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In this paper we extend the Balian--Low theorem, which is a version of theuncertainty principle for Gabor (Weyl--Heisenberg) systems, to functions ofseveral variables. In particular, we first prove the Balian--Low theorem forarbitrary quadratic forms. Then we generalize further and prove the Balian--Lowtheorem for differential operators associated with a symplectic basis for thesymplectic form on ${\mathbb R}^{2d}$.Comment: Latex, 25 page

The Balian–Low theorem for the symplectic form on R2d