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

• Published in: Transactions of the American Mathematical Society 2016-06, Vol.368 (6), p.4439-4459
• Main Authors: Alexander, James C., Fitzpatrick, Patrick M.
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/6474

To be more precise, let I \lambda _0 an open ball about the origin of a real, separable Hilbert space H. \psi \colon I\times B\to \ R functions. For \lambda \in I, \nabla _x\psi (\lambda ,0)=0, \mathop {\rm Hessian}\nolimits _x\psi (\lambda ,\,0)\equiv L_\lambda . is invertible if \lambda \ne \lambda _0 L_{\lambda _0} L\colon I\to {\mathcal L}(H) \lambda _0 (\lambda _0,\,0) (\lambda ,\,x), for which \nabla _x\psi (\lambda ,x)=0. L\colon I\to {\mathcal L}(H) is invertible for \lambda \ne \lambda _0, L_{\lambda _0} L\colon I\to {\mathcal L}(H) \lambda _0 that contains the point \lambda _0 about the origin, and a family \psi \colon J\times B\to \ R functions such that, for each \lambda \in J, \nabla _x\psi (\lambda ,0)=0 \mathop {\rm Hessian}\nolimits _x\psi (\lambda ,\,0)= L_\lambda , \nabla _x\psi (\lambda ,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.]]>