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

Johannes Sjöstrand1, Martin Vogel2
1IMB, Université de Bourgogne, Dijon Cedex, France
2Institut de Recherche Mathématique Avancée - UMR 7501, CNRS et Université de Strasbourg, Strasbourg Cedex, France

Tóm tắt

We study the small singular values of the 2-dimensional semiclassical differential operator $$P = 2\textrm{e}^{-\phi /h}\circ hD_{\overline{z}}\circ \textrm{e}^{\phi /h}$$ 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 this to determine a Weyl law for the exponentially small singular values of P.

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