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Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

For the Schrödinger equation, ( i ∂ t + ∇ ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: u t = 0 L 2 ( 𝕋 2 ) ≤ K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i ∂ t + ∇ - V ) u = 0 where V ∈ L 2 ( 𝕋 2 ) , and 𝕋 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V ∈ C ( 𝕋 2 ) and conjectured for V ∈ L ∞ ( 𝕋 2 ) . The higher dimensional generalization remains open for V ∈ L ∞ ( 𝕋 n ) .

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