Asymptotics of the Berezin transform and quantization on planar domains (Q1900906)

Let \(\Omega\subset \mathbb{C}\) be a domain whose complement contains at least two points, \(ds= w(\xi)^{- 1} |d\xi|\) the Poincaré metric on \(\Gamma\), \(q\) a positive integer, and \({\mathcal A}^2_q(\Omega)\) the Bergman space of all holomorphic functions on \(\Omega\) square-integrable against the measure \(w(\xi)^{2q- 2} dE(\xi)\), where \(dE\) is the Lebesgue (area) measure. For \(q\geq 2\) the space \({\mathcal A}^2_q(\Omega)\) is nontrivial and admits a reproducing kernel \(K_q(\xi, \eta)\); the Berezin transform \(B_q\) is the integral operator on functions on \(\Omega\) defined by \[ B_q f(\eta)= \int_\Omega |K_q(\xi, \eta)|^2 K_q(\eta, \eta)^{- 1} w(\xi)^{2q- 2} f(\xi) dE(\xi). \] It is shown that as \(q\to \infty\), the reproducing kernels satisfy \[ |K_q(\xi, \eta) w(\xi)^q w(\eta)^q|\sim \pi^{- 1}(2q- 1) [1- d(\xi, \eta)^2]^q, \] where \(d\) is the Kobayashi distance. From this it follows that the Berezin transform has an asymptotic expansion \(B_q f= \sum^\infty_{k= 0} q^{- k} L_k f\) as \(q\to \infty\), where the differential operators \(L_k= P_k(D_\Omega)\) are polynomials of the Laplace-Beltrami operator \(D_\Omega\) corresponding to the Poincaré metric (in particular, \(L_0= I\) and \(L_1= D_\Omega/2\)). This has hitherto been known only for \(\Omega\) a bounded symmetric domain in \(\mathbb{C}^n\). It seems remarkable that the polynomials \(P_k\) are independent of the planar domain \(\Omega\). The result has immediate applications to the Berezin quantization on \(\Omega\).