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Toeplitz operators with analytic symbols

Abstract : We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact Kähler manifold. These objects depend on an integer N and we study, in the limit N → +∞, situations in which one can control them up to an error O(e^{-cN}) for some c > 0. We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to Kähler man-ifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to O(e^{-cN}) on any real-analytic Kähler manifold as N → +∞. We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to O(e^{-cN}). As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.
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Contributor : Alix Deleporte Connect in order to contact the contributor
Submitted on : Tuesday, April 14, 2020 - 12:00:51 PM
Last modification on : Friday, April 1, 2022 - 3:56:37 AM


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Alix Deleporte. Toeplitz operators with analytic symbols. The Journal of Geometric Analysis, 2021, 31, pp.3915-3967. ⟨10.1007/s12220-020-00419-w⟩. ⟨hal-01957594v2⟩



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