  We study the small singular values of the $2$-dimensional semiclassical
differential operator $P = 2\,\mathrm{e}^{-\phi/h}\circ hD_{\overline{z}}\circ
\mathrm{e}^{\phi/h}$ on $S^1+iS^1$ and on $S^1+i\mathbb{R}$ where $\phi$ is
given by $\sin y$ and by $y^3/3$, respectively. The key feature of this model
is the fact that we can pinpoint precisely where in phase space the Poisson
bracket $\{p,\overline{p}\}=0$, where $p$ is the semiclassical symbol of $P$.
  We give a precise asymptotic description of the exponentially small singular
values of $P$ by studying the tunneling effects of an associated Witten
complex. We use these asymptotics to determine a Weyl law for the exponentially
small singular values of $P$.

Spectral Theory