Maslov index in the infinite dimension and a splitting formula for a spectral flow

Details Maslov index in the infinite dimension and a splitting formula for a spectral flow Maslov index in the infinite dimension and a splitting formula for a spectral flow

2003-06-28

arxiv.org/abs/math/0306409v1

Japanese Journal of Mathematics, Vol. 28, No. 2, pp. 215-243(2002)

Mathematics

Differential Geometry

Scientific paper

First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic operators on a closed manifold, which we decompose into two parts with commom boundary. Then the formula says that the spectral flow is a sum of two spectral flows on each part of the separated manifold with naturally introduced elliptic boundary conditions. In the course of proving this formula, we investigate a property of the Maslov index for paths of Fredholm pairs of Lagrangian subspaces

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