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In this paper we characterize global regularity in the sense of Shubin oftwisted partial differential operators of second order in dimension $2$. Theseoperators form a class containing the twisted Laplacian, and in bi-uniquecorrespondence with second order ordinary differential operators withpolynomial coefficients and symbol of degree $2$. This correspondence isestablished by a transformation of Wigner type. In this way the globalregularity of twisted partial differential operators turns out to be equivalentto global regularity and injectivity of the corresponding ordinary differentialoperators, which can be completely characterized in terms of the asymptoticbehavior of the Weyl symbol. In conclusion we observe that we have obtained anew class of globally regular partial differential operators which is disjointfrom the class of hypo-elliptic operators in the sense of Shubin.

Global regularity of second order twisted differential operators