Observability estimates for the Schrödinger equation in the plane with periodic bounded potentials from measurable sets
Abstract
The goal of this article is to obtain observability estimates for Schrödinger equations in the plane R 2. More precisely, considering a 2πZ 2-periodic potential V ∈ L ∞ (R 2), we prove that the evolution equation i∂tu = −∆u + V (x)u, is observable from any 2πZ 2-periodic measurable set, in any small time T > 0. We then extend Taüffer's recent result [Täu22] in the two-dimensional case to less regular observable sets and general bounded periodic potentials. The methodology of the proof is based on the use of the Floquet-Bloch transform, Strichartz estimates and semiclassical defect measures for the obtention of observability inequalities for a family of Schrödinger equations posed on the torus R 2 /2πZ 2 .
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