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On the logarithm component in trace defect formulas. (English) Zbl 1087.35100

To begin with we remind that in asymptotic expansions of resolvent traces \(\text{Tr}(A(P-\lambda)^{-1})\) for classical pseudodifferential operators on closed manifolds, the coefficient \(C_0(A, P)\) of \((-\lambda)^{-1}\) is of a special interest. In the paper under consideration it is shown how the trace defect formulas can be obtained by the resolvents in a simple way, avoiding the use of complex powers of \(P\) as in the original proofs (Okikioly, Kontsevich and Vishik, Melrose and Nistor). Moreover, the author gives a simple direct proof of a recent residue formula of Scott for \(C_0(I,P)\) [see S. Scott, ibid. 30, 483–507 (2005; Zbl 1236.58037)]. Using only resolvents, trace defect residue formulas for operators on manifolds with boundary are established. A generalization of Scott’s formulae to boundary problems is given at the end of the paper.

MSC:

35S15 Boundary value problems for PDEs with pseudodifferential operators
58J42 Noncommutative global analysis, noncommutative residues
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 1236.58037
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