2019/12/15 by Benevieri, Pierluigi, Calamai, Alessandro, Furi, Massimo +1
#Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1912.07021
We consider the nonlinear eigenvalue problem Lx + ε N(x) = λCx, ‖x‖=1, where ε,λ are real parameters, L, C\colon G → H are bounded linear operators between separable real Hilbert spaces, and N\colon S → H is a continuous map defined on the unit sphere of G. We prove a global persistence result regarding the set Σ of the solutions (x,ε,λ) ∈ S × \mathbb R× \mathbb R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p_*=(x_*,0,λ_*) of the unperturbed problem, we prove that the connected component of Σ containing p_* is either unbounded or meets a triple p^*=(x^*,0,λ^*) with p^* \not= p_*. When C is the identity and G=H is finite dimensional, the assumptions on (x_*,0,λ_*) mean that x_* is an eigenvector of L whose corresponding eigenvalue λ_* is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.