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

Claudio Dappiaggi1,2,3,a, Alessio Marta4,5,6,b,
1 Dipartimento di Fisica – Università di Pavia, Via Bassi 6, 27100 Pavia,
Italy.
2 INFN, Sezione di Pavia – Via Bassi 6, 27100 Pavia, Italy.
3 Istituto Nazionale di Alta Matematica – Sezione di Pavia, Via Ferrata, 5,
27100 Pavia, Italy.
4 Dipartimento di Matematica – Università di Milano, Via Cesare Saldini, 50 –
I-20133 Milano, Italy.
5 INFN, Sezione di Milano – Via Celoria, 16 – I-20133 Milano, Italy.
6 Istituto Nazionale di Alta Matematica – Sezione di Milano, Via Saldini, 50,
I-20133 Milano, Italy.
a<EMAIL_ADDRESS>, b<EMAIL_ADDRESS>
###### Abstract

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.

##  1 Introduction

The $n$-dimensional anti-de Sitter spacetime (AdSn) 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 AdSn 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 $b$-pseudodifferential
operator $\Theta\in\Psi^{k}_{b}(\partial M)$ with $k\leq 2$, 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:

######  Theorem 1.1.

Let $P_{\Theta}$ be the Klein-Gordon operator as per Equation (20) where
$\Theta$ abides to Hypothesis 4.1. Then there exist unique retarded $(+)$ and
advanced $(-)$ propagators, that is continuous operators
$G_{\Theta}^{\pm}:\dot{\mathcal{H}}^{-1,m+1}_{\pm}(M)\rightarrow\mathcal{H}^{1,m}_{\pm}(M)$
such that $P_{\Theta}G_{\Theta}^{\pm}=\mathbb{I}$ on
$\dot{\mathcal{H}}^{-1,m+1}_{\pm}(M)$ and
$G_{\Theta}^{\pm}P_{\Theta}=\mathbb{I}$ on
$\mathcal{H}^{1,m}_{\pm,\Theta}(M)$. Furthermore, $G_{\Theta}^{\pm}$ is a
continuous map from $\dot{\mathcal{H}_{0}^{-1,\infty}}(M)$ to
$\mathcal{H}_{loc}^{1,\infty}(M)$ where the subscript $0$ indicates that we
consider only functions of compact support.

Here the spaces $\dot{\mathcal{H}}^{-1,s+1}_{\pm}(M)$,
$\mathcal{H}^{1,s}_{\pm}(M)$ as well as
$\dot{\mathcal{H}_{0}^{-1,\infty}}(M)$, $\mathcal{H}_{loc}^{1,\infty}(M)$ and
$\mathcal{H}^{1,m}_{\pm,\Theta}(M)$ are characterized in Definition 3.5 and in
Section 4, see in particular Equations (41b), (41a) and (42).

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
$\Theta_{K}$ be a static and physically admissible boundary condition so that
the Klein-Gordon operator $P_{\Theta_{K}}$ on $(M_{S},g_{S})$ admits a
Hadamard two-point 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 $\Theta_{K}$.

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.

##  2 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
$\dim M=n\geq 2$ equipped with a smooth Lorentzian manifold $g$ of signature
$(-,+,\dots,+)$. With $C^{\infty}(M)$ (resp. $C^{\infty}_{0}(M)$) we indicate
the space of smooth (resp. smooth and compactly supported) functions on $M$,
while $\dot{C}^{\infty}(M)$ (resp. $\dot{C}^{\infty}_{0}(M)$) stands for the
collection of all smooth (resp. smooth and compactly supported) functions
vanishing at $\partial M$ 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 $\iota:\partial M\to M$. We
say that $(M,g)$

  1. 1.

has a timelike boundary if $(\partial M,\iota^{*}g)$ is a smooth, Lorentzian
manifold,

  2. 2.

is globally hyperbolic if it does not contain closed causal curves and if, for
every $p,q\in M$, $J^{+}(p)\cap J^{-}(q)$ is either empty or compact.

If both conditions are met, we call $(M,g)$ a globally hyperbolic spacetime
with timelike boundary and we indicate with $\mathring{M}=M\setminus\partial
M$ the interior of $M$.

Observe that, for simplicity, we assume throughout the paper that also
$\partial M$ is connected. Notice in addition that Definition 2.1 reduces to
the standard notion of globally hyperbolic spacetimes when $\partial
M=\emptyset$. 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 $\partial M=\emptyset$.

######  Theorem 2.1.

Let $(M,g)$ be an $n$-dimensional globally hyperbolic spacetime with timelike
boundary. Then it is isometric to a Cartesian product $\mathbb{R}\times\Sigma$
where $\Sigma$ is an $(n-1)$-dimensional Riemannian manifold. The associated
line element reads

$ds^{2}=-\beta d\tau^{2}+\kappa_{\tau},$ (1)

where $\beta\in C^{\infty}(\mathbb{R}\times\Sigma;(0,\infty))$ while
$\tau:\mathbb{R}\times\Sigma\to\mathbb{R}$ plays the rôle of time coordinate.
In addition $\mathbb{R}\ni\tau\mapsto\kappa_{\tau}$ identifies a family of
Riemmannian metrics, smoothly dependent on $\tau$ and such that, calling
$\Sigma_{\tau}\doteq\\{\tau\\}\times\Sigma$, each
$(\Sigma_{\tau},\kappa_{\tau})$ is a Cauchy surface with non empty boundary.

######  Remark 2.1.

Observe that a notable consequence of this theorem is that, calling
$\iota_{\partial M}:\partial M\to M$ the natural embedding map, then
$(\partial M,h)$ where $h=\iota^{*}_{\partial M}g$ is a globally hyperbolic
spacetime. In particular the associated line element reads

$ds^{2}|_{\partial M}=-\beta|_{\partial M}d\tau^{2}+\kappa_{\tau}|_{\partial
M}.$

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 $\partial M$.
Suppose that $\mathring{M}=M\setminus\partial M$ is equipped with a smooth
Lorentzian metric $g$ and that

  * a)

If $x\in\mathcal{C}^{\infty}(M)$ is a boundary function, then
$\widehat{g}=x^{2}g$ extends smoothly to a Lorentzian metric on $M$.

  * b)

The pullback $h=\iota^{*}_{\partial M}\widehat{g}$ via the natural embedding
map $\iota_{\partial M}:\partial M\to M$ individuates a smooth Lorentzian
metric.

  * c)

$\widehat{g}^{-1}(dx,dx)=1$ on $\partial M$.

Then $(M,g)$ is called an asymptotically anti-de Sitter (AdS) spacetime. In
addition, if $(M,\widehat{g})$ 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 $\Omega\in C^{\infty}(M)$. 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 $p\in\partial M$, it is possible to
find a neighbourhood $U\subset\partial M$ containing $p$ and $\epsilon>0$ such
that on $U\times[0,\epsilon)$ the line element associated to $g$ reads

$ds^{2}=\frac{-dx^{2}+h_{x}}{x^{2}}$ (2)

where $h_{x}$ is a family of Lorentzian metrics depending smoothly on $x$ such
that $h_{0}\equiv h$.

######  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 $\partial M$ is required to be diffeomorphic to
$\mathbb{R}\times\mathbb{S}^{n-2}$. 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 $\tau$ and $x$ we shall always indicate
respectively the time coordinate as in Equation (1) and the spatial coordinate
as in Equation (2).

###  2.1 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
[Sch01, AGN16, GS13, GOW17] as well as in [DDF18, Sec. 2.1 & 2.2].

######  Definition 2.3.

A Riemannian manifold $(N,h)$ with empty boundary is of bounded geometry if

  * a)

The injectivity radius $r_{inj}(N)$ is strictly positive,

  * b)

$N$ is of totally bounded curvature, namely for all
$k\in\mathbb{N}\cup\\{0\\}$ there exists a constant $C_{k}>0$ such that
$\|\bigtriangledown^{k}R\|_{L^{\infty}(M)}<C_{k}$.

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
$(Y,\iota_{Y})$ be a codimension $k$, closed, embedded smooth submanifold with
an inward pointing, unit normal vector field $\nu_{Y}$. The submanifold
$(Y,\iota^{*}_{Y}g)$ is of bounded geometry if:

  * a)

The second fundamental form $II$ of $Y$ in $N$ and all its covariant
derivatives along $Y$ are bounded,

  * b)

There exists $\varepsilon_{Y}>0$ such that the map
$\phi_{\nu_{Y}}:Y\times(-\varepsilon_{Y},\varepsilon_{Y})\rightarrow N$
defined as $(x,z)\mapsto\phi_{\nu_{Y}}(x,z)\doteq exp_{x}(z\nu_{Y,x})$ is
injective.

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 $\partial N\neq\emptyset$. We say
that $(N,h)$ is of bounded geometry if there exists a Riemannian manifold of
bounded geometry $(N^{\prime},h^{\prime})$ of the same dimension as $N$ such
that:

  * a)

$N\subset N^{\prime}$ and $h=h^{\prime}|_{N}$

  * b)

$(\partial N,\iota^{*}h^{\prime})$ is a bounded geometry submanifold of
$N^{\prime}$, where $\iota:\partial N\rightarrow N^{\prime}$ is the embedding
map.

######  Remark 2.4.

Observe that Definition 2.5 is independent from the choice of $N^{\prime}$.
For completeness, we stress that an equivalent definition which does not
require introducing $N^{\prime}$ 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
$BT^{m}_{m^{\prime}}(B_{n}(0,\frac{r_{inj}(N)}{2}),\delta_{E})$, the space of
all bounded tensors on the ball $B_{n}(0,\frac{r_{inj}(N)}{2})$ centered at
the origin of the Euclidean space $(\mathbb{R}^{n},\delta_{E})$ where
$\delta_{E}$ stands for the flat metric. For every
$m,m^{\prime}\in\mathbb{N}\cup\\{0\\}$, we denote with
$BT^{m}_{m^{\prime}}(N)$ the space of all rank $(m,m^{\prime})$ tensors $T$ on
$N$ such that, for any $p\in M$, calling $T_{p}\doteq(\exp_{p}\circ
e_{p})^{*}T$ where $e_{p}:(\mathbb{R}^{n},\delta)\to(T_{p}N,h_{p})$ is a
linear isometry, the family $\\{T_{p}\\}_{p\in M}$ is bounded on
$BT^{m}_{m^{\prime}}(B_{n}(0,\frac{r_{inj}(N)}{2}),\delta_{E})$.

######  Definition 2.6.

A smooth Lorentzian manifold $(M,g)$ is of bounded geometry if there exists a
Riemannian metric $\widehat{g}$ on $M$ such that:

  * a)

$(M,\widehat{g})$ is of bounded geometry.

  * b)

$g\in BT^{0}_{2}(M,\widehat{g})$ and $g^{-1}\in BT^{2}_{0}(M,\widehat{g})$.

On top of a Riemannian (or of a Lorentzian) manifold of bounded geometry
$(N,h)$ we can introduce $H^{k}(N)\equiv W^{2,k}(N)$ which is the completion
of

$\mathcal{E}^{k}(N)\doteq\\{f\in C^{\infty}(N)\;|\;f,\nabla
f,\dots,(\nabla)^{k}f\in L^{2}(N)\\},$

with respect to the norm

$\|f\|_{W^{2,k}(N)}=\left(\sum\limits_{i=0}^{k}\|(\nabla)^{i}f\|_{L^{2}(N)}\right)^{\frac{1}{2}},$

where $\nabla$ is the covariant derivative built out of the Riemannian metric
$h$, while $(\nabla)^{i}$ indicates the $i$-th covariant derivative. This
notation is employed to disambiguate with $\nabla^{i}=h^{ij}\nabla_{j}$.

######  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.

##  3 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].

###  3.1 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 ${}^{b}TM$ we indicate the $b$-tangent bundle which is a
vector bundle whose fibres are

${}^{b}T_{p}M=\left\\{\begin{array}[]{ll}T_{p}M&p\in\mathring{M}\\\
\textrm{span}_{\mathbb{R}}(x\partial_{x},T_{p}\partial M)&p\in\partial
M\end{array}\right.,$

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 are

${}^{b}T^{*}_{p}M=\left\\{\begin{array}[]{ll}T^{*}_{p}M&p\in\mathring{M}\\\
\textrm{span}_{\mathbb{R}}(\frac{dx}{x},T^{*}_{p}\partial M)&p\in\partial
M\end{array}\right.$

For future convenience, whenever we fix a chart $U$ of $M$ centered at a point
$p\in\partial M$, we consider $(x,y_{i},\xi,\eta_{i})$ and
$(x,y_{i},\zeta,\eta_{i})$, $i=1,\dots,n-1=\dim\partial M$, 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 $\tau$, cf. Equation (1), we identify implicitly
$\eta_{n-1}\equiv\tau$. In addition, observe that there exists a natural
projection map

$\pi:T^{*}M\to{}^{b}T^{*}M,\quad(x,y_{i},\xi,\eta_{i})\mapsto\pi(x,y_{i},\xi,\eta_{i})=(x,y_{i},x\xi,\eta_{i}),$

which is non-injective. This feature prompts the definition of a very
important structure in our investigation, namely the compressed $b$-cotangent
bundle

${}^{b}\dot{T}^{*}M\doteq\pi[T^{*}M],$ (3)

which is a vector sub-bundle of ${}^{b}T^{*}M$, such that
${}^{b}\dot{T}^{*}_{p}M\equiv T^{*}_{p}M$ whenever $p\in\mathring{M}$. 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 $T^{*}_{b}M\setminus\\{0\\}$, namely

${}^{b}S^{*}M\doteq{\raisebox{1.99997pt}{${}^{b}T^{*}M\setminus\\{0\\}$}\left/\raisebox{-1.99997pt}{$\mathbb{R}^{+}$}\right.}.$
(4)

We remark that, if we consider a local chart $U\subset M$ such that
$U\cap\partial M\neq\emptyset$ and the local coordinates
$(x,y_{i},\zeta,\eta_{i})$, $i=1,\dots,n-1=\dim\partial M$, on
${}^{b}T^{*}_{U}M\doteq{}^{b}T^{*}M|_{U}$, we can build a natural counterpart
on ${}^{b}S^{*}_{U}M$, namely $(x,y_{i},\widehat{\zeta},\widehat{\eta}_{i})$
where $\widehat{\zeta}=\frac{\zeta}{|\eta_{n-1}|}$ and
$\widehat{\eta_{i}}=\frac{\eta_{i}}{|\eta_{n-1}|}$. 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

  * •

$\textbf{Diff}_{b}(M)\doteq\bigoplus_{k=0}^{\infty}\textbf{Diff}^{k}_{b}(M)$
the graded, differential operator algebra generated by $\Gamma({}^{b}TM)$, the
space of smooth section of the $b$-tangent bundle.

  * •

$\Psi_{b}^{m}(M)$ the set of properly supported $b$-pseudodifferential
operators ($b-\Psi$DOs) of order $m$, $m\in\mathbb{R}$.

The notion of $b-\Psi$DOs 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

$\sigma_{b,m}:\Psi_{b}^{m}(M)\to
S^{m}({}^{b}T^{*}M)/S^{m-1}({}^{b}T^{*}M),\quad A\mapsto a=\sigma_{b,m}(A),$
(5)

which gives rise to an isomorphism

$\Psi_{b}^{m}(M)/\Psi_{b}^{m-1}(M)\simeq
S^{m}({}^{b}T^{*}M)/S^{m-1}({}^{b}T^{*}M).$

In addition we can endow the space of symbols $S^{m}({}^{b}T^{*}M)$ with a
Fréchet topology induced by the family of seminorms

$\|a\|_{N}\ =\sup_{(z,k_{z})\in
K_{i}\times\mathbb{R}^{n}}\max_{|\alpha|+|\gamma|\leq
N}\dfrac{|\partial_{z}^{\alpha}\partial_{\zeta}^{\gamma}a(z,k_{z})|}{\langle
k_{z}\rangle^{m-|\gamma|}},$

where $\langle k_{z}\rangle=(1+|k_{z}|^{2})^{\frac{1}{2}}$, while
$\\{K_{i}\\}_{i\in I}$, $I$ 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

$d(a,b)=\sum_{N\in\mathbb{N}}2^{-N}\dfrac{\|a-b\|_{N}}{1+\|a-b\|_{N}}.\quad\forall
a,b\in S^{m}({}^{b}T^{*}M)$

In view of these data the following definition is natural

######  Definition 3.2.

A subset of $\Psi_{b}^{m}(M)$ is called bounded if such is the associated
subset of $S^{m}({}^{b}T^{*}M)$ with respect to the Fréchet topology.

Finally we can recall the notion of elliptic $b-\Psi$DO and of wavefront set
both of a single and of a family of pseudodifferential operators, cf. [Hör03]:

######  Definition 3.3.

A b-pseudodifferential operator $A\in\Psi^{m}_{b}(M)$ is elliptic at a point
$q_{0}\in\ {}^{b}T^{*}M\setminus\\{0\\}$ if there exists $c\in
S^{-m}(^{b}T^{*}M)$ such that

$\sigma_{b,m}(A)\cdot c-1\in S^{-1}(^{b}T^{*}M),$

in a conic neighbourhood of $q_{0}$. We call $ell_{b}(A)$ the (conic) subset
of ${}^{b}T^{*}M\setminus\\{0\\}$ in which $A$ is elliptic.

######  Definition 3.4.

For any $P\in\Psi^{m}_{b}(M)$, we say that $(z_{0},k_{z_{0}})\notin
WF^{\prime}_{b}(P)$ if the associated symbol $p(z,k_{z})$ is such that, for
every multi-indices $\gamma$ and for every $N\in\mathbb{N}$, there exists a
constant $C_{N,\alpha,\gamma}$ such that

$|\partial_{z}^{\alpha}\partial^{\gamma}_{k_{z}}p(z,k_{z})|\leq
C_{N,\alpha,\gamma}\langle k_{z}\rangle^{-N},$

for $z$ in a neighbourhood of $z_{0}$ and $k_{z}$ in a conic neighbourhood of
$k_{z_{0}}$.

Similarly, if $\mathcal{A}$ is a bounded subset of $\Psi_{b}^{m}(M)$ and
$q\in{}^{b}T^{*}M$. We say that $q\not\in WF_{b}^{\prime}(\mathcal{A})$ if
there exists $B\in\Psi_{b}(M)$, elliptic at $q$, such that
$\\{BA:A\in\mathcal{A}\\}$ is a bounded subset of $\Psi_{b}^{-\infty}(M)$.

To conclude this part of the section, we stress that, in order to study the
behavior of a b-pseudodifferential operator at the boundary, it is useful to
introduce the notion of indicial family, [GW18]. Let $A\in\Psi_{b}^{m}(M)$.
For a fixed boundary function $x$, cf. Definition 2.2, and for any
$v\in\mathcal{C}^{\infty}(\partial M)$ we define the indicial family
$\widehat{N}(A)(s):C^{\infty}(\partial M)\to C^{\infty}(\partial M)$ as:

$\widehat{N}(A)(s)v=x^{-is}A\left(x^{is}u\right)|_{\partial M}$ (6)

where $u\in\mathcal{C}^{\infty}(M)$ is any function such that $u|_{\partial
M}=v$.

###  3.2 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 $P\doteq\Box_{g}-m^{2}$, where $m^{2}$ plays the rôle of a mass term,
while $\Box_{g}$ is the D’Alembert wave operator built out of the metric $g$.
It is convenient to introduce the parameter

$\nu=\frac{1}{2}\sqrt{(n-1)^{2}+4m^{2}},$ (7)

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

$\textbf{Diff}^{1}_{\nu}(M)\doteq\\{x^{\nu_{-}}Dx^{-\nu_{-}}\;|\;D\in\textbf{Diff}^{1}(M)\\},$

where $\nu_{-}=\frac{n-1}{2}-\nu$, $n=\dim M$. Starting from these data, and
calling with $x$ and $d\mu_{g}$ respectively the global boundary function, cf.
Definition 2.2, and the metric induced volume measure we set

$\mathcal{L}^{2}(M)\doteq
L^{2}(M;x^{2}d\mu_{g})\;\;\textrm{and}\;\mathcal{H}^{1}(M)\doteq\\{u\in\mathcal{L}^{2}(M)\;|\;Qu\in\mathcal{L}^{2}(M)\;\forall
Q\in\textbf{Diff}^{1}_{\nu}(M)\\}.$ (8)

The latter is a Sobolev space if endowed with the norm

$\|u\|^{2}_{\mathcal{H}^{1}(M)}=\|u\|^{2}_{\mathcal{L}^{2}(M)}+\sum_{i=1}^{n}\|Q_{i}u\|^{2}_{\mathcal{L}^{2}(M)},$

where $\\{Q_{i}\\}_{i=1\dots n}$ is a generating set of
$\textbf{Diff}^{1}_{\nu}(M)$. In addition we shall be using
$\mathcal{L}^{2}_{loc}(M)$, the space of locally square integrable functions
over $M$ with respect to the measure $x^{2}d\mu_{g}$ and
$\dot{\mathcal{L}^{2}}_{loc}(M)$ the counterpart built starting from
$\dot{C}^{\infty}(M)$ in place of $C^{\infty}(M)$. Starting from these spaces
we can build the first order Sobolev spaces $\mathcal{H}^{1}_{loc}(M)$ and
$\dot{\mathcal{H}}^{1}_{loc}(M)$ as well as their respective topological
duals, $\dot{\mathcal{H}}^{-1}_{loc}(M)$ and $\mathcal{H}^{-1}_{loc}(M)$.
Finally, calling $\mathcal{E}^{\prime}(M)$ the topological dual space of
$\dot{C}^{\infty}(M)$, we set

$\mathcal{H}^{1}_{0}(M)=\mathcal{H}^{1}_{loc}(M)\cap\mathcal{E}^{\prime}(M),$
(9)

while, similarly, we define $\mathcal{H}^{-1}_{0}(M)$.

We discuss succinctly the interactions between $\Psi^{m}_{b}(M)$ and
$\textbf{Diff}_{\nu}^{1}(M)$. We begin by studying the action of
pseudodifferential operators of order zero on the spaces
$\mathcal{H}^{k}_{loc/0}(M)$, $k=\pm 1$, just defined. Every
$A\in\Psi^{0}_{b}(M)$ is a bounded operator thereon, as stated in the
following lemma.

######  Lemma 3.1 ([GW18], Lemma 3.8 and [Vas08], Lemma 3.2, Corollary 3.4).

Let $A\in\Psi^{0}_{b}(M)$. Then $A$ is a continuous linear map

$\mathcal{H}^{1}_{loc/0}(M)\rightarrow\mathcal{H}^{1}_{loc/0}(M),\ \ \
\dot{\mathcal{H}}^{1}_{loc/0}(M)\rightarrow\dot{\mathcal{H}}^{1}_{loc/0}(M),$

which extends per duality to a continuous map

$\dot{\mathcal{H}}^{-1}_{0/loc}(M)\rightarrow\dot{\mathcal{H}}^{-1}_{0/loc}(M),\
\ \ \mathcal{H}^{-1}_{0/loc}(M)\rightarrow\mathcal{H}^{-1}_{0/loc}(M).$

The proof of this lemma gives a useful information. Let $A\in\Psi^{0}_{b}(M)$
be with compact support $U\subset M$. Then there exists
$\chi\in\mathcal{C}_{0}^{\infty}(U)$ such that

$\|Au\|_{\mathcal{H}^{k}(M)}\leq C\|\chi u\|_{\mathcal{H}^{k}(M)},$ (10)

for every $u\in\mathcal{H}^{k}(M)$ with $k=\pm 1$. The constant $C$ is bounded
by a seminorm of $A$.

To study in full generality the interactions between $\Psi^{m}_{b}(M)$ and
$\textbf{Diff}_{\nu}^{1}(M)$, we need to introduce one last class of relevant
spaces

######  Definition 3.5.

Let $k=-1,0,1$ and let $m\geq 0$. Given $u\in\mathcal{H}_{loc}^{k}(M)$ (resp.
$\mathcal{H}^{k}(M)$), we say that $u\in\mathcal{H}_{loc}^{k,m}(M)$ (resp.
$\mathcal{H}^{k,m}(M)$) if $Au\in\mathcal{H}_{loc}^{k}(M)$ (resp.
$\mathcal{H}^{k}(M)$) for all $A\in\Psi^{m}_{b}(M)$. Furthermore, we define
$\mathcal{H}^{k,\infty}(M)$ as:

$\mathcal{H}^{k,\infty}(M)\doteq\bigcap_{m=0}^{\infty}\mathcal{H}^{k,m}(M).$
(11)

######  Remark 3.1.

As observed in [Vas08], whenever $m$ is finite, it is enough to check that
both $u$ and $Au$ lie in $\mathcal{H}^{k}_{loc}(M)$ for a single elliptic
operator $A\in\Psi^{m}_{b}(M)$.

Observe that, in full analogy to Definition 3.5, we define similarly
$\mathcal{H}^{k,m}_{0}(M)$ and $\dot{\mathcal{H}}^{k,m}_{loc}(M)$. In the
following definition, we extend the notion of wavefront set to the spaces
$\mathcal{H}_{loc}^{k,m}(M)$.

######  Definition 3.6.

Let $k=0,\pm 1$ and let $u\in\mathcal{H}^{k,m}_{loc}(M)$, $m\in\mathbb{R}$.
Given $q\in{}^{b}T^{*}M\setminus\\{0\\}$, we say that $q\not\in
WF_{b}^{k,m}(u)$ if there exists $A\in\Psi_{b}^{m}(M)$ such that $q\in
ell_{b}(A)$ and $Au\in\mathcal{H}^{k}_{loc}(M)$, where $ell_{b}$ stands for
the elliptic set as per Definition 3.3. When $m=+\infty$, we say that
$q\not\in WF_{b}^{k,\infty}(M)$ if there exists $A\in\Psi^{0}_{b}(M)$ such
that $q\in ell_{b}(A)$ and $Au\in\mathcal{H}^{k,\infty}_{loc}(M)$.

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 $\nu>0$, cf. Equation (7). Then there
exists a continuous map
$\widetilde{\gamma}_{-}:\mathcal{H}^{1}_{0}(M)\to\mathcal{H}^{\nu}(\partial
M)$, which can be extended to a continuous map

$\gamma_{-}:\mathcal{H}^{1,m}_{loc}(M)\to\mathcal{H}^{\nu+m}_{loc}(\partial
M),\quad\forall m\leq 0.$

######  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
$\mathbb{R}^{n}_{+}\doteq[0,\infty)\times\mathbb{R}^{n-1}$. In this setting,
any $u\in\mathcal{H}^{1}(\mathbb{R}^{n}_{+})$ can be restricted to the subset
$[0,\epsilon)\times\mathbb{R}^{n-1}$, $\epsilon>0$ admitting an asymptotic
expansion $u=x^{\nu_{-}}u_{-}+x^{r+1}u_{0}$ where $2r=n-2$, while
$u_{-}\in\mathcal{H}^{\nu}(\mathbb{R}^{n})$ and
$u_{0}\in\mathcal{H}^{1}([0,\epsilon);L^{2}(\mathbb{R}^{n-1}))$. In this
context it holds that $\gamma_{-}(u)=u_{-}$.

At last we recall from [GW18] a notable property of the trace $\gamma_{-}$
related to its boundedness. Let $u\in\mathcal{H}(M)$, then for every
$\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that

$\|\gamma_{-}u\|^{2}_{L^{2}(\partial
M)}\leq\varepsilon\|u\|^{2}_{\mathcal{H}^{1}(M)}+C_{\varepsilon}\|u\|^{2}_{\mathcal{L}^{2}(M)}.$
(12)

###  3.3 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 $P=\Box_{g}-m^{2}$ 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
$m\in\mathbb{R}\cup\\{\pm\infty\\}$, we define the Frechét spaces

$\mathcal{X}^{m}(M)=\\{u\in\mathcal{H}^{1,m}_{loc}(M)\;|\;Pu\in
x^{2}\mathcal{H}^{0,m}_{loc}(M)\\},$ (13)

with respect to the seminorms

$\norm{u}_{\mathcal{X}^{m}(M)}=\norm{\phi
u}_{\mathcal{H}^{1,m}(M)}+\norm{x^{-2}\phi Pu}_{\mathcal{H}^{0,m}(M)},$ (14)

where $\phi\in C^{\infty}_{0}(M)$.

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, the collection of $F\in x^{\nu_{-}}C^{\infty}(M)$ such that

  1. 1.

$x^{-\nu_{-}}F>0$ on $M$,

  2. 2.

$S_{F}\doteq F^{-1}P(F)\in x^{2}L^{\infty}(M)$ where $P$ is the Klein-Gordon
operator.

For any such function, we can define a twisted differential

$d_{F}\doteq F\circ d\circ
F^{-1}:\dot{C}^{\infty}(M)\to\dot{C}^{\infty}(M;T^{*}M),\quad v\mapsto
d_{F}(v)=dv+vF^{-1}(dF).$ (15)

Accordingly we can introduce the twisted Dirichlet (energy) form

$\mathcal{E}_{0}(u,v)\doteq-\int\limits_{M}g(d_{F}u,d_{F}\overline{v})d\mu_{g}.\quad\forall
u,v,\in\mathcal{L}^{2}_{loc}(M)$ (16)

Starting from these data, we are ready to introduce a second trace map. More
precisely we start from

$\widetilde{\gamma}_{+}:\mathcal{X}^{\infty}(M)\to\mathcal{H}^{1,\infty}_{loc}(M)\quad
u\mapsto\widetilde{\gamma}_{+}(u)=x^{1-2\nu}\partial_{x}(F^{-1}u)|_{\partial
M}.$

Calling $d^{\dagger}_{F}$ the formal adjoint of $d_{F}$ as in Equation (15)
with respect to the inner product on $L^{2}(M;d\mu_{g})$ we observe that, on
account of the identity $P=-d^{\dagger}_{F}d_{F}+F^{-1}P(F)$, the following
Green’s formula holds true for all $u\in\mathcal{X}^{\infty}(M)$ and for all
$v\in\mathcal{H}^{1}_{0}(M)$:

$\int Pu\cdot\overline{v}\ d\mu_{g}=\mathcal{E}_{0}(u,v)+\int
S_{F}u\cdot\bar{v}\ d\mu_{g}+\int\gamma_{+}u\cdot\gamma_{-}\bar{v}\ d\mu_{h}.$
(17)

With these premises the following result holds true, cf. [GW18, Lemma 4.8]:

######  Lemma 3.2.

The map $\widetilde{\gamma}_{+}$ can be extended to a bounded linear map

$\gamma_{+}:\mathcal{X}^{k}(M)\to\mathcal{H}^{k-\nu}_{loc}(\partial
M),\quad\forall k\in\mathbb{R}$

and, if $u\in\mathcal{X}^{k}(M)$, the Green’s formula (17) holds true for
every $v\in\mathcal{H}^{1,-k}_{0}(M)$.

######  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
$\mathbb{R}^{n}_{+}\doteq[0,\infty)\times\mathbb{R}^{n-1}$ endowed with a
metric whose line element reads in standard Cartesian coordinates

$ds^{2}=\frac{-dx^{2}+h_{ab}dy^{a}dy^{b}}{x^{2}},$

where $h$ is a smooth Lorentzian metric on $\mathbb{R}^{n-1}$. Consider an
admissible twisting function $F$ such that $\lim\limits_{x\to
0^{+}}x^{-\nu_{-}}F=1$ and $u\in\mathcal{H}^{1,k}_{0}(\mathbb{R}^{n}_{+})$
such that $Pu\in x^{2}\mathcal{H}^{0,k}_{0}(\mathbb{R}^{n}_{+})$ for $k\geq
0$. Then, for every $\epsilon>0$, the restriction of $u$ to
$[0,\epsilon)\times\mathbb{R}^{n}$ admits an asymptotic expansion of the form
$u=Fu_{-}+x^{\nu_{+}}u_{+}+x^{r+2}\mathcal{H}_{b}^{k+2}([0,\epsilon);\mathcal{H}^{k-3}(\mathbb{R}^{n-1}))$
where $2r=n-2$ while $u_{-}\in\mathcal{H}^{\nu+k}(\mathbb{R}^{n-1})$ and
$u_{+}\in\mathcal{H}^{k-1-2\nu}(\mathbb{R}^{n-1})$. In this context it holds
that $\gamma_{+}(u)=2\nu u_{+}$.

###  3.4 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 $u\in\mathcal{H}^{1}_{loc}(M)$ such that

$\left\\{\begin{array}[]{l}Pu=(\Box_{g}-m^{2})u=f\\\
\gamma_{+}u=\Theta\gamma_{-}u\end{array}\right.,$ (18)

where $\Theta\in\Psi^{k}_{b}(\partial M)$ while $\gamma_{-},\gamma_{+}$ 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 $\Theta\in\Psi^{k}_{b}(\partial M)$ , we assume that there exists an
admissible twisting function $F$ and we define the energy functional

$\mathcal{E}_{\Theta}(u,v)=\mathcal{E}_{0}(u,v)+\int\limits_{M}S_{F}u\cdot\overline{v}\,d\mu_{g}+\int\limits_{\partial
M}\Theta\gamma_{-}u\cdot\gamma_{-}\overline{v},$ (19)

where $S_{F}=F^{-1}P(F)$, $\mathcal{E}_{0}$ is the twisted Dirichlet form, cf.
Equation (16), $u\in\mathcal{H}^{1}_{loc}(M)$, while
$v\in\mathcal{H}^{1}_{0}(M)$. Hence, we can introduce
$P_{\Theta}:\mathcal{H}^{1}_{loc}(M)\rightarrow\dot{\mathcal{H}}^{-1}_{loc}(M)$
by

$\langle P_{\Theta}u,v\rangle=\mathcal{E}_{\Theta}(u,v).$ (20)

Observe that, on account of the regularity of $\gamma_{-}u$, we can extend
$P_{\Theta}$ as an operator
$P_{\Theta}:\mathcal{H}^{1,m}_{loc}(M)\rightarrow\dot{\mathcal{H}}^{-1,m}_{loc}(M)$,
$m\in\mathbb{R}$ [GW18].

######  Remark 3.4.

The reader might be surprised by the absence of $\gamma_{+}$ 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
$\Theta\in\Psi^{k}_{b}(\partial M)$ with $k\leq 2$. As a preliminary step, we
introduce two notable geometric structures. More precisely, since the
principal symbol of $x^{-2}P$ reads $\widehat{p}\doteq\widehat{g}(X,X)$, where
$X\in\Gamma(T^{*}M)$, the associated characteristic set is

$\mathcal{N}=\left\\{(q,k_{q})\in
T^{*}M\setminus\\{0\\}\;|\;\widehat{g}^{ij}(k_{q})_{i}(k_{q})_{j}=0\right\\},$
(21)

while the compressed characteristic set is

$\dot{\mathcal{N}}=\pi[\mathcal{N}]\subset{}^{b}\dot{T}(M),$ (22)

where $\pi$ 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 $I\subset\mathbb{R}$ be an interval. A continuous map
$\gamma:I\rightarrow\dot{\mathcal{N}}$ is a generalized broken
bicharacteristic (GBB) if for every $s_{0}\in I$ the following conditions hold
true:

  * a)

If $q_{0}=\gamma(s_{0})\in\mathcal{G}$, then for every
$\omega\in\Gamma^{\infty}(^{b}T^{*}M)$,

$\frac{d}{ds}(\omega\circ\gamma)=\\{\widehat{p},\pi^{*}\omega\\}(\eta_{0}),$
(23)

where $\eta_{0}\in\mathcal{N}$ is the unique point for which
$\pi(\eta_{0})=q_{0}$, while $\pi:T^{*}M\to{}^{b}T^{*}M$ and $\\{,\\}$ are the
Poisson brackets on $T^{*}M$.

  * b)

If $q_{0}=\gamma(s_{0})\in\mathcal{H}$, then there exists $\varepsilon>0$ such
that $0<|s-s_{0}|<\varepsilon$ implies $x(\gamma(s))\neq 0$, where $x$ is the
global boundary function, cf. Definition 2.2.

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]:

######  Theorem 3.2.

Let $\Theta\in\Psi_{b}^{k}(\partial M)$ with $0<k\leq 2$. If
$u\in\mathcal{H}_{loc}^{1,m}(M)$ for $m\leq 0$ and
$s\in\mathbb{R}\cup\\{+\infty\\}$, then
$WF_{b}^{1,s}(u)\setminus\left(WF_{b}^{-1,s+1}(P_{\Theta}u)\cup
WF_{b}^{-1,s+1}(\Theta u)\right)$ is the union of maximally extended
generalized broken bicharacteristics within the compressed characteristic set
$\dot{\mathcal{N}}$.

In full analogy it holds also

######  Theorem 3.3.

Let $\Theta\in\Psi_{b}^{k}(M)$ with $k\leq 0$. If
$u\in\mathcal{H}_{loc}^{1,m}(M)$ for $m\leq 0$ and
$s\in\mathbb{R}\cup\\{+\infty\\}$, then it holds that
$WF_{b}^{1,s}(u)\setminus WF_{b}^{-1,s+1}(P_{\Theta}u)$ is the union of
maximally extended GBBs within the compressed characteristic set
$\dot{\mathcal{N}}$.

##  4 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-$\Psi$DOs $\Theta\in\Psi_{b}^{k}(\partial M)$ 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.

We recall that the case of Dirichlet boundary condition was already analysed
in [Vas12], while the generalization to Robin boundary conditions was studied
in [War13] and [GW18], that we follow closely. We introduce a cutoff function
playing an important rôle in the following theorems. Consider

$\chi_{0}(s)=\begin{cases}exp(s^{-1})\ if\ s>0\\\ 0\ \ \ \ \ \ \ \ \ \ \ if\
s\leq 0\end{cases},$

and let $\chi_{1}\in C^{\infty}(\mathbb{R})$ be such that $\chi_{1}(s)=0$ for
all $s\in(-\infty,0]$ while $\chi_{1}(s)=1$ if $s\in[1,+\infty)$. For any but
fixed $\tau_{0},\tau_{1}\in\mathbb{R}$ with $\tau_{0}<\tau_{1}$, we call
$\chi:(\tau_{0},\tau_{1})\rightarrow\mathbb{R}$ the smooth function

$\chi(s)\doteq\chi_{0}(-\delta^{-1}(s-\tau_{1}))\chi_{1}((s-\tau_{0})/\varepsilon),$
(24)

where $\delta\gg 1$ while $\varepsilon\in(0,\tau_{1}-\tau_{0})$. Under these
hypotheses, calling $\chi^{\prime}_{0}=\frac{d\chi_{0}}{ds}$, it holds that,
cf. [Vas12]

$\chi\leq-\delta^{-1}(\tau_{1}-\tau_{0})^{2}\chi^{\prime}\;\;\textrm{with}\;\;\chi^{\prime}=-\delta^{-1}\chi_{0}^{\prime}(-\delta^{-1}(s-\tau_{1})).$
(25)

Consider $u_{loc}\in\mathcal{H}^{1,1}(M)$ such that its support lies in
$[\tau_{0}+\varepsilon,\tau_{1}]\times\Sigma$, cf. Definition 2.1. As
discussed in [GW18], one can use the cutoff function introduced to prove a
twisted version of the Poincaré inequality proved in [Vas12, Proposition 2.5]:

$\|(-\chi^{\prime})^{1/2}u\|^{2}_{\mathcal{L}^{2}(M)}\leq
C\|(-\chi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)},$ (26)

where $d_{F}$ is the twisted differential as per Equation (15).

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 $\partial M$ with its natural extension on $M$, i.e. constant in
$x$, the global boundary function. As starting point we need a preliminary
definition:

######  Definition 4.1.

Let $\Theta\in\Psi^{k}_{b}(M)$. We call it local in time if, for every $u$ in
the domain of $\Theta$, $\tau(\textrm{supp}(\Theta
u))\subseteq\tau(\textrm{supp}(u))$ where
$\tau:\mathbb{R}\times\Sigma\to\mathbb{R}$ 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

######  Hypothesis 4.1.

We consider $\Theta\in\Psi^{k}_{b}(M)$ with $k\leq 2$, only if it is local in
time, see Definition 4.1, and if $\Theta=\Theta^{*}$.

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 $u\in\mathcal{H}^{1,1}_{loc}(M)$ and let $\Theta\in\Psi^{k}_{b}(\partial
M)$ be such that its canonical extension to $M$ abides to the Hypothesis 4.1.
Then there exists a compact subset $K\subset M$ and a real positive constant
$C$ such that

$\|(-\phi^{\prime})^{1/2}u\|_{\mathcal{H}^{1}(K)}\leq
C\|P_{\Theta}u\|_{\mathcal{H}^{-1,1}(K)},$

where $\phi=\tau\chi$, $\chi$ being the same as in Equation (26), while
$P_{\Theta}$ 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
$supp(u)\subset[\tau_{0}+\varepsilon,\tau_{1}]\times\Sigma$. We start by
computing a twisted version of the energy form considered in [Vas12]. Consider
$\langle-i[(V^{\prime})^{*}P_{\Theta}-P_{\Theta}V^{\prime}]u,u\rangle$, with
$V^{\prime}=FVF^{-1}\in\textit{Diff}_{b}^{\ 1}(M)$ and
$V\in\mathcal{V}_{b}(M)$ with compact support. Note that, since $\Theta$ is
self-adjoint, i.e., $\Theta=\Theta^{*}$, then
$i[(V^{\prime})^{*}P_{\Theta}-P_{\Theta}V^{\prime}]$ is a second order
formally self-adjoint operator, the purpose of $V^{\prime*}$ being to remove
zeroth order terms. Let $V=-\phi W$ with
$W=\bigtriangledown_{\widehat{g}}\tau$. It belongs to $\mathcal{V}_{b}(X)$
because $\widehat{g}(dx,dt)=0$. A direct computation shows that

$\begin{split}\langle-i[(V^{\prime})^{*}P_{\Theta}-P_{\Theta}V^{\prime}]u,u\rangle=2Re\langle
P_{\Theta}u,V^{\prime}u\rangle=\\\
=2Re\mathcal{E}_{0}(u,V^{\prime}u)+2Re\langle
S_{F}u,V^{\prime}u\rangle+2Re\langle\Theta\gamma_{-}u,\gamma_{-}V^{\prime}u\rangle,\end{split}$
(27)

where $\mathcal{E}_{0}$ is the twisted Dirichlet energy form, cf. Equation
(16), $S_{F}$ is defined in Section 3.3, while $\gamma_{+}$ and $\gamma_{-}$
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

$2Re\mathcal{E}_{0}(u,V^{\prime}u)=\langle B^{ij}Q_{i}u,Q_{j}u\rangle,$

where $Q_{i}$, $i=1,\dots,n$ is a generating set of
$\textbf{Diff}^{1}_{\nu}(M)$, while the symmetric tensor $B$ is

$\begin{split}B=-(\phi\cdot div_{\widehat{g}}W+2F\phi V(F^{-1})+(n-2)\phi
x^{-1}W(x))\widehat{g}^{-1}+\\\
+\phi\mathcal{L}_{W}\widehat{g}^{-1}+2T(W,\bigtriangledown_{\widehat{g}}\phi).\end{split}$
(28)

Here $T(W,\bigtriangledown_{\widehat{g}}\phi)$ is the stress-energy tensor,
with respect to $\widehat{g}$, see Definition 2.2, of a scalar field
associated with $W$ and $\bigtriangledown_{\widehat{g}}\phi$, that is,
denoting with $\odot$ the symmetric tensor product,

$T(W,\bigtriangledown_{\widehat{g}}\phi)=(\bigtriangledown_{\widehat{g}}\phi)\odot
W-\frac{1}{2}\widehat{g}(\bigtriangledown_{\widehat{g}}\phi,W)\cdot\widehat{g}^{-1}.$
(29)

Focusing on this term and using that
$\bigtriangledown_{\widehat{g}}\phi=\chi^{\prime}\bigtriangledown_{\widehat{g}}\tau$,
a direct computation yields:

$T_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)=\frac{1}{2}(\chi^{\prime}\circ\tau)\big{[}2(\bigtriangledown_{\widehat{g}}\tau)\otimes(\bigtriangledown_{\widehat{g}}\tau)-\widehat{g}(\bigtriangledown_{\widehat{g}}\tau,\bigtriangledown_{\widehat{g}}\tau)\cdot\widehat{g}^{-1}\big{]}.$
(30)

Since $\bigtriangledown_{\widehat{g}}\phi$ and
$\bigtriangledown_{\widehat{g}}\tau$ are respectively past- and future-
pointing timelike vectors, then
$T_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)$ is negative definite.
Hence we can rewrite Equation (27) as

$\begin{split}\langle-T^{ij}_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)Q_{i}u,Q_{j}u\rangle=\langle-i[(V^{\prime})^{*}P_{\Theta}-P_{\Theta}V^{\prime}]u,u\rangle+2Re\mathcal{E}_{0}(K^{ij}Q_{i}u,Q_{j}u)+\\\
+2Re\langle
S_{F}u,V^{\prime}u\rangle+2Re\langle\Theta\gamma_{-}u,\gamma_{-}V^{\prime}u\rangle,\end{split}$
(31)

with

$K=-(F\phi V(F^{-1})+(n-2)\phi
x^{-1}W(x))\widehat{g}^{-1}+\phi\mathcal{L}_{W}\widehat{g}^{-1}.$

Since $-T_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)^{ij}$ is
positive definite, then $\mathcal{Q}(u,u)\doteq\langle-
T_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)^{ij}Q_{i}u,Q_{j}u\rangle\geq
0$. This can be seen by direct inspection from the explicit form

$\begin{split}\mathcal{Q}(u,u)=\int_{M}\phi^{\prime}\left((\bigtriangledown_{\widehat{g}}\tau)^{i}(\bigtriangledown_{\widehat{g}}\tau)^{j}-\dfrac{1}{2}\widehat{g}((\bigtriangledown_{\widehat{g}}\tau)^{i}(\bigtriangledown_{\widehat{g}}\tau)^{j})\right)Q_{i}u\
\overline{Q_{j}u}\ x^{2}d\mu_{g}\\\
=\int_{M}H((-\phi^{\prime})^{\/2}d_{F}u,(-\phi^{\prime})^{1/2}d_{F}\overline{u})x^{2}d\mu_{g},\end{split}$
(32)

where $H$ is the sesquilinear pairing between $1$-forms induced by the metric.
Focusing then on the term $\langle K^{ij}Q_{i}u,Q_{j}u\rangle$, we observe
that, as a consequence of our choice for the functions $f$ and $W$, we have
$V(x)=\widehat{g}(\bigtriangledown_{\widehat{g}}\tau,\bigtriangledown_{\widehat{g}}x)=0$
on $\partial M$. In addition it holds that $x^{-1}W(x)=\mathcal{O}(1)$ near
$\partial M$, and
$\mathcal{L}_{V}\widehat{g}^{-1}=2\bigtriangledown_{\widehat{g}}(\bigtriangledown_{\widehat{g}}\tau)=2\widehat{\Gamma}^{i}_{\tau\tau}\partial_{i}$.
These observations allow us to establish the following bound, cf. [Vas12] and
[GW18]:

$|\langle K^{ij}Q_{i}u,Q_{j}u\rangle|\leq
C\|\phi^{1/2}d_{F}u\|_{\mathcal{L}^{2}(M)}\leq
C\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)},$
(33)

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:

$\displaystyle|\langle-i[(V^{\prime})^{*}P_{\Theta}-P_{\Theta}V^{\prime}]u,u\rangle|\leq$
$\displaystyle
C\left(\|\phi^{1/2}FWF^{-1}P_{\Theta}u\|_{\dot{\mathcal{H}}^{-1}(M)}^{2}+\|\phi^{1/2}u\|_{\mathcal{H}^{1}(M)}^{2}\right)+C\left(\|\phi^{1/2}P_{\Theta}u\|_{\mathcal{L}^{2}(M)}^{2}+\|\phi^{1/2}FWF^{-1}u\|_{\mathcal{L}^{2}(M)}^{2}\right)\leq$
$\displaystyle\leq
C\big{(}\|FWF^{-1}P_{\Theta}u\|_{\dot{\mathcal{H}}^{-1}(M)}^{2}+\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}u\|_{\mathcal{H}^{1}(M)}^{2}+$
$\displaystyle+\|P_{\Theta}u\|_{\mathcal{L}^{2}(M)}^{2}+\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}FWF^{-1}u\|_{\mathcal{L}^{2}(M)}^{2}\Big{)},$
(34)

where in the last inequality we used Equation (25). As for the second term in
Equation (31), using that $S_{F}\in x^{2}L^{\infty}(M)$, we establish the
bound

$2|Re\langle S_{F}u,V^{\prime}u\rangle|\leq\widetilde{C}\left(\|\phi^{1/2}\
u\|^{2}_{\mathcal{L}^{2}(M)}+\|\phi^{1/2}\
d_{F}u\|^{2}_{\mathcal{L}^{2}(M)}\right),$

for a suitable constant $\widetilde{C}>0$. Using Equation (25) and the
Poincaré inequality, this last bound becomes

$2|Re\langle S_{F}u,V^{\prime}u\rangle|\leq
C\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)}.$
(35)

At last we give a give a bound for the last term in Equation (27), containing
the pseudodifferential operator $\Theta$ which implements the boundary
conditions. Recalling Hypothesis 4.1, it is convenient to consider the
following three cases separately

  * a)

$\Theta\in\Psi^{k}_{b}(\partial M)$ with $k\leq 1$,

  * b)

$\Theta\in\Psi^{k}_{b}(\partial M)$ with $1<k\leq 2$.

Now we give a bound case by case.

  * a)

It suffices to focus on $\Theta\in\Psi^{1}_{b}(\partial M)$ recalling that,
for $k<1$, $\Psi^{k}_{b}(\partial M)\subset\Psi^{1}_{b}(\partial M)$. If with
a slight abuse of notation we denote with $\Theta$ both the operator on the
boundary and its trivial extension to the whole manifold, we can write

$\langle\Theta\gamma_{-}u,\gamma_{-}V^{\prime}u\rangle=\langle\widehat{N}(\Theta)(-i\nu_{-})\gamma_{-}u,\gamma_{-}V^{\prime}u\rangle=\langle\gamma_{-}\Theta
u,\gamma_{-}V^{\prime}u\rangle,$

where $\widehat{N}(\Theta)(-i\nu_{-})$ is the indicial family as in Equation
(6). We recall that any $A\in\Psi^{s}_{b}(\partial M)$, $s\in\mathbb{N}$, can
be decomposed as $\sum\limits_{i=1}^{n}Q_{i}A_{i}+B$, with
$A_{i},B\in\Psi^{s-1}_{b}(\partial M)$, while $Q_{i}$, $i=1,\dots,n$ is a
generating set of $\mathbf{Diff}^{1}_{\nu}(M)$. Hence we can rewrite $\Theta$
as

$\Theta=\sum_{i}Q_{i}\Theta_{i}+\Theta^{\prime}=\sum_{i}\left(\Theta_{i}Q_{i}+[Q_{i},\Theta_{i}]\right)+\Theta^{\prime},$

where $\Theta_{i},\Theta^{\prime}$ and $[Q_{i},\Theta_{i}]$ are in
$\Psi^{0}(\partial M)$. Therefore

$|\langle\gamma_{-}\Theta
u,\gamma_{-}V^{\prime}u\rangle|\leq|\langle\gamma_{-}\left(\sum_{i}\Theta_{i}Q_{i}u\right),\gamma_{-}V^{\prime}u\rangle|+|\langle\gamma_{-}\left(\left([Q_{i},\Theta_{i}]+\Theta^{\prime}\right)u\right),\gamma_{-}V^{\prime}u\rangle|.$

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,

$\begin{split}|\langle\gamma_{-}\left(\sum_{i}\Theta_{i}Q_{i}u\right),\gamma_{-}V^{\prime}u\rangle|\leq\varepsilon\left(\sum_{i}\|\phi^{1/2}\Theta_{i}Q_{i}u\|_{\mathcal{H}^{1}(M)}^{2}+\|\phi^{1/2}FWF^{-1}u\|_{\mathcal{H}^{1}(M)}^{2}\right)+\\\
+C_{\varepsilon}\left(\sum_{i}\ \|\phi^{1/2}\
Q_{i}u\|_{\mathcal{L}^{2}(M)}^{2}+\|\phi^{1/2}FWF^{-1}u\|_{\mathcal{L}^{2}(M)}^{2}\right)\leq
C_{\varepsilon}\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)},\end{split}$

for a suitable constant $C_{\varepsilon}>0$. As for the second term, since
$u\in\mathcal{H}^{1,1}(M)$ we can proceed as above using that the operator
$\Theta^{\prime}+[Q_{i},\Theta_{i}]$ is of order $0$ and we can conclude that

$\left|\langle\gamma_{-}\left(\left([Q_{i},\Theta_{i}]+\Theta^{\prime}\right)u\right),\gamma_{-}V^{\prime}u\rangle\right|\leq\widetilde{C}_{\varepsilon}\|\phi^{1/2}\
u\|_{\mathcal{H}^{1}(M)}^{2}\leq
C_{\varepsilon}\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|_{\mathcal{L}^{2}(M)}^{2},$

for suitable positive constants $C_{\varepsilon}$ and
$\widetilde{C}_{\varepsilon}$. Therefore, it holds a bound of the form

$|Re\langle\Theta\gamma_{-}u,\gamma_{-}V^{\prime}u\rangle|\leq
C^{\prime}_{\epsilon}\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|_{\mathcal{L}^{2}(M)}^{2}.$

  * b)

Since $\Psi^{k}_{b}(\partial M)\subset\Psi^{k^{\prime}}_{b}(\partial M)$ if
$k<k^{\prime}$, it is enough to consider $\Theta\in\Psi^{2}_{b}(\partial M)$
and to observe that, we can decompose $\Theta$ as

$\Theta=\sum\limits_{i=1}^{n}Q_{i}\left(\sum\limits_{j=1}^{n}Q_{j}A_{ij}\right)+B_{i},$

where $B_{i}\in\Psi^{1}_{b}(\partial M)$ while $A_{ij}\in\Psi^{0}_{b}(\partial
M)$. 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

$\langle-
T_{\widehat{g}}^{ij}(W,\bigtriangledown_{\widehat{g}}\phi)Q_{i}u,Q_{j}u\rangle\leq
C\Big{(}\|P_{\Theta}u\|_{\dot{\mathcal{H}}^{-1,1}(M)(}^{2}+C\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|_{\mathcal{L}^{2}(M)}^{2}.$
(36)

Since the inner product $H$ defined by the left hand side of Equation (32) is
positive definite, then for $\delta$ large enough

$\langle-T^{ij}_{\widehat{g}}(W,\bigtriangledown_{\widehat{g}}\phi)Q_{i}u,Q_{j}u\rangle-C\delta^{-1}(\tau_{1}-\tau_{0})^{2}\|(-\phi^{\prime})^{1/2}d_{F}u\|_{\mathcal{L}^{2}(M)}^{2}\geq
0,$

and the associated Dirichlet form $\widetilde{\mathcal{Q}}$ defined as

$\widetilde{\mathcal{Q}}(u,u)=\int_{M}\left[H((-\phi^{\prime})^{\/2}d_{F}u,(-\phi^{\prime})^{1/2}d_{F}\overline{u})-C\delta^{-1}(\tau_{1}-\tau_{0})^{2}|(-\phi^{\prime})^{1/2}d_{F}u|^{2}\right]x^{2}d\mu_{g},$
(37)

bounds $\|(-\phi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)}$. We
conclude the proof by observing that, once we have an estimate for
$\|(-\phi^{\prime})^{1/2}d_{F}u\|^{2}_{\mathcal{L}^{2}(M)}$, with the Poincaré
inequality we can also bound
$\|(-\phi^{\prime})^{1/2}u\|_{\mathcal{L}^{2}(M)}$. Therefore, considering the
support of $\chi$ and $u$, there exists a compact subset $K\subset M$ such
that

$\|(-\phi^{\prime})^{1/2}u\|_{\mathcal{L}^{2}(M)}\leq
C\|(-\phi^{\prime})^{1/2}P_{\Theta}u\|_{\dot{\mathcal{H}}^{-1,1}(K)},$ (38)

from which the sought thesis descends.

∎

######  Remark 4.1.

The case with $\Theta\in\Psi^{k}(M)$ of order $k\leq 0$, 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
$P_{\Theta}$ individuated in Equation (20). Mutatis mutandis, since we assume
that $\Theta$ 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 $f\in\dot{\mathcal{H}}^{-1,1}(M)$ be vanishing whenever
$\tau<\tau_{0}$, $\tau_{0}\in\mathbb{R}$. Suppose in addition that $\Theta$
abides to the Hypothesis 4.1. Then there exists at most one
$u\in\mathcal{H}^{1}_{0}(M)$ such that $supp(u)\subset\\{q\in M\ |\
\tau(q)\geq\tau_{0}\\}$ and it is a solution of $P_{\Theta}u=f$

At the same time the following statement holds true.

######  Lemma 4.2.

Let $M$ be a globally hyperbolic, asymptotically anti-de Sitter spacetime, cf.
Definition 2.2 and let $f\in\dot{\mathcal{H}}^{-1,1}(M)$ be vanishing whenever
$\tau<\tau_{0}$, $\tau_{0}\in\mathbb{R}$. Then there exists
$u\in\mathcal{H}^{1,-1}(M)$ of the problem $P_{\Theta}u=f$, cf. Equation (20),
such that $\tau(\textrm{supp}(u))\geq\tau_{0}$.

The proof follows the one given in [Vas12, Prop. 4.15], but we feel worth
sketching the main ideas. The first step consists of proving a local version
of the lemma, namely that given a compact set $I\subset\mathbb{R}$, there
exists $\sigma>0$ such that for every $\tau_{0}\in I$ there exists
$u\in\mathcal{H}^{1,-1}(M)$ such that $supp(u)=\\{p\in M\ |\ \tau(p)\geq 0\\}$
and $P_{\Theta}u=f$ for $\tau<\tau_{0}+\sigma$. The main point of this part of
the proof consists of applying Lemma 4.1 to ensure that the adjoint of the
Klein-Gordon operator, say $P^{*}_{\Theta}$, is invertible over the set of
smooth functions supported in suitable compact subsets of $M$ – see [Vas12,
Lem. 4.14] for further details. With this result in hand, one divides the time
direction into sufficiently small intervals $[\tau_{j},\tau_{j+1}]$ and uses a
partition of unity along the time coordinate to build a global solution for
$P_{\Theta}u=f$.

At last we extend our results for $u\in\mathcal{H}^{1,m}_{loc}(M)$ and for
$f\in\dot{\mathcal{H}}^{-1,m+1}_{loc}(M)$. Let us consider
$\Theta\in\Psi^{k}_{b}(\partial M)$ with $k\leq 0$, the proof for the positive
cases being the same. If $m\geq 0$, Lemma 4.1 entails that Equation (18)
admits a unique solution lying in $\mathcal{H}^{1}_{loc}(M)$. By the
propagation of singularities theorem, cf. Theorem 3.3 and using Hypothesis
4.1, the solution lies in $\mathcal{H}^{1,m}_{loc}(M)$ and the following
generalization of the bound in Lemma 4.1 holds true:

$\|u\|_{\mathcal{H}^{1,m}(M)}\leq C\|f\|_{\dot{\mathcal{H}}^{-1,m+1}(M)}.$

If $m<0$ we can draw the same conclusion considering, as in [Vas12, Thm.
8.12],

$P_{\Theta}u_{j}=f_{j}\\\ $ (39)

where $f_{j}\in\dot{\mathcal{H}}^{-1,m+1}(M)$ is sequence converging to $f$ as
$j\to\infty$. Each of these equations has a unique solution
$u_{j}\in\mathcal{H}^{1}(M)$. In addition the propagation of singularities
theorem, cf. Theorem (3.3) yields the bound

$\|u_{k}-u_{j}\|_{\mathcal{H}^{1,m}(K)}\leq
C\|f_{k}-f_{j}\|_{\dot{\mathcal{H}}^{-1,m+1}(L)}$

for suitable compact sets $K,L\subset M$ and for every $j,k\in\mathbb{N}$.
Since $f_{j}\rightarrow f$ in $\dot{\mathcal{H}}^{-1,m+1}(L)$, we can conclude
that the sequence $u_{j}$ is converging to $u\in\mathcal{H}^{1,m}(K)$.
Considering $f_{j}$ such that each $f_{j}$ vanishes if $\\{\tau<\tau_{0}\\}$,
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 $m,\tau_{0}\in\mathbb{R}$ while
$f\in\dot{\mathcal{H}}^{-1,m+1}_{loc}(M)$. Assume in addition that $\Theta$
abides to Hypothesis 4.1. If $f$ vanishes for $\tau<\tau_{0}$,
$\tau_{0}\in\mathbb{R}$ being arbitrary but fixed, then there exists a unique
$u\in\mathcal{H}^{1,m}_{loc}(M)$ such that

$P_{\Theta}u=f,$ (40)

where $P_{\Theta}$ 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
$P_{\Theta}$, cf. Equation (20). To this end let us define the following
notable subspaces of $\mathcal{H}^{k,m}(M)$, $k=0,\pm 1$,
$m\in\mathbb{N}\cup\\{0\\}$:

$\mathcal{H}^{k,m}_{-}(M)=\\{u\in\mathcal{H}^{k,m}(M)\;|\;\exists\tau_{-}\in\mathbb{R}\;\textrm{such
that}\;p\notin\textrm{supp}(u),\;\textrm{if}\,\tau(p)<\tau_{-}\\},$ (41a)
$\mathcal{H}^{k,m}_{+}(M)=\\{u\in\mathcal{H}^{k,m}(M)\;|\;\exists\tau_{+}\in\mathbb{R}\;\textrm{such
that}\;p\notin\textrm{supp}(u)\;\textrm{if}\,\tau(p)>\tau_{+}\\},$ (41b)
$\mathcal{H}^{k,m}_{tc}(M)\doteq\mathcal{H}^{k,m}_{+}(M)\cap\mathcal{H}^{k,m}_{-}(M),$
(41c)

where the subscript $tc$ stands for timelike compact. In addition we call

$\mathcal{H}^{1,m}_{\pm,\Theta}(M)\doteq\\{u\in
H^{1,m}_{\pm}(M)\;|\;\gamma_{+}(u)=\Theta\gamma_{-}(u)\\},$ (42)

where $\gamma_{-},\gamma_{+}$ are the trace maps introduced in Theorem 3.1 and
in Lemma 3.2, while $\Theta$ 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 $G_{\Theta}^{\pm}$
associated to the Klein-Gordon operator $P_{\Theta}$, cf. Equation (20).

######  Theorem 4.1.

Let $P_{\Theta}$ be the Klein-Gordon operator as per Equation (20) where
$\Theta$ abides to Hypothesis 4.1. Then there exist unique retarded $(+)$ and
advanced $(-)$ propagators, that is continuous operators
$G_{\Theta}^{\pm}:\dot{\mathcal{H}}^{-1,m+1}_{\pm}(M)\rightarrow\mathcal{H}^{1,m}_{\pm}(M)$
such that $P_{\Theta}G_{\Theta}^{\pm}=\mathbb{I}$ on
$\dot{\mathcal{H}}^{-1,m+1}_{\pm}(M)$ and
$G_{\Theta}^{\pm}P_{\Theta}=\mathbb{I}$ on
$\mathcal{H}^{1,m}_{\pm,\Theta}(M)$. Furthermore, $G_{\Theta}^{\pm}$ is a
continuous map from $\dot{\mathcal{H}}_{0}^{-1,\infty}(M)$ to
$\mathcal{H}_{loc}^{1,\infty}(M)$ where the subscript $0$ indicates that we
consider only functions of compact support.

Observe that the restriction to $\mathcal{H}^{1,m}_{\pm,\Theta}(M)$ is
necessary since, per construction an element in the range of
$G^{\pm}_{\Theta}P_{\Theta}$ abides to the boundary conditions as in Equation
(18).

######  Remark 4.2.

Associated to the advanced and to retarded propagators, one can define the
causal propagator
$G_{\Theta}:\dot{\mathcal{H}}_{0}^{-1,m+1}(M)\rightarrow\mathcal{H}^{1,m}_{loc}(M)$
as $G_{\Theta}=G_{\Theta}^{+}-G_{\Theta}^{-}$.

Since $G^{\pm}_{\Theta}$ are continuous maps, cf. Theorem 4.1, one can apply
Schwartz kernel theorem to infer that one can associate to them a bi-
distribution $\mathcal{G}^{\pm}_{\Theta}\in\mathcal{D}^{\prime}(M\times M)$.
To conclude the section we highlight a standard and important application of
the fundamental solutions and in particular of the causal propagator cf.
Remark 4.2.

######  Proposition 4.2.

Let $P_{\Theta}$ be the Klein-Gordon operator as per Equation (20) and let
$G_{\Theta}$ be its associated causal propagator, cf. Remark 4.2. Then the
following is an exact sequence:

$\displaystyle
0\to\mathcal{H}^{1,\infty}_{tc,\Theta}(M)\stackrel{{\scriptstyle
P_{\Theta}}}{{\longrightarrow}}\dot{\mathcal{H}}^{-1,\infty}_{tc}(M)\stackrel{{\scriptstyle
G_{\Theta}}}{{\longrightarrow}}\mathcal{H}^{1,\infty}_{\Theta}(M)\stackrel{{\scriptstyle
P_{\Theta}}}{{\longrightarrow}}\dot{\mathcal{H}}^{-1,\infty}(M)\to 0\,.$ (43)

###### Proof.

To prove that the sequence is exact, we start by establishing that
$P_{\Theta}$ is injective on $\mathcal{H}^{1,\infty}_{tc,\Theta}(M)$. This is
a consequence of Theorem 4.1 which guarantees that, if $P_{\Theta}(h)=0$ for
$h\in\mathcal{H}^{1,\infty}_{tc,\Theta}(M)$, then $G^{+}P_{\Theta}(h)=h=0$.

Secondly, on account of Theorem 4.1 and in particular of the identity
$G^{\pm}_{\Theta}P_{\Theta}=\mathbb{I}$ on $\mathcal{H}^{1}_{\pm,\Theta}(M)$,
it holds that $G_{\Theta}P_{\Theta}(f)=0$ for all
$f\in\mathcal{H}^{1,\infty}_{tc,\Theta}(M)$. Hence
$\mathrm{Im}(P_{\Theta})\subseteq\ker(P_{\Theta})$. Assume that there exists
$f\in\dot{\mathcal{H}}^{-1,\infty}_{tc}(M)$ such that $G_{\Theta}(f)=0$. It
descends that
$G^{+}_{\Theta}(f)=G^{-}_{\Theta}(f)\in\mathcal{H}^{1,\infty}_{tc,\Theta}(M)$.
Applying $P_{\Theta}$ it holds that $f=P_{\Theta}G^{+}_{\Theta}(f)$, that is
$f\in P_{\Theta}[\mathcal{H}^{1,\infty}_{tc,\Theta}(M)]$.

The third step consists of recalling that, per construction,
$P_{\Theta}G_{\Theta}=0$ and that, still on account of Theorem 4.1,
$\textrm{Im}(G_{\Theta})\subseteq\ker(P_{\Theta})$. To prove the opposite
inclusion, suppose that $u\in\ker(P_{\Theta})$. Let $\chi\equiv\chi(\tau)$ be
a smooth function such that there exists $\tau_{0},\tau_{1}\in\mathbb{R}$ such
that $\chi=1$ if $\tau>\tau_{1}$ and $\chi=0$ if $\tau<\tau_{0}$. Since
$\Theta$ is a static boundary condition and, therefore, it commutes with
$\chi$, it holds that $\chi u\in\mathcal{H}^{1,\infty}_{+,\Theta}(M)$. Hence
setting $f\doteq P_{\Theta}\chi u$, a direct calculation shows that
$G_{\Theta}f=u$

To conclude we need to show that the map $P_{\Theta}$ on the before last arrow
is surjective. To this end, let $j\in\dot{\mathcal{H}}^{-1,\infty}(M)$ and let
$\chi\equiv\chi(\tau)$ be as above. Let $h\doteq G^{+}_{\Theta}\left(\chi
j\right)+G^{-}_{\Theta}\left((1-\chi)j\right)$. Per construction
$h\in\mathcal{H}^{1,\infty}(M)$ and $P_{\Theta}(h)=j$. ∎

Mainly for physical reasons we individuate the following special classes of
boundary conditions. Recall that, according to Theorem 2.1 $M$ is isometric to
$\mathbb{R}\times\Sigma$ and $\partial M$ to $\mathbb{R}\times\partial\Sigma$.

######  Definition 4.2.

Let $\Theta\in\Psi^{k}_{b}(M)$ with $k\leq 2$ and let $\Theta=\Theta^{*}$ We
call $\Theta$

  * •

physically admissible if $WF_{b}^{-1,s+1}(\Theta u)\subseteq
WF_{b}^{-1,s+1}(P_{\Theta}u)$ for all $u\in\mathcal{H}^{1,m}_{loc}(M)$ with
$m\leq 0$ and $s\in\mathbb{R}\cup\\{\infty\\}$.

  * •

a static boundary condition if $\Theta\equiv\Theta_{K}$ is the natural
extension to $\Psi^{k}_{b}(M)$ of a pseudodifferential operator
$K=K^{*}\in\Psi_{b}^{k}(\partial\Sigma)$ with $k\leq 2$.

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
$\mathcal{W}_{b}^{-\infty}(M)$ the space of bounded operators from
$\dot{\mathcal{H}}_{0}^{-1,-\infty}(M)$ to $\mathcal{H}^{1,\infty}_{loc}(M)$
and we give a definition of wavefront set complementary to that of Definition
3.4.

######  Definition 4.3 (Operatorial wavefront set $WF_{b}^{Op}(M)$).

Let
$\Lambda:\dot{\mathcal{H}}^{-1,-\infty}_{0}(M)\rightarrow\mathcal{H}^{1,\infty}_{loc}(M)$
be a continuous map. A point
$(q_{1},q_{2})\in{}^{b}S^{*}M\times{}^{b}S^{*}(M)\not\in WF_{b}^{Op}(M)$ if
there exists two b-pseudodifferential operators $B_{1}$ and $B_{2}$ in
$\Psi_{b}^{0}(M)$ elliptic at $q_{1}$ and $q_{2}$ respectively, such that
$B_{1}\Lambda B_{2}^{*}\in\mathcal{W}_{b}^{-\infty}(M)$.

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.

######  Theorem 4.2.

Let $\Delta$ denote the diagonal in ${}^{b}S^{*}M\times{}^{b}S^{*}M$ and let
$\Theta$ be physically admissible as per Definition 4.2. Then

$WF_{b}^{Op}(G_{\Theta}^{\pm})\setminus\Delta\subset\\{(q_{1},q_{2})\in{}^{b}S^{*}M\times{}^{b}S^{*}M\
|\ q_{1}\dot{\sim}q_{2},\ \pm t(q_{1})>\pm t(q_{2})\\},$

where $q_{1}\dot{\sim}q_{2}$ means that $q_{1},q_{2}$ are two points in
$\dot{\mathcal{N}}$, cf. Equation (22) connected by a generalized broken
bicharacteristic, cf. Definition 3.8.

######  Remark 4.3.

The reason for the hypothesis on $\Theta$ lies in the fact that we do not want
to alter the microlocal behavior of the system in $\mathring{M}$. More
precisely, if no restriction on the wavefront set of $\Theta u$ is placed,
then by the propagation of singularities theorem, cf. Theorem 3.2, in addition
to the singularities propagating along the generalized broken
bicharacteristics of the Klein-Gordon operator we should account also for
those of $\Theta u$. On the one hand this would be in sharp contrast with what
happens if $M$ were a globally hyperbolic spacetime without boundary. On the
other hand, in concrete applications such as the construction of Hadamard two-
point functions, one seeks for bi-distributions with a prescribed form of the
wave front set and whose antisymmetric part coincides with the difference
between the advanced and retarded fundamental solutions associated to the
Klein-Gordon operator with boundary condition implemented by $\Theta$, see
e.g. [DF16, DF18, DW19, Wro17, GW18].

In addition one can infer the following localization property which is
sometimes referred to as time-slice axiom.

######  Corollary 4.2.

Let
$\mathcal{H}^{-1,\infty}_{tc,[\tau_{1},\tau_{2}]}(M)\subset\mathcal{H}^{-1,\infty}_{tc}(M)$
be the collection of all $u\in\mathcal{H}^{-1,\infty}_{tc}(M)$ such that
$p\notin\textrm{supp}(u)$ whenever $\tau(p)\notin[\tau_{1},\tau_{2}]$,
$\tau_{1},\tau_{2}\in\mathbb{R}$. Then, if $\Theta$ is a static boundary
condition as per Definition 4.2, the inclusion map
$\iota_{\tau_{1},\tau_{2}}:\dot{\mathcal{H}}^{-1,\infty}_{tc,[\tau_{1},\tau_{2}]}(M)\rightarrow\dot{\mathcal{H}}^{-1,\infty}_{tc}(M)$
induces the isomorphism

$[\iota_{\tau_{1},\tau_{2}}]:\dfrac{\dot{\mathcal{H}}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}\rightarrow\dfrac{\dot{\mathcal{H}}_{tc}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc}^{1,\infty}(M)}.$
(44)

###### Proof.

By direct inspection one can realize that the map $\iota_{\tau_{1},\tau_{2}}$
descends to the quotient space
$\dfrac{\dot{\mathcal{H}}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}$.
The ensuing application $[\iota_{\tau_{1},\tau_{2}}]$ is manifestly injective.
We need to show that it is also surjective. Consider therefore any
$[f]\in\dfrac{\dot{\mathcal{H}}_{tc}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc}^{1,\infty}(M)}$
and let $G_{\Theta}(f)$ be the associated solution of the Klein-Gordon
equation, cf. Equation (43). Let $\chi\equiv\chi(\tau)$ be a smooth function
such that $\chi=1$ if $\tau>\tau_{2}$ while $\chi=0$ if $\tau<\tau_{1}$. The
function $h\doteq P_{\Theta}\left(\chi
G_{\Theta}(f)\right)\in\dot{\mathcal{H}}^{-1,\infty}_{tc,[\tau_{1},\tau_{2}]}(M)$,
where $G_{\Theta}$ is the causal propagator, cf. Remark 4.2 and Proposition
4.2. Per construction the map $P_{\Theta}\circ\chi\circ G_{\Theta}$ descends
to an application from
$\dfrac{\dot{\mathcal{H}}_{tc}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc}^{1,\infty}(M)}$
to
$\dfrac{\dot{\mathcal{H}}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}{P_{\Theta}\mathcal{H}_{tc,[\tau_{1},\tau_{2}]}^{1,\infty}(M)}$
which is both a left and a right inverse of $[\iota_{\tau_{1},\tau_{2}}]$. ∎

##  5 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.

###  5.1 Fundamental solutions on static spacetimes

In this section we give a concrete example of advanced and retarded
fundamental solutions for the Klein-Gordon operator $P_{\Theta}$, 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 $\nu=(n-1)/2$, 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 $\chi\in\Gamma(TM)$, such
that $\mathcal{L}_{\chi}(x)=0$ where $x$ is the global boundary function,

  * 2)

$(M,\hat{g})$ is isometric to a standard static spacetime, that is a warped
product $\mathbb{R}\times_{\beta}S$ with line element
$ds^{2}=-\alpha^{2}dt^{2}+h_{S}$ where $h_{S}$ is a $t$-independent Riemannian
metric on $S$, while $\alpha\neq\alpha(t)$ 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 $\chi$ coincides with the
vector field $\partial_{\tau}$, cf. Theorem 2.1. Hence the underlying line-
element reads as $ds^{2}=-\beta d\tau^{2}+\kappa$ where both $\beta$ and
$\kappa$ are $\tau$-independent and $S$ can be identified with the Cauchy
surface $\Sigma$ in Theorem 2.1. For convenience we also remark that, in view
of this characterization of the metric, the associated Klein-Gordon equation
$Pu=0$ with $P=\Box_{g}$ reads

$\left(-\partial^{2}_{\tau}+E\right)u=0,$ (45)

where $E=\beta\Delta_{\kappa}$, being $\Delta_{\kappa}$ the Laplace-Beltrami
operator associated to the the Riemannian metric $\kappa$.

Henceforth we consider only static boundary conditions as per Definition 4.2
which we indicate with the symbol $\Theta_{K}$ to recall that they are induced
from $K\in\Psi^{k}_{b}(\partial M)$. Since the underlying spacetime is static,
in order to construct the advanced and retarded fundamental solutions, we can
focus our attention on
$\mathcal{G}_{\Theta_{K}}\in\mathcal{D}^{\prime}(\mathring{M}\times\mathring{M})$
, the bi-distribution associated to the causal propagator $G_{\Theta_{K}}$,
cf. Remark 4.2. It satisfies the following initial value problem, see also
[DDF18]:

$\begin{cases}(P_{\Theta_{K}}\otimes\mathbb{I})\mathcal{G}_{\Theta_{K}}=(\mathbb{I}\otimes
P_{\Theta_{K}})\mathcal{G}_{\Theta_{K}}=0\\\
\mathcal{G}_{\Theta_{K}}|_{\tau=\tau^{\prime}}=0\quad\\\
\partial_{\tau}\mathcal{G}_{\Theta_{K}}|_{\tau=\tau^{\prime}}=-\partial_{\tau^{\prime}}\mathcal{G}_{\Theta_{K}}|_{\tau=\tau^{\prime}}=\delta\end{cases}$
(46)

where $\delta$ is the Dirac distribution on the diagonal of
$\mathring{M}\times\mathring{M}$. Starting from $\mathcal{G}_{\Theta_{K}}$ one
can recover the advanced and retarded fundamental solutions
$\mathcal{G}^{\pm}_{\Theta_{K}}$ via the identities:

$\mathcal{G}^{-}_{\Theta_{K}}=\vartheta(\tau-\tau^{\prime})\mathcal{G}_{\Theta_{K}}\quad\textrm{and}\quad\mathcal{G}^{+}_{\Theta_{K}}=-\vartheta(\tau^{\prime}-\tau)\mathcal{G}_{\Theta_{K}},$
(47)

where $\vartheta$ is the Heaviside function. The existence and the properties
of $\mathcal{G}_{\Theta_{K}}$ 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 $\mathbb{C}$ and let $S:D(S)\subset
H\rightarrow H$ be a closed, linear and symmetric operator. A boundary triple
for the adjoint operator $S^{*}$ is a triple
$(\mathsf{h},\gamma_{0},\gamma_{1})$, where $\mathsf{h}$ is a separable
Hilbert space over $\mathbb{C}$ while
$\gamma_{0},\gamma_{1}:D(S^{*})\rightarrow\mathsf{h}$ are two linear maps
satisfying

  * 1)

For every $f,f^{\prime}\in D(P^{*})$ it holds

$(S^{*}f|f^{\prime})_{H}-(f|S^{*}f^{\prime})_{H}=(\gamma_{1}f|\gamma_{0}f^{\prime})_{\mathsf{h}}-(\gamma_{0}f|\gamma_{1}f^{\prime})_{\mathsf{h}}$
(48)

  * 2)

The map $\gamma:D(S^{*})\rightarrow\mathsf{h}\times\mathsf{h}$ defined by
$\gamma(f)=(\gamma_{0}f,\gamma_{1}f)$ is surjective.

One of the key advantages of this framework is encoded in the following
proposition, see [Mal92]

######  Proposition 5.1.

Let $S$ be a linear, closed and symmetric operator on $H$. Then an associated
boundary triple $(\mathsf{h},\gamma_{0},\gamma_{1})$ exists if and only if
$S^{*}$ has equal deficiency indices. In addition, if
$\Theta:D(\Theta)\subseteq\mathsf{h}\rightarrow\mathsf{h}$ is a closed and
densely defined linear operator, then $S_{\Theta}\doteq
S^{*}|_{ker(\gamma_{1}-\Theta\gamma_{0})}$ is a closed extension of $S$ with
domain

$D(S_{\Theta})\doteq\\{f\in D(S^{*})\;|\;\gamma_{0}(f)\in
D(\Theta),\;\textrm{and}\;\gamma_{1}(f)=\Theta\gamma_{0}(f)\\}$

The map $\Theta\mapsto S_{\Theta}$ is one-to-one and
$S^{*}_{\Theta}=S_{\Theta^{*}}$. In other word there is a one-to-one
correspondence between self-adjoint operators $\Theta$ on $\mathsf{h}$ and
self-adjoint extensions of $S$.

Noteworthy is the application of this framework to the case where the rôle of
$S$ is played by a second order elliptic partial differential operator $E$.
Observe that this symbol is employed having in mind the subsequent application
to Equation (45). To construct a boundary triple associated with $E^{*}$, let
$n$ be the unit, outward pointing, normal of $\partial\Sigma$ and let

$\Gamma_{0}\colon H^{2}(\Sigma)\ni f\mapsto\Gamma f\in
H^{3/2}(\Sigma),\,\qquad\Gamma_{1}\colon H^{2}(\Sigma)\ni
f\mapsto-\Gamma\nabla_{n}f\in H^{1/2}(\Sigma)\,,$

where $H^{k}(\Sigma)$ indicates the Sobolev space associated to the Riemannian
manifold $(\Sigma,\kappa)$ introduced at the end of Section 2.1. Here
$\Gamma:H^{s}(\Sigma)\to H^{s-\frac{1}{2}}(\Sigma)$, $s>\frac{1}{2}$ is the
continuous surjective extension of the restriction map from
$C^{\infty}_{0}(\Sigma)$ to $C^{\infty}_{0}(\partial\Sigma)$, cf. [GS13, Th.
4.10 & Cor. 4.12]. In addition, since the inner product
$(\,|\,)_{L^{2}(\partial\Sigma)}$ on $L^{2}(\partial\Sigma)\equiv
L^{2}(\partial\Sigma;\iota^{*}_{\Sigma}d\mu_{g})$,
$\iota_{\Sigma}:\partial\Sigma\hookrightarrow\Sigma$, extends continuously to
a pairing on $H^{-1/2}(\partial\Sigma)\times H^{1/2}(\partial\Sigma)$ as well
as on $H^{-3/2}(\partial\Sigma)\times H^{3/2}(\partial\Sigma)$, there exist
isomorphisms

$\iota_{\pm}\colon H^{\pm 1/2}(\partial\Sigma)\to L^{2}(\partial\Sigma),\qquad
j_{\pm}\colon H^{\pm 3/2}(\partial\Sigma)\to L^{2}(\partial\Sigma)\,,$

such that, for all $(\psi,\phi)\in H^{1/2}(\partial\Sigma)\times
H^{-1/2}(\partial\Sigma)$ and for all $(\widetilde{\psi},\widetilde{\phi})\in
H^{3/2}(\partial\Sigma)\times H^{-3/2}(\partial\Sigma)$,

$\displaystyle(\psi,\phi)_{(1/2,-1/2)}=(\iota_{+}\psi|\,\iota_{-}\phi)_{L^{2}(\partial\Sigma)}\,,\quad(\widetilde{\psi},\widetilde{\phi})_{(3/2,-3/2)}=(j_{+}\widetilde{\psi}|\,j_{-}\widetilde{\phi})_{L^{2}(\partial\Sigma)}\,,$

where $(,)_{(1/2,-1/2)}$ and $(,)_{(3/2,-3/2)}$ stand for the duality pairings
between the associated Sobolev spaces.

######  Remark 5.2.

Note that in the massless case, the two trace operators $\Gamma_{0}$ and
$\Gamma_{1}$ coincide respectively with the restriction to $H^{2}(M)$ of the
traces $\gamma_{-}$ and $\gamma_{+}$ introduced in Theorem 3.1 and in Lemma
3.2.

Gathering all the above ingredients, we can state the following proposition,
cf. [DDF18, Thm. 24 & Rmk 25]:

######  Proposition 5.2.

Let $E^{*}$ be the adjoint of a second order, elliptic, partial differential
operator on a Riemannian manifold $(\Sigma,\kappa)$ with boundary and of
bounded geometry. Let

$\displaystyle\gamma_{0}\colon H^{2}(M)\ni f\mapsto\iota_{+}\Gamma_{0}f\in
L^{2}(\partial M)\,,$ (49) $\displaystyle\gamma_{1}\colon H^{2}(M)\ni f\mapsto
j_{+}\Gamma_{1}f\in L^{2}(\partial M)\,,$ (50)

Then $(L^{2}(\partial M),\gamma_{0},\gamma_{1})$ is a boundary triple for
$E^{*}$.

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 $(\gamma_{0},\gamma_{1},L^{2}(\partial M))$ 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 $L^{2}(\partial\Sigma)$ 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
$E_{K}\doteq E^{*}|_{D(E_{K})}$, where
$D(E_{K})\doteq\ker(\gamma_{1}-K\gamma_{0})$. Furthermore, let assume that the
spectrum of $E_{K}$ is bounded from below.
Then, calling $\Theta_{K}$ the associated boundary condition, the advanced and
retarded Green’s operators $\mathsf{G}^{\pm}_{\Theta_{K}}$ associated to the
wave operator $\partial_{t}^{2}+E_{K}$ exist and they are unique. They are
completely determined in terms of
$\mathcal{G}^{\pm}_{\Theta_{K}}\in\mathcal{D}^{\prime}(\mathring{M}\times\mathring{M})$.
These are bidistributions such that
$\mathcal{G}^{-}_{\Theta_{K}}=\vartheta(t-t^{\prime})\mathcal{G}_{\Theta_{K}}$
and
$\mathcal{G}^{+}_{\Theta_{K}}=-\vartheta(t^{\prime}-t)\mathcal{G}_{\Theta_{K}}$
where
$\mathcal{G}_{\Theta_{K}}\in\mathcal{D}^{\prime}(\mathring{M}\times\mathring{M})$
is such that, for all $f\in\mathcal{D}(\mathring{M})$

$\displaystyle\mathcal{G}_{\Theta_{K}}(f_{1},f_{2})\doteq\int_{\mathbb{R}^{2}}\textrm{d}t\textrm{d}t^{\prime}\,\bigg{(}f_{1}(t)\bigg{|}(-E_{K})^{-\frac{1}{2}}\sin[(-E_{K})^{\frac{1}{2}}(t-t^{\prime})\big{]}f_{2}(t^{\prime})\bigg{)},$
(51)

where $f(t)\in H^{2}(\Sigma)$ denotes the evaluation of $f$, regarded as an
element of $C_{\textrm{c}}^{\infty}(\mathbb{R},H^{\infty}(\Sigma))$ and
$E_{K}^{-\frac{1}{2}}\sin[E_{K}^{\frac{1}{2}}(t-t^{\prime})]$ is defined
exploiting the functional calculus for $E_{K}$. Moreover it holds that

$\mathsf{G}^{\pm}_{\Theta_{K}}\colon\mathcal{D}(\mathring{M})\to
C^{\infty}(\mathbb{R},H^{\infty}_{\Theta_{K}}(\Sigma))\,,$

where $H^{\infty}_{\Theta_{K}}(\Sigma)\doteq\bigcap_{k\geq
0}D(E_{\Theta_{K}}^{k})$. In particular,

$\displaystyle\gamma_{1}\big{(}\mathsf{G}^{\pm}_{\Theta_{K}}f\big{)}=\Theta_{K}\gamma_{0}\big{(}\mathsf{G}^{\pm}_{\Theta_{K}}f\big{)}\qquad\forall
f\in C^{\infty}_{0}(\mathring{M})\,.$ (52)

######  Remark 5.3.

Observe that, in Theorem 5.1 we have constructed the advanced and retarded
fundamental solutions $\mathcal{G}^{\pm}_{\Theta}$ as elements of
$\mathcal{D}^{\prime}(\mathring{M}\times\mathring{M})$. 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 to
$\mathring{M}$ coincides with $\mathcal{G}^{\pm}_{\Theta_{K}}$. With a slight
abuse of notation we shall refer to these extended fundamental solutions with
the same symbol.

###  5.2 Existence of Hadamard States on Static Spacetimes

In this section, we discuss the existence of Hadamard two-point functions. We
stress that the so-called 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 $\lambda_{2}\in\mathcal{D}^{\prime}(M\times
M)$ is called of Hadamard form if its restriction to $\mathring{M}$ has the
following wavefront set

$WF(\lambda_{2})=\left\\{(p,k,p^{\prime},-k^{\prime})\in
T^{*}(\mathring{M}\times\mathring{M})\setminus\\{0\\}\;|\;(p,k)\sim(p^{\prime},k^{\prime})\;\textrm{and}\;k\triangleright
0\right\\},$ (53)

where $\sim$ entails that $(p,k)$ and $(p^{\prime},k^{\prime})$ are connected
by a generalized broken bicharactersitic, while $k\triangleright 0$ means that
the co-vector $k$ at $p\in\mathring{M}$ is future-pointing. Furthermore we
call $\lambda_{2,\Theta}\in\mathcal{D}^{\prime}(M\times M)$ a Hadamard two-
point function associated to $P_{\Theta}$, if, in addition to Equation (53),
it satisfies

$(P_{\Theta}\otimes\mathbb{I})\lambda_{2,\Theta}=(\mathbb{I}\otimes
P_{\Theta})\lambda_{2,\Theta}=0,$

and, for all $f,f^{\prime}\in\mathcal{D}(\mathring{M})$,

$\lambda_{2,\Theta}(f,f)\geq
0,\quad\textrm{and}\quad\lambda_{2,\Theta}(f,f^{\prime})-\lambda_{2,\Theta}(f^{\prime},f)=i\mathcal{G}_{\Theta}(f,f^{\prime}),$
(54)

where $P_{\Theta}$ is the Klein-Gordon operator as in Equation (20), while
$\mathcal{G}_{\Theta}$ 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 $\lambda_{2,\Theta}$, we can identify the
following associated bidistributions:

  * •

the bulk-to-bulk two-point function
$\mathring{\lambda}_{2,\Theta}\in\mathcal{D}^{\prime}(\mathring{M}\times\mathring{M})$
such that
$\mathring{\lambda}_{2,\Theta}\doteq\left.\lambda_{2,\Theta}\right|_{\mathring{M}}$
is the restriction of the Hadamard two-point function to
$\mathring{M}\times\mathring{M}$.

  * •

the boundary-to-boundary two-point function
$\lambda_{2,\partial,\Theta}\in\mathcal{D}^{\prime}(\partial M\times\partial
M)$ such that
$\lambda_{2,\partial,\Theta}\doteq(\iota_{\partial}^{*}\otimes\iota_{\partial}^{*})\lambda_{2,\Theta}$
where $\iota_{\partial}:\partial M\to M$ is the embedding map of the boundary
in $M$.

Observe that $\lambda_{2,\partial,\Theta}$ is well-defined on account of
Equation (53) and of [Hör03, Thm. 8.2.4].

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 $\Theta$ 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 $\lambda_{2,\Theta}$, it corresponds
$\Lambda_{\Theta}:\dot{\mathcal{H}}_{0}^{-k,-\infty}(M)\rightarrow\mathcal{H}^{k,-\infty}_{loc}(M)$,
with $k=\pm 1$. Recalling Definition 3.3 and 4.3, the following lemma holds
true, cf. [Wro17, Lem. 5.3]:

######  Lemma 5.1.

For any $q_{1},q_{2}\in{}^{b}S^{*}M$, $(q_{1},q_{2})\not\in
WF^{Op}(\Lambda_{\Theta})$ if and only if there exist neighbourhoods
$\Gamma_{i}$ of $q_{i}$, $i=1,2$, such that for all $B_{i}\in\Psi_{b}^{0}(M)$
elliptic at $q_{i}$ satisfying $WF_{b}^{Op}(B_{i})\subset\Gamma_{i}$,
$B_{1}\Lambda B_{2}\in\mathcal{W}^{-\infty}_{b}(M)$.

Observe that this lemma entails in particular that, given any $f_{i}\in
C^{\infty}(M)$, $i=1,2$ such that $\textrm{supp}(f_{i})\subset\mathring{M}$
then $f_{1}\Lambda_{\Theta}f_{2}$ has a smooth kernel over
$\mathring{M}\times\mathring{M}$. In addition the following also holds true,
cf. [Wro17, Prop. 5.6]:

######  Proposition 5.3.

Let $\Lambda_{\Theta}$ identify an Hadamard two-point function. If
$(q_{1},q_{2})\in WF_{b}^{Op}(\Lambda_{\Theta})$ for $q_{1},q_{2}\in
T^{*}M\setminus\\{0\\}$, then $(q_{1},q_{1})\in WF_{b}^{Op}(\Lambda_{\Theta})$
or $(q_{2},q_{2})\in WF_{b}^{Op}(\Lambda_{\Theta})$.

Given any two points $q_{1}$ and $q_{2}$ in the cosphere bundle
${}^{b}S^{*}M$, cf. Equation (4) we shall write $q_{1}\dot{\sim}q_{2}$ if both
$q_{1}$ and $q_{2}$ lie in the compressed characteristic bundle
$\dot{\mathcal{N}}$ 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:

######  Proposition 5.4.

Let
$\Lambda_{\Theta}:\dot{\mathcal{H}}_{0}^{-1,-\infty}(M)\rightarrow\mathcal{H}^{1,-\infty}_{loc}(M)$
and suppose that $(q_{1},q_{2})\in WF^{Op}_{b}(\Lambda_{\Theta})$. If
$P_{\Theta}\Lambda_{\Theta}=0$, then $q_{1}\in\dot{\mathcal{N}}$ and
$(q_{1}^{\prime},q_{2})\in WF_{b}^{Op}(\Lambda_{\Theta})$ for every
$q_{1}^{\prime}$ such that $q_{1}^{\prime}\dot{\sim}q_{1}$. Similarly, if
$\Lambda_{\Theta}P_{\Theta}=0$, then $q_{2}\in\dot{\mathcal{N}}$ and
$(q_{1},q_{2}^{\prime})\in WF_{b}^{Op}(\Lambda_{\Theta})$ for all
$q_{2}^{\prime}$ such that $q_{2}^{\prime}\dot{\sim}q_{2}$.

Our next step consists of refining Theorem 4.2 in $\mathring{M}$, cf. for
similarities with [DF18, Cor. 4.5].

######  Corollary 5.1.

Let
$G_{\Theta}:\mathcal{H}^{-1,-\infty}(\mathring{M})\rightarrow\mathcal{H}^{1,-\infty}(\mathring{M})$
be the restriction to $\mathring{M}$ of the causal propagator as per Remark
4.2 . Then

$WF_{b}^{Op}(G_{\Theta})=\\{(q_{1},q_{2})\in{}^{b}S^{*}\mathring{M}\times{}^{b}S^{*}\mathring{M}\
|\ q_{1}\dot{\sim}q_{2}\\}.$

###### Proof.

A direct application of Theorem 4.2 yields

$WF^{Op}(G_{\Theta})\subseteq\\{(q_{1},q_{2})\in{}^{b}S^{*}\mathring{M}\times{}^{b}S^{*}\mathring{M}\;|\;q_{1}\dot{\sim}q_{2}\\}$

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 in $\mathring{M}$. Since ${}^{b}T^{*}\mathring{M}\simeq
T^{*}\mathring{M}$, 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 $P_{\Theta}$ be the Klein-Gordon
operator as per Equation (20) and let $\Theta\equiv\Theta_{K}$ 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 $P_{\Theta_{K}}$ 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 $P_{\Theta}$,
$\lambda_{2,\Theta_{K}}\in\mathcal{D}^{\prime}(M\times M)$ such that, for all
$f_{1},f_{2}\in\mathcal{D}(M)$

$\displaystyle\lambda_{2,\Theta_{K}}(f_{1},f_{2})\doteq
2i\int_{\mathbb{R}^{2}}\textrm{d}t\textrm{d}t^{\prime}\,\bigg{(}f_{1}(t)\bigg{|}\frac{\exp[iE_{\Theta_{K}}^{\frac{1}{2}}(t-t^{\prime})\big{]}}{(-E_{\Theta_{K}})^{\frac{1}{2}}}f_{2}(t^{\prime})\bigg{)},$
(55)

###### Proof.

Observe that, per construction $\lambda_{2,\Theta_{k}}$ is a bi-solution of
the Klein-Gordon equation associated to the operator $P_{\Theta_{K}}$ 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 of $\mathring{\lambda}_{2,\Theta_{K}}$, 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 $\Theta_{K}$.

###  5.3 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
$\mathcal{O}(x^{3})$ close to $\partial M$ if $h(x)=h_{0}+x^{2}h_{1}(x)$ where
$h_{1}$ is a symmetric two-tensor, see [Wro17, Def. 4.3].

######  Lemma 5.2.

Suppose $(M,g)$ is a globally hyperbolic, asymptotically anti-de Sitter
spacetime. For any $\tau_{2}\in\mathbb{R}$ there a static, globally hyperbolic
asymptotically AdS spacetime $(M,g^{\prime})$ as well as $\tau_{0},\tau_{1}$
with $\tau_{0}<\tau_{1}<\tau_{2}$ such that $g^{\prime}=g$ if
$\\{\tau\geq\tau_{1}\\}$, while, if $\\{\tau\leq\tau_{0}\\}$, $(M,g^{\prime})$
is isometric to a standard static asymptotically AdS spacetime $(M,g_{S})$
which is even modulo $\mathcal{O}(x^{3})$ and in which $C\leq\beta\leq C^{-1}$
for some $C>0$, with $\beta$ as in Equation (1).

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 $\tau\in[\tau_{1},\tau_{2}]$. 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 $\Lambda_{\Theta}\in\mathcal{D}^{\prime}(M\times M)$ is a bi-
solution of the Klein-Gordon equation ruled by $P_{\Theta}$ abiding to
Equation (54) and with a wavefront set of Hadamard form in the region of $M$
such that $\tau_{1}<\tau<\tau_{2}$. Then $\Lambda_{\Theta}$ 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
$\Theta_{K}$ be a static and physically admissible boundary condition so that
the Klein-Gordon operator $P_{\Theta_{K}}$ on $(M_{S},g_{S})$ admits a
Hadamard two-point 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 $\Theta_{K}$.

###### 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 $(M,g^{\prime})$
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
$\mathbb{R}\times\Sigma$.

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
$\Theta_{K}$. We indicate it with $\lambda_{2,S}$ omitting any reference to
$\Theta_{K}$ since it plays no explicit rôle in the analysis.

Focusing the attention on $(M,g^{\prime})$, Lemma 5.2 guarantees that, if
$\tau<\tau_{0}$, $\tau$ being the time coordinate along $\mathbb{R}$, then
therein $(M,g^{\prime})$ is isometric to $(M,g_{S})$. Calling this region
$M_{0}$, the restriction $\lambda_{2,S}|_{M_{0}\times M_{0}}$ identifies a
two-point distribution of Hadamard form. Notice that we have omitted to write
explicitly the underlying isometries for simplicity of notation.

Using the time-slice axiom in Corollary 4.2, for any pair of test-functions
$f,f^{\prime}\in\mathcal{D}(M^{\prime})$ such that for all
$p\in\textrm{supp}(f)\cup\textrm{supp}(f^{\prime})$, $\tau(p)>\tau_{0}$, we
set $h=P_{\Theta_{K}}\chi G_{\Theta_{K}}(f)$ and
$h^{\prime}=P_{\Theta_{K}}\chi G_{\Theta_{K}}(f^{\prime})$ where
$G_{\Theta_{K}}$ is the causal propagator associated to $P_{\Theta_{K}}$ in
$(M,g^{\prime})$, while $\chi=\chi(\tau)$ is any smooth function such that
there exists $\tau_{1},\tau_{2}<\tau_{0}$ for which $\tau=0$ if
$\tau<\tau_{1}$ while $\chi=1$ if $\tau>\tau_{2}$. We define

$\lambda^{\prime}_{2}(f,f^{\prime})=\lambda_{2,S}(h,h^{\prime}).$

Observe that $h,h^{\prime}\in C^{\infty}_{tc}(M)$ and therefore the right-hand
side of this identity is well-defined. In addition, since $G_{\Theta_{K}}$ is
continuous on $\mathcal{D}(M)$, sequential continuity entails that
$\lambda_{2}^{\prime}\in\mathcal{D}(M^{\prime}\times M^{\prime})$. In
addition, per construction, it is a solution of the Klein-Gordon equation
ruled by $P_{\Theta_{K}}$ on $(M^{\prime},g^{\prime})$ and abiding to Equation
(54).

Furthermore Lemma 5.3 yields that $\lambda_{2}^{\prime}$ is of Hadamard form.

To conclude it suffices to focus on $(M,g)$ recalling that there exists
$\tau_{1}\in\mathbb{R}$ such that, in the region
$(M_{1},g^{\prime})\subset(M,g^{\prime})$ for which $\tau>\tau_{1}$,
$(M,g^{\prime})$ is isometric to $(M,g)$. Hence,we can repeat the argument
given above. More precisely we consider
$\lambda_{2}^{\prime}|_{M^{\prime}\times M^{\prime}}$ and, using the time-
slice axiom, see Corollary 4.2, we can identify
$\lambda_{2}\in\mathcal{D}^{\prime}(M\times M)$ which is a solution of the
Klein-Gordon equation ruled by $P_{\Theta_{K}}$ and it abides to Equation
(54). Lemma 5.3 entails also that it is of Hadamard form, hence proving the
sought result. ∎

## Acknowledgments

We are grateful to Benito Juarez Aubry for the useful discussions which
inspired the beginning of this project and to Nicolò Drago both for the useful
discussions and for pointing out references [GM20, GMP14]. We are also
grateful to Simone Murro and to MichałWrochna for the useful discussions. The
work of A. Marta is supported by a fellowship of the Università Statale di
Milano, which is gratefully acknowledged. C. Dappiaggi is grateful to the
Department of Mathematics of the Università Statale di Milano for the kind
hospitality during the realization of part of this work.

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