解决数学物理学中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发挥着核心作用，其解决方案将导致加拿大数学家在这些领域中取得重大进展并发挥领导作用。 I. 受扰动的布朗运动。奇异向量场（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，搜索漂移的最大允许奇异性。也就是说，相应的 SDE 具有独特的解决方案，吸引了许多数学家的兴趣，但仍远未完成，我打算通过应用最近允许的新算子理论技术来大幅推进这一搜索，达到临界阶奇点。我第一次将漂移的临界点和临界超曲面奇点结合起来（在这个问题的较弱变体中，即构建相关的费勒过程）接下来，我打算开发研究此类解决方案所需的仪器。 SDE，包括相应 Kolmogorov 后向算子 II 的基本解的（非高斯）两侧界，我继续致力于解决 R^d 上薛定谔算子缺乏正特征值的长期存在的问题。在维度 d=3 或更高的情况下，以及薛定谔算子本征函数的唯一连续（UC）的相关问题，我打算在缺少 的问题上获得新的、接近最优的结果。通过利用使用 UC 链接的算子理论技术（扩展了我与 L. Shartser 的早期工作）以及不依赖于 UC 的技术（一种新方法）来获得正特征值。 II 是将现代算子理论技术引入扩散过程和唯一延拓领域 III 最近，我（与 A. Brudnyi 一起）在某些 Fréchet 代数中建立了复函数理论的基本结果。通过将嘉当定理 A 和 B（奥卡嘉当理论）扩展到这些代数谱上的相干型滑轮，研究斯坦因流形覆盖上的全纯函数（模型示例：全纯几乎周期函数，出现在分析和数学物理的各种问题中） ，例如在安德森定位中）这项工作表明，奥卡嘉当理论作为研究 d 杆方程的复函数理论方法，是超越经典的。我打算将所开发的技术扩展到全纯函数的代数，这些函数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，例如多盘上的哈代代数的某些子代数（获得这些代数），旨在确定奥卡嘉当理论的“自然域”。