QUANTUM
   INTEGRABLE
   SYSTEMS and
   GEOMETRY
   2012









 3-7 SEPTEMBER 2012
 OLHÃO, PORTUGAL

Symplectic dimensional extensions of a pseudo-differential operator

Maurice de Gosson

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We consider the pseudo-differential operator $\widetilde{A}_{\widetilde{s}}:\mathcal{S}(\mathbb{R}^{n+k})\longrightarrow \mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbol $\widetilde{a}_S=(a\otimes 1_{2k})\circ \widetilde{s}$ where $1_{2k}(x)=1$ for all $x\in \mathbb{R}^{2k}$ and $\widetilde{s}$ is a linear symplectomorphism of $\mathbb{R}^{2(n+k)}$. Here $a\in \mathcal{S}^{\prime}(\mathbb{R}^{2n})$ and $A:\mathcal{S}(\mathbb{R}^n)\longrightarrow \mathcal{S}^{\prime}(\mathbb{R}^n)$ is the pseudo-differential operator with Weyl symbol $a$. We will call the operator $\widetilde{A}_{\widetilde{s}}$ a dimensional extension of $A$. We show that there exists a family of partial isometries which intertwine the operators $\widetilde{A}_{\widetilde{s}}$ and $A$. These isometries allow us to completely determine the spectrum and the eigenfunctions of $\widetilde{A}_{\widetilde{s}}$ from those of $A$. Moreover, they can be used to obtain new classes of pseudo-differential operators with specific spectral properties. We will illustrate this point by constructing an extension of the Shubin class $HG_{\rho}^{m_1,m_0}$ of globally hypoelliptic operators. This talk is based on joint work with N. C. Dias and J. N. Prata.