Tunneling for the $\bar{\mathrm{\partial }}$-operator

We study the small singular values of the $2$-dimensional semiclassical differential operator $P=2{\mathrm{e}}^{-\phi \mathrm{/}h}\circ h{D}_{\bar{z}}\circ {\mathrm{e}}^{\phi \mathrm{/}h}$ on $S^1+i{S}^1$ and on $S^1+i\mathbb{R}$ where $\phi$ is given by $\sin y$ and by $y^3\mathrm{/}3$, respectively. The key feature of this model is the fact that we can pinpoint precisely where in phase space the Poisson bracket $\{p,\bar{p}\}=0$, 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 $P$.