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October 2019 Interface asymptotics of Partial Bergman kernels around a critical level
Steve Zelditch, Peng Zhou
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Ark. Mat. 57(2): 471-492 (October 2019). DOI: 10.4310/ARKIV.2019.v57.n2.a12

Abstract

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.

Funding Statement

Research partially supported by NSF grant DMS-1541126 and by the Stefan Bergman trust.

Citation

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Steve Zelditch. Peng Zhou. "Interface asymptotics of Partial Bergman kernels around a critical level." Ark. Mat. 57 (2) 471 - 492, October 2019. https://doi.org/10.4310/ARKIV.2019.v57.n2.a12

Information

Received: 16 May 2019; Revised: 2 June 2019; Published: October 2019
First available in Project Euclid: 16 April 2020

zbMATH: 07114516
MathSciNet: MR4018764
Digital Object Identifier: 10.4310/ARKIV.2019.v57.n2.a12

Rights: Copyright © 2019 Institut Mittag-Leffler

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Vol.57 • No. 2 • October 2019
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