Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

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dc.creatorGrubb, Gerd
dc.creatorSchrohe, Elmar
dc.date2001-06-05
dc.date.accessioned2026-07-07T04:41:59Z
dc.date.available2026-07-07T04:41:59Z
dc.descriptionFor a pseudodifferential boundary operator A of integer order νand class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(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 Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+ν-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero 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 Trace(A(B-λ)^{-k}) in powers of λ^{-j/2} and log-powers λ^{-j/2} log λ, where the noncommutative residue equals the coefficient of the highest log-power. There is a related expansion for Trace(A exp(-tB)).
dc.description37 pages, to appear in J. Reine Angew. Math
dc.identifierhttps://arxiv.org/abs/math/0106030
dc.identifierhttp://arxiv.org/abs/math/0106030
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61591
dc.subjectAnalysis of PDEs
dc.subjectSpectral Theory
dc.subject58J42; 35S15
dc.titleTrace Expansions and the Noncommutative Residue for Manifolds with Boundary
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