Qualitative Properties of Eigenfunctions for some Selfadjoint and Non-selfadjoint partial differential equations

这个项目考虑本征函数，它是给出信息的对象，例如，响起的鼓头的振动。在其他情况下，这些特征函数可以描述，例如，流体如何在极小的长度尺度上混合，潜在地应用于提高医疗药物输送和透析的效率。这些特征函数是理解数学、物理、化学甚至生物医学工程中问题的基本构件。正是这些拟议的研究与STEM主题之间的重要基础联系，使本研究在STEM领域的更大背景下具有更广泛的意义。PI一直在为学生和年轻的研究人员开发与他的大部分研究相关的简单的教学模型问题。这个项目涉及偏微分方程解、微分几何、动力系统和数学物理之间的深层关系的研究。这些不同的数学领域经常通过频谱理论和微局部分析中的问题联系在一起；也就是涉及特征值、特征函数、相空间局部化及其推广的问题。本文的研究分为自伴特征函数问题和非自伴特征函数问题。对偏微分方程解的性态的研究与基本几何和经典相空间动力学有着丰富的联系。例如，众所周知，本征函数倾向于沿测地线集中。如果有孤立的周期测地线，人们可能会期望特征函数的子序列沿着这样的测地线集中。理解浓缩率是量子混沌中一个极其重要的问题。另一方面，如果测地线流足够混沌，人们可能会期望本征函数在相空间中均匀分布。通过相空间估计、约束估计和扰动来研究这些现象的健壮性是很重要的。例如，在混沌的情况下，PI和他的合作者正在努力理解本征函数对超曲面的限制。通过对超曲面的温和几何假设，他们猜想这种限制的质量是上下有界的，与本征值无关。研究方法将引入新的微观局部能量技术，演示如何将以前仅在算术曲面上使用数论来理解的问题推广到非常一般的几何情况。特征函数的一些性质在小的复杂扰动下趋于稳定，这意味着人们也可以理解一些非自伴问题。对于虚分量较大的非自伴问题，摄动方法不再严格有效。PI正在致力于发展一种适用于任何阶次退化的对流-扩散型方程的几何控制的一般理论。这对福克-普朗克方程和统计力学中的类似方程有直接的应用。这一理论也适用于理论微流体学中的某些模型，在有效的几何局域药物输送和透析方面具有潜在的应用。

This project considers eigenfunctions, which are objects that give information about, for example, the vibrations of a ringing drum head. In other contexts, these eigenfunctions can describe, for example, how fluids mix on extremely small length scales, with potential applications to improved efficiency in medical drug delivery and dialysis. These eigenfunctions are fundamental building blocks to understand problems in mathematics, physics, chemistry, and even biomedical engineering. It is precisely these important fundamental connections between the proposed research and STEM subjects that makes this research have broader significance within the larger context of the STEM areas. The PI has been developing simple, instructional model problems related to most of his research for students and younger researchers. This project concerns research in the deep relationships between solutions to partial differential equations, differential geometry, dynamical systems, and mathematical physics. These different areas of mathematics are often tied together by problems in spectral theory and microlocal analysis; that is, problems concerned with eigenvalues, eigenfunctions, phase space localization, and the generalizations thereof. The research in this proposal is divided between selfadjoint and non-selfadjoint eigenfunction problems. The study of the behaviour of solutions to partial differential equations has a rich connection to the underlying geometry and classical phase space dynamics. For example, it is well known that eigenfunctions tend to concentrate along geodesics. If there are isolated periodic geodesics, one might expect a subsequence of eigenfunctions to concentrate along such geodesics. Understanding the rate of concentration is an extremely important question in quantum chaos. On the other hand, if the geodesic flow is sufficiently chaotic, one might expect the eigenfunctions to be equidistributed in phase space. It is important to investigate how robust these phenomena are, through phase space estimates, restriction estimates, and perturbations. For example, in the chaotic case, the PI and his collaborators are working to understand restrictions of eigenfunctions to hypersurfaces. With mild geometric assumptions on a hypersurface, they conjecture the mass of such restrictions is bounded above and below, independent of the eigenvalue. The methods of investigation will introduce new microlocal energy techniques, demonstrating how to generalize a problem, previously only understood on arithmetic surfaces using number theory, to very general geometric situations. Some properties of eigenfunctions tend to be stable under small complex perturbations, which means one can also understand some non-selfadjoint problems. For non-selfadjoint problems with larger imaginary component, perturbation techniques are no longer strictly valid. The PI is working to develop a general theory of geometric control adapted to degenerate advection-diffusion type equations of any order. This has straightforward applications to the Fokker-Planck equation and similar equations from statistical mechanics. This theory also has applications to certain models in theoretic micro-fluidics, with potential applications to efficient geometrically localized drug delivery and dialysis.