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=ξj(x) of the equation p(x, ξ)=0, where p(x, ξ) is the (Weyl) symbol of P. © 2013 Elsevier Inc.
Nicola, F., & Rodino, L. (2013). Global regularity for ordinary differential operators with polynomial coefficients. Journal of Differential Equations, 255(9), 2871–2890. https://doi.org/10.1016/j.jde.2013.07.022