Spectral properties of two parameter eigenvalue problems II

Abstract

SynopsisWe consider eigenvalues λ =(λ1, λ2) ∈R2 for the problem W(λ)x = 0, x ≠ 0, x ∈ H, where W(λ) = R + λ1V1 + λ2V2), and R, V1, V2 are self-adjoint operators on a separable Hilbert space H, R being bounded below with compact resolvent and V1, V2 being bounded. The i-th eigencurve Z1 is the set of eigenvalues λ, for which the i-th eigenvalue (counted according to multiplicity and in increasing order) of W(λ) vanishes. We study monotonic and asymptotic properties of Zi, and we give formulae for any asymptotes that exist. Additional results are given in the finite dimensional case.