Spectral flow is a complete invariant for detecting bifurcation of critical points

Abstract:

Given a one-parameter path of equations for which there is a trivial branch of solutions, to determine the points on the branch from which there bifurcate nontrivial solutions, there is the heuristic principle of linearization. That is to say, at each point on the branch, linearize the equation, and justify the inference that points on the branch that are bifurcation points for the path of linearized equations are also bifurcation points for the original path of equations. In quite general circumstances, for the bifurcation of critical points, we show that, at isolated singular points of the path of linearizations, a property of the path that is known to be sufficient to force bifurcation of nontrivial critical points is also necessary.

To be more precise, let $I$ be an open interval of real numbers that contains the point $\lambda _0$ and $B$ an open ball about the origin of a real, separable Hilbert space $H.$ Let $\psi \colon I\times B\to \ R$ be a family of $C^2$ functions. For $\lambda \in I,$ assume $\nabla _x\psi (\lambda ,0)=0,$ and set $\operatorname {Hessian} _x\psi (\lambda ,\,0)\equiv L_\lambda .$ Assume $L_\lambda$ is invertible if $\lambda \ne \lambda _0$ and $L_{\lambda _0}$ is Fredholm. It is known that if the spectral flow of $L\colon I\to {\mathcal L}(H)$ across $\lambda _0$ is nonzero, then in each neighborhood of $(\lambda _0,\,0)$ there are pairs $(\lambda ,\,x),$ $x\ne 0,$ for which $\nabla _x\psi (\lambda ,x)=0.$ We prove that if $L\colon I\to {\mathcal L}(H)$ is a continuous path of symmetric operators for which $L_\lambda$ is invertible for $\lambda \ne \lambda _0,$ $L_{\lambda _0}$ is Fredholm, and the spectral flow of $L\colon I\to {\mathcal L}(H)$ across $\lambda _0$ is zero, then there is an open interval $J$ that contains the point $\lambda _0$, an open ball $B$ about the origin, and a family $\psi \colon J\times B\to \ R$ of $C^2$ functions such that, for each $\lambda \in J,$ $\nabla _x\psi (\lambda ,0)=0$ and $\operatorname {Hessian} _x\psi (\lambda ,\,0)= L_\lambda ,$ but $\nabla _x\psi (\lambda ,x)\ne 0$ if $x\ne 0.$ Therefore, at an isolated singular point of the path of linearizations of the gradient, under the sole further assumption that the linearization at the singular point is Fredholm, spectral flow is a complete invariant for the detection of bifurcation of nontrivial critical points.