Asymptotic expansion of polyanalytic Bergman kernels

Abstract

Abstract. We consider the q-analytic functions on a given planar domain Ω, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if∂ q f = 0 for the given positive integer q.We also obtain asymptotic formulae for the q-analytic Bergman kernel in the setting of degenerating power weights e −2mQ , as the positive real parameter m tends to infinity. This is only known for q = 1 in view of the work of Tian, Yau, Zelditch, and Catlin. We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality. OverviewIn Section 2, we define, in the one-variable context, the weighted Bergman spaces and their polyanalytic extensions, and in Section 3, we consider the various possible ramifications for Bergman metrics. It should be remarked that the polyanalytic Bergman spaces can be understood as vector-valued (analytic) Bergman spaces with singular local inner product matrix.Generally speaking, Bergman kernels are difficult to obtain in explicit form. However, it is sometimes possible to obtain an asymptotic expansion for them, for instance as the weight degenerates in a power fashion. In Sections 5-9, we extend the asymptotic expansion to the polyanalytic context. Our analysis is based on the microlocal PDE approach of Berman, Berndtsson, Sjöstrand [8]. We focus mainly on the biholomorphic (bianalytic) case, and obtain the explicit form of the first few terms of the expansion. In Section 4, we estimate the norm of point evaluations on bianalytic Bergman spaces, which is later needed to estimate the bianalytic Bergman kernel along the diagonal.Finally, in Section 10, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics introduced in Section 3, and after suitable blow-up, the resulting metrics turn out to be independent of the given potential (which defines the power weight). We interpret this as an instance of geometric universality.2. Weighted polyanalytic Bergman spaces and kernels 2.1. Basic notation. We let C denote the complex plane and R the real line. For z 0 ∈ C and positive real r, let D(z 0 , r) be the open disk centered at z 0 with radius r; moreover, we let T(z 0 , r)

Generalized Wehrl entropies and Euclidean Landau levels

“…is a Rakhmanov probability density of Laguerre-type [20]. We also observe that Q (m) j coincides with the random variable, denoted S (m,1) j , and defined by Shirai [21] in the context of determinantal processes for higher Landau levels (see [22] for an alternative approach and [23,24] for finitedimensional versions) :…”

On the existence of polyanalytic functions

“…There are several classical and well known applications of mathematical analysis which require the theory of polyanalytic functions. In some recent studies we can find some connections of this theory with wavelet theory (see, e.g., ), applications in Physics (see, e.g., ) and universality‐like properties of rather general bi‐analytic Bergman kernels .…”

Bulk asymptotics for polyanalytic correlation kernels

“…Next, we will prove an approximate reproducing identity for polyanalytic functions. For the case q = 2, this was already done in [22]. Here we present the argument for general q ≥ 1.…”

“…We then show how to extend this modified algorithm to polyanalytic functions. This provides a simplification of the method of [22], which was based on the original microlocal analysis technique.…”

“…satisfies ∂ q w L q,m (z, w) = 0), it will be called a local q-analytic Bergman kernel. In [22], we presented an algorithm producing local q-analytic Bergman kernels mod(m −k ) for arbitrary k, based on a microlocal analysis technique of Berman-Berndtsson-Sjöstrand [9] in the analytic case q = 1. We will next show that when q = 1, Taylor expansion and partial integration is enough.…”