Tunneling for the \(\overline{\partial }\)-Operator

We study the small singular values of the 2-dimensional semiclassical differential operator \(P = 2\textrm{e}^{-\phi /h}\circ hD_{\overline{z}}\circ \textrm{e}^{\phi /h}\) $P=2{\text{e}}^{-\varphi /h}\circ h{D}_{\overline{z}}\circ {\text{e}}^{\varphi /h}$ on \(S^1+iS^1\) $S^1+i{S}^1$ and on \(S^1+i\mathbb {R}\) $S^1+i\mathbb{R}$ , where \(\phi \) $\varphi$ is given by \(\sin y\) $siny$ and by \(y^3/3\) $y^3/3$ , respectively. The key feature of this model is the fact that we can pinpoint precisely where in phase space the Poisson bracket \(\{p,\overline{p}\}=0\) $\{p,\overline{p}\}=0$ , 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 this to determine a Weyl law for the exponentially small singular values of P.