On a formula for the spectral flow and its applications

Abstract

ABSTRACT. We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

“…By Proposition 4.3 the path s → H P (s) is smooth, and by [5,Proposition 4.9] this implies the continuity as closed subspaces in the sense of Section 2; by Proposition 4.4 the symmetric bilinear forms C s are essentially positive and by Lemma 4.5 the initial contribution is always zero, so that the index theorem follows forÎ 1 and of course for I = I 1 restricted to W P (1) × W P (1) (we observe that W P (1) = H P (1)). By Proposition 3.11 the dimensions of the kernel of I s restricted to W P (s) × W P (s) coincides with the number of (P, Y )-pseudo focal points counted with multiplicity.…”

“…Weaker notions of continuity may also be considered (see Appendix A). Given a projection 1 P ∈ L(H), we will denote by Im(P ) the image P (H), which is a closed subspace of H. The following lemma can be found in [5,Lemma 4.7].…”

Spectral flow and iteration of closed semi-Riemannian geodesics

“…Details on the definition and the basic properties of the spectral flow can be found, for instance, in Refs. [7,11,20]. There are several conventions on how to compute the contribution of the endpoints of the path, in case of degenerate endpoints; although making a specific choice is irrelevant in the context of the present paper, we will follow the convention in [20].…”

“…2. As to the point z = 1, where Corollary 2.3 does not apply, we use a certain finite dimensional reduction formula for the spectral flow (Proposition 4.5), which was proved recently in [7] to show that the spectral flow function has in z = 1 a sort of artificial discontinuity when γ is nondegenerate. The reduction formula is used also in last section, where we obtain the iteration formula for the spectral flow (Proposition 6.1) and we prove estimates on its growth.…”

“…As to the periodic case, the notion of spectral flow sf(γ ) of a closed geodesic γ has been introduced only recently [7]; this is a generalization of the Morse index of the geodesic action functional in the Riemannian case. For the reader's convenience, in Sect.…”