Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

Details Trace Expansions and the Noncommutative Residue for Manifolds with Boundary Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

2001-06-05

arxiv.org/abs/math/0106030v1

Mathematics

Analysis of PDEs

37 pages, to appear in J. Reine Angew. Math

Scientific paper

For a pseudodifferential boundary operator A of integer order \nu and 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+\nu-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-\lambda)^{-k}) in powers of \lambda^{-j/2} and log-powers \lambda^{-j/2} log \lambda, where the noncommutative residue equals the coefficient of the highest log-power. There is a related expansion for Trace(A exp(-tB)).

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