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.