The spectral flow, the Fredholm index, and the spectral shift function

TL;DR

This paper explores the relationship between the Fredholm index, spectral flow, and spectral shift functions, providing a new interpretation of the well-known theorem as a limit case of spectral shift identities.

Contribution

It introduces a novel perspective by linking the Fredholm index and spectral flow to spectral shift functions through a limiting process.

Findings

The Fredholm index equals the spectral flow in certain contexts.

Spectral shift functions can be used to interpret the Fredholm index as a limit case.

The paper offers a new conceptual framework connecting these spectral invariants.

Abstract

We discuss the well known ``Fredholm index=spectral flow'' theorem and show that it can be interpreted as a limit case of an identity involving two spectral shift functions.