Doll N. Spectral Flow. A Functional Analytic and Index-Theoretic Approach 2023

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This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.
Presents a self-contained functional analytic treatment of spectral flow
Includes a detailed analysis of the homotopy theory of the set of unbounded selfadjoint Fredholm operators
Provides an analysis of Bott-Maslov index via spectral flow
Preface
Acknowledgment
Spectral flow in finite dimension
Applications of finite-dimensional spectral flow
Bounded Fredholm operators
Spectral flow for bounded self-adjoint Fredholm operators
Fredholm pairs and their index
Unbounded Fredholm operators
Spectral flow for unbounded self-adjoint Fredholm operators
Homotopy theory of Fredholm operators
Bott–Maslov index via spectral flow
Index pairings and spectral localizer
Spectral flow in semifinite von Neumann algebras
Spectral flow in bifurcation theory
A Collection of technical elements
Acronyms and notations
Bibliography
Index