Uncertainty principles for discrete Schrödinger evolutions
Also appears in collections : Annual conference of the functional analysis, harmonic analysis and probability Gdr research group / Journées du Gdr analyse Fonctionnelle, harmonique et probabilités, Exposés de recherche
We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss different extensions of this result to equations with bounded potentials. The continuous case was studied in a series of articles by L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega. The talk is mainly based on joint work with Ph. Jaming, Yu. Lyubarskii, and K.-M. Perfekt.
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