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

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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Correspondence to Xiaonan Ma.

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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