nLab spectral flow

Ides

In operator algebra, by spectral flow one refers to the change of spectrum of an operator as that operator continuously varies with a parameter.

Concretely, for closed paths (loops) of self-adjoint Fredholm operators their spectral flow is essentially the number of eigenvalues (which are all real, by self-adjointness) that crosses from positive to negative as one goes around the loop.

References

On spectral flow for self-adjoint Fredholm operators:

The concept is first mentioned, in the context of introducing the eta-invariant, on p. 47-48 of:

• Michael F. Atiyah, Vijay K. Patodi, Isadore M. Singer: Spectral asymmetry and Riemannian Geometry. I, Mathematical Proceedings of the Cambridge Philosophical Society 77 1 (1975) 43-69 [doi:10.1017/S0305004100049410]

Monograph:

• Nora Doll, Hermann Schulz-Baldes, Nils Waterstraat: Spectral Flow — A Functional Analytic and Index-Theoretic Approach, Studies in Mathematics 94, De Gruyter (2023) [doi:10.1515/9783111172477]

Further discussion:

• John Phillips: Self-Adjoint Fredholm Operators And Spectral Flow, Canadian Mathematical Bulletin 39 4 (1996) 460-467 [doi:10.4153/CMB-1996-054-4]

• Alan Carey, John Phillips: Fredholm modules and spectral flow, J. Canadian Math. Soc. 50 (1998) 673-718 [cms:10.4153/CJM-1998-038-x]

• Bernhelm Booß-Bavnbek, Matthias Lesch, John Phillips: Unbounded Fredholm Operators and Spectral Flow, Canadian Journal of Mathematics 57 2 (2005) 225-250 [doi:10.4153/CJM-2005-010-1, arXiv:math/0108014]

• Nils Waterstraat: Fredholm Operators and Spectral Flow [arXiv:1603.02009]

• Ping Wong Ng, Arindam Sutradhar, Cangyuan Wang: On spectral flow for operator algebras [arXiv:2312.12061]