Rough controls for Schrödinger operators on 2-tori

The purpose of this note is to use the results and methods of [BBZ13] and [BZ12] to obtain control and observability by rough functions and sets on 2-tori, 𝕋 2 = ℝ 2 / ℤ ⊕ γ ℤ . We show that for a non-trivial W ∈ L ∞ ( 𝕋 2 ) , solutions to the Schrödinger equation, ( i ∂ t + Δ ) u = 0 , satisfy ∥ u | t = 0 ∥ L 2 ( 𝕋 2 ) ≤ K T ∥ W u ∥ L 2 ( [ 0 , T ] × 𝕋 2 ) . In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions in L 2 ( 𝕋 2 ) and controls in L 4 ( [ 0 , T ] × 𝕋 2 ) . For continuous W this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for 𝕋 2 = ℝ 2 / ( 2 π ℤ ) 2 (the rational torus) and T > π this can be deduced from the results of Jakobson [Jak97].