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On the Spectral Flow of Families of Dirac Operators with Constant Symbol
Author(s) -
Bunke Ulrich
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941650113
Subject(s) - mathematics , dirac operator , constant (computer programming) , invariant (physics) , dirac (video compression format) , atiyah–singer index theorem , spectrum (functional analysis) , constant coefficients , mathematical analysis , projection (relational algebra) , elliptic operator , operator (biology) , pure mathematics , mathematical physics , quantum mechanics , physics , biochemistry , chemistry , algorithm , repressor , computer science , transcription factor , neutrino , gene , programming language
Abstract We consider families of generalized Dirac operators D t with constant principal symbol and constant essential spectrum such that the endpoints are gauge equivalent, i.e., D 1 = W * D 0 W . The spectral flow un any gap in the essential spectrum we express as the Fredholm index of 1 + ( W ‐ 1) P where P is the spectral projection on the interval d , ∞) with respect to D 0 and d is in the gap. We reduce the computation of this index to the Atiyah‐Singer index theorem for elliptic pseudodifferential operators. We find an invariant of the Riemannian geometry for odd dimensional spin manifolds estimating the length of gaps in the spectrum of the Dirac operator.