Spectrum of a linear differential equation with constant coefficients

TL;DR

This paper computes the spectrum of ultrametric linear differential equations with constant coefficients over affinoid domains, revealing it as a finite union of disks and establishing its continuity properties.

Contribution

It introduces a method to determine the spectrum of such differential equations in the Berkovich setting, highlighting its geometric structure.

Findings

Spectrum is a finite union of disks or their closures

Spectrum satisfies a continuity property

Provides a geometric description of the spectrum

Abstract

In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differential equation with constant coefficients, defined over an affinoid domain of the analytic affine line $A_k^{1,an}$. We show that it is a finite union of either closed disks or topological closures of open disks and that it satisfies a continuity property.