Bounded H∞H_\infty-calculus for cone differential operators

We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities

Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

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)).Comment: 37 pages, to appear in J. Reine Angew. Mat

Traces and Quasi-traces on the Boutet de Monvel Algebra

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.Comment: Final version to appear in Ann. Inst. Fourie

Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties

Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.Comment: 68 pages, Latex, no figures, minor changes in the text, 2 references adde