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