Details
| Original language | English |
|---|---|
| Pages (from-to) | 167-207 |
| Number of pages | 41 |
| Journal | Journal fur die Reine und Angewandte Mathematik |
| Volume | 536 |
| Publication status | Published - 2001 |
| Externally published | Yes |
Abstract
For a pseudodifferential boundary operator A of order ν ∈ ℤ and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB-s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB-s) has a meromorphic extension to ℂ with poles at the half-integers s = (n + ν - j)/2, j ∈ ℕ (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Tr(A(B - λ)-k) in powers λ-1/2 and log-powers λ-1/2 log λ, where the noncommutative residue equals the coefficient of the highest order log-power. There is a related expansion of Tr(Ae-tB).
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal fur die Reine und Angewandte Mathematik, Vol. 536, 2001, p. 167-207.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Trace expansions and the noncommutative residue for manifolds with boundary
AU - Grubb, Gerd
AU - Schrohe, Elmar
N1 - Copyright: Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2001
Y1 - 2001
N2 - For a pseudodifferential boundary operator A of order ν ∈ ℤ and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB-s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB-s) has a meromorphic extension to ℂ with poles at the half-integers s = (n + ν - j)/2, j ∈ ℕ (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Tr(A(B - λ)-k) in powers λ-1/2 and log-powers λ-1/2 log λ, where the noncommutative residue equals the coefficient of the highest order log-power. There is a related expansion of Tr(Ae-tB).
AB - For a pseudodifferential boundary operator A of order ν ∈ ℤ and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB-s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB-s) has a meromorphic extension to ℂ with poles at the half-integers s = (n + ν - j)/2, j ∈ ℕ (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Tr(A(B - λ)-k) in powers λ-1/2 and log-powers λ-1/2 log λ, where the noncommutative residue equals the coefficient of the highest order log-power. There is a related expansion of Tr(Ae-tB).
UR - http://www.scopus.com/inward/record.url?scp=0035593093&partnerID=8YFLogxK
U2 - 10.1515/crll.2001.055
DO - 10.1515/crll.2001.055
M3 - Article
AN - SCOPUS:0035593093
VL - 536
SP - 167
EP - 207
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
SN - 0075-4102
ER -