Spectral description of a cell growth and division equation

TL;DR

This paper analyzes the spectral properties of a non-local operator modeling cell growth and division, providing criteria for the spectrum to consist of explicit simple eigenvalues based on weighted space conditions.

Contribution

It offers a refined spectral description of a cell growth operator, combining asymptotic analysis with Weyl's theorem to identify conditions for simple eigenvalues.

Findings

Spectral characterization in half-planes beyond the first eigenvalue accumulation point.

Criteria on weighted spaces for explicit simple eigenvalues.

Application of long-time asymptotic expansion and Weyl theorem.

Abstract

We give a refined description of the dominant spectrum of a non-local operator that models growth and equal mitosis of cells. More precisely we look at the spectrum in half planes at the right hand side of the first accumulation point of eigenvalues and give criteria on the weight of weighted $L^1$ spaces for this spectrum to be made of explicit simple eigenvalues. The method relies on a high order long time asymptotic expansion of the solutions to the associated evolution equation obtained in [Zaidi, van Brunt, Wake, Proc. A, R. Soc. Lond., 2015] combined with a Weyl theorem taken from [Mischler, Scher, Ann. Inst. Henri Poincar\'e, Anal. Non Lin\'eaire, 2016].