On the Perturbed Eigenvalue Problem

The perturbed eigenvalue problem L(x)v(x) = λ(x)v(x) is considered near 0 = x ∈ R with L(x) a Fredholm operator of index zero. Using the Implicit Function Theorem and the Bordering Lemma, a necessary and sufficient smoothness condition is derived for the characteristic pairs (λ(x), υ(x)) with eigenvalue λ(x) emanating from a semi-simple eigenvalue λ(0) of L(0).