Unique Continuation for the Schrödinger Equation

Abstract: The first part of the talk will be a short history on unique continuation, Carleman estimates and how Hardy’s Uncertainty Principle is related to a unique continuation result for the free Schrödinger equation. In a series of work by Escauriaza, Kenig, Ponce and Vega this result was extended to the Schrödinger equation with potential and to the NLS. I will present the main steps of the proof for this result for the Schrödinger equation with potential. The proof is based on Carleman estimates, which formally is based on calculus and convexity arguments. However, going from a formal level to a rigorous one is not straight forward, and if we do not justify the computations rigorously, we can prove wrong results. In particular I will present an example of a formal Carleman argument for which the corresponding inequalities leads to a false statement.