Harmonic Analysis Methods Related to the Schrodinger Equation

本计画主要研究线性、非线性及离散薛定谔方程的动力学性质。 一个目标是确定一组最小的条件（关于环境力、底层空间的几何形状等）。在此条件下，解具有与自由线性情形相同的色散和平滑特性。 这样的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性和长时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互连的方式。 从调和分析、傅立叶分析和微分算子谱理论中得到的技术将在所有这些工作中发挥重要作用。薛定谔方程是量子力学中基本运动定律的表述。 这是描述从化学相互作用到沿着光纤电缆的信息传输的各种现象的首选数学模型。 因此，了解该模型在不同时间尺度和不同空间分辨率水平上的表现是有益的。 该项目将这些问题作为微分方程研究中的一个抽象问题来解决。 因为这些方程是用来表示实际的物理系统的，在这个层次上的发现应该可以将量子力学原理应用于更广泛的科学问题。

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.