The Callias Index Formula Revisited

Fritz Gesztesy, Marcus Waurick

Research output: Book/ReportBook

Abstract

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.
Original languageEnglish
Place of PublicationSwitzerland
PublisherSpringer
Number of pages191
ISBN (Print)9783319299778
Publication statusPublished - 20 Jun 2016

Publication series

NameLecture Notes in Mathematics
PublisherSpringer
Volume2157
ISSN (Print)0075-8434

Fingerprint

Fredholm Index
Partial Differential Operators
Physics
Index Theorem
Linear Differential Operator
Singular Integral Operator
Quantum Field Theory
Quantum Mechanics
Paul Adrien Maurice Dirac
Differential operator
Branch
First-order
Computing
Operator

Keywords

  • Callias index formula
  • Fredholm indices
  • linear partial differential operators

Cite this

Gesztesy, F., & Waurick, M. (2016). The Callias Index Formula Revisited. (Lecture Notes in Mathematics; Vol. 2157). Switzerland: Springer.
Gesztesy, Fritz ; Waurick, Marcus. / The Callias Index Formula Revisited. Switzerland : Springer, 2016. 191 p. (Lecture Notes in Mathematics).
@book{d0b269636b5e4ef8bf04857f39f54e44,
title = "The Callias Index Formula Revisited",
abstract = "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-H{\"o}rmander Index Theorem. As a byproduct of our proof we also offer a glimpse at special non-Fredholm situations employing a generalized Witten index.",
keywords = "Callias index formula, Fredholm indices, linear partial differential operators",
author = "Fritz Gesztesy and Marcus Waurick",
year = "2016",
month = "6",
day = "20",
language = "English",
isbn = "9783319299778",
series = "Lecture Notes in Mathematics",
publisher = "Springer",

}

Gesztesy, F & Waurick, M 2016, The Callias Index Formula Revisited. Lecture Notes in Mathematics, vol. 2157, Springer, Switzerland.

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

Switzerland : Springer, 2016. 191 p. (Lecture Notes in Mathematics; Vol. 2157).

Research output: Book/ReportBook

TY - BOOK

T1 - The Callias Index Formula Revisited

AU - Gesztesy, Fritz

AU - Waurick, Marcus

PY - 2016/6/20

Y1 - 2016/6/20

N2 - 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.

AB - 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.

KW - Callias index formula

KW - Fredholm indices

KW - linear partial differential operators

UR - http://www.springer.com/gb/book/9783319299761

M3 - Book

SN - 9783319299778

T3 - Lecture Notes in Mathematics

BT - The Callias Index Formula Revisited

PB - Springer

CY - Switzerland

ER -

Gesztesy F, Waurick M. The Callias Index Formula Revisited. Switzerland: Springer, 2016. 191 p. (Lecture Notes in Mathematics).