Off-Spectral Analysis of Bergman Kernels

Haakan Hedenmalm,Aron Wennman

doi:10.1007/s00220-019-03667-2

Haakan Hedenmalm, Aron Wennman

Open Access

https://doi.org/10.1007/s00220-019-03667-2

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Abstract

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behavior is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying weights on the complex plane. We obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off-spectral point, which we also refer to as the root function at the point in question. The expansion is valid in the entire off-spectral component containing the root point, and protrudes into the spectrum as well. This allows us to obtain error function transition behavior of the density of states along the smooth interface. Previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum, which contrasts with our non-local expansions.

Highlights

The study of Bergman kernel asymptotics has a sizeable literature
The majority of the contributions have the flavor of local asymptotics near a given point w0, under a positive curvature condition
In the study of partial Bergman kernels for the subspace of all functions vanishing to a given order at the point w0, the assumption of vanishing has the effect of introducing a negative point mass for the curvature form at w0

Summary

Introduction

It concerns the expansion of the coherent state kn(z, w0) of depth n rooted at a given off-spectral point w0 This is the normalized reproducing kernel function at the point w0 for the Bergman space defined by vanishing to order n at w0. In [18], Ross and Singer investigate the partial Bergman kernels associated to spaces of holomorphic sections vanishing along a divisor, and obtain error function transition behavior under the assumption that the setup is invariant under a holomorphic S1-action This result was later extended by Zelditch and Zhou [24], in the context of S1-symmetry.

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