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. W. Junker and E. SchroheAnn. Henri Poincaré can be extended to interacting fields, in close analogy to the Hadamard states. It turned out that the notion of the Sobolev (or H s -) wavefront set is the appropriate mathematical tool for this purpose. In Appendix B we review this notion and the calculus related to it.After an introduction to the structure of the algebra of observables of the Klein-Gordon quantum field on a globally hyperbolic spacetime manifold (M, g) in Section 2 we present our definition of adiabatic states of order N (Definition 3.2) in Section 3. It contains the Hadamard states as a special case: They are adiabatic states "of infinite order". To decide which order of adiabatic vacuum is physically admissible we investigate the algebraic structure of the corresponding GNS-representations. Haag, Narnhofer & Stein [23] suggested as a criterion for physical representations that they should locally generate von Neumann factors that have all the same set of normal states (in other words, the representations are locally primary and quasiequivalent). We show in Section 4.1 (Theorem 4.5 and Theorem 4.7) that this is generally the case if N > 5/2. For the case of pure states on a spacetime with compact Cauchy surface, which often occurs in applications, we improve the admissible order to N > 3/2. In addition, in Section 4.2 we show that adiabatic vacua of order N > 5/2 satisfy the properties of local definiteness (Corollary 4.13) and those of order N > 3/2 Haag duality (Theorem 4.15). These results extend corresponding statements for adiabatic vacuum states on Robertson-Walker spacetimes due to Lüders & Roberts [35], and for Hadamard states due to Verch [49]; for their discussion in the framework of algebraic quantum field theory we refer to [21]. In Section 5 we explicitly construct pure adiabatic vacuum states on an arbitrary spacetime manifold with compact Cauchy surface (Theorem 5.10). In Section 6 we show that our adiabatic states are indeed a generalization of the well-known adiabatic vacua on Robertson-Walker spaces: Theorem 6.3 states that the adiabatic vacua of order n (according to the definition of [35]) on a Robertson-Walker spacetime with compact spatial section are adiabatic vacua of order 2n in the sense of our microlocal Definition 3.2. We conclude in Section 7 by summarizing the physical interpretation of our mathematical analysis and calcula...

We construct a trace on the algebra of classical elements in Boutet de Monvel's calculus on a compact manifold with boundary of dimension n>2. This trace coincides with Wodzicki's noncommutative residue if the boundary is reduced to the empty set. Moreover, we show that it is the unique continuous trace on this algebra up to multiplication by a constant.

Abstract. We study closed extensions A of an elliptic differential operator A on a manifold with conical singularities, acting as an unbounded operator on a weighted L p -space. Under suitable conditions we show that the resolvent (λ − A) −1 exists in a sector of the complex plane and decays like 1/|λ| as |λ| → ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A.As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problemu − ∆u = f , u(0) = 0.

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A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: The algebra G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e., Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and There is a holomorphic functional calculus for the elements of G in several complex variables.