Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

We consider the Klein-Gordon operator on an $n$-dimensional asymptotically anti-de Sitter spacetime $(M,g)$ together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on $\partial M$ of order up to $2$. Using techniques from $b$-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.


Introduction
The n-dimensional anti-de Sitter spacetime (AdS n ) is a maximally symmetric solution of the vacuum Einstein equations with a negative cosmological constant. From a geometric viewpoint it is noteworthy since it is not globally hyperbolic and it possesses a timelike conformal boundary. Due to these features the study of hyperbolic partial differential equations on top of this background becomes particularly interesting, especially since the initial value problem does not yield a unique solution unless suitable boundary conditions are assigned. Therefore several authors have investigated the properties of the Klein-Gordon equation on an AdS spacetime, see e.g. [Bac11,EnKa13,Hol12,War13,Vas12] to quote some notable examples, which have inspired our analysis.
A natural extension of the framework outlined in the previous paragraph consists of considering a more general class of geometries, namely the so-called n-dimensional asymptotically AdS spacetimes, which share the same behaviour of AdS n in a neighbourhood of conformal infinity. In this case the analysis of partial differential equations such as the Klein-Gordon one becomes more involved due to admissible class of backgrounds and, in particular, due to the lack of isometries of the metric. Noteworthy has been the recent analysis by Gannot and Wrochna,[GW18], in which, using techniques proper of b-calculus they have investigated the structural properties of the Klein-Gordon operator with Robin boundary conditions. In between the several results proven, we highlight in particular the theorem of propagation of singularities and the existence of advanced and retarded fundamental solutions.
Yet, as strongly advocated in [DDF18], the class of boundary conditions which are of interest in concrete models is greater than the one considered in [GW18], a notable example in this direction being the so-called Wentzell boundary conditions, see e.g. [Coc14,DFJ18,FGGR02,Ue73,Za15]. For this reason in [DM20], we started an investigation aimed at generalizing the results of [GW18] proving a theorem of propagation of singularities for the Klein-Gordon operator on an asymptotically anti-de Sitter spacetime M such that the boundary condition is implemented by a bpseudodifferential operator Θ ∈ Ψ k b (∂M ) with k ≤ 2, see Section 3.1 for the definitions. Starting from this result, in this work we proceed with our investigation and, still using techniques proper of b-calculus, we discuss the existence of advanced and retarded fundamental solutions for the Klein-Gordon operator with prescribed boundary conditions. The first main result that we prove is the following: In addition, we characterize the wavefront set of the advanced (−) and of the retarded (+) fundamental solutions as well as their wavefront set, thanks to the theorem of propagation of singularities proven in [DM20]. This result allows us to discuss a notable application which is strongly inspired by the so-called algebraic approach to quantum field theory, see e.g. [BDFY15] for a recent review. In this framework a key rôle is played by the so-called Hadamard two-point distributions, which are positive bi-distributions on the underlying background which are characterized by the following defining properties: they are bi-solutions of the underlying equations of motion, their antisymmetric part is proportional to the difference between the advanced and retarded fundamental solutions and their wavefront set has a prescribed form, see e.g. [KM13]. If the underlying background is globally hyperbolic and with empty boundary, the existence of these two-point distributions is a by-product of the standard Hörmander propagation of singularities theorem and of a deformation argument due to Fulling, Narcovich and Wald, see [FNW81].
In the scenarios investigated in this work this conclusion does no longer apply since we are considering asymptotically AdS spacetimes which possess in particular a conformal boundary. At the level of Hadamard two-point distributions this has long-standing consequences since even the standard form of the wavefront set has to be modified to take into account reflection of singularities at the boundary, see [DF18] and Definition 5.3 below. Our second main result consists of showing that, under a suitable restriction on the allowed class of boundary conditions, see Hypothesis 4.1 in the main body of this work, it is possible to prove existence of Hadamard two-point distributions. First we focus on static spacetimes and, using spectral techniques, we construct explicitly an example, which, in the language of theoretical physics, is often referred to as the ground state. Subsequently we show that, starting from this datum and using the theorem of propagation of singularities proven in [DM20], we can use also in this framework a deformation argument to infer the existence of an Hadamard two-point distribution on a generic n-dimensional asymptotically AdS spacetime. It is important to observe that this result is in agreement and it complements the one obtained in [Wro17]. To summarize our second main statement is the following, see also Definition 4.2 for the notion of static and of physically admissible boundary conditions: Theorem 1.2. Let (M, g) be a globally hyperbolic, asymptotically anti-de Sitter spacetime and let (M S , g S ) be its static deformation as per Lemma 5.2. Let Θ K be a static and physically admissible boundary condition so that the Klein-Gordon operator P Θ K on (M S , g S ) admits a Hadamard twopoint function as per Proposition 5.5. Then there exists a Hadamard two point-function on (M, g) for the associated Klein-Gordon operator with boundary condition ruled by Θ K .
It is important to stress that the deformation argument forces us to restrict in the last part of the paper the class of admissible boundary conditions and notable examples such as those of Wentzell type are not included. They require a separate analysis of their own [ADM21].
The paper is structured as follows. In Section 2 we recollect the main geometric data, particularly the notions of globally hyperbolic spacetime with timelike boundary and that of asymptotically AdS spacetime. In Section 3 we discuss the analytic data at the heart of our analysis. We start from a succinct review of b-calculus in Section 3.1, followed by one of twisted Sobolev spaces and energy forms. In Section 3.4 we formulate the dynamical problem, we are interested in, both in a strong and in a weak sense. In Section 4 we obtain our first main result, namely the existence of advanced and retarded fundamental solutions for all boundary conditions abiding to Hypothesis 4.1. In addition we investigate the structural properties of these propagators and we characterize their wavefront set. In Section 5 we investigate the existence of Hadamard two-point distributions in the case of vanishing mass. First, in Section 5.1 and 5.2, using spectral techniques we prove their existence on static spacetimes though for a restricted class of admissible boundary conditions, see Hypothesis 4.1 and Definition 4.2. Subsequently, in Section 5.3, we extend to the case in hand a deformation argument due to Fulling, Narcowich and Wald proving existence of Hadamard two-point distributions on a generic n-dimensional asymptotically AdS spacetime.

Geometric Data
In this section our main goal is to fix notations and conventions as well as to introduce the three main geometric data that we shall use in our analysis, namely globally hyperbolic spacetimes with timelike boundary, asymptotically anti-de Sitter spacetimes and manifolds of bounded geometry. We assume that the reader is acquainted with the basic notions of Lorentzian geometry, cf. [ON83]. Throughout this paper with spacetime, we indicate always a smooth, connected, oriented and time oriented Lorentzian manifold M of dimension dim M = n ≥ 2 equipped with a smooth Lorentzian manifold g of signature (−, +, . . . , +). With C ∞ (M ) (resp. C ∞ 0 (M )) we indicate the space of smooth (resp. smooth and compactly supported) functions on M , whileĊ ∞ (M ) (resp.Ċ ∞ 0 (M )) stands for the collection of all smooth (resp. smooth and compactly supported) functions vanishing at ∂M with all their derivatives. In between all spacetimes, the following class plays a notable rôle [AFS18].
Definition 2.1. Let (M, g) be a spacetime with non empty boundary ι : ∂M → M . We say that (M, g) 1. has a timelike boundary if (∂M, ι * g) is a smooth, Lorentzian manifold, 2. is globally hyperbolic if it does not contain closed causal curves and if, for every p, q ∈ M , J + (p) ∩ J − (q) is either empty or compact.
If both conditions are met, we call (M, g) a globally hyperbolic spacetime with timelike boundary and we indicate withM = M \ ∂M the interior of M .
Observe that, for simplicity, we assume throughout the paper that also ∂M is connected. Notice in addition that Definition 2.1 reduces to the standard notion of globally hyperbolic spacetimes when ∂M = ∅. The following theorem, proven in [AFS18], gives a more explicit characterization of the class of manifolds, we are interested in and it extends a similar theorem valid when ∂M = ∅.
Remark 2.1. Observe that a notable consequence of this theorem is that, calling ι ∂M : ∂M → M the natural embedding map, then (∂M, h) where h = ι * ∂M g is a globally hyperbolic spacetime. In particular the associated line element reads In addition to Definition 2.1 we consider another notable class of spacetimes introduced in [GW18].
Definition 2.2. Let M be an n-dimensional manifold with non empty boundary ∂M . Suppose thatM = M \ ∂M is equipped with a smooth Lorentzian metric g and that a) If x ∈ C ∞ (M ) is a boundary function, then g = x 2 g extends smoothly to a Lorentzian metric on M .
b) The pullback h = ι * ∂M g via the natural embedding map ι ∂M : ∂M → M individuates a smooth Lorentzian metric.
Then (M, g) is called an asymptotically anti-de Sitter (AdS) spacetime. In addition, if (M, g) is a globally hyperbolic spacetime with timelike boundary, cf. Definition 2.1, then we call (M, g) a globally hyperbolic asymptotically AdS spacetime.
Observe that conditions a), b) and c) are actually independent from the choice of the boundary function x and the pullback h is actually determined up to a conformal multiple since there exists always the freedom of multiplying the boundary function x by any nowhere vanishing Ω ∈ C ∞ (M ). Such freedom plays no rôle in our investigation and we shall not consider it further. Hence, for definiteness, the reader can assume that a global boundary function x has been fixed once and for all.
As a direct consequence of the collar neighbourhood theorem and of the freedom in the choice of the boundary function in Definition 2.2, this can always be engineered in such a way, that, given any p ∈ ∂M , it is possible to find a neighbourhood U ⊂ ∂M containing p and ǫ > 0 such that on U × [0, ǫ) the line element associated to g reads where h x is a family of Lorentzian metrics depending smoothly on x such that h 0 ≡ h.
Remark 2.2. It is important to stress that the notion of asymptotically AdS spacetime given in Definition 2.2 is actually more general than the one given in [AD99], which is more commonly used in the general relativity and theoretical physics community. Observe in particular that h x in Equation (2) does not need to be an Einstein metric nor ∂M is required to be diffeomorphic to R × S n−2 . Since we prefer to make a close connection to both [GW18] and [DM20] we stick to their nomenclature.
Remark 2.3. Throughout the paper, with the symbols τ and x we shall always indicate respectively the time coordinate as in Equation (1) and the spatial coordinate as in Equation (2).

Manifolds of bounded geometry
To conclude this section we introduce the manifolds of bounded geometry since they are the natural arena where one can define Sobolev spaces when the underlying background has a non empty boundary. In this section we give a succinct survey of the main concepts and of those results which will play a key rôle in our analysis. An interested reader can find more details in [ This definition cannot be applied slavishly to a manifold with non empty boundary and, to extend it, we need to introduce a preliminary concept.
Definition 2.4. Let (N, h) be a Riemannian manifold of bounded geometry and let (Y, ι Y ) be a codimension k, closed, embedded smooth submanifold with an inward pointing, unit normal vector field ν Y . The submanifold (Y, ι * Y g) is of bounded geometry if: a) The second fundamental form II of Y in N and all its covariant derivatives along Y are bounded, These last two definitions can be combined to introduce the following notable class of Riemannian manifolds Definition 2.5. Let (N, h) be a Riemannian manifold with ∂N = ∅. We say that (N, h) is of bounded geometry if there exists a Riemannian manifold of bounded geometry (N ′ , h ′ ) of the same dimension as N such that: Remark 2.4. Observe that Definition 2.5 is independent from the choice of N ′ . For completeness, we stress that an equivalent definition which does not require introducing N ′ can be formulated, see for example [Sch01].
Definition 2.5 applies to a Riemannian scenario, but we are particularly interested in Lorentzian manifolds. In this case the notion of bounded geometry can be introduced as discussed in [GOW17] for the case of a manifold without boundary, although the extension is straightforward. More precisely let us start from (N, h) a Riemannian manifold of bounded geometry such that dim N = n. In addition we call BT m m ′ (B n (0, ), δ E ), the space of all bounded tensors on the ball B n (0, ) centered at the origin of the Euclidean space (R n , δ E ) where δ E stands for the flat metric. For every m, m ′ ∈ N ∪ {0}, we denote with BT m m ′ (N ) the space of all rank (m, m ′ ) tensors T on N such that, for any p ∈ M , calling T p .
Definition 2.6. A smooth Lorentzian manifold (M, g) is of bounded geometry if there exists a Riemannian metric g on M such that: a) (M, g) is of bounded geometry.
On top of a Riemannian (or of a Lorentzian) manifold of bounded geometry (N, h) we can introduce H k (N ) ≡ W 2,k (N ) which is the completion of with respect to the norm where ∇ is the covariant derivative built out of the Riemannian metric h, while (∇) i indicates the i-th covariant derivative. This notation is employed to disambiguate with ∇ i = h ij ∇ j .
Remark 2.5. One might wonder why the assumption of bounded geometry is necessary since it seems to play no rôle in above characterization. The reason is actually two-fold. On the one hand it is possible to give a local definition of Sobolev spaces via a suitable choice of charts, which yields in turn a global counterpart via a partition of unity argument. Such definition is a prior different from the one given above unless one assumes to work with manifolds of bounded geometry, see [GS13].
In addition such alternative characterization of Sobolev spaces allows for introducing a suitable generalization to manifolds of bounded geometry of the standard Lions-Magenes trace, which will play an important rôle especially in Section 5.1.
Observe that, henceforth, we shall always assume implicitly that all manifolds that we consider are of bounded geometry.

Analytic Preliminaries
In this section we introduce the main analytic tools that play a key rôle in our investigation. We start by recollecting the main results from [DM20] which are, in turn, based on [GW18] and [Vas10,Vas12].

On b-pseudodifferential operators
In the following we assume for definiteness that (M, g) is a globally hyperbolic, asymptotically AdS spacetime of bounded geometry as per Definition 2.2 and Definition 2.6. In addition we assume that the reader is familiar with the basic ideas and tools behind b-geometry, first introduced by R. Melrose in [Mel92]. Here we limit ourselves to fix notations and conventions, following the presentation of [GMP14].
In the following with b T M we indicate the b-tangent bundle which is a vector bundle whose where x is the global boundary function introduced in Definition 2.2, here promoted to coordinate. Similarly we can define per duality the b-cotangent bundle, b T * M which is a vector bundle whose fibers For future convenience, whenever we fix a chart U of M centered at a point p ∈ ∂M , we consider (x, y i , ξ, η i ) and (x, y i , ζ, η i ), i = 1, . . . , n − 1 = dim ∂M , local coordinates respectively of T * M | U and of b T * M | U . Since we are considering globally hyperbolic spacetimes, hence endowed with a distinguished time direction τ , cf. Equation (1), we identify implicitly η n−1 ≡ τ . In addition, observe that there exists a natural projection map which is non-injective. This feature prompts the definition of a very important structure in our investigation, namely the compressed b-cotangent bundle which is a vector sub-bundle of b T * M , such that bṪ * p M ≡ T * p M whenever p ∈M . The last geometric structure that we shall need in this work is the b-cosphere bundle which is realized as the quotient manifold obtained via the action of the dilation group on We remark that, if we consider a local chart U ⊂ M such that U ∩ ∂M = ∅ and the local coordinates On top of these geometric structures we can define two natural classes of operators.
Definition 3.1. Let (M, g) be a globally hyperbolic, asymptotically AdS spacetime. We call The notion of b − ΨDOs is strictly intertwined with S m ( b T * M ) the set of all symbols of order m on b T * M and in particular there exists a principal symbol map which gives rise to an isomorphism In addition we can endow the space of symbols S m ( b T * M ) with a Fréchet topology induced by the family of seminorms where k z = (1 + |k z | 2 ) 1 2 , while {K i } i∈I , I being an index set, is an exhaustion of M by compact subsets. Hence one can endow S m ( b T * M ) with a metric d as follows In view of these data the following definition is natural Finally we can recall the notion of elliptic b − ΨDO and of wavefront set both of a single and of a family of pseudodifferential operators, cf. [Hör03]: Definition 3.4. For any P ∈ Ψ m b (M ), we say that (z 0 , k z 0 ) / ∈ W F ′ b (P ) if the associated symbol p(z, k z ) is such that, for every multi-indices γ and for every N ∈ N, there exists a constant C N,α,γ such that for z in a neighbourhood of z 0 and k z in a conic neighbourhood of To conclude this part of the section, we stress that, in order to study the behavior of a bpseudodifferential operator at the boundary, it is useful to introduce the notion of indicial family, [GW18]. Let A ∈ Ψ m b (M ). For a fixed boundary function x, cf. Definition 2.2, and for any v ∈ C ∞ (∂M ) we define the indicial family N (A)(s) : C ∞ (∂M ) → C ∞ (∂M ) as: where u ∈ C ∞ (M ) is any function such that u| ∂M = v.

Twisted Sobolev Spaces
In this section we introduce the second main analytic ingredient that we need in our investigation.
To this end, once more we consider (M, g) a globally hyperbolic, asymptotically AdS spacetime and the associated Klein-Gordon operator P .
= ✷ g − m 2 , where m 2 plays the rôle of a mass term, while ✷ g is the D'Alembert wave operator built out of the metric g. It is convenient to introduce the parameter which is constrained to be positive. This is known in the literature as the Breitenlohner-Freedman bound [BF82]. In the spirit of [GW18] and [DM20, Sec. 3.2] we introduce the following, finitely generated, space of twisted differential operators where ν − = n−1 2 − ν, n = dim M . Starting from these data, and calling with x and dµ g respectively the global boundary function, cf. Definition 2.2, and the metric induced volume measure we set The latter is a Sobolev space if endowed with the norm while, similarly, we define H −1 0 (M ). We discuss succinctly the interactions between Ψ m b (M ) and Diff 1 ν (M ). We begin by studying the action of pseudodifferential operators of order zero on the spaces H k loc/0 (M ), k = ±1, just defined.
is a bounded operator thereon, as stated in the following lemma.
which extends per duality to a continuous maṗ The proof of this lemma gives a useful information.
To study in full generality the interactions between Ψ m b (M ) and Diff 1 ν (M ), we need to introduce one last class of relevant spaces Remark 3.1. As observed in [Vas08], whenever m is finite, it is enough to check that both u and Au lie in H k loc (M ) for a single elliptic operator A ∈ Ψ m b (M ). Observe that, in full analogy to Definition 3.5, we define similarly H k,m 0 (M ) andḢ k,m loc (M ). In the following definition, we extend the notion of wavefront set to the spaces H k,m loc (M ).
Definition 3.6. Let k = 0, ±1 and let u ∈ H k,m and Au ∈ H k loc (M ), where ell b stands for the elliptic set as per Definition 3.3. When m = +∞, we say that With all these data, we can define two notable trace maps which will be a key ingredient in the next section. The following proposition summarizes the content of [GW18, Lemma 3.3] and [DM20, Lemma 3.4]: Theorem 3.1. Let (M, g) be a globally hyperbolic, asymptotically AdS spacetime of bounded geometry with n = dim M and let ν > 0, cf. Equation (7). Then there exists a continuous map , which can be extended to a continuous map Remark 3.2. In order to better grasp the rôle of the trace map defined in Theorem 3.1, it is convenient to focus the attention on R n At last we recall from [GW18] a notable property of the trace γ − related to its boundedness. Let u ∈ H(M ), then for every ε > 0 there exists C ε > 0 such that

Twisted Energy Form
In this section we focus the attention on discussing the last two preparatory key concepts before stating the boundary value problem, we are interested in. We recall that P = ✷ g − m 2 is the Klein-Gordon operator and, following [GW18], we can individuate a distinguished class of spaces whose elements enjoy additional regularity with respect to P : Definition 3.7. Let (M, g) be a globally hyperbolic, asymptotically anti-de Sitter spacetime and let P be the Klein-Gordon operator. For all m ∈ R ∪ {±∞}, we define the Frechét spaces with respect to the seminorms where φ ∈ C ∞ 0 (M ).
At this point we are ready to introduce a suitable energy form. The standard definition must be adapted to the case in hand, in order to avoid divergences due to the behaviour of the solutions of the Klein-Gordon equation at the boundary. To this end it is convenient to make use of the so-called admissible twisting functions, that is, calling x the global boundary function as per Definition 2.2, For any such function, we can define a twisted differential Accordingly we can introduce the twisted Dirichlet (energy) form Starting from these data, we are ready to introduce a second trace map. More precisely we start from (15) with respect to the inner product on L 2 (M ; dµ g ) we observe that, on account of the identity P = −d † F d F + F −1 P (F ), the following Green's formula holds true for all u ∈ X ∞ (M ) and for all v ∈ H 1 0 (M ): With these premises the following result holds true, cf. [GW18, Lemma 4.8]: Lemma 3.2. The map γ + can be extended to a bounded linear map and, if u ∈ X k (M ), the Green's formula (17) holds true for every v ∈ H 1,−k 0 (M ).
Remark 3.3. In order to better grasp the rôle of the second trace map characterized in Lemma 3.2, it is convenient to focus once more the attention on R n + . = [0, ∞) × R n−1 endowed with a metric whose line element reads in standard Cartesian coordinates where h is a smooth Lorentzian metric on R n−1 . Consider an admissible twisting function F such that lim Then, for every ǫ > 0, the restriction of u to [0, ǫ) × R n admits an asymptotic expansion of the form . In this context it holds that γ + (u) = 2νu + .

The boundary value problem
In this section we use the ingredients introduced in the previous analysis to formulate the dynamical problem we are interested in. At a formal level we look for u ∈ H 1 loc (M ) such that where Θ ∈ Ψ k b (∂M ) while γ − , γ + are the trace maps introduced in Theorem 3.1 and in Lemma 3.2 respectively. It is not convenient to look for strong solutions of Equation (18). More precisely, for any Θ ∈ Ψ k b (∂M ) , we assume that there exists an admissible twisting function F and we define the energy functional where Observe that, on account of the regularity of γ − u, we can extend P Θ as an operator P Θ : . Remark 3.4. The reader might be surprised by the absence of γ + in the weak formulation of the boundary value problem as per Equation (20). This is only apparent since the last term in the right hand side of Equation (20) is a by-product of the Green's formula as per Equation (17) together with the boundary condition introduced in Equation (18).
We are now in position to recollect the two main results proved in [DM20] concerning a propagation of singularities theorem for the Klein-Gordon operator with boundary conditions ruled by a pseudo-differential operator Θ ∈ Ψ k b (∂M ) with k ≤ 2. As a preliminary step, we introduce two notable geometric structures. More precisely, since the principal symbol of x −2 P reads p . = g(X, X), where X ∈ Γ(T * M ), the associated characteristic set is while the compressed characteristic set iṡ where π is the projection map from T * M to the compressed cotangent bundle, cf. Equation (3). A related concept is the following: Definition 3.8. Let I ⊂ R be an interval. A continuous map γ : I →Ṅ is a generalized broken bicharacteristic (GBB) if for every s 0 ∈ I the following conditions hold true: where η 0 ∈ N is the unique point for which π(η 0 ) = q 0 , while π : T * M → b T * M and {, } are the Poisson brackets on T * M .
With these structures and recalling in particular the wavefront set introduced in Definition 3.6 we can state the following two theorems, whose proof can be found in [DM20]: (Θu) is the union of maximally extended generalized broken bicharacteristics within the compressed characteristic setṄ .
In full analogy it holds also is the union of maximally extended GBBs within the compressed characteristic setṄ .

Fundamental Solutions
In this section we prove the first of the main results of our work. We start by investigating the existence of fundamental solutions associated to the boundary value problem as in Equation (18). We shall uncover that a positive answer can be found, though we need to restrict suitably the class of admissible b-ΨDOs Θ ∈ Ψ k b (∂M ) in comparison to that of Theorem 3.2 and 3.3. We stress that, from the viewpoint of applications, these additional conditions play a mild rôle since all scenarios of interest are included in our analysis.
Since we deal with a larger class of boundary conditions than those considered in [Vas12] and in [GW18], we need to make an additional hypothesis. Recall that, as in the previous sections, we are identifying a pseudodifferential operator on ∂M with its natural extension on M , i.e. constant in x, the global boundary function. As starting point we need a preliminary definition: . We call it local in time if, for every u in the domain of Θ, τ (supp(Θu)) ⊆ τ (supp(u)) where τ : R × Σ → R is the time coordinate individuated in Theorem 2.1.
Recalling [Jos99,Sec. 6] for the definition of the adjoint of a pseudodifferential operator, we can now formulate the following hypothesis The next step in the analysis of the problem in hand lies in proving the following lemma which generalizes a counterpart discussed in [GW18] for the case of Robin boundary conditions. Lemma 4.1. Let u ∈ H 1,1 loc (M ) and let Θ ∈ Ψ k b (∂M ) be such that its canonical extension to M abides to the Hypothesis 4.1. Then there exists a compact subset K ⊂ M and a real positive where φ = τ χ, χ being the same as in Equation (26), while P Θ is defined in Equation (20).
Proof. The proof is a generalization of those in [Vas12] and [GW18] to the case of boundary conditions encoded by pseudodifferential operators. Therefore we shall sketch the common part of the proof, focusing on the terms introduced by the boundary conditions. Adopting the same conventions as at the beginning of the section, assume that supp(u) ⊂ [τ 0 + ε, τ 1 ] × Σ. We start by computing a twisted version of the energy form considered in [Vas12].
is a second order formally self-adjoint operator, the purpose of V ′ * being to remove zeroth order terms.
where E 0 is the twisted Dirichlet energy form, cf. Equation (16), S F is defined in Section 3.3, while γ + and γ − are the trace maps introduced in Theorem 3.1 and in Lemma 3.2. We analyze each term in the above sum separately. Starting form the first one and proceeding as in [GW18], we rewrite where Q i , i = 1, . . . , n is a generating set of Diff 1 ν (M ), while the symmetric tensor B is Here T (W, ▽ g φ) is the stress-energy tensor, with respect to g, see Definition 2.2, of a scalar field associated with W and ▽ g φ, that is, denoting with ⊙ the symmetric tensor product, Focusing on this term and using that ▽ g φ = χ ′ ▽ g τ , a direct computation yields: Since ▽ g φ and ▽ g τ are respectively past-and future-pointing timelike vectors, then T g (W, ▽ g φ) is negative definite. Hence we can rewrite Equation (27) as with K = −(F φV (F −1 ) + (n − 2)φx −1 W (x)) g −1 + φL W g −1 .
Since −T g (W, ▽ g φ) ij is positive definite, then Q(u, u) . = −T g (W, ▽ g φ) ij Q i u, Q j u ≥ 0. This can be seen by direct inspection from the explicit form where H is the sesquilinear pairing between 1-forms induced by the metric. Focusing then on the term K ij Q i u, Q j u , we observe that, as a consequence of our choice for the functions f and W , we have V (x) = g(▽ g τ, ▽ g x) = 0 on ∂M . In addition it holds that x −1 W (x) = O(1) near ∂M , and L V g −1 = 2 ▽ g (▽ g τ ) = 2 Γ i τ τ ∂ i . These observations allow us to establish the following bound, cf. [Vas12] and [GW18]: with C a suitable, positive constant. Now we focus on establishing a bound for the terms on the right hand side of Equation (31). We estimate the first one as follows: where in the last inequality we used Equation (25). As for the second term in Equation (31), using that S F ∈ x 2 L ∞ (M ), we establish the bound for a suitable constant C > 0. Using Equation (25) and the Poincaré inequality, this last bound becomes 2|Re At last we give a give a bound for the last term in Equation (27), containing the pseudodifferential operator Θ which implements the boundary conditions. Recalling Hypothesis 4.1, it is convenient to consider the following three cases separately a) Θ ∈ Ψ k b (∂M ) with k ≤ 1, b) Θ ∈ Ψ k b (∂M ) with 1 < k ≤ 2. Now we give a bound case by case.
a) It suffices to focus on Θ ∈ Ψ 1 b (∂M ) recalling that, for k < 1, Ψ k b (∂M ) ⊂ Ψ 1 b (∂M ). If with a slight abuse of notation we denote with Θ both the operator on the boundary and its trivial extension to the whole manifold, we can write where N (Θ)(−iν − ) is the indicial family as in Equation (6). We recall that any A ∈ Ψ s b (∂M ), s ∈ N, can be decomposed as . . , n is a generating set of Diff 1 ν (M ). Hence we can rewrite Θ as To begin with, we focus on the first term on the right hand side of this inequality. Using Equations (12) and (25) together with the Poincaré inequality (26) and Lemma 3.1, for a suitable constant C ε > 0. As for the second term, since u ∈ H 1,1 (M ) we can proceed as above using that the operator Θ ′ + [Q i , Θ i ] is of order 0 and we can conclude that for suitable positive constants C ε and C ε . Therefore, it holds a bound of the form it is enough to consider Θ ∈ Ψ 2 b (∂M ) and to observe that, we can decompose Θ as . At this point one can apply twice consecutively the same reasoning as in item a) to draw the sought conclusion.
Finally, considering Equation (31) and collecting all bounds we proved, we obtain Since the inner product H defined by the left hand side of Equation (32) is positive definite, then for δ large enough and the associated Dirichlet form Q defined as bounds (−φ ′ ) 1/2 d F u 2 L 2 (M ) . We conclude the proof by observing that, once we have an estimate for (−φ ′ ) 1/2 d F u 2 L 2 (M ) , with the Poincaré inequality we can also bound (−φ ′ ) 1/2 u L 2 (M ) . Therefore, considering the support of χ and u, there exists a compact subset K ⊂ M such that from which the sought thesis descends.
Remark 4.1. The case with Θ ∈ Ψ k (M ) of order k ≤ 0, can also be seen as a corollary of the well-posedness result of [GW18].
The following two statements guarantee uniqueness and existence of the solutions for the Klein-Gordon equation associated to the operator P Θ individuated in Equation (20). Mutatis mutandis, since we assume that Θ is local in time, the proof of the first statement is identical to the counterpart in [Vas12] and therefore we omit it.
Corollary 4.1. Let M be a globally hyperbolic, asymptotically anti-de Sitter spacetime, cf. Definition 2.2 and let f ∈Ḣ −1,1 (M ) be vanishing whenever τ < τ 0 , τ 0 ∈ R. Suppose in addition that Θ abides to the Hypothesis 4.1. Then there exists at most one u ∈ H 1 0 (M ) such that supp(u) ⊂ {q ∈ M | τ (q) ≥ τ 0 } and it is a solution of P Θ u = f At the same time the following statement holds true. If m < 0 we can draw the same conclusion considering, as in [Vas12,Thm. 8.12], is sequence converging to f as j → ∞. Each of these equations has a unique solution u j ∈ H 1 (M ). In addition the propagation of singularities theorem, cf. Theorem (3.3) yields the bound for suitable compact sets K, L ⊂ M and for every j, k ∈ N. Since f j → f inḢ −1,m+1 (L), we can conclude that the sequence u j is converging to u ∈ H 1,m (K). Considering f j such that each f j vanishes if {τ < τ 0 }, one obtains the desired support property of the solution. To conclude this analysis we summarize the final result which combines Corollary 4.1 and Lemma 4.2.
Proposition 4.1. Let M be a globally hyperbolic, asymptotically anti-de Sitter spacetime, cf. Definition 2.2 and let m, τ 0 ∈ R while f ∈Ḣ −1,m+1 loc (M ). Assume in addition that Θ abides to Hypothesis 4.1. If f vanishes for τ < τ 0 , τ 0 ∈ R being arbitrary but fixed, then there exists a unique u ∈ H 1,m loc (M ) such that where P Θ is the operator in Equation (20).
We have gathered all ingredients to prove the existence of advanced and retarded fundamental solutions associated to the Klein-Gordon operator P Θ , cf. Equation (20). To this end let us define the following notable subspaces of H k,m (M ), k = 0, ±1, m ∈ N ∪ {0}: where the subscript tc stands for timelike compact. In addition we call where γ − , γ + are the trace maps introduced in Theorem 3.1 and in Lemma 3.2, while Θ is a pseudodifferential abiding to Hypothesis 4.1. Exactly as in [GW18] from Lemma 4.1 and from Proposition 4.1, it descends the following result on the advanced and retarded propagators G ± Θ associated to the Klein-Gordon operator P Θ , cf. Equation (20). Proposition 4.2. Let P Θ be the Klein-Gordon operator as per Equation (20) and let G Θ be its associated causal propagator, cf. Remark 4.2. Then the following is an exact sequence: Proof. To prove that the sequence is exact, we start by establishing that P Θ is injective on H 1,∞ tc,Θ (M ). This is a consequence of Theorem 4.1 which guarantees that, if P Θ (h) = 0 for h ∈ H 1,∞ tc,Θ (M ), then Secondly, on account of Theorem 4.1 and in particular of the identity G ± Θ P Θ = I on H 1 ±,Θ (M ), it holds that G Θ P Θ (f ) = 0 for all f ∈ H 1,∞ tc,Θ (M ). Hence Im(P Θ ) ⊆ ker(P Θ ). Assume that there exists . The third step consists of recalling that, per construction, P Θ G Θ = 0 and that, still on account of Theorem 4.1, Im(G Θ ) ⊆ ker(P Θ ). To prove the opposite inclusion, suppose that u ∈ ker(P Θ ). Let χ ≡ χ(τ ) be a smooth function such that there exists τ 0 , τ 1 ∈ R such that χ = 1 if τ > τ 1 and χ = 0 if τ < τ 0 . Since Θ is a static boundary condition and, therefore, it commutes with χ, it holds that χu ∈ H 1,∞ +,Θ (M ). Hence setting f . = P Θ χu, a direct calculation shows that G Θ f = u To conclude we need to show that the map P Θ on the before last arrow is surjective. To this end, let j ∈Ḣ −1,∞ (M ) and let χ ≡ χ(τ ) be as above. Let h .
Mainly for physical reasons we individuate the following special classes of boundary conditions. Recall that, according to Theorem 2.1 M is isometric to R × Σ and ∂M to R × ∂Σ. • a static boundary condition if Θ ≡ Θ K is the natural extension to Ψ k b (M ) of a pseudodifferential operator K = K * ∈ Ψ k b (∂Σ) with k ≤ 2.
Observe that any static boundary condition is automatically local in time, see Definition 4.1. Starting from these premises we can investigate further properties of the fundamental solutions, starting from the singularities of the advanced and retarded propagators. To this end let us introduce  Recalling Equation (4), we can state the following theorem characterizing the singularities of the advanced and of the retarded fundamental solutions. The proof is a direct application of Theorem 3.2 or of Theorem 3.3.
where q 1∼ q 2 means that q 1 , q 2 are two points inṄ , cf. Equation (22) connected by a generalized broken bicharacteristic, cf. Definition 3.8. In addition one can infer the following localization property which is sometimes referred to as time-slice axiom. .
Proof. By direct inspection one can realize that the map ι τ 1 ,τ 2 descends to the quotient spacė . The ensuing application [ι τ 1 ,τ 2 ] is manifestly injective. We need to show that it is also surjective. Consider therefore any [f ] ∈Ḣ

Hadamard States
In this section, we discuss a specific application of the results obtained in the previous section, namely we prove existence of a family of distinguished two-point correlation functions for a Klein-Gordon field on a globally hyperbolic, asymptotically AdS spacetime, dubbed Hadamard two-point distributions. These play an important rôle in the algebraic formulation of quantum field theory, particularly when the underlying background is a generic globally hyperbolic spacetime with or without boundary, see e.g. [KM13] for a review as well as [DF16,DF18,DFM18] for the analysis on anti-de Sitter spacetime and [Wro17] for an that on a generic asymptotically AdS spacetime, though only in the case of Dirichlet boundary conditions. Here our goal is to prove that such class of two-point functions exists even if one considers more generic boundary conditions. To prove this statement, the strategy that we follow is divided in three main steps, which we summarize for the reader's convenience. To start with, we restrict our attention to static, asymptotically anti-de Sitter and globally hyperbolic spacetimes and to boundary conditions which are both physically acceptable and static, see Definition 4.2. In this context, by means of spectral techniques, we give an explicit characterization of the advanced and retarded fundamental solutions. To this end we use the theory of boundary triples, a framework which is slightly different, albeit connected, to the one employed in the previous sections, see [DDF18].
Subsequently we show that, starting from the fundamental solutions and from the associated causal propagator, it is possible to identify a distinguished two-point distributions of Hadamard form.
To conclude, we adapt and we generalize to the case in hand a deformation argument due to Fulling, Narcowich and Wald, [FNW81] which, in combination with the propagation of singularities theorem, allows to infer the existence of Hadamard two-point distributions for a Klein-Gordon field on a generic globally hyperbolic and asymptotically AdS spacetime starting from those on a static background.

Fundamental solutions on static spacetimes
In this section we give a concrete example of advanced and retarded fundamental solutions for the Klein-Gordon operator P Θ , cf. Equation (20) on a static, globally hyperbolic, asymptotically AdS spacetime. For the sake of simplicity, we consider a massless scalar field, corresponding to the case ν = (n − 1)/2, see Equation 7. Observe that, since the detailed analysis of this problem has been mostly carried out in [DDF18], we refer to it for the derivation and for most of the technical details.
Here we shall limit ourselves to giving a succinct account of the main results.
As a starting point, we specify precisely the underlying geometric structure: Definition 5.1. Let (M, g) be an n-dimensional Lorentzian manifold. We call it a static globally hyperbolic, asymptotically AdS spacetime if it abides to Definition 2.2 and, in addition, 1) There exists an irrotational, timelike Killing field χ ∈ Γ(T M ), such that L χ (x) = 0 where x is the global boundary function, 2) (M,ĝ) is isometric to a standard static spacetime, that is a warped product R × β S with line element ds 2 = −α 2 dt 2 + h S where h S is a t-independent Riemannian metric on S, while α = α(t) is a smooth, positive function.
Remark 5.1. In the following, without loss of generality, we shall assume that, whenever we consider a static globally hyperbolic, asymptotically flat spacetime if it abides to Definition 2.2, the timelike Killing field χ coincides with the vector field ∂ τ , cf. Theorem 2.1. Hence the underlying lineelement reads as ds 2 = −βdτ 2 + κ where both β and κ are τ -independent and S can be identified with the Cauchy surface Σ in Theorem 2.1. For convenience we also remark that, in view of this characterization of the metric, the associated Klein-Gordon equation P u = 0 with P = ✷ g reads where E = β∆ κ , being ∆ κ the Laplace-Beltrami operator associated to the the Riemannian metric κ.
Henceforth we consider only static boundary conditions as per Definition 4.2 which we indicate with the symbol Θ K to recall that they are induced from K ∈ Ψ k b (∂M ). Since the underlying spacetime is static, in order to construct the advanced and retarded fundamental solutions, we can focus our attention on G Θ K ∈ D ′ (M ×M ) , the bi-distribution associated to the causal propagator G Θ K , cf. Remark 4.2. It satisfies the following initial value problem, see also [DDF18]: where δ is the Dirac distribution on the diagonal ofM ×M . Starting from G Θ K one can recover the advanced and retarded fundamental solutions G ± Θ K via the identities: where ϑ is the Heaviside function. The existence and the properties of G Θ K have been thoroughly analyzed in [DDF18] using the framework of boundary triples, cf. [Gru68]. Here we recall the main structural aspects.
Definition 5.2. Let H be a separable Hilbert space over C and let S : D(S) ⊂ H → H be a closed, linear and symmetric operator. A boundary triple for the adjoint operator S * is a triple (h, γ 0 , γ 1 ), where h is a separable Hilbert space over C while γ 0 , γ 1 : D(S * ) → h are two linear maps satisfying 1) For every f, f ′ ∈ D(P * ) it holds 2) The map γ : One of the key advantages of this framework is encoded in the following proposition, see [Mal92] Proposition The map Θ → S Θ is one-to-one and S * Θ = S Θ * . In other word there is a one-to-one correspondence between self-adjoint operators Θ on h and self-adjoint extensions of S.
Remark 5.2. Note that in the massless case, the two trace operators Γ 0 and Γ 1 coincide respectively with the restriction to H 2 (M ) of the traces γ − and γ + introduced in Theorem 3.1 and in Lemma 3.2.
Gathering all the above ingredients, we can state the following proposition, cf. [DDF18, Thm.
Combining all data together, particularly Proposition 5.1 and Proposition 5.2 we can state the following theorem, whose proof can be found in [DDF18,Thm 29] Theorem 5.1. Let (M, g) be a static, globally hyperbolic, asymptotically AdS spacetime as per Definition 5.1. Let (γ 0 , γ 1 , L 2 (∂M )) be the boundary triple as in Proposition 5.2 associated with E * , the adjoint of the elliptic operator defined in (45) and let K be a densely defined self-adjoint operator on L 2 (∂Σ) which individuates a static and physically admissible boundary condition as per Definition 4.2. Let E K be the self-adjoint extension of E defined as per Proposition 5.1 by = ker(γ 1 − Kγ 0 ). Furthermore, let assume that the spectrum of E K is bounded from below. Then, calling Θ K the associated boundary condition, the advanced and retarded Green's operators G ± Θ K associated to the wave operator ∂ 2 t + E K exist and they are unique. They are completely determined in terms of G ± Θ K ∈ D ′ (M ×M ). These are bidistributions such that where f (t) ∈ H 2 (Σ) denotes the evaluation of f , regarded as an element of C ∞ c (R, H ∞ (Σ)) and E − 1 2 K sin [E 1 2 K (t − t ′ )] is defined exploiting the functional calculus for E K . Moreover it holds that In particular, Remark 5.3. Observe that, in Theorem 5.1 we have constructed the advanced and retarded fundamental solutions G ± Θ as elements of D ′ (M ×M ). Yet we can combine this result with Theorem 4.1 to conclude that there must exist unique and advanced retarded propagators on the whole M whose restriction toM coincides with G ± Θ K . With a slight abuse of notation we shall refer to these extended fundamental solutions with the same symbol.

Existence of Hadamard States on Static Spacetimes
In this section, we discuss the existence of Hadamard two-point functions. We stress that the socalled Hadamard condition and its connection to microlocal analysis have been first studied and formulated under the assumption that the underlying spacetime is without boundary and globally hyperbolic. We shall not enter into the details and we refer an interested reader to the survey in [KM13].
As outlined in the introduction, if the underlying background possesses a timelike boundary, the notion of Hadamard two-point function needs to be modified accordingly. Here we follow the same rationale advocated in [DF16,DF17] and also in [DW19,Wro17].
Definition 5.3. Let (M, g) be a globally hyperbolic, asymptotically AdS spacetime as per Definition 2.2. A bi-distribution λ 2 ∈ D ′ (M × M ) is called of Hadamard form if its restriction toM has the following wavefront set where ∼ entails that (p, k) and (p ′ , k ′ ) are connected by a generalized broken bicharactersitic, while k ⊲0 means that the co-vector k at p ∈M is future-pointing. Furthermore we call λ 2,Θ ∈ D ′ (M ×M ) a Hadamard two-point function associated to P Θ , if, in addition to Equation (53), it satisfies (P Θ ⊗ I)λ 2,Θ = (I ⊗ P Θ )λ 2,Θ = 0, and, for all f, f ′ ∈ D(M ), where P Θ is the Klein-Gordon operator as in Equation (20), while G Θ is the associated causal propagator, cf. Remark 4.2.
Remark 5.4. To make contact with the terminology often used in theoretical physics, given a Hadamard two-point function λ 2,Θ , we can identify the following associated bidistributions: • the bulk-to-bulk two-point functionλ 2,Θ ∈ D ′ (M ×M ) such thatλ 2,Θ . = λ 2,Θ |M is the restriction of the Hadamard two-point function toM ×M .
The existence of Hadamard two-point functions is not a priori obvious and it represents an important question at the level of applications. Here we address it in two steps. First we focus on static, globally hyperbolic, asymptotically anti-de Sitter spacetimes and subsequently we drop the assumption that the underlying background is static, proving existence of Hadamard two-point functions via a deformation argument.
Let us focus on the first step. To this end, on the one hand we need the boundary condition Θ to abide to Hypothesis 4.1, while, on the other hand we make use of some auxiliary results from [Wro17], specialized to the case in hand. In the next statements it is understood that to any Hadamard two-point function λ 2,Θ , it corresponds Λ Θ :Ḣ −k,−∞ Lemma 5.1. For any q 1 , q 2 ∈ b S * M , (q 1 , q 2 ) ∈ W F Op (Λ Θ ) if and only if there exist neighbourhoods Γ i of q i , i = 1, 2, such that for all Observe that this lemma entails in particular that, given any f i ∈ C ∞ (M ), i = 1, 2 such that supp(f i ) ⊂M then f 1 Λ Θ f 2 has a smooth kernel overM ×M . In addition the following also holds true, cf. [Wro17, Prop. 5.6]: Proposition 5.3. Let Λ Θ identify an Hadamard two-point function.
Given any two points q 1 and q 2 in the cosphere bundle b S * M , cf. Equation (4) we shall write q 1∼ q 2 if both q 1 and q 2 lie in the compressed characteristic bundleṄ and they are connected by a generalized broken bicharacteristic, cf. Definition 3.8. With these data and using [Wro17,Prop. 5.9] together with Hypothesis 4.1 and with Theorems 3.2 and 3.3, we can establish the following operator counterpart of the propagation of singularities theorem: Our next step consists of refining Theorem 4.2 inM , cf. for similarities with [DF18, Cor. 4.5].
Proof. A direct application of Theorem 4.2 yields From this inclusion, it descends that every pair of points in the singular support of G is connected by a generalized broken bicharacteristic completely contained inM . Since b T * M ≃ T * M , we can apply [BF09, Ch.4, Thm. 16] and the sought statement is proven.
With these data, we are ready to address the main question of this section. Suppose that (M, g) is a static, globally hyperbolic, asymptotically AdS spacetime, cf. Definition 2.2 and 5.1. Let P Θ be the Klein-Gordon operator as per Equation (20) and let Θ ≡ Θ K be a static boundary condition as per Theorem 5.1. For simplicity we also assume that the spectrum of E K is contained in the positive real axis. Then the following key result holds true: Proposition 5.5. Let (M, g) be a static, globally hyperbolic asymptotically AdS spacetime and let P Θ K be the Klein-Gordon operator with a static and physically admissible boundary condition as per Definition 4.2 Then there exists a Hadamard two-point function associated to P Θ , λ 2,Θ K ∈ D ′ (M × M ) such that, for all f 1 , f 2 ∈ D(M ) Proof. Observe that, per construction λ 2,Θ k is a bi-solution of the Klein-Gordon equation associated to the operator P Θ K and it abides to Equation (54). We need to show that Equation (53) holds true. To this end it suffices to combine the following results. From [SV00] one can infer that, the restriction ofλ 2,Θ K , the bulk-to-bulk two-point distribution, to every globally hyperbolic submanifold of M not intersecting the boundary is consistent with Equation (53). At this point it suffices to invoke Proposition 5.3 and 5.5 to draw the sought conclusion.
Remark 5.5. Observe that, from a physical viewpoint, in the preceding theorem, we have individuated the two-point function of the so-called ground state with boundary condition prescribed by Θ K .

A Deformation Argument
In order to prove the existence of Hadamard two-point functions on a generic asymptotically anti-de Sitter spacetime for a Klein-Gordon field with prescribed static boundary condition, we shall employ a a deformation argument akin to that first outlined in [FNW81] on globally hyperbolic spacetimes with empty boundary.
To this end we need the following lemma, see [Wro17,Lem. 4.6], slightly adapted to the case in hand. In anticipation, recalling Equation (2), we say that a globally hyperbolic, asymptotically AdS spacetime is even modulo O(x 3 ) close to ∂M if h(x) = h 0 + x 2 h 1 (x) where h 1 is a symmetric two-tensor, see [Wro17,Def. 4.3].
Consider now a generic, globally hyperbolic, asymptotically anti-de Sitter spacetime (M, g) and a deformation as per Lemma 5.2. Observe that, per construction, all generalized broken bicharacteristics reach the region of M with τ ∈ [τ 1 , τ 2 ]. This observation leads to the following result which is a direct consequence of the propagation of singularities theorem 3.3 and 3.2. Mutatis mutandis, the proof is as that of [Wro17, Lem. 5.10] and, thus, we omit it.
Lemma 5.3. Suppose that Λ Θ ∈ D ′ (M × M ) is a bi-solution of the Klein-Gordon equation ruled by P Θ abiding to Equation (54) and with a wavefront set of Hadamard form in the region of M such that τ 1 < τ < τ 2 . Then Λ Θ is a Hadamard two-point function.
To conclude, employing Corollary 4.2 we can prove the sought result: Theorem 5.2. Let (M, g) be a globally hyperbolic, asymptotically anti-de Sitter spacetime and let (M S , g S ) be its static deformation as per Lemma 5.2. Let Θ K be a static and physically admissible boundary condition so that the Klein-Gordon operator P Θ K on (M S , g S ) admits a Hadamard twopoint function as per Proposition 5.5. Then there exists a Hadamard two point-function on (M, g) for the associated Klein-Gordon operator with boundary condition ruled by Θ K .
Proof. Let (M, g) be as per hypothesis and let (M, g S ) be a static, globally hyperbolic, asymptotically AdS spacetime such that there exists a third, globally hyperbolic, asymptotically AdS spacetime (M, g ′ ) interpolating between (M, g) and (M, g S ) in the sense of Lemma 5.2. On account of Theorem 2.1, in all three cases M is isometric to R × Σ.
On account of Proposition 5.5, on (M, g S ) we can identify an Hadamard two-point function as in Equation (55) subordinated to the boundary condition Θ K . We indicate it with λ 2,S omitting any reference to Θ K since it plays no explicit rôle in the analysis.
Focusing the attention on (M, g ′ ), Lemma 5.2 guarantees that, if τ < τ 0 , τ being the time coordinate along R, then therein (M, g ′ ) is isometric to (M, g S ). Calling this region M 0 , the restriction λ 2,S | M 0 ×M 0 identifies a two-point distribution of Hadamard form. Notice that we have omitted to write explicitly the underlying isometries for simplicity of notation.
Observe that h, h ′ ∈ C ∞ tc (M ) and therefore the right-hand side of this identity is well-defined. In addition, since G Θ K is continuous on D(M ), sequential continuity entails that λ ′ 2 ∈ D(M ′ × M ′ ). In addition, per construction, it is a solution of the Klein-Gordon equation ruled by P Θ K on (M ′ , g ′ ) and abiding to Equation (54). Furthermore Lemma 5.3 yields that λ ′ 2 is of Hadamard form. To conclude it suffices to focus on (M, g) recalling that there exists τ 1 ∈ R such that, in the region (M 1 , g ′ ) ⊂ (M, g ′ ) for which τ > τ 1 , (M, g ′ ) is isometric to (M, g). Hence,we can repeat the argument given above. More precisely we consider λ ′ 2 | M ′ ×M ′ and, using the time-slice axiom, see Corollary 4.2, we can identify λ 2 ∈ D ′ (M × M ) which is a solution of the Klein-Gordon equation ruled by P Θ K and it abides to Equation (54). Lemma 5.3 entails also that it is of Hadamard form, hence proving the sought result.