This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'.
Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes 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 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes 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 history of the subject and earlier references are provided accordingly. The book is aimed at experts in index theory as well as newcomers to the field.
Author(s): Alan Carey, Galina Levitina
Series: Lecture Notes in Mathematics, 2323
Publisher: Springer
Year: 2022
Language: English
Pages: 185
City: Cham
Preface
Acknowledgements
Notations
Contents
1 Introduction
1.1 Motivation and Background
1.2 An Overview of Recent Results
1.3 Discussion of the Methods and the Applications in These Notes
1.4 Summary of the Exposition
2 Double Operator Integrals
2.1 Double Operator Integrals in the Discrete Setting
2.2 Double Operator Integrals in the General Setting
2.3 Double Operator Integrals for Resolvent Comparable Operators
2.4 Continuity of Double Operator Integrals with Respect to the Operator Parameters
3 The Model Operator and Its Approximants
3.1 The Class of p-Relative Trace-Class Perturbations
3.2 Main Setting and Assumptions
4 The Spectral Shift Function
4.1 An Introduction to the Theory of the Spectral Shift Function
4.1.1 Perturbation Determinants
4.1.2 M. G. Krein' s Construction of the Spectral Shift Function
4.1.3 Properties of the Spectral Shift Function
4.2 More General Classes of Perturbations
4.2.1 Spectral Shift Function for Unitary Operators
4.2.2 Spectral Shift Function for Resolvent Comparable Operators
4.2.3 Invariance Principle
4.2.4 Spectral Shift Function for m-Resolvent Comparable Operators
4.3 Continuity of the Spectral Shift Function with Respect to the Operator Parameter
4.4 Representation of the Spectral Shift Function via a Regularised Perturbation Determinant
4.5 Spectral Shift Functions for the Pairs (A+,A-), (H2,H1)
5 Spectral Flow
5.1 Phillips' Definition of Spectral Flow and Analytic Formulas
5.1.1 The Variation of eta Formula
5.1.2 A Review of Analytic Formulas for Spectral Flow
5.2 The Relation Between the Spectral Shift Function and the Spectral Flow
5.3 Generalised Spectral Flow
6 The Principal Trace Formula and Its Applications
6.1 A Brief History of the Principal Trace Formula
6.2 Proving the Principal Trace Formula
6.3 A Generalised Pushnitski Formula
6.4 The Witten Index
6.4.1 Preliminaries
6.4.2 The Formula in Terms of the Spectral Shift Function
6.5 Cyclic Homology and Invariance
6.5.1 How the Witten Index Relates to This
6.5.2 Higher Schatten Classes
6.6 The Anomaly in Terms of the Spectral Shift Function
6.6.1 The Origin of the Notion of an `Anomaly'
6.6.2 Relationship to the Spectral Shift Function
7 Examples
7.1 The Dirac Operator in Rd
7.1.1 The Setting
7.1.2 Verification of Hypothesis 3.2.5
7.1.3 The Index of DA
7.1.4 Behaviour of the Spectral Shift Function for the Massless Dirac Operator
7.2 A Compact One-Dimensional Example
7.2.1 The Setting
7.2.2 Spectral Shift Function of the Pair (Dα+φ, Dα)
7.2.3 The Index of the Operator DA
7.2.4 Spectral Flow Along the path {Dα+θ(t)φ}tR
7.2.5 The Anomaly for the Operator DA
References
