Pollicott–Ruelle spectrum and Witten Laplacians

We study the asymptotic behavior of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott–Ruelle spectral data of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten–Helffer–Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.