LEAPS-MPS: Elliptic theory for the Schrodinger operator

拉普拉斯方程是典型的二阶椭圆型偏微分方程。因此，拉普拉斯方程的解，称为调和函数，是偏微分方程组理论的基本组成部分。但这些函数对科学和工程的许多其他领域也很重要，如复分析、调和分析、几何、物理和工程。因此，调和函数已被广泛研究并被很好地理解。虽然拉普拉斯方程模拟了均匀环境中的稳态现象，但我们生活的世界并不是各向同性的真空。支配许多自然现象的数学方程，如电磁学、天文学和流体动力学，往往比拉普拉斯方程复杂。例如，薛定谔方程描述了量子力学波的行为，而它的推广描述了更复杂的环境。因此，有必要了解此类一般椭圆型偏微分方程解的性质。该项目将调和分析中的数学追求与促进包容和保留不同数学社区的目标结合在一起。后一个目标将通过为即将入学的研究生开设的定向课程和课外辅导课程来实现。在这个项目中，PI将探索低阶项和可变系数的存在如何以及在多大程度上影响椭圆型方程的解的行为。本文以薛定谔方程为例，从唯一性延拓、齐次化和可解性三个不同的角度来考察这些效应。调和函数具有如下独特的延拓性质：局部，它们不能消失到无限级；如果定义为全局，刘维尔定理断言，它们不可能处处有界。在兰迪斯猜想的启发下，本程序的一个方面寻求精确地量化广义薛定谔方程解的这种局部和整体行为。通过进一步考虑具有周期系数的椭圆型方程，本程序还探索了齐次化理论和唯一延拓之间的相互作用。Carleman估计和复杂的分析技术将与紧凑性论证相结合来完成这一壮举。拉普拉斯方程Dirichlet边值问题和Neumann边值问题的可解性研究使偏微分方程组和调和分析理论得到了极大的发展。PI以前的工作将被用来探索具有低阶项的一般椭圆型偏微分方程组的可解性问题，并将在汇集不同数学领域的想法的同时获得进一步的知识。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.