Index Theory Beyond the Fredholm Case

This book discusses extending index theory to non-Fredholm operators, focusing on the Witten index and introducing spectral flow beyond the Fredholm case. It reviews Double Operator Integrals, p-relative trace class perturbations, and the analytic approach to spectral flow, culminating in the principal trace formula. Examples are provided using explicit computations on two examples.

Format: Paperback / softback
Length: 175 pages
Publication date: 02 December 2022
Publisher: Springer International Publishing AG

This book delves into the realm of extending index theory to instances where non-Fredholm operators emerge. It specifically focuses on one aspect of the challenge of substituting the concept of spectral flow and the Fredholm index when the operators under consideration possess zero in their essential spectrum. The majority of research in this field can be traced back to the so-called Witten index, which is extensively discussed herein. The novel approach outlined in these notes involves the introduction of "spectral flow beyond the Fredholm case." This work aims to provide a comprehensive overview of numerous investigations and scattered notions from the past 50 years, carefully introducing spectral flow, the Witten index, and the spectral shift function. It then describes their interconnections in detail.

After the introduction, Chapter 2 provides a thorough review of Double Operator Integrals. Chapter 3 introduces the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 examines the analytic approach to spectral flow, culminating in Chapter 6 with the book's main abstract result, known as the principal trace formula. Chapter 7 concludes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd and a differential operator on an interval. Throughout, attention is paid to the historical background of the subject, and appropriate earlier references are provided.

This book is designed to appeal to both experts in index theory and newcomers to the field.

Weight: 302g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031194351
Edition number: 1st ed. 2022