The Callias Index Formula Revisited
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.
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3 Functional Analytic Preliminaries
4 On Schattenvon Neumann Classes and Trace Class Estimates
5 Pointwise Estimates for Integral Kernels
6 DiracType Operators
The Trace Class Result
9 The Case n3
10 The Index Theorem and Some Consequences
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ˇ ˇ admissible potentials analytic assertion follows BL.z bounded linear operator Callias Index Formula Chap chapter closed linear operator compact open topology computes converges Definition 6.11 derivatives diagonal differentiability Dirac algebra Dirac operators Dirac-type operators dnx1 estimate exists Fredholm index Fredholm operator Fredholm property function Gesztesy given by 7.1 Hence Hilbert space ind(L Index Formula Revisited integral kernel International Publishing Switzerland L D Q C Lecture Notes Lemma linear operator locally bounded Mathematics 2157 Montel's Theorem Moreover nC1X kD2 non-Fredholm Notes in Mathematics observes proof of Theorem Proposition A.8 prove Publishing Switzerland 2016 qb(x R1+z R1Cz Re(z recall Remark respectively Riº satisfies ſº Springer International Publishing strong operator topology Theorem 7.1 tr2Ond trace class trand tria triz trº Waurick Witten index