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Über dieses Buch

The book deals with the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fact that there is actually a fundamental localization principle underlying all these solutions has mostly passed unnoticed. The ignorance of this general principle has often necessitated using various artificial tricks and hindered the solution of new important problems in index theory. So far, the localization principle has been only scarcely covered in journal papers and not covered at all in monographs. The suggested book is intended to fill the gap. So far, it is the first and only monograph dealing with the topic. Both the general localization principle and its applications to specific problems, existing and new, are covered. The book will be of interest to working mathematicians as well as graduate and postgraduate university students specializing in differential equations and related topics.​



Chapter 0. Introduction

The main subject of this book is the index locality principle, which can more precisely be called the

superposition principle for the relative index

. This introduction, which is partly based on the paper [60] by Nazaikinskii and Sternin, is intended as an elementary introduction to this principle. We discuss it and give some examples of its consequences and applications showing that it often proves to be a powerful tool for obtaining index formulas in various situations. Here we try to keep things as clear as possible and often give only the simplest versions of the results. A more detailed exposition, as well as the proofs, is given in subsequent chapters. Readers are also encouraged to consult the bibliographical remarks at the end of the introduction, which in particular recommend some further reading.

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin

Superposition Principle


Chapter 1. Superposition Principle for the Relative Index

The superposition principle outlined in the introduction deals with modifications (surgeries) of elliptic operators performed on two disjoint closed subsets, say




, of the manifold


on which these operators are defined and says that the index increments ∆


and ∆


resulting from these surgeries are independent (i.e., are just summed if both surgeries are carried out simultaneously).

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin

Chapter 2. Superposition Principle for K-Homology

In Chapter 1, we proved the superposition principle for the relative index in the fairly broad setting of collar spaces. However, although the index is possibly the most important homotopy invariant of elliptic operators, it is by no means the only one, and it is natural to ask whether the principle can be generalized to cover other invariants as well. This and the next chapter deal exactly with this topic. The results of the present chapter were obtained in [63].

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin

Chapter 3. Superposition Principle for KK-Theory

In the present chapter, we prove a relative index type theorem that compares certain elements of the Kasparov




. Just as the superposition principle in Chapter 2 permits one to obtain relative index theorems for elliptic operators, the theorem given here implies relative index theorems for elliptic operators over a




in the sense of Mishchenko–Fomenko [47], where the index is an element of the







Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin



Chapter 4. Elliptic Operators on Noncompact Manifolds

In this chapter, we discuss various relative index theorems of Gromov–Lawson type, which have important applications in geometry and topology. We, however, do not dwell on these applications (for which we refer the reader to the literature) but focus our attention on the relationship between these theorems and the general relative index theorems given in Part I of the book.

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin

Chapter 5. Applications to Boundary Value Problems

Here we introduce some notation and present technical tools that come in handy when applying the general superposition principle for the relative index to the theory of boundary value problems for elliptic differential operators.

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin

Chapter 6. Spectral Flow for Families of Dirac Type Operators with Classical Boundary Conditions

In this chapter, we discuss the results obtained in [38] with the use of the relative index superposition principle for the spectral flow of families of self-adjoint Dirac type operators with classical elliptic boundary conditions on a compact Riemannian manifold with boundary. In the two-dimensional case, such operators arise, for example, when describing electron states in graphene [39] or in topological insulators [33,68], and their spectral flow has an important physical meaning, being related to the creation of electron–hole (or, more generally, particle–antiparticle) pairs.

Vladimir Nazaikinskii, Bert-Wolfgang Schulze, Boris Sternin


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