This talk is based on a joint work with Michael Hitrik and Martin Vogel. Let

$P ( x , h D_{x} ; h ) = p ( x , h D_{x} ) + h p_{1} ( x , h D_{x} ) + h^{2} p_{2} ( x , h D_{x} ) + \cdots$

be a differential operator in the semi-classical limit $0$<$h \ll 1$. If the Poisson bracket

$i^{- 1} \{ p , \overline{p} \} ( x , \xi ) = i^{-1} ( p_{\xi}^{\prime} \cdot \overline{p}^{\prime}_{x} - p^{\prime}_{x} \cdot \overline{p}^{\prime}_{\xi} ) ( x , \xi )$

does not vanish identically on the zero set of $p$, then $P$ is non-self-adjoint and we know (cf. Hörmander 1960, ...) that $P$ has singular values that are ${\mathcal O} (h^{N} )$ , $\forall N \geq 0$, and even ${\mathcal O} ( \exp - 1 / ( C h ) )$ for some $C$>$0$ in the analytic case. Also, $P$ has approximate null solutions concentrated to points in phase space where $p = 0$, $\{ p , \overline{p} \} / i$>$0$ and similarly $P^{*}$ has approximate null solutions concentrated to points in phase space where $p = 0$, $\{ p , \overline{p} \} / i$<$0$.

One may ask for the precise exponential decay rate of the small singular values. We obtain such results for the $\overline{\partial}$ operator on certain exponentially weighted $L^{2}$ spaces on a torus or on a cylinder, that can be interpretated in terms of quantum tunneling between the regions in the energy surface $p = 0$ where $\{ p , \overline{p} \} / i$>$0$ and $\{ p , \overline{p} \} / i$<$0$ respectively.