Unique continuation is one of the most important subjects in PDEs, whose aim is to recover globally a solution from its partial information, while observability can be treated as a special kind of unique continuation, which has wide applications in control theory. In this paper, we build up an observability inequality at two time instants for the Schr?dinger equation. With the aid of this, we obtain several unique continuation inequalities at one time instant. These provide new applications in controls of the Schr?dinger equation. The main contribution of this paper is to find connection between the aforementioned observability and the Nazarov type uncertainty inequality which comes from the well-known Heisenberg uncertainty principle.

About this paper, AMS Math. Reviews says: To conclude, the results here are new and interesting......

J. EUR. MATH. SOC.

https://arxiv.org/pdf/1606.05861.pdf