  In this paper we consider the Witten Laplacian on 0-forms and give sufficient
conditions under which the Witten Laplacian admits a compact resolvent. These
conditions are imposed on the potential itself, involving the control of high
order derivatives by lower ones, as well as the control of the positive
eigenvalues of the Hessian matrix. This compactness criterion for resolvent is
inspired by the one for the Fokker-Planck operator. Our method relies on the
nilpotent group techniques developed by Helffer-Nourrigat [Hypoellipticit\'e
maximale pour des op\'erateurs polyn\^omes de champs de vecteurs, 1985].
