03188cam a2200433 i 4500991003531769707536m o d cr cn ---mpcbr180807s2016 sz a ob 001 0 eng d9783319299778331929977810.1007/978-3-319-29977-8doib14348184-39ule_instBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematicaeng514.7423AMS 47-02LC QA614.92Gesztesy, Fritz66693The Callias index formula revisited[e-book] /Fritz Gesztesy, Marcus WaurickCham, Switzerland :Springer,20161 online resource (ix, 192 pages) :illustrationtexttxtrdacontentcomputercrdamediaonline resourcecrrdacarrierLecture notes in mathematics,0075-8434 ;2157Includes bibliographical references and indexIntroduction.-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 ; IndexThese 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 indexIndex theoremsDifferential equations, PartialWaurick, Marcusauthorhttp://id.loc.gov/vocabulary/relators/aut732989Springer eBooksPrinted edition:9783319299761https://link.springer.com/book/10.1007/978-3-319-29977-8An electronic book accessible through the World Wide Web.b1434818403-03-2207-08-18991003531769707536Callias index formula revisited1420415UNISALENTOle01307-08-18m@ -engsz 40