Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

TL;DR

This paper investigates how eigenvalues and eigenvectors in Hilbert spaces persist under perturbations, especially when the unperturbed problem has an odd-dimensional kernel, using topological degree theory.

Contribution

It extends global continuation results to eigenvalue problems with odd multiplicity kernels in infinite-dimensional Hilbert spaces using topological methods.

Findings

Established Rabinowitz-type global continuation for eigenvectors with odd multiplicity

Developed a topological degree framework for Fredholm maps in Banach manifolds

Proved persistence of eigenvalues and eigenvectors under perturbations in Hilbert spaces

Abstract

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.