  We consider a discrete Schr\"odinger operator $ H_\varepsilon=
-\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon
\mathbb Z^d)$, where $\varepsilon>0$ is a small parameter and the potential
$V_\varepsilon$ is defined in terms of a multiwell energy landscape $f$ on
$\mathbb R^d$. This operator can be seen as a discrete analog of the
semiclassical Witten Laplacian of $\mathbb R^d$. It is unitarily equivalent to
the generator of a diffusion on $\varepsilon \mathbb Z^d$, satisfying the
detailed balance condition with respect to the Boltzmann weight
$\exp{(-f/\varepsilon)}$. These type of diffusions exhibit metastable behaviour
and arise in the context of disordered mean field models in Statistical
Mechanics. We analyze the bottom of the spectrum of $H_\varepsilon$ in the
semiclassical regime $\varepsilon\ll1$ and show that there is a one-to-one
correspondence between exponentially small eigenvalues and local minima of $f$.
Then we analyze in more detail the bistable case and compute the precise
asymptotic splitting between the two exponentially small eigenvalues. Through
this purely spectral-theoretical analysis of the discrete Witten Laplacian we
recover in a self-contained way the Eyring-Kramers formula for the metastable
tunneling time of the underlying stochastic process.
