  We study the asymptotic behaviour 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 spectrum of the corresponding
gradient flow acting on appropriate anisotropic Sobolev spaces. In particular,
our results relate the approach of Laudenbach and Harvey--Lawson to Morse
theory using currents, which was discussed in previous work of the authors, and
Witten's point of view based on semiclassical analysis and tunneling. As an
application of our methods, we also construct a natural family of quasimodes
satisfying the Witten-Helffer-Sj\"ostrand tunneling formulas and the Fukaya
conjecture on Witten deformation of the wedge product.
