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We show that the same result remains true for $ (i \\partial_t + \\Delta - V) u = 0 $ where $ V \\in L^2 (\\T^2) $, and $ \\T^2 $ is a (rational or irrational) torus. That extends the results of \\cite{AM}, and \\cite{BZ4} where the observability was proved for $ V \\in C (\\T^2) $ and conjectured for $ V \\in L^\\infty (\\T^2) $. 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