Morse-type inequalities for dynamical systems and the Witten Laplacian

We provide an analytic proof of Morse-type inequalities for vector fields determining a Morse decomposition with normally hyperbolic dynamics. In the demonstration we reduce the problem to the gradient case using a Morse–Bott Lyapunov function and then apply Schrödinger operator techniques. This yields an explicit construction of the cohomology complex of the manifold in terms of the invariant sets of the Morse decomposition associated with the vector field.