Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 1641-1696 |
| Fachzeitschrift | Annales de l'Institut Fourier |
| Jahrgang | 54 |
| Ausgabenummer | 5 |
| Frühes Online-Datum | 31 Okt. 2003 |
| Publikationsstatus | Veröffentlicht - 2004 |
Abstract
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in: Annales de l'Institut Fourier, Jahrgang 54, Nr. 5, 2004, S. 1641-1696.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung
}
TY - JOUR
T1 - Traces and Quasi-traces on the Boutet de Monvel Algebra
AU - Grubb, Gerd
AU - Schrohe, Elmar
N1 - Final version to appear in Ann. Inst. Fourier
PY - 2004
Y1 - 2004
N2 - We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-\lambda)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-\lambda)^{-N}) (with N large) in powers and log-powers of \lambda as \lambda tends to infinity in a suitable sector of the complex plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series for the meromorphic extension of the generalized zeta function \zeta(A,B,s)= Tr(AB^{-s}) at s=0, when B is invertible. We show that C_0(A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A'] modulo local terms, and has a specific value independent of B modulo local terms; and we single out particular cases where the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A. Our main tool is a precise analysis of the asymptotic expansion of the resolvent trace, based on pseudodifferential calculations involving rational functions (in particular Laguerre functions) of the normal variable.
AB - We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-\lambda)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-\lambda)^{-N}) (with N large) in powers and log-powers of \lambda as \lambda tends to infinity in a suitable sector of the complex plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series for the meromorphic extension of the generalized zeta function \zeta(A,B,s)= Tr(AB^{-s}) at s=0, when B is invertible. We show that C_0(A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A'] modulo local terms, and has a specific value independent of B modulo local terms; and we single out particular cases where the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A. Our main tool is a precise analysis of the asymptotic expansion of the resolvent trace, based on pseudodifferential calculations involving rational functions (in particular Laguerre functions) of the normal variable.
KW - math.AP
KW - math.SP
KW - 58J42, 35S15
U2 - 10.5802/aif.2062
DO - 10.5802/aif.2062
M3 - Article
VL - 54
SP - 1641
EP - 1696
JO - Annales de l'Institut Fourier
JF - Annales de l'Institut Fourier
SN - 0373-0956
IS - 5
ER -