Off-spectral analysis of Bergman kernels

TL;DR

This paper develops a unified approach to analyze the asymptotic behavior of Bergman kernels near interfaces where the limiting density vanishes, revealing transition phenomena in off-spectral regions for complex planar measures.

Contribution

It introduces uniform asymptotic expansions of root functions for off-spectral points, extending previous local analyses to entire off-spectral components and interfaces.

Findings

Derived uniform asymptotic expansions valid across off-spectral regions

Identified error function transition behavior along interfaces

Unified treatment of negatively curved and positively curved metric cases

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 behaviour 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 planar measures on the complex plane. We obtain uniform asymptotic expansions of root functions, which are essentially normalized partial Bergman kernels at an off-spectral point, valid in the entire off-spectral component and protruding into the spectrum as well, which allows us to show error function transition behaviour of the original kernel along the interface. In contrast, previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum.