Weighted Estimates for the Solution to the Schrodinger Equation

Description

The Schrodinger equation is a fundamental element of the quantum mechanics. The solution to this equation reflects the wave-like behavior of a particle (such as an electron) and describes the change in the particle's probability density over time. Given the probability density of a particle at time 0, a natural problem is to predict the probability density of the particle at a given time. Fortunately, there is a mathematical formula for the solution to the Schrodinger equation provided that the initial data (i.e., the initial probability density of the particle) is sufficiently smooth. A subtle problem involves determining the minimal required smoothness of the initial data. In particular, what is the minimal smoothness required in order for the solution to converge pointwise almost everywhere to the initial data as time approaches 0? This problem was first proposed by Carleson in the 1980s. It was only recently solved in all dimensions, using powerful tools and techniques from harmonic analysis. The proposed research project aims to strengthen and extend known results related to Carleson's problem. For Carleson's problem, one must have a good estimate on the size of a related maximal function. This maximal function takes the maximum value of the solution to the Schrodinger equation during a time interval (e.g., from 0 to 1). Such an estimate can be seen as a specific case of space-time weighted estimates for the solution, where the weight is lower dimensional in a certain sense. These weighted estimates are not well-understood, except for weights of certain dimensions. Our goal is to address this issue and extend the validity of the pointwise convergence phenomena to a larger class of initial data. The project is expected to involve and strengthen several tools from harmonic analysis.