The Callias Index Formula Revisited / by Fritz Gesztesy, Marcus Waurick.

Format:

Book

Author/Creator:

Gesztesy, Fritz., Author.

Waurick, Marcus., Author.

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2157

Language:

English

Subjects (All):

Differential equations, Partial.

Operator theory.

Functional analysis.

Partial Differential Equations.

Operator Theory.

Functional Analysis.

Local Subjects:

Partial Differential Equations.

Operator Theory.

Functional Analysis.

Physical Description:

1 online resource (IX, 192 p. 1 illus.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Summary:

These lecture notes aim at providing a purely analytical and accessible proof of the Callias index formula. In various branches of mathematics (particularly, linear and nonlinear partial differential operators, singular integral operators, etc.) and theoretical physics (e.g., nonrelativistic and relativistic quantum mechanics, condensed matter physics, and quantum field theory), there is much interest in computing Fredholm indices of certain linear partial differential operators. In the late 1970’s, Constantine Callias found a formula for the Fredholm index of a particular first-order differential operator (intimately connected to a supersymmetric Dirac-type operator) additively perturbed by a potential, shedding additional light on the Fedosov-Hörmander Index Theorem. As a byproduct of our proof we also offer a glimpse at special non-Fredholm situations employing a generalized Witten index.

Contents:

Introduction.-Notational Conventions

Functional Analytic

On Schatten–von Neumann Classes and Trace Class

Pointwise Estimates for Integral Kernels

Dirac-Type

Derivation of the Trace Formula – The Trace Class Result

Derivation of the Trace Formula – Diagonal Estimates

The Case n = 3

The Index Theorem and Some Consequences

Perturbation Theory for the Helmholtz Equation

The Proof of Theorem 10.2: The Smooth Case

The Proof of Theorem 10.2: The General Case

A Particular Class of Non-Fredholm Operators L and Their Generalized Witten Index

A: Construction of the Euclidean Dirac Algebra

B: A Counterexample to [22, Lemma 5]

References

Index.

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-29977-8

OCLC:

952973283