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Traces and Determinants of Pseudodifferential Operators$
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Simon Scott

Print publication date: 2010

Print ISBN-13: 9780198568360

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780198568360.001.0001

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Computations, Transition Formulae, and the Local Index Formula

Computations, Transition Formulae, and the Local Index Formula

(p.271) III Computations, Transition Formulae, and the Local Index Formula
Traces and Determinants of Pseudodifferential Operators

Simon Scott

Oxford University Press

In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the literature and which have acted as guide posts to the development of trace and determinant methods in geometric analysis over the past few decades. There are basic equivalences, indispensable in both theoretical developments and exact computations, between coefficients in the asymptotic expansions of zeta traces, heat traces and resolvent traces. The middle part of the chapter presents and proves these identifications, and provides an application to the computation of relative zeta determinants. The final part of the chapter turns to residue trace and residue determinant computations. Based on a residue determinant formula for the spectral zeta function at zero, an elementary proof of the local Atiyah Singer index theorem is given.

Keywords:   Regularized trace, regularized determinants, explicit computations, resolvent trace, heat trace, complex powers, pole structure, asymptotic expansion, Atiyah‐Singer, index theorem, Riemann Roch Hirzebruch

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