Journal of Mathematical Physics, Analysis, Geometry
2021, vol. 17, No 4, pp. 422-440   https://doi.org/10.15407/mag17.04.422     ( to contents , go back )
https://doi.org/10.15407/mag17.04.422

Berezin Transforms Attached to Landau Levels on the Complex Projective Space Pn(ℂ)

Nizar Demni

Aix-Marseille Université CNRS Centrale Marseille I2M-UMR 7373. 39 rue F. Joliot Curie, 13453 Marseille, France
E-mail: nizar.demni@univ-amu.fr

Zouhaïr Mouayn

Department of Mathematics, Faculty of Sciences and Technics (M'Ghila), Sultan Moulay Slimane University, P.O. Box. 523, Béni Mellal, Morocco

Department of Mathematics, KTH Royal Institute of Technology, SE-10044, Stockholm, Sweden
E-mail: mouayn@usms.ma, mouayn@kth.se

Houda Yaqine

Department of Mathematics, Faculty of Sciences and Technics (M'Ghila), Sultan Moulay Slimane University, P.O. Box. 523, Béni Mellal, Morocco
E-mail: yaqinehou@gmail.com

Received October 4, 2020, revised October 29, 2021.

Abstract

We construct coherent states for each eigenspace of a magnetic Laplacian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini-Study Laplacian in a closed form involving of a terminating Kampé de Fériet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself.

Mathematics Subject Classification 2010: 81Q10, 47G10, 58C40, 46E22
Key words: complex projective space, coherent states, Berezin transform, magnetic Laplacians, Fubini-Study Laplacian, Koornwinder's formula, Clebsh-Gordan type relation, Kampé de Fériet function

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