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Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds

The Journal of Geometric Analysis volume 30, pages 2615–2646 (2020)Cite this article

Abstract

We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu–Paris Rive Gauche, UFR de Mathématiques, Université Paris Diderot - Paris 7, Case 7012, 75205, Paris Cedex 13, France

    Louis Ioos & Xiaonan Ma

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wen Lu

  3. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wen Lu

  4. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    George Marinescu

  5. Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania

    George Marinescu

Authors
  1. Louis Ioos
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  2. Wen Lu
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  3. Xiaonan Ma
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  4. George Marinescu
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Corresponding author

Correspondence to Xiaonan Ma.

Additional information

To Gennadi Henkin, in memoriam.

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Louis Ioos was supported by grants from Région Ile-de-France. Wen Lu was partially supported by the National Natural Science Foundation of China (Grant No. 11401232). Xiaonan Ma was partially supported by NNSFC 11528103, ANR-14-CE25-0012-01, and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative. George Marinescu was partially supported by DFG-funded project SFB TRR 191.

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Ioos, L., Lu, W., Ma, X. et al. Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds. J Geom Anal 30, 2615–2646 (2020). https://doi.org/10.1007/s12220-017-9977-y

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  • DOI: https://doi.org/10.1007/s12220-017-9977-y

Keywords

  • Toeplitz operator
  • Berezin-Toeplitz quantization
  • Generalized Bergman kernel
  • Bochner Laplacian

Mathematics Subject Classification

  • 53D50
  • 81S10
  • 47B35
  • 58J20