Compactness of the resolvent for the Witten Laplacian

TL;DR

This paper establishes conditions under which the Witten Laplacian on 0-forms has a compact resolvent, extending techniques inspired by the Fokker-Planck operator and using nilpotent group methods.

Contribution

It provides new sufficient conditions involving potential derivatives and Hessian eigenvalues for the compactness of the Witten Laplacian's resolvent.

Findings

Derived a compactness criterion for the Witten Laplacian

Connected the conditions to control of derivatives and eigenvalues

Extended nilpotent group techniques to this setting

Abstract

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].