Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue

Abstract

In the Euclidean space $\mathbb R^k$, we consider the perturbed eigenvalue problem $Lx + \varepsilon N(x) = \lambda x$, $\|x\| = 1$, where $\varepsilon,\lambda$ are real parameters, $L$ is a linear endomorphism of $\mathbb R^k$, and $N\colon S^{k-1} \to \mathbb R^k$ is a continuous map defined on the unit sphere of $\mathbb R^k$. We prove a global continuation result for the solutions $(x,\varepsilon,\lambda)$ of this problem. Namely, under the assumption that $x_* \in S^{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a~simple eigenvalue $\lambda_* \in \mathbb R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,\lambda_*)$ is either unbounded or meets a~solution $(x^*,0,\lambda^*)$ having $x^* \not= x_*$. Our result is inspired by a paper of R. Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space.