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 R, i.e. u∈S^'(R) and Pu∈S(R) imply u∈S(R) (this can be regarded as a global version of the Schwartz' hypoellipticity notion). The condition involves the asymptotic behavior, at infinity, of the roots ξ=ξ_j(x) of the equation p(x,ξ)=0, where p(x,ξ) is the (Weyl) symbol of P.