LEAPS-MPS: Elliptic theory for the Schrodinger operator

LEAPS-MPS：薛定谔算子的椭圆理论

基本信息

批准号：
2137743
负责人：
Blair Davey
金额：
$ 11.96万
依托单位：
Montana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137743&HistoricalAwards=false
关键词：
LEAPS MPS Elliptic theory Schrodinger

项目摘要

The Laplace equation is the prototypical second-order elliptic partial differential equation (PDE). Consequently, solutions to the Laplace equation, known as harmonic functions, are a fundamental component of PDE theory. But these functions are also important to many other areas of science and engineering, like complex analysis, harmonic analysis, geometry, physics, and engineering. As such, harmonic functions have been extensively studied and are well understood. While the Laplace equation models steady-state phenomena in a uniform environment, the world that we live in is not an isotropic vacuum. The mathematical equations that govern many natural phenomena like electromagnetism, astronomy, and fluid dynamics are often more complicated than Laplace’s equation. For example, the Schrodinger equation describes the behavior of quantum-mechanical waves, while its generalizations describe even more complex settings. Therefore, there is a need to understand the properties of solutions to such general elliptic PDEs. This project combines mathematical pursuits in harmonic analysis with the goal of promoting the inclusion and retention of a diverse mathematical community. The latter objective will be achieved through an orientation program for incoming graduate students along with extra-curricular mentorship programs.In this project, the PI will explore how and to what extent the presence of lower-order terms and variable coefficients affects the behavior of solutions to elliptic equations. With the Schrodinger equation serving as the standard example, these effects will be examined through the three distinct perspectives of unique continuation, homogenization, and solvability. Harmonic functions have the following unique continuation properties: locally, they cannot vanish to infinite order; and if defined globally, Liouville’s Theorem asserts that they cannot be bounded everywhere. Motivated by Landis’ conjecture, one facet of this program seeks to precisely quantify these kinds of local and global behaviors for solutions to generalized Schrodinger equations. By going further and considering elliptic equations with periodic coefficients, this program also explores the interplay between homogenization theory and unique continuation. Carleman estimates and complex analysis techniques will be combined with compactness arguments to accomplish this feat. Work on the solvability of the Dirichlet and Neumann boundary value problems for the Laplace equation led to a huge development in the theory of PDEs and harmonic analysis. The PI’s previous work will be used to explore the questions of solvability for general systems of elliptic PDEs with lower order terms, and further knowledge will be gained while bringing together ideas from distinct areas of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

拉普拉斯方程是典型的二阶椭圆偏微分方程（PDE）。因此，拉普拉斯方程的解决方案（称为谐波函数）是PDE理论的基本组成部分。但是，这些功能对于许多其他科学和工程领域也很重要，例如复杂分析，谐波分析，几何，物理和工程。因此，谐波功能已被广泛研究，并且被充分理解。虽然拉普拉斯方程在统一环境中模拟稳态现象，但我们生活的世界不是各向同性真空。控制许多自然现象（例如电磁，天文学和流体动力学）的数学方程通常比拉普拉斯方程更为复杂。例如，Schrodinger方程描述了量子力波的行为，而其概括描述了更复杂的设置。因此，有必要了解这种通用椭圆PDE的解决方案的特性。该项目将数学追求与谐波分析结合在一起，以促进和保留潜水员的数学社区的包容和保留。以后的目标将通过针对入学生的定向计划以及课外心态计划来实现。在该项目中，PI将探讨如何以及在何种程度上以及在何种程度上影响椭圆方程的解决方案的行为。通过Schrodinger方程作为标准示例，将通过独特的延续，同质化和解决性的三种不同观点来检查这些效果。谐波功能具有以下独特的延续属性：在本地，它们不能消失到无限顺序；如果在全球范围内定义，Liouville的定理断言它们在任何地方都不能被束缚。受兰迪斯（Landis）概念的促进，该计划的一个方面试图精确地量化这些类型的本地和全球行为，以解决通用的schrodinger方程的解决方案。通过进一步考虑并考虑具有定期系数的椭圆方程，该程序还探讨了均化理论与唯一延续之间的相互作用。卡尔曼估计和复杂的分析技术将与紧凑的参数相结合，以实现这一壮举。 dirichlet和neumann边界价值问题的可溶性的工作导致PDE和谐波分析理论的巨大发展。 PI先前的工作将用于探索具有较低阶段的椭圆形PDE的通用系统的解决性问题，并将获得进一步的知识，同时从数学领域的不同领域中汇总出想法。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力功能和广泛影响的评估来评估CRETIRIA的评估。