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Journal of Spectral Theory

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Volume 9, Issue 1, 2019, pp. 137–170
DOI: 10.4171/JST/243

Published online: 2018-10-23

Spectral flow for skew-adjoint Fredholm operators

Alan L. Carey[1], John Phillips[2] and Hermann Schulz-Baldes[3]

(1) Australian National University, Canberra, Australia and University of Wollongong, Australia
(2) University of Victoria, Canada
(3) Universität Erlangen-Nürnberg, Germany

An analytic definition of a $\mathbb Z_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through 0 along the path. The $\mathbb Z_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $\mathbb Z_2$-index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the $\mathbb Z_2$-polarization in these models.

Keywords: Spectral flow, skew-symmetric operators, topological insulators

Carey Alan, Phillips John, Schulz-Baldes Hermann: Spectral flow for skew-adjoint Fredholm operators. J. Spectr. Theory 9 (2019), 137-170. doi: 10.4171/JST/243