Berezin transforms attached to Landau levels on the complex projective space CPn

TL;DR

This paper constructs Berezin transforms for Landau levels on complex projective spaces using coherent states and provides a variational formula linking it to the Fubini-Study Laplacian, generalizing known results for the sphere.

Contribution

It introduces a new method to define Berezin transforms on complex projective spaces via coherent states and derives a variational formula involving the Fubini-Study Laplacian.

Findings

Derived a variational formula for Berezin transforms on CPn.

Reduced the formula to known cases when n=1 and for the lowest Landau level.

Extended the Berezin transform framework to higher-dimensional complex projective spaces.

Abstract

In this paper, we construct coherent states for each generalized Bergman space on the n-dimensional complex projective space in order to apply a coherent states quantization method. Doing so allows to define the Berezin transform for these spaces. In particular, we provide a variational formula for this transform by means of the Fubini-Study Laplace operator which reduces when n = 1 and for the lowest spherical Landau level to the well-known formula previously given by Berezin himself.