Trace Formulas for a Class of non-Fredholm Operators: A Review

TL;DR

This paper reviews spectral flow and index theory for non-Fredholm operators, focusing on spectral shift functions and Witten index extensions in a model setting with essential spectrum.

Contribution

It provides a comprehensive review of spectral flow and index theory for non-Fredholm operators, including new results using spectral shift functions and Witten index extensions.

Findings

Spectral shift functions describe spectral flow in non-Fredholm cases.

Extensions of index theory to operators with essential spectrum.

Illustrative model setup demonstrating the approach.

Abstract

We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.