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 = 0. This result can be reformulated in terms of solutions to the free Schrödinger equation. In a series of work Escauriaza, Kenig, Ponce and Vega extended this result to the Schrödinger equation with potential and to NLS by the use of Carleman estimates. More precisely, it was proven that if u is a solution to the Schrödinger equation with potential, which at two times has Gaussian decay, and given the right conditions on the potential, then u = 0.

The proof is based on Carleman estimates, which formally relies 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.

We have adapted these techniques to the hyperbolic Schrödinger equation, which is physically relevant in for example Fluid mechanics. We will go through the main ideas of the proof with the Carleman estimates, and where the main changes are when going from the Laplace operator to the hyperbolic Laplace operator in the Schrödinger equation.