Interface asymptotics of Partial Bergman kernels around a critical level

Steve Zelditch Northwestern University Peng Zhou Northwestern University

Complex Variables and Complex Analysis mathscidoc:1912.43048

Arkiv for Matematik, 57, (2), 471 – 492, 2019
In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels $\Pi_{k, [E_1, E_2]} (z,w)$ and the associated DOS (density of states) $\Pi_{k, [E_1, E_2]} (z)$ across the interface $\mathcal{C}$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H : M \to \mathbb{R}$ on a Kähler manifold. The allowed region is $H^{-1} ([E_1, E_2])$ and the interface $\mathcal{C}$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\frac{1}{2}}$ tube around $\mathcal{C}$ for regular interfaces, one obtains $\delta$-asymptotics in $k^{-\frac{1}{4}}$ tubes around singular points of a critical interface. In $k^{-\frac{1}{2}}$ tubes, the transition law is given by the osculating metaplectic propagator.
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@inproceedings{steve2019interface,
  title={Interface asymptotics of Partial Bergman kernels around a critical level},
  author={Steve Zelditch, and Peng Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204142836993389604},
  booktitle={Arkiv for Matematik},
  volume={57},
  number={2},
  pages={471 – 492},
  year={2019},
}
Steve Zelditch, and Peng Zhou. Interface asymptotics of Partial Bergman kernels around a critical level. 2019. Vol. 57. In Arkiv for Matematik. pp.471 – 492. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204142836993389604.
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