Abstract. We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein-Gordon equations on a class of spacetimes whose metric is well-behaved at spatial infinity. In particular on this class of space-times, we construct all pure Hadamard states whose two-point function (expressed in terms of Cauchy data on a Cauchy surface) is a matrix of pseudo-differential operators. We also study their covariance under symplectic transformations.As an aside, we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times which is an alternative to the classical construction by Fulling, Narcowich and Wald.

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularising terms, of parametrices with prescribed b-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the b-wavefront set.

Abstract. We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a oneparameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.2010 Mathematics Subject Classification. 81T20, 35S05, 35L05, 58J40, 53C50.The detailed results are stated in Thm. 7.8 and 7.10, see also Prop. 7.6 for the arguments that allow to get two-point functions for the original Klein-Gordon equation on the full spacetime (M, g) rather than for the reduced equation (1.3) on I × Σ.Since one can get many regular states out of a given one by applying suitable Bogoliubov transformations as in [GW1], Thm. 1.2 yields in fact a large class of Hadamard states.1.3. From quantum fields to spacetime geometry. In our approach, microlocal splittings are obtained by settingis defined by formula (1.6) with b ± (t) constructed for t ∈ I as approximate solutions (i.e. modulo smoothing terms) of the operatorial equation (1.7)∂ t + ib ± + r • ∂ t − ib ± = ∂ 2 t + r∂ t + a, and satisfying some additional conditions, see Sect. 6 (in particular Thm. 6.1) for details. We note that the approximate factorization (1.7) was already used by Junker in his construction of Hadamard states [Ju1, Ju2]. h 1,sϕ ∈ S 0 KdS , resp. ∈ S −1,0 K

We consider the massive Klein-Gordon equation on asymptotically Minkowski spacetimes, in the sense that the manifold is R 1+d and the metric approaches that of Minkowski space at infinity in a short-range way (jointly in time and space variables). In this setup we define Feynman and anti-Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer type boundary conditions at infinite times. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and Hörmander.2010 Mathematics Subject Classification. 81T13, 81T20, 35S05, 35S35.

Abstract. We construct Hadamard states for the Yang-Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal R d .We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein-Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs.The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald.As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

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We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfies the Hadamard condition. Using pseudodifferential calculus on the cone $C$ we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in $M_{0}$.Comment: 34 p.; v3, references added, matches published versio