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}$^{n})$withWeylsymbol$a\in$\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 $\tilde{a}=(a\otimes 1_{2k})\circ s$, 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 $\tilde{A}$ symplectic dimensional extensions of $A$. In this paper we study the relation between $A$ and $\tilde{A}$ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of $\tilde{A}$ 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 $H{G}_{\rho }^{m_1,m_0}$ 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.