Tunneling for the $\overline{\partial }$-operator

Abstract

We study the small singular values of the $2$-dimensional semiclassical differential operator $P=2{\mathrm{e}}^{-\varphi /h}\circ h{D}_{\overline{z}}\circ {\mathrm{e}}^{\varphi /h}$ on $S^1+i{S}^1$ and on $S^1+i\mathbb{R}$ where $\varphi$ 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 $P$.

24 pages

References in corpus (1)

• Semi-classical States for Non-self-adjoint Schrodinger Operators