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The sharp Hardy uncertainty principle for Schrödinger evolutions. (English) Zbl 1220.35008
The authors study $$u$$ in $$C([0, T],L^2(\mathbb{R}^n))$$ satisfying the Schrödinger equation $\partial_t u= i(\Delta u+ V(x,t)u)\quad\text{in }\mathbb{R}^n\times [0,T].\tag{1}$ They give conditions in terms of the potential $$V$$ and the behaviour of $$u(0)$$ and $$u(T)$$ at infinity which are sufficient in order that $$u\equiv 0$$. For instance they prove that $$\equiv 0$$, provided $$V$$ is bounded, $$\lim_{R\to\infty}\| V\|_{L^1([0, T], L^\infty(\mathbb{R}^n\setminus B_R))}= 0$$ and there exist positive $$\alpha$$, $$\beta$$ such that $\| e^{|x|^2/\beta^2} u(0)\|_{L^2(\mathbb{R}^n)}<\infty\quad\text{and}\quad \| e^{|x|^2/\alpha^2} u(T)\|_{L^2(\mathbb{R}^n)}< \infty$ and $$T/\alpha\beta> 1/4$$. They show that this condition is sharp in the sense that there exist $$V$$ and a nonzero function $$u$$ in $$C^\infty([0,T],{\mathcal S}(\mathbb{R}^n))$$ satisfying above conditions in the case $$T/\alpha\beta= 1/4$$. As a byproduct they derive estimates for the Gaussian decay of solutions of (1).

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B60 Continuation and prolongation of solutions to PDEs
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