Global regularity for ordinary differential operators with polynomial coefficients

TL;DR

This paper characterizes when certain polynomial coefficient differential operators are globally regular on the real line, based on the asymptotic behavior of roots of their symbols, extending Schwartz's hypoellipticity concept.

Contribution

It provides a necessary and sufficient condition for global regularity of polynomial coefficient differential operators in terms of root asymptotics.

Findings

Characterization of global regularity via root asymptotics

Extension of Schwartz hypoellipticity to polynomial operators

Necessary and sufficient condition established

Abstract

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 $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply $u\in \cS(\R)$ (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 $P$.