On the spectral flow theorem of Robbin-Salamon for finite intervals

TL;DR

This paper extends the spectral flow theorem of Robbin-Salamon to finite intervals by defining boundary conditions that make certain operators Fredholm, with the index given by the spectral flow of the operator path.

Contribution

It introduces a method to impose boundary conditions on operators of the form ∂_s + A(s) over finite intervals, linking their Fredholm index to spectral flow.

Findings

Operators become Fredholm with suitable boundary conditions

Fredholm index equals spectral flow of A(s)

Extension of spectral flow theorem to finite intervals

Abstract

In this article we consider operators of the form $\partial_s\xi+A(s)\xi$ where $s$ lies in an interval $[-T,T]$ and $s\mapsto A(s)$ is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path $A$.