# A note on the critical points of the localization landscape

## TL;DR

This paper investigates the number of critical points of a specific harmonic function related to elliptic problems in complex domains, providing bounds for special classes of domains and exploring implications for eigenfunction localization.

## Contribution

It introduces bounds on the critical points of the function in certain complex domains, extending classical results with new bounds for rational and quadrature domains.

## Key findings

- Bound on critical points for domains with boundary in a specific algebraic form
- Examples where the bound is attained
- Bound on critical points for quadrature domains

## Abstract

Let $\Omega\subset\mathbb{C}$ be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function $v=v_\Omega$ that solves the elliptic problem $\Delta v = -2$ in $\Omega,$ with boundary values $v=0$ on $\partial\Omega.$ This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of $v$ when $\Omega$ belongs to a special class of domains in the plane, namely, domains for which the boundary $\partial\Omega$ is contained in $\{z:|z|^2 = f(z) + \overline{f(z)}\},$ where $f'(z)$ is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when $\Omega$ is a quadrature domain, and conclude the note by stating some open problems and conjectures.