Spectral flow and bifurcation of critical points of strongly indefinite functionals I

Spectral flow is a well–known homotopy invariant of paths of self-adjoint Fredholm operators. We describe here a new construction of this invariant and prove the following theorem: Let f : I × U -> R be a C2 function defined on the product of a real interval I = [a, b] with a neighborhood U of the origin of a real separable Hilbert space H and such that for each t in I, 0 is a critical point of the functional f (t, ·). Assume that the Hessian L of f at 0 is Fredholm and moreover that L_a and L_b are nonsingular. If the spectral flow of the path L does not vanish, then the interval I contains at least one bifurcation point from the trivial branch for solutions of the equation Grad f (t, x) = 0. Equivalently: every neighborhood of I × {0} contains points of the form (t,x) where x is a critical point of f_t different from 0 .