  On any compact manifold of dimension greater than 4, we prescribe the volume
and any finite part of the spectrum of the Witten Laplacian acting on $p$-form
for $0<p<n$. In particular, we prescribe the multiplicity of the first
eigenvalues. On 3-dimensional manifolds, we give examples of multiple first
eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of
embedded surfaces all of whose 1-cohomology is induced by the cohomology of the
manifold. In particular, this multiplicity is at least 3.
