A generalization of Yoshida-Nicolaescu theorem using partial signatures
The authors give an alternative proof of the Yoshida-Nicolaescu theorem which works in a quite general situation, both in the degenerate and the non-degenerate case. The technique used employs the theory of partial signatures as in Giambò, Piccione and Portaluri, which has its roots in the work of Farber and Ievine and makes the proof simple and direct. The theorem gives the equality of the spectral flow of a family of first order self-adjoint differential operators defined on sections of a Hermitian vector bundle over a partitioned manifold and the Maslov index of the corresponding pair of Cauchy data spaces. No nondegeneracy assumption is made on the endpoints of the path of differential operators. They prove that, if the path of differential operators is analytic, then at each singularity, the contribution to the spectral flow and the Maslov index coincise; this is proved by an explicit calculation of the partial signatures. Moreover, the general case is obtained by a direct perturbation argument.
scientific article; zbMATH DE number 5706070