  This article shows that counting or computing the small eigenvalues of the
Witten Laplacian in the semi-classical limit can be done without assuming that
the potential is a Morse function as the authors did in [LNV]. In connection
with persistent cohomology, we prove that the rescaled logarithms of these
small eigenvalues are asymptotically determined by the lengths of the bar code
of the function f. In particular, this proves that these quantities are stable
in the C 0 topology on the space of functions. Additionally, our analysis
provides a general method for computing the subexponential corrections in a
large number of cases.
