A symplectic extension map and a new Shubin class of pseudo-differential operators

For an arbitrary pseudo-differential operator $A:\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with Weyl symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$, we consider the pseudo-differential operators $\widetilde{A}:\mathcal{S}(\mathbb{R}^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbols $\widetilde{a}=(a\otimes1_{2k})\circ{s}$, where $1_{2k}(x)=1$ for all $x\in\mathbb{R}^{2k}$ and ${s}$ is a linear symplectomorphism of $\mathbb{R}^{2(n+k)}$. We call the operators $\widetilde{A}$ symplectic dimensional extensions of $A$. In this paper we study the relation between $A$ and $\widetilde{A}$ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of $\widetilde{A}$ in terms of those of $A$. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes $HG_{\rho}^{m_{1},m_{0}}$ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in $HG_{\rho }^{m_{1},m_{0}}$ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.Comment: 28 pages, new version, accepted for publication in JF

On the Weyl Representation of Metaplectic Operators

We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non-trivial fixed point, and justify a formula suggested in earlier work of Mehlig and Wilkinson. We give precise calculations of the associated Maslov-type indices; these indices intervene in a crucial way in Gutzwiller's formula of semiclassical mechanics, and are simply related to an index defined by Conley and Zehnder.Comment: To appear in Lett. Math. Physiqu