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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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