SMALL EIGENVALUES OF THE WITTEN LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS: THE CASE WITH CRITICAL POINTS ON THE BOUNDARY

We give sharp asymptotic equivalents in the limit h -> 0 of the small eigenvalues of the Witten Laplacian, that is, the operator associated with the quadratic form psi is an element of H-0(1)(Omega)-> - h(2) integral(Omega)vertical bar del(e(1/hf)psi)(2)e(-2/hf), where (Omega) over bar = Omega boolean OR partial derivative Omega is an oriented C-infinity compact and connected Riemannian manifold with nonempty boundary partial derivative Omega and f : (Omega) over bar -> R is a C-infinity Morse function. The function f is allowed to admit critical points on @center dot, which is the main novelty of this work in comparison with the existing literature.