Annales Henri Poincaré

, Volume 19, Issue 4, pp 1259–1282 | Cite as

Compactness of the Resolvent for the Witten Laplacian



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 (Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, 1985).


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics and Computational Science Hubei Key LaboratoryWuhan UniversityWuhanChina

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