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

✍ Scribed by Patrick M Fitzpatrick; Jacobo Pejsachowicz; Lazaro Recht

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 274 KB
• Volume: 162
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3366

✦ Synopsis

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: Let : I_U Ä R be a C 2 function 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 * in I, 0 is a critical point of the functional * # (*, } ). Assume that the Hessian L * of * at 0 is Fredholm and moreover that L a and L b are nonsingular. If the spectral f low of the path [L * ] * # I does not vanish, then the interval I contains at least one bifurcation point from the trivial branch for solutions of the equation { x (*, x)=0. Equivalently: every neighborhood of I_[0] contains points of the form (*, x) where x{0 is a critical point of * .