On unique continuation for the Schrödinger equation
Spyridon Filippas1 ; Camille Laurent2 ; Matthieu Léautaud3
1 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
2 CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
3 Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Saclay, CNRS, Bâtiment 307, 91405 Orsay Cedex France & Institut Universitaire de France, Paris, France
Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 15, 13 p.

Motivated by applications to approximate and exact controllability, we are interested in the following unique continuation question: assume the solution of the linear Schrödinger equation on a domain vanishes on a very small open set for a very short time interval, then is this solution identically zero? In the situation where the Schrödinger operator includes a potential, the answer to this question depends on the regularity of the latter. We present a result proved in [FLL24] which assumes that the potential is Gevrey 2 in time and bounded in space, relaxing in this context the analyticity assumption of the Tataru-Robbiano-Zuily-Hörmander theorem. We also give a sketch of proof.

Publié le :
DOI : 10.5802/slsedp.172
Spyridon Filippas 1 ; Camille Laurent 2 ; Matthieu Léautaud 3

1 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
2 CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
3 Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Saclay, CNRS, Bâtiment 307, 91405 Orsay Cedex France & Institut Universitaire de France, Paris, France 
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Spyridon Filippas; Camille Laurent; Matthieu Léautaud. On unique continuation for the Schrödinger equation. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 15, 13 p. doi : 10.5802/slsedp.172. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.172/

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