A note on Berezin-Toeplitz quantization of the Laplace operator

Alberto Della Vedova

Complex Manifolds (2015)

  • Volume: 2, Issue: 1, page 131-139, electronic only
  • ISSN: 2300-7443

Abstract

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Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

How to cite

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Alberto Della Vedova. "A note on Berezin-Toeplitz quantization of the Laplace operator." Complex Manifolds 2.1 (2015): 131-139, electronic only. <http://eudml.org/doc/275930>.

@article{AlbertoDellaVedova2015,
abstract = {Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.},
author = {Alberto Della Vedova},
journal = {Complex Manifolds},
keywords = {Hodge manifold; Berezin-Toeplitz quantization map},
language = {eng},
number = {1},
pages = {131-139, electronic only},
title = {A note on Berezin-Toeplitz quantization of the Laplace operator},
url = {http://eudml.org/doc/275930},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Alberto Della Vedova
TI - A note on Berezin-Toeplitz quantization of the Laplace operator
JO - Complex Manifolds
PY - 2015
VL - 2
IS - 1
SP - 131
EP - 139, electronic only
AB - Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.
LA - eng
KW - Hodge manifold; Berezin-Toeplitz quantization map
UR - http://eudml.org/doc/275930
ER -

References

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  2. [2] M. Bordemann, E. Meinrenken and M. Schlichenmaier. Toeplitz quantization of K¨ahler manifolds and gl(N), N → 1 limits. Comm. Math. Phys. 165 (1994), no. 2, 281–296. Zbl0813.58026
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  12. [12] M. Schlichenmaier. Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact K¨ahler manifolds. In ‘Proceedings of the XVIIth workshop on geometric methods in physics, Bia lowie˙za, Poland, July 3 – 10, 1998’ (M. Schlichenmaier, et. al. Eds.), Warsaw University Press, 45–56. arXiv:math/9902066v2 [math.QA]. 
  13. [13] M. Schlichenmaier. Berezin-Toeplitz quantization and star products for compact K¨ahler manifolds. In ‘Mathematical Aspects of Quantization’, S. Evens, M. Gekhtman, B. C. Hall, X. Liu, C. Polini Eds. Contemporary Mathematics 583. AMS (2012). arXiv:1202.5927v3 [math.QA]. 
  14. [14] G. Tian. On a set of polarized K¨ahler metrics on algebraic manifolds J. Differential Geometry 32 (1990) 99–130. Zbl0706.53036
  15. [15] S. Zelditch. Szeg˝o kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331. [Crossref] Zbl0922.58082

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