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，解的拉普拉斯方程，被称为调和函数，是一个基本组成部分的偏微分方程理论。但是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得到很好的理解。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。控制许多自然现象如电磁学、天文学和流体动力学的数学方程往往比拉普拉斯方程更复杂。例如，薛定谔方程描述了量子力学波的行为，而它的推广则描述了更复杂的设置。因此，有必要了解这些一般椭圆型偏微分方程的解的性质。这个项目结合了数学追求谐波分析的目标，促进包容和保留一个多样化的数学社区。后一个目标将通过为即将入学的研究生沿着的指导计划以及课外导师计划来实现。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方程解的行为。薛定谔方程作为标准的例子，这些影响将通过独特的连续性，均匀化和可解性三个不同的角度进行检查。调和函数具有以下独特的连续性质：局部地，它们不能消失到无限阶;如果全局地定义，刘维尔定理断言它们不能处处有界。受兰迪斯猜想的启发，该计划的一个方面是寻求精确量化广义薛定谔方程解的局部和全局行为。通过进一步考虑具有周期系数的椭圆方程，该程序还探索了均匀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参数相结合，以完成这一壮举。关于拉普拉斯方程Dirichlet和Neumann边值问题的可解性的工作导致了偏微分方程和调和分析理论的巨大发展。PI以前的工作将用于探索具有低阶项的椭圆偏微分方程的一般系统的可解性问题，并在汇集不同数学领域的思想的同时获得进一步的知识。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.