Abstract
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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Acknowledgements
We thank M. Hitrik and M. Zworski for interesting discussions around this work and related work in progress. M. Vogel was partially funded by the Agence Nationale de la Recherche, through the project ADYCT (ANR-20-CE40-0017).
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Dedicated to Carlos Kenig on the occasion of his birthday.
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Sjöstrand, J., Vogel, M. Tunneling for the \(\overline{\partial }\)-Operator. Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-024-00692-0
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DOI: https://doi.org/10.1007/s10013-024-00692-0