  We discuss semiclassical asymptotics for the eigenvalues of the Witten
Laplacian for compact manifolds with boundary in the presence of a general
Riemannian metric. To this end, we modify and use the variational method
suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of
quadratic forms instead of the operators. We also utilize some important ideas
and technical elements from Helffer and Nier (2006), who were the first to
supply a complete proof of the full semi-classical asymptotic expansions for
the eigenvalues with fixed numbers.
