Counting zeros of holomorphic functions of exponential growth

TL;DR

This paper generalizes existing results on counting zeros of holomorphic functions with exponential growth, providing geometric interpretations and refined estimates, which are crucial for understanding eigenvalue distributions of perturbed elliptic operators.

Contribution

The paper extends previous zero-counting results to more general settings with natural geometric formulations and improved remainder estimates.

Findings

Zeros count approximately as (2πh)^{-1} times the integral of the Laplacian of the exponent.

Generalization of earlier results to broader classes of functions.

Provides natural geometric statements and sharper remainder estimates.

Abstract

We consider the number of zeros of holomorphic functions in a bounded domain that depend on a small parameter and satisfy an exponential upper bound near the boundary of the domain and similar lower bounds at finitely many points along the boundary. Roughly the number of such zeros is $(2\pi h)^{-1}$ times the integral over the domain of the laplacian of the exponent of the dominating exponential. Such results have already been obtained by M. Hager and by Hager and the author and they are of importance when studying the asymptotic distribution of eigenvalues of elliptic operators with small random perturbations. In this paper we generalize these results and arrive at geometrically natural statements and natural remainder estimates.