The Hardy Uncertainty Principle and Unique Continuation for the Schrödinger Equation

Abstract

The Hardy Uncertainty Principle states that if both a function f and its Fourier transform decay faster than the Gaussian function with a specific weight, then f is the zero function. This result can be reformulated for solutions of the free Schrödinger equation, which implies a unique continuation result for this equation. By the use of Carleman estimates, Escauriaza, Kenig, Ponce and Vega extended this result to the Schrödinger equation with potential and to the nonlinear Schrödinger equation. More precisely, the authors proved that if u is a solution of the Schrödinger equation with potential, which at two times has Gaussian decay, and given the right conditions on the potential, then u is 0.

The formal arguments of the proof, relying on Carleman estimates, are based on calculus and convexity arguments. However, these computations are not straightforward to justify rigorously. In particular, if u is in L^2 for all time between 0 and 1, it is not guaranteed that e^{\phi}u is in L^2 for a specific weight function phi, for all time. This is a fundamental step when we want to prove the main result.

In this thesis, we will study the proof of the result by Escauriaza, Kenig, Ponce and Vega, which provides a rigorous strategy to justify the use of the Carleman estimates.