Some unique continuation results for Schrodinger equations

The unique continuation property is a fundamental concept in the field of partial differential equations, which describes propagation of the zeros of solutions to PDEs. Essentially, it asks what condition is required to guarantee that if a solution to a PDE vanishes on a certain subset of the spatial domain, then it must also vanish on a larger subset of the domain. 

In a series of papers by Escauriaza, Kenig, Ponce and Vega, motivated by Hardy's uncertainty principle, they were able to asked the question from a different perspective. That is, they showed that if a linear Schr\”odinger solution decays sufficiently fast at two different times, the solution must be trivial. In this talk, we will discuss unique continuation properties of solutions to higher-order Schr\"odinger equations and variable-coefficient Schr\”odinger equations, and extend the classical Escauriaza-Kenig-Ponce-Vega type of result to these models. This is based on joint works with S. Federico- Z. Li, and Z. Lee.