Global regularity for ordinary differential operators with polynomial coefficients

For a class of ordinary differential operators $P$ with polynomial coefficients, we give a necessary and sufficient condition for $P$ to be globally regular in $\mathbb{R}$, i.e. $u\in {\text{cS}}^{\mathrm{\prime }}(\mathbb{R})$ and $Pu\in \text{cS}(\mathbb{R})$ imply $u\in \text{cS}(\mathbb{R})$ (this can be regarded as a global version of the Schwartz’ hypoellipticity notion). The condition involves the asymptotic behaviour, at infinity, of the roots $\xi ={\xi }_j(x)$ of the equation $p(x,\xi )=0$, where $p(x,\xi )$ is the (Weyl) symbol of $P$.