Unique Continuation for Schr\"odinger Evolutions, with applications to profiles of concentration and traveling waves

We prove unique continuation properties for solutions of the evolution Schr\"odinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schr\"odinger equations.Comment: 23 page

Uncertainty Principle of Morgan type and Schr\"odinger Evolutions

We prove unique continuation properties for solutions of evolution Schr\"odinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an application of our method we also obtain results concerning the possible concentration profiles of solutions of semi-linear Schr\"odinger equations

Null-Control and Measurable Sets

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.Comment: Two remarks adde

Hardy uncertainty principle, convexity and parabolic evolutions

We give a new proof of the $L^2$ version of Hardy’s uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time.We extend the result to heat equations with lower order variable coefficient.IT641-13 (GIC12/96), DMS-0968472, DMS-126524

Hardy's Uncertainty Principle, Convexity and Schr\"odinger Evolutions

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schr\"odinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions