On the logarithm component in trace defect formulas
| dc.creator | Grubb, Gerd | |
| dc.date | 2004-11-22 | |
| dc.date.accessioned | 2026-07-07T05:14:34Z | |
| dc.date.available | 2026-07-07T05:14:34Z | |
| dc.description | In asymptotic expansions of resolvent traces $\Tr(A(P-λ)^{-1})$ for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-λ)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems. | |
| dc.description | 41 pages | |
| dc.identifier | https://arxiv.org/abs/math/0411483 | |
| dc.identifier | http://arxiv.org/abs/math/0411483 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73321 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Spectral Theory | |
| dc.subject | 35S15, 58J42 | |
| dc.title | On the logarithm component in trace defect formulas | |
| dc.type | text |