1209.1852 (Nuno Costa Dias et al.)

Dimensional Extension of Pseudo-Differential Operators: Properties and
Spectral Results    [PDF]

Nuno Costa Dias, Maurice A. de Gosson, João Nuno Prata

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 operator $\widetilde{A}_{s}:\mathcal{S}% (\mathbb{R}^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbol $\widetilde{a}_{s}=(a\otimes1_{2k}%)\circ {s}$ 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 operator $\widetilde{A}_{s}$ a dimensional extension of $A$. We construct a family of partial isometries which intertwine the operators $\widetilde{A}_{s}$ and $A$. These maps allow us to completely determine the spectrum and the eigenfunctions of $\widetilde{A}%_{s}$ from those of $A$. Moreover, these maps can be used to obtain new classes of pseudo-differential operators with specific spectral properties. We will use this procedure to construct an extension of the Shubin class $HG_{\rho}^{m_{1},m_{0}}$ of globally hypoelliptic operators.

View original: http://arxiv.org/abs/1209.1852