Observability estimates for the Schr{ö}dinger equation in the plane with periodic bounded potentials from measurable sets

Abstract

The goal of this article is to obtain observability estimates for Schr{ö}dinger equations in the plane R 2. More precisely, considering a 2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any 2$\pi$Z 2-periodic measurable set, in any small time T > 0. We then extend Ta{ü}ffer's recent result [T{ä}u22] 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 Schr{ö}dinger equations posed on the torus R 2 /2$\pi$Z 2 .