Spectral Flow and Bifurcation of Critical Points of Strongly-Indefinite Functionals Part I. General Theory

Abstract

Spectral flow is a well-known homotopy invariant of paths of selfadjoint Fredholm operators. We describe here a new construction of this invariant and prove the following theorem:Letψ;:I×U→Rbe aC2function on the product of a real intervalI=[a,b]with a neighborhood U of the origin of a real separable Hilbert space H and such that for each λ in I, 0is a critical point of the functional ψλ≡ψ(λ,·).Assume that the Hessian Lλof ψλat0is Fredholm and moreover that Laand Lbare nonsingular. If the spectral flow of the path{Lλ}λ∈Idoes not vanish, then the interval I contains at least one bifurcation point from the trivial branch for solutions of the equation∇xψ(λ,x)=0.Equivalently: every neighborhood of I×{0}contains points of the form(λ,x) wherex≠0 is a critical point ofψλ.