Pseudodifferential operators and a canonical operator in general symplectic manifolds

A calculus of $h$-pseudodifferential operators with symbols on $\mathfrak X$ is defined modulo $O(h^2)$ on a closed symplectic manifold $(\mathfrak X,\omega)$ under the condition that $[\omega]/(2\pi h)-\varkappa/4 \in H^2(\mathfrak X,\mathbf Z)$. The class $\varkappa\in H^2(\mathfrak X,\mathbf Z)$ is described. On Lagrangian submanifolds $\Lambda\subset\mathfrak X$ a class in $H^1(\Lambda,\mathbf U(1))$ obstructing the definition of a canonical operator on $\Lambda$ is found. It is shown that an analogus calculus of pseudodifferential operators can be constructed with respect to homogeneity from an action of the group $\mathbf R_+$ on $\mathfrak X$.
Bibliography: 22 titles.