Spectral flow for skew-adjoint Fredholm operators

Alan L. Carey; John Phillips; Hermann Schulz-Baldes

doi:10.4171/JST/243

Spectral flow for skew-adjoint Fredholm operators

Alan L. Carey, John Phillips, Hermann Schulz-Baldes

Research output: Contribution to journal › Article › peer-review

18 Citations (Scopus)

Abstract

An analytic definition of a 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 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 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 Z ₂ -polarization in these models.

Original language	English
Pages (from-to)	137-170
Number of pages	34
Journal	Journal of Spectral Theory
Volume	9
Issue number	1
DOIs	https://doi.org/10.4171/JST/243
Publication status	Published - 2019

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title = "Spectral flow for skew-adjoint Fredholm operators",

abstract = " An analytic definition of a 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 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 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 Z 2 -polarization in these models.",

keywords = "Skew-symmetric operators, Spectral flow, Topological insulators",

author = "Carey, {Alan L.} and John Phillips and Hermann Schulz-Baldes",

note = "Publisher Copyright: {\textcopyright} European Mathematical Society.",

year = "2019",

doi = "10.4171/JST/243",

language = "English",

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journal = "Journal of Spectral Theory",

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T1 - Spectral flow for skew-adjoint Fredholm operators

AU - Carey, Alan L.

AU - Phillips, John

AU - Schulz-Baldes, Hermann

N1 - Publisher Copyright: © European Mathematical Society.

PY - 2019

Y1 - 2019

N2 - An analytic definition of a 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 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 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 Z 2 -polarization in these models.

AB - An analytic definition of a 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 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 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 Z 2 -polarization in these models.

KW - Skew-symmetric operators

KW - Spectral flow

KW - Topological insulators

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U2 - 10.4171/JST/243

DO - 10.4171/JST/243

M3 - Article

SN - 1664-039X

VL - 9

SP - 137

EP - 170

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

IS - 1

ER -

Spectral flow for skew-adjoint Fredholm operators

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