The usual Weyl calculus is intimately associated with the choice of thestandard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In thispaper we will show that the replacement of this structure by an arbitrarysymplectic structure leads to a pseudo-differential calculus of operatorsacting on functions or distributions defined, not on $\mathbb{R}^{n}$ butrather on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. These operators are intertwinedwith the standard Weyl pseudo-differential operators using an infinite familyof partial isometries of $L^{2}(\mathbb{R}^{n})\longrightarrowL^{2}(\mathbb{R}^{2n})$ \ indexed by $\mathcal{S}(\mathbb{R}^{n})$. This allowsus obtain spectral and regularity results for our operators using Shubin'ssymbol classes and Feichtinger's modulation spaces.
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