Harmonic Analysis Methods Related to the Schrodinger Equation

本项目旨在研究线性、非线性和离散薛定谔方程的动力学性质。一个目标是确定最小的条件集(关于环境作用力、底层空间的几何形状等)。在这种情况下，解具有与自由线性情形相同的色散和光滑性。这种估计对于非线性薛定谔方程解的短时存在性以及它们的散射和长时间渐近性都是至关重要的。当底层域是离散图而不是连续流形时，可能的分散程度应该反映各个顶点相互连接的方式。采用调和和傅立叶分析以及微分算符频谱理论的技术，将在所有这些努力中发挥重要作用。薛定谔方程是量子力学中基本运动定律的陈述。这是描述从化学相互作用到沿光缆传输信息的各种现象的首选数学模型。因此，了解模型在不同时间尺度和不同空间分辨率水平上的行为是有指导意义的。该项目将这些问题作为微分方程式研究中的一个抽象问题来解决。由于这些方程旨在表示实际的物理系统，在这个层面上的发现应该会使将量子力学原理应用于更广泛的科学问题成为可能。

This project aims to study the dynamical properties of linear, nonlinear, and discrete Schrödinger equations. One goal is to identify a minimal set of conditions (on ambient forces, geometry of the underlying space, etc.) under which solutions enjoy the same dispersive and smoothing properties as in the free linear case. Such estimates are crucial to both the short-time existence of solutions to nonlinear Schrödinger equations as well as their scattering and long-time asymptotics. When the underlying domain is a discrete graph instead of a continuous manifold, the level of possible dispersion should reflect the manner in which individual vertices are interconnected. Techniques adapted from harmonic and Fourier analysis, and from the spectral theory of differential operators, will play a prominent role in all these efforts.Schrödinger equations are a statement of the basic laws of motion in quantum mechanics. This is the mathematical model of first choice for describing diverse phenomena from chemical interactions to the transmission of information along a fiber-optic cable. It is therefore instructive to know how the model might behave across different time-scales and different levels of spatial resolution. The project addresses these issues as an abstract question in the study of differential equations. Because the equations are intended to represent actual physical systems, discoveries at this level should make it possible to apply quantum mechanical principles to a broader range of scientific problems.