Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of H\'erau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.Comment: Lemma 4.6 is correcte

Is f1(1420)f_1(1420) the partner of f1(1285)f_1(1285) in the 3P1^3P_1 qqˉq\bar{q} nonet?

Based on a $2\times 2$ mass matrix, the mixing angle of the axial vector states $f_1(1420)$ and $f_1(1285)$ is determined to be $51.5^{\circ}$, and the theoretical results about the decay and production of the two states are presented. The theoretical results are in good agreement with the present experimental results, which suggests that $f_1(1420)$ can be assigned as the partner of $f_1(1285)$ in the $^3P_1$ $q\bar{q}$ nonet. We also suggest that the existence of $f_1(1510)$ needs further experimental confirmation.Comment: Latex, 6 pages, to be published in Chin. Phys. let