# The absolute spectrum revisited from a topological viewpoint

## TL;DR

This paper revisits the spectral analysis of linear operators related to traveling wave stability, using topological methods to understand the absolute spectrum and its relation to eigenvalue accumulation.

## Contribution

It provides new topological proofs connecting the spectral behavior of operators with the geometry of Grassmannian manifolds and Schubert cycles.

## Key findings

- Eigenfunctions induce curves on Grassmannian manifolds.
- Decomposition of Grassmannian into submanifolds aids spectral analysis.
- Topological framework offers new insights into spectral accumulation sets.

## Abstract

We consider the topological relation behind the spectral behavior of a linear operator that arises in the stability problem of traveling waves on a large bounded domain. When the domain size tends to infinity, the absolute and asymptotic essential spectra appears as accumulation sets of eigenvalues under separated and periodic boundary conditions, respectively. We present new proofs of Sandstede and Scheel [Theorems 4 and 5 of [14]] in a topological framework. The eigenfunction induces a curve on the Grassmannian manifold. To extract topological information from them, we decompose the Grassmannian into the submanifolds using the Schubert cycles, and analyze the curves on each submanifolds.