# Perturbations of non-autonomous second-order abstract Cauchy problems

Christian Budde  University of the Free State, Department of Mathematics and
Applied Mathematics, P.O. Box 339, 9300 Bloemfontein, South Africa
<EMAIL_ADDRESS>and  Christian Seifert  Technische Universität Hamburg,
Institut für Mathematik, Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany
<EMAIL_ADDRESS>
###### Abstract.

In this paper we present time-dependent perturbations of second-order non-
autonomous abstract Cauchy problems associated to a family of operators with
constant domain. We make use of the equivalence to a first-order non-
autonomous abstract Cauchy problem in a product space, which we elaborate in
full detail. As an application we provide a perturbed non-autonomous wave
equation.

###### Key words and phrases:

non-autonomous, second-order, abstract Cauchy problem, perturbations

###### 2020 Mathematics Subject Classification:

47D09, 47Dxx, 34G10, 47A55, 35L05

Declarations of interest: none.

##  1\. Introduction

Autonomous second-order abstract Cauchy problems, which are of the form

(ACP2) $\displaystyle\begin{cases}\ddot{u}(t)=Au(t),&\quad t>0,\\\ u(0)=x,\\\
\dot{u}(0)=y,\end{cases}$

for some (unbounded) operator $(A,\mathrm{D}(A))$, which often occur in the
context of wave equations, have been studied intensively by several authors in
the past, e.g., Sova [39], Da Prato and Giusti [5], Fattorini [8], Neubrander
[27], Xio and Jin [49] as well as Xiao and Liang [48]. One can also find more
information in the monographs by Arendt et al. [3, Sect. 3.14 & 3.15],
Melnikova and Filinkov [24, Sect. 1.7] or Vasil’ev and Piskarev [44]. In
contrast to the first-order problem, where (classical) solutions are given by
$C_{0}$-semigroups, one needs another solution concept for (ACP2), the so-
called cosine and sine families. Similar to the Hille–Yosida generation
theorem for strongly continuous semigroups, one can also characterize
generators of cosine families, cf. [3, Thm. 3.15.3], [24, Thm. 1.7.2] or [49,
Thm. A]. The classical operator theoretical approach to (ACP2) is to reduce
these to first-order ones, where one can apply the theory of
$C_{0}$-semigroups. For a detailed overview on $C_{0}$-semigroups, we refer
for example to the monographs by Engel and Nagel [6], Goldstein [9] or Pazy
[30]. In [17], Kisyński gives an explicit correspondence between generators of
cosine families and strongly continuous semigroups, see also [3, Thm.
3.14.11].

In contrast to the autonomous second-order problems, one has the non-
autonomous second-order abstract Cauchy problems of the form

(nACP2) $\displaystyle\begin{cases}\ddot{u}(t)=A(t)u(t),&\quad
t\in\left(0,T\right],\\\ u(0)=x,\\\ \dot{u}(0)=y,\end{cases}$

for some fixed $T>0$, where $(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$
is a family of operators. These non-autonomous second-order abstract Cauchy
problems have been studied first by Kozak [18, 19, 20] and later on by
Bochneck [4], Winiarska [47, 46] and Lan [21], just to mention a few. The same
idea as for (ACP2) helps to reduce (nACP2) again to a first-order problem.
Solutions of non-autonomous first-order abstract Cauchy problems have been
studied exhaustively by means of evolution families for example by
Acquistapace and Terreni [1] and Kato and Tanabe [41, 15, 42]. A semigroup
approach by so-called evolution semigroups was firstly introduced by Howland
[13] and later on studied by several authors, e.g., Evans [7], Nagel [25],
Nickel [28], Rhandi [26] and Schnaubelt [34].

The goal of this paper is to establish a perturbation result for (nACP2).
Perturbation theorems for cosine families associated to (ACP2) have been
developed for example by Piskarev and Shaw [31, 32], Miyadera [38], Takenaka
and Okazawa [40] as well as Travis and Webb [43]. Also time-dependent
perturbations have been studied, cf. [37, 22, 23]. We want to perturb (nACP2)
in a time-dependent way as it has been done for first-order non-autonomous
problems by Räbinger et al. [33, 35]. Especially, we want to cover time-
dependent perturbations of bounded type.

The paper is structured as follows. Section 2 consists of preliminary
definitions regarding solutions of non-autonomous abstract Cauchy problems.
The following Section 3 provides a relation between existence of fundamental
solutions to first- and second-order non-autonomous Cauchy problems. This can
be viewed as a non-autonomous version of the generation theorem of Kisyński in
[17], cf. Theorem 3.4. Section 4 provides our main result regarding
perturbations of second-order non-autonomous Cauchy problems, cf. Theorem 4.2.
The final Section 5 provides an example of a non-autonomous wave equation.

##  2\. Preliminaries on Non-Autonomous Abstract Cauchy Problems

Let $X$ be a Banach space, $T>0$ and
$(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ be a family of closed
operators on $X$.

We make the following crucial assumption throughout this article.

######  Assumption 2.1.

The domains $\mathrm{D}(A(t))$ do not depend on time, i.e., there exists
$D\subseteq X$ such that $\mathrm{D}(A(t))=D$ for all $t\in[0,T]$.

We write $\Delta:=\Delta_{T}:=\\{(t,s)\in\left[0,T\right]^{2}:\ t\geq s\\}$.

###  2.1. Non-Autonomous First-Order Cauchy Problems

Non-autonomous first-order abstract Cauchy problems are of the form

(nACP) $\displaystyle\begin{cases}\dot{v}(t)=A(t)v(t),&\quad
t\in\left(0,T\right],\\\ v(0)=x,\end{cases}$

where $x\in X$.

######  Definition 2.2.

A function $v\colon\left[0,T\right]\to X$ is called a _(classical) solution_
to (nACP) if $v$ is continuously differentiable, $v(t)\in D$ for all
$t\in\left(0,T\right]$ and $v$ satisfies (nACP).

The following solution concept is important for non-autonomous first-order
problems.

######  Definition 2.3.

A _fundamental solution_ to (nACP) associated with
$(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ satisfying Assumption 2.1 is
a family of bounded linear operators $(U(t,s))_{(t,s)\in\Delta}$ on a Banach
space $X$ satisfying the following conditions:

  * (U1)

$U(t,t)=I$ and $U(t,s)U(s,r)=U(t,r)$ for all $(t,s),(s,r)\in\Delta$.

  * (U2)

The mapping $\Delta\ni(t,s)\mapsto U(t,s)$ is strongly continuous on $X$.

  * (U3)

$U(t,s)D\subseteq D$

  * (U4)

For all $(t,s)\in\Delta$ and $x\in D$ one has that
$\frac{\partial}{\partial{t}}U(t,s)x$ and
$\frac{\partial}{\partial{s}}U(t,s)x$ exist and

(2.1) $\displaystyle\frac{\partial}{\partial{t}}U(t,s)x=A(t)U(t,s)x\text{ and
}\frac{\partial}{\partial{s}}U(t,s)x=-U(t,s)A(s)x.$

Note that (U4) requires (U3).

######  Remark 2.4.

  1. (a)

Observe that a fundamental solution to (nACP) is bounded in $\mathcal{L}(X)$,
which follows from (U2), compactness of $\Delta$ and the uniform boundedness
principle.

  2. (b)

Fundamental solutions of (nACP) as defined in Definition 2.3 are also often
called evolution families.

######  Proposition 2.5.

Let $(U(t,s))_{(t,s)\in\Delta}$ be a fundamental solution of (nACP). If $x\in
D$ then $v(t):=U(t,0)x$ defines a classical solution of (nACP).

###### Proof.

The statement follows easily from the properties (U1)–(U4). ∎

###  2.2. Non-Autonomous Second-Order Cauchy Problems

Non-autonomous second-order abstract Cauchy problems are of the form

(nACP2) $\displaystyle\begin{cases}\ddot{u}(t)=A(t)u(t),&\quad
t\in\left(0,T\right],\\\ u(0)=x,\\\ \dot{u}(0)=y,\end{cases}$

where $x,y\in X$.

######  Definition 2.6.

A function $u\colon\left[0,T\right]\to X$ is called a _(classical) solution_
to (nACP2) if $u$ is twice continuously differentiable, $u(t)\in D$ for all
$t\in\left(0,T\right]$ and $u$ satisfies (nACP2).

The solution concept of (nACP2) is vastly more involved than this of (nACP).
In fact, it goes back to Kozak [20, Def. 2.1, Def. 3.1]. Note that our
definition is slightly different from the one of Kozak, see also Remark 2.8
below.

######  Definition 2.7.

A _fundamental solution_ to (nACP2) associated with
$(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ satisfying Assumption 2.1 is
a family of bounded linear operators $(S(t,s))_{(t,s)\in\Delta}$ on $X$
satisfying the following conditions:

  * (S1)

    1. (a)

$S(t,t)=0$ for all $t\in\left[0,T\right]$.

    2. (b)

The mapping $\Delta\ni(t,s)\mapsto S(t,s)$ in strongly continuous on $X$.

    3. (c)

For all $x\in X$ and $s\in[0,T]$ the mapping $[s,T]\ni t\mapsto S(t,s)x$ is
continuously differentiable, and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{t}}}S(t,s)x$ is continuous with

$\left.\frac{\partial}{\partial{t}}S(t,s)x\right|_{t=s}=x.$

    4. (d)

For all $x\in D$ and $t\in[0,T]$ the mapping $[0,t]\ni s\mapsto S(t,s)x$ is
continuously differentiable, and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ is continuous with

$\left.\frac{\partial}{\partial{s}}S(t,s)x\right|_{t=s}=-x.$

  * (S2)

$S(t,s)D\subseteq D$ for all $(t,s)\in\Delta$, for $x\in D$ the mapping
$\Delta\ni(t,s)\mapsto S(t,s)x$ is twice continuously differentiable and

    1. (a)

$\displaystyle{\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)x=A(t)S(t,s)x.}$

    2. (b)

$\displaystyle{\frac{\partial^{2}}{\partial{s^{2}}}S(t,s)x=S(t,s)A(s)x.}$

    3. (c)

$\displaystyle{\left.\frac{\partial}{\partial{t}}\frac{\partial}{\partial{s}}S(t,s)x\right|_{t=s}=0}$

  * (S3)

For all $(t,s)\in\Delta$, if $x\in D$ then $\frac{\partial}{\partial
s}S(t,s)x\in D$, there exist
$\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x$ and
$\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x$ and
the following properties hold

    1. (a)

$\displaystyle{\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x=A(t)\frac{\partial}{\partial{s}}S(t,s)x.}$

    2. (b)

$\displaystyle{\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x=\frac{\partial}{\partial{t}}S(t,s)A(s)x.}$

    3. (c)

The mapping $\Delta\ni(t,s)\mapsto A(t)\frac{\partial}{\partial{s}}S(t,s)x$ is
continuous.

Moreover, we call a fundamental solution $(S(t,s))_{(t,s)\in\Delta}$
_evolutionary_ if additionally

  * (S4)

For all $(t,s),(s,r)\in\Delta$ and $x\in D$ one has

$\displaystyle\left(-\frac{\partial}{\partial{s}}S(t,s)\right)S(s,r)x+S(t,s)\frac{\partial}{\partial{s}}S(s,r)x=S(t,r)x.$

######  Remark 2.8.

  1. (a)

Note that our Definition of fundamental solutions is slightly different from
[20, Def. 2.1, Def. 3.1] in the sense that we do not assume $(t,s)\mapsto
S(t,s)x$ to be continuously differentiable for all $x\in X$ in (S1), but allow
for the partial derivative $\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ only
for $x\in D$ in (S1)(d). Thus, our definition is also different from the ones
used in [4, 10, 11, 12]. The reason we adjusted the definition stems from [19,
Theorem 4.1, 5)] which seems to be only possible for $x\in\mathrm{D}(B)$
instead of $x\in X$ there. In fact, the reasoning in [19, Remark 4.2] does not
apply for $\mathop{\frac{\partial}{\partial{s}}}S(t,s)$ in the topology of
$X$, but only in the topology of $\mathrm{D}(B)$ there.

  2. (b)

Moreover, in contrast to [19, 20, 4, 10, 11, 12], we only define the
fundamental solutions on $\Delta$ instead of $[0,T]^{2}$, see also [18].

######  Lemma 2.9.

Let $(S(t,s))_{(t,s)\in\Delta}$ be a fundamental solution of (nACP2) in $X$.
Then $(S(t,s))_{(t,s)\in\Delta}$ and $(\frac{\partial}{\partial
t}S(t,s))_{(t,s)\in\Delta}$ are bounded in $\mathcal{L}(X)$.

###### Proof.

Since (S1)(b)–(c) yield that $(t,s)\mapsto S(t,s)x$ and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{t}}}S(t,s)x$ are continuous on
the compact set $\Delta$ for all $x\in X$, we observe that
$(S(t,s))_{(t,s)\in\Delta}$ and $(\frac{\partial}{\partial
t}S(t,s))_{(t,s)\in\Delta}$ are pointwise bounded. The uniform boundedness
principle then yields boundedness in $\mathcal{L}(X)$. ∎

######  Proposition 2.10.

Let $(S(t,s))_{(t,s)\in\Delta}$ be a fundamental solution of (nACP2). If
$x,y\in D$, then $u(t):=-\mathop{\frac{\partial}{\partial{s}}}S(t,0)x+S(t,0)y$
defines a classical solution of (nACP2).

###### Proof.

The statement follows easily from the properties (S1)–(S3). ∎

##  3\. Existence of Fundamental Solutions: A Generation Type Result

Let $X$ be a Banach space, $T>0$,
$(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ a family of closed linear
operators in $X$ satisfying Assumption 2.1 with $D\subseteq X$ dense. We will
further assume the following.

######  Assumption 3.1.

There exists $C>0$ such that

$\frac{1}{C}\|\cdot\|_{A(s)}\leq\|\cdot\|_{A(t)}\leq
C\|\cdot\|_{A(s)}\quad(s,t\in[0,T]).$

We equip $D$ with the graph norm of $A(0)$ (equivalently, with the graph norm
of any $A(t)$) such that $D$ is a Banach space.

It is worth to mention, that the equivalence of the graph norms is an
assumption made regularly throughout the literature, see for example [36, Rem.
4.5], [45, Rem. 4.2] and [2, Sect. 7], just to mention a few.

Moreover, the following regularity assumption on $A(\cdot)$ will be made.

######  Assumption 3.2.

Assume $A(\cdot)x\in\mathrm{C}^{1}([0,T];X)$ for all $x\in D$.

Note that the first-order non-autonomous abstract Cauchy problem (nACP) is
well-posed according to [6, Chapter VI, Def. 9.1] if Assumption 2.1 and
Assumption 3.2 are satisfied and every operator generates a contractive
$C_{0}$-semigroup, cf. [14, 16].

The following lemma will be useful in what follows.

######  Lemma 3.3.

Let $Z,X$ be Banach spaces, $Z\subseteq X$, $B\in\mathcal{L}(X)$.

  1. (a)

Assume $BX\subseteq Z$. Then $B\in\mathcal{L}(X,Z)$.

  2. (b)

Assume $BZ\subseteq Z$. Then $B\in\mathcal{L}(Z)$.

###### Proof.

  1. (a)

Let $(x_{n})$ in $X$, $x\in X$, $y\in Z$ such that $x_{n}\to x$ in $X$ and
$Bx_{n}\to y$ in $Z$. Since $Z\subseteq X$ we also have $Bx_{n}\to y$ in $X$.
Hence, by continuity of $B$ on $X$, we have $Bx=y$, so $B\colon X\to Z$ is
closed and hence bounded by the closed graph theorem.

  2. (b)

Let $(x_{n})$ in $Z$, $x\in Z$, $y\in Z$ such that $x_{n}\to x$ in $Z$ and
$Bx_{n}\to y$ in $Z$. Since $Z\subseteq X$ we also have $x_{n}\to x$ in $X$
and $Bx_{n}\to y$ in $X$. Hence, by continuity of $B$ on $X$, we have $Bx=y$,
so $B\colon Z\to Z$ is closed and hence bounded by the closed graph theorem.

∎

Let $Z$ be a Banach space such that $D\subseteq Z$ dense and $Z\subseteq X$
dense. Let
$(\mathcal{A}(t),\mathrm{D}(\mathcal{A}(t)))_{t\in\left[0,T\right]}$ in
$\mathcal{Z}:=Z\times X$ be defined by

$\displaystyle\mathcal{A}(t):=\begin{pmatrix}0&I\\\
A(t)&0\end{pmatrix},\quad\mathrm{D}(\mathcal{A}(t)):=\mathcal{D}:=D\times Z.$

######  Theorem 3.4.

The following assertions are equivalent:

  1. (a)

There exists an evolutionary fundamental solution $(S(t,s))_{(t,s)\in\Delta}$
on $X$ of (nACP2) associated to $(A(t),D)_{t\in\left[0,T\right]}$ such that
for all $(t,s)\in\Delta$ we have

     * •

$S(t,s)X\subseteq Z$, $S(t,s)Z\subseteq D$, $(t,s)\mapsto S(t,s)x\in Z$ is
continuous for all $x\in X$,

     * •

$\mathop{\frac{\partial}{\partial{t}}}S(t,s)Z\subseteq Z$,
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x$ exists for all $x\in Z$ and
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x=A(t)S(t,s)x$,

     * •

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ exists for all $x\in Z$,
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)Z\subseteq Z$ and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in Z$ is continuous
for all $x\in Z$,

     * •

$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)D\subseteq
Z$,
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$
exists for all $x\in Z$ and there exists $C\geq 0$ such that
$\|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\|_{X}\leq
C\|x\|_{Z}$ for all $x\in Z$ and $(t,s)\in\Delta$.

  2. (b)

There exists a fundamental solution $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ on
$\mathcal{L}(\mathcal{Z})$ of (nACP) associated to
$(\mathcal{A}(t),\mathcal{D})_{t\in\left[0,T\right]}$.

###### Proof.

We first prove the implication (b)$\Longrightarrow$(a).

Define $(S(t,s))_{(t,s)\in\Delta}$ on $X$ by

(3.1) $\displaystyle S(t,s)x:=\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix},\quad x\in X,$

where $\pi_{1}\colon Z\times X\to Z$ denotes the projection on the first
component. Hence, $S(t,s)\in\mathcal{L}(X,Z)$ and since $Z\subseteq X$ we also
have $S(t,s)\in\mathcal{L}(X)$ for all $(t,s)\in\Delta$. Since
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is strongly continuous on $\mathcal{Z}$,
also $(t,s)\mapsto S(t,s)\in\mathcal{L}(X,Z)$ strongly continuous.

Let us provide some useful formulas. First, we note that
$\pi_{1}\mathcal{A}(t)=\pi_{2}|_{\mathcal{D}}$, where $\pi_{2}\colon Z\times
X\to X$ is the projection on the second component. Second, for $x\in Z$ we
have that $\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}=\begin{pmatrix}x\\\ 0\end{pmatrix}$ for all $s\in[0,T]$. In
particular, for $x\in D$ we have $\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}\in\mathcal{D}$ for all $s\in[0,T]$.

For $x\in Z$ we have $\begin{pmatrix}0\\\ x\end{pmatrix}\in\mathcal{D}$, so by
(U3) we obtain $S(t,s)x\in D$ for all $(t,s)\in\Delta$. Moreover, $t\mapsto
S(t,s)x$ is differentiable and

$\mathop{\frac{\partial}{\partial{t}}}S(t,s)x=\pi_{1}\mathop{\frac{\partial}{\partial{t}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}.$

By (U3) we observe $\mathop{\frac{\partial}{\partial{t}}}S(t,s)x\in Z$.

Let $x\in Z$. Then, by (U4), we have that $\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}$ is differentiable with respect to $s$ and we obtain

$\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\mathcal{U}(t,s)\begin{pmatrix}x\\\ 0\end{pmatrix}.$

Thus, $\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ exists,
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in Z$ and

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x=\pi_{1}\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\ 0\end{pmatrix}$

for all $(t,s)\in\Delta$. As $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is strongly
continuous, also $(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in
Z$ is continuous for all $x\in Z$.

We now show existence of the mixed derivative of $(S(t,s))_{(t,s)\in\Delta}$.
First, let $x\in D$. Then, by (U4) and the above, we have that
$\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}$ is differentiable with respect to $t$ and

$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\mathop{\frac{\partial}{\partial{t}}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}=-\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}.$

Thus,
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$
exists and

$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x=-\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}x\\\ 0\end{pmatrix}.$

By (U3) this shows that
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in
Z$.

Let $x\in Z$. Then $\begin{pmatrix}0\\\ x\end{pmatrix}\in\mathcal{D}$, so as
above

$\mathop{\frac{\partial}{\partial{t}}}S(t,s)x=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}.$

As the right-hand side is differentiable with respect to $t$,
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x$ exists and

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}S(t,s)x$
$\displaystyle=\pi_{2}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{2}\mathcal{A}(t)\begin{pmatrix}\pi_{1}\\\
\pi_{2}\end{pmatrix}\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}$
$\displaystyle=A(t)\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=A(t)S(t,s)x.$

Now, since $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is bounded on $\mathcal{Z}$,
there exists $C\geq 0$ such that
$\|\mathcal{U}(t,s)\|_{\mathcal{L}(\mathcal{Z})}\leq C$ for all
$(t,s)\in\Delta$. Thus,

(3.2)
$\|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\|_{X}\leq
C\|x\|_{Z}$

for all $(t,s)\in\Delta$ and $x\in Z$. Let $x\in Z$ and choose $(x_{n})$ in
$D$ such that $x_{n}\to x$ in $Z$. Then
$t\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x_{n}$ is continuously
differentiable for all $n\in\mathbb{N}$ and these functions converge uniformly
to $t\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$, by boundedness of
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$. Moreover,
$t\mapsto\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x_{n}$
is continuous for all $n\in\mathbb{N}$ and these functions converge uniformly
by the estimate above. Thus,
$t\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ is continuously
differentiable and (3.2) holds for all $x\in Z$.

We are now ready to prove (S1)–(S4) similarly as in [18, Sect. 3] and [19,
Thm. 4.1].

(S1): We prove (a)–(d) step by step.

  1. (a)

For $t\in[0,T]$ we have $\mathcal{U}(t,t)=I$ and therefore $S(t,t)=0$.

  2. (b)

Strong continuity of $(t,s)\mapsto S(t,s)$ follows from strong continuity of
$(t,s)\mapsto\mathcal{U}(t,s)$ and continuity of $\pi_{1}$.

  3. (c)

Let $s\in[0,T]$ and $x\in Z$. Then $\begin{pmatrix}0\\\
x\end{pmatrix}\in\mathcal{D}$ and (U4) yields that $[s,T]\ni
t\mapsto\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}$ is continuously
differentiable. Thus, also $[s,T]\ni t\mapsto S(t,s)x$ is continuously
differentiable, and by (U4) we compute

$\frac{\partial}{\partial{t}}S(t,s)x=\frac{\partial}{\partial{t}}\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{1}\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}.$

Now, for $x\in X$ we choose $(x_{n})$ in $Z$ such that $x_{n}\to x$ in $X$.
Then $t\mapsto S(t,s)x_{n}$ is continuously differentiable for all
$n\in\mathbb{N}$ and these functions converge uniformly to $t\mapsto S(t,s)x$,
by boundedness of $(S(t,s))_{(t,s)\in\Delta}$. Moreover,
$t\mapsto\mathop{\frac{\partial}{\partial{t}}}S(t,s)x_{n}$ is continuous for
all $n\in\mathbb{N}$ and these functions converge uniformly by boundedness of
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$. Thus, $t\mapsto S(t,s)x$ is
continuously differentiable with

$\frac{\partial}{\partial{t}}S(t,s)x=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}.$

Since the right-hand side is continuous in $(t,s)$, we have that
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{t}}}S(t,s)x$ is continuous.
Moreover,

$\left.\mathop{\frac{\partial}{\partial{t}}}S(t,s)x\right|_{t=s}=\pi_{2}\mathcal{U}(s,s)\begin{pmatrix}0\\\
x\end{pmatrix}=x.$

  4. (d)

We have already established that
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ exists for all $x\in Z$ and
$(t,s)\in\Delta$, and that

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x=-\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}.$

Since the right-hand side is continuous in $s$, $s\mapsto S(t,s)x$ is
continuously differentiable with

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x=-\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}.$

Since the right-hand side is continuous in $(t,s)$, we have that
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ is continuous.
Moreover,

$\left.\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\right|_{t=s}=-\pi_{1}\mathcal{U}(s,s)\begin{pmatrix}x\\\
0\end{pmatrix}=-x.$

(S2): We have already established $S(t,s)Z\subseteq D$, so since $D\subseteq
Z$ we have $S(t,s)D\subseteq D$ for all $(t,s)\in\Delta$.

Let $x\in D$. By (U4) and the formulas at the beginning of the proof we have
that $\Delta\ni(t,s)\mapsto\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}$
is twice continuously differentiable. Thus, also $\Delta\ni(t,s)\mapsto
S(t,s)x=\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\ x\end{pmatrix}$ is twice
continuously differentiable. For the second derivatives, with (U4) we compute:

  1. (a)

$\displaystyle\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)x$
$\displaystyle=\frac{\partial^{2}}{\partial{t^{2}}}\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\frac{\partial}{\partial{t}}\pi_{1}\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\frac{\partial}{\partial{t}}\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}$
$\displaystyle=\frac{\partial}{\partial{t}}\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{2}\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{2}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=A(t)S(t,s)x.$

  2. (b)

$\displaystyle\frac{\partial^{2}}{\partial{s^{2}}}S(t,s)x$
$\displaystyle=\pi_{1}\frac{\partial^{2}}{\partial{s^{2}}}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\pi_{1}\frac{\partial}{\partial{s}}\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\pi_{1}\frac{\partial}{\partial{s}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}$
$\displaystyle=\pi_{1}\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\
0\end{pmatrix}=\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}0\\\
A(s)x\end{pmatrix}=S(t,s)A(s)x.$

  3. (c)

$\displaystyle\frac{\partial}{\partial{s}}\frac{\partial}{\partial{t}}S(t,s)x$
$\displaystyle=\frac{\partial}{\partial{s}}\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}x\\\ 0\end{pmatrix}$

This also yields the last assertion by (U1).

(S3): Let $(t,s)\in\Delta$ and $x\in D$. Then by (U4) we have
$\frac{\partial}{\partial{s}}S(t,s)x=-\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}\in D$ by (U3).

Moreover, by (U4) we observe that
$\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x$ and
$\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x$
exist and we can compute these derivatives:

  1. (a)

$\displaystyle\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x$
$\displaystyle=-\frac{\partial^{2}}{\partial{t^{2}}}\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}=-\frac{\partial}{\partial{t}}\pi_{1}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}=-\frac{\partial}{\partial{t}}\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}$
$\displaystyle=-\pi_{2}\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}=A(t)\Bigl{(}-\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}\Bigr{)}=A(t)\frac{\partial}{\partial{s}}S(t,s)x.$

  2. (b)

$\displaystyle\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x$
$\displaystyle=\frac{\partial^{2}}{\partial{s^{2}}}\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\frac{\partial}{\partial{s}}\pi_{2}\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}0\\\
x\end{pmatrix}=-\frac{\partial}{\partial{s}}\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}x\\\
0\end{pmatrix}$
$\displaystyle=\pi_{2}\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\
0\end{pmatrix}=\pi_{2}\mathcal{U}(t,s)\begin{pmatrix}0\\\
A(s)x\end{pmatrix}=\frac{\partial}{\partial{t}}S(t,s)A(s)x,$

where the last equality follows from (S1)(c).

  3. (c)

The continuity of $\Delta\ni(t,s)\mapsto
A(t)\frac{\partial}{\partial{s}}S(t,s)x$ follows from strong continuity of
$t\mapsto A(t)y$ for all $y\in D$, continuity of
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ for all $x\in D$
and the fact that $\mathop{\frac{\partial}{\partial{s}}}S(t,s)D\subseteq D$.

(S4): Let $(t,s),(s,r)\in\Delta$ and $x\in D$. Then, by (U1), we compute

$\displaystyle\left(-\frac{\partial}{\partial{s}}S(t,s)\right)S(s,r)x+S(t,s)\frac{\partial}{\partial{s}}S(s,r)x$
$\displaystyle=\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}\pi_{1}\\\
\pi_{2}\end{pmatrix}\mathcal{U}(s,r)\begin{pmatrix}0\\\ x\end{pmatrix}$
$\displaystyle=\pi_{1}\mathcal{U}(t,s)\mathcal{U}(s,r)\begin{pmatrix}0\\\
x\end{pmatrix}=\pi_{1}\mathcal{U}(t,r)\begin{pmatrix}0\\\
x\end{pmatrix}=S(t,r)x.$

Let us prove the converse implication (a)$\Longrightarrow$(b).

For $(t,s)\in\Delta$ and $\begin{pmatrix}x\\\ y\end{pmatrix}\in D\times X$ we
define

$\displaystyle\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}:=\begin{pmatrix}-\frac{\partial}{\partial s}S(t,s)x+S(t,s)y\\\
-\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)x+\frac{\partial}{\partial
t}S(t,s)y\end{pmatrix}=\begin{pmatrix}-\frac{\partial}{\partial
s}S(t,s)&S(t,s)\\\ -\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)&\frac{\partial}{\partial t}S(t,s)\end{pmatrix}\begin{pmatrix}x\\\
y\end{pmatrix}\in Z\times X.$

We first show that $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ can be extended to a
family in $\mathcal{L}(\mathcal{Z})$. Let $x\in Z$. Then there exists
$(x_{n})$ in $D$ such that $x_{n}\to x$ in $Z$. Since
$S(t,s)\in\mathcal{L}(X,Z)$ by Lemma 3.3(a), by (S2) and $D\subseteq Z$ we
have that the functions $s\mapsto S(t,s)x_{n}\in Z$ converge pointwise to
$s\mapsto S(t,s)x\in Z$. Moreover, we have

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x_{n}=\mathop{\frac{\partial}{\partial{s}}}S(s,s)x_{n}+\int_{s}^{t}\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(\tau,s)x_{n}\,\mathrm{d}\tau=-x_{n}+\int_{s}^{t}\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(\tau,s)x_{n}\,\mathrm{d}\tau$

for all $n\in\mathbb{N}$, by (S1)(c). By assumption, we have that the
functions
$t\mapsto\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x_{n}$
converge uniformly to
$t\mapsto\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$.
Thus, also the functions
$t\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x_{n}\in Z$ converge
uniformly. Thus, $t\mapsto S(t,s)x$ is differentiable. Lemma 3.3(b) now yields
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)\in\mathcal{L}(Z)$ for all
$(t,s)\in\Delta$. As
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)\in\mathcal{L}(Z,X)$
for all $(t,s)\in\Delta$ by assumption, we can continuously extend
$\mathcal{U}(t,s)$ to $\mathcal{Z}$ for all $(t,s)\in\Delta$.

We now check (U1)–(U4).

(U1): By (S1)(c)–(d) and (S2)(c), for $\begin{pmatrix}x\\\ y\end{pmatrix}\in
D\times X$ we obtain $\mathcal{U}(t,t)\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}x\\\ y\end{pmatrix}$ for all $t\in[0,T]$. Thus,
by continuity and denseness of $D$ in $Z$, $\mathcal{U}(t,t)=I$ for all
$t\in[0,T]$.

Let $\begin{pmatrix}x\\\ y\end{pmatrix}\in D\times D$ and
$(t,s),(s,r)\in\Delta$. By (S3) we have
$\mathop{\frac{\partial}{\partial{r}}}S(s,r)x\in D\subseteq Z$, and
$S(s,r)y\in Z$ by (S2). Thus, by (S4), we observe

$\displaystyle\mathcal{U}(t,s)\mathcal{U}(s,r)\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}-\frac{\partial}{\partial s}S(t,s)&S(t,s)\\\
-\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)&\frac{\partial}{\partial
t}S(t,s)\end{pmatrix}\begin{pmatrix}-\frac{\partial}{\partial
r}S(s,r)x+S(s,r)y\\\ -\frac{\partial}{\partial s}\frac{\partial}{\partial
r}S(s,r)x+\frac{\partial}{\partial s}S(s,r)y\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\left(\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)\mathop{\frac{\partial}{\partial{r}}}S(s,r)x-S(t,s)\mathop{\frac{\partial}{\partial{s}}}\mathop{\frac{\partial}{\partial{r}}}S(s,r)x+\left(-\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)S(s,r)y+S(t,s)\mathop{\frac{\partial}{\partial{s}}}S(s,r)y\\\
\left(\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)\right)\frac{\partial}{\partial r}S(s,r)x-\frac{\partial}{\partial
t}S(t,s)\frac{\partial}{\partial s}\frac{\partial}{\partial
r}S(s,r)x-\left(\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)\right)S(s,r)y+\frac{\partial}{\partial
t}S(t,s)\frac{\partial}{\partial s}S(s,r)y\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-\mathop{\frac{\partial}{\partial{r}}}\left(\left(-\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)S(s,r)x+S(t,s)\mathop{\frac{\partial}{\partial{s}}}S(s,r)x\right)+\left(-\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)S(s,r)y+S(t,s)\mathop{\frac{\partial}{\partial{s}}}S(s,r)y\\\
-\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{r}}}\left(\left(-\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)S(s,r)x+S(t,s)\mathop{\frac{\partial}{\partial{s}}}S(s,r)x\right)+\mathop{\frac{\partial}{\partial{t}}}\left(\left(-\mathop{\frac{\partial}{\partial{s}}}S(t,s)\right)S(s,r)+S(t,s)\mathop{\frac{\partial}{\partial{s}}}S(s,r)\right)y\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}-\frac{\partial}{\partial
r}S(t,r)x+S(t,r)y\\\ -\frac{\partial}{\partial t}\frac{\partial}{\partial
r}S(t,r)x+\frac{\partial}{\partial
t}S(t,r)y\end{pmatrix}=\mathcal{U}(t,r)\begin{pmatrix}x\\\ y\end{pmatrix}.$

Since $D\times D$ is dense in $\mathcal{Z}$ and $\mathcal{U}(t,r)$,
$\mathcal{U}(t,s)$ and $\mathcal{U}(s,r)$ are continuous, we obtain the
assertion.

(U2): We show componentwise the strong continuity of
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$. To do so for the first component, let
$\begin{pmatrix}x\\\ y\end{pmatrix}\in Z\times X$, then

$\pi_{1}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\frac{\partial}{\partial{s}}S(t,s)x+S(t,s)y\in Z.$

By assumption on $(S(t,s))_{(t,s)\in\Delta}$, we obtain strong continuity in
the first component.

Likewise, for $\begin{pmatrix}x\\\ y\end{pmatrix}\in D\times X$ we observe
that

$\pi_{2}\mathcal{U}(t,s)\binom{x}{y}=-\frac{\partial}{\partial
t}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial}{\partial t}S(t,s)y.$

By (S2) we know that $(t,s)\mapsto S(t,s)x$ is twice continuously
differentiable. As we also assumed that
$\|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\|_{X}\leq
C\|x\|_{Z}$ for all $x\in Z$ and $(t,s)\in\Delta$ we also conclude the strong
continuity of the first term. For the second term, (S1)(c) directly yields the
strong continuity. Thus, we obtain strong continuity in the second component.

(U3): Let $\begin{pmatrix}x\\\ y\end{pmatrix}\in\mathcal{D}$ and
$(t,s)\in\Delta$. Then $x\in D$ and therefore $\frac{\partial}{\partial
s}S(t,s)x\in D$ by (S3). Moreover, $S(t,s)y\in D$ by assumption, and
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in
Z$ and $\mathop{\frac{\partial}{\partial{t}}}S(t,s)y\in Z$ by assumption.
Hence,

$\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}\frac{\partial}{\partial s}S(t,s)x+S(t,s)y\\\
-\frac{\partial}{\partial t}\frac{\partial}{\partial
s}S(t,s)x+\frac{\partial}{\partial t}S(t,s)y\end{pmatrix}\in\mathcal{D}.$

(U4): Let $\begin{pmatrix}x\\\ y\end{pmatrix}\in\mathcal{D}$. Then
$\Delta\ni(t,s)\mapsto S(t,s)x$ is two times continuously differentiable by
(S2), and
$\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x$
exists by (S3). Moreover, by assumption
$\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)y$ exists and
$\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)y=A(t)S(t,s)y$. Thus, we observe
that

$\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}-\frac{\partial}{\partial{t}}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial}{\partial{t}}S(t,s)y\\\
-\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)y\end{pmatrix}$

exists, and by assumption and (S3)(a) we conclude

$\displaystyle\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\frac{\partial}{\partial{t}}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial}{\partial{t}}S(t,s)y\\\
-\frac{\partial^{2}}{\partial{t^{2}}}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial^{2}}{\partial{t^{2}}}S(t,s)y\end{pmatrix}=\begin{pmatrix}-\frac{\partial}{\partial{t}}\frac{\partial}{\partial{s}}S(t,s)x+\frac{\partial}{\partial{t}}S(t,s)y\\\
-A(t)\frac{\partial}{\partial{s}}S(t,s)x+A(t)S(t,s)y\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&I\\\
A(t)&0\end{pmatrix}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\mathcal{A}(t)\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}.$

To show the other statement, we first let $\begin{pmatrix}x\\\
y\end{pmatrix}\in D\times D$. Then $\Delta\ni(t,s)\mapsto S(t,s)x$ and
$\Delta\ni(t,s)\mapsto S(t,s)y$ are two times continuously differentiable by
(S2). Moreover,
$\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x$
exists by (S3). Thus,

$\frac{\partial}{\partial{s}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}-\frac{\partial^{2}}{\partial{s^{2}}}S(t,s)x+\frac{\partial}{\partial{s}}S(t,s)y\\\
-\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x+\frac{\partial}{\partial{s}}\frac{\partial}{\partial{t}}S(t,s)y\end{pmatrix}$

exists, and by (S2)(b) and (S3)(b) we conclude

$\displaystyle\frac{\partial}{\partial{s}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\frac{\partial^{2}}{\partial{s^{2}}}S(t,s)x+\frac{\partial}{\partial{s}}S(t,s)y\\\
-\frac{\partial^{2}}{\partial{s^{2}}}\frac{\partial}{\partial{t}}S(t,s)x+\frac{\partial}{\partial{s}}\frac{\partial}{\partial{t}}S(t,s)y\end{pmatrix}=\begin{pmatrix}-S(t,s)A(s)x+\frac{\partial}{\partial{s}}S(t,s)y\\\
-\frac{\partial}{\partial{t}}S(t,s)A(s)x+\frac{\partial}{\partial{s}}\frac{\partial}{\partial{t}}S(t,s)y\end{pmatrix}$
$\displaystyle=-\mathcal{U}(t,s)\begin{pmatrix}0&I\\\
A(s)&0\end{pmatrix}\begin{pmatrix}x\\\
y\end{pmatrix}=-\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\
y\end{pmatrix}.$

Now, let $\begin{pmatrix}x\\\ y\end{pmatrix}\in\mathcal{D}$. There exists
$(y_{n})$ in $D$ such that $y_{n}\to y$ in $Z$. Then the sequence of
continuous functions $s\mapsto\mathcal{U}(t,s)\begin{pmatrix}x\\\
y_{n}\end{pmatrix}$ converge uniformly to
$s\mapsto\mathcal{U}(t,s)\begin{pmatrix}x\\\ y\end{pmatrix}$ by boundedness of
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$. Moreover, the sequence of functions
$t\mapsto\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y_{n}\end{pmatrix}=\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\
y_{n}\end{pmatrix}=\mathcal{U}(t,s)\begin{pmatrix}y_{n}\\\ A(s)x\end{pmatrix}$
converges uniformly to
$s\mapsto\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\ y\end{pmatrix}$,
since $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is bounded and $s\mapsto A(s)x$ is
continuous. Thus,
$\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}$ exists and

$\mathop{\frac{\partial}{\partial{s}}}\mathcal{U}(t,s)\begin{pmatrix}x\\\
y\end{pmatrix}=\mathcal{U}(t,s)\mathcal{A}(s)\begin{pmatrix}x\\\
y\end{pmatrix}.$

∎

######  Remark 3.5.

If Theorem 3.4(a) is true the statement in Proposition 2.10 can be extended to
$y\in Z$.

######  Remark 3.6 (The Choice of the Space $Z$).

Let us remark on the choice of the space $Z$.

  1. (a)

Let us consider the autonomous case, i.e., let $A$ be a densely defined closed
operator and $A(t)=A$ for all $t\in[0,T]$. Then $Z$ is the Kisyński-space
given by

$Z=\\{x\in X:\ S(\cdot,\cdot)x\in C(\Delta;D)\\},$

cf. [17] or [3, Theorem 3.14.11]. Note that in this case the space is uniquely
defined. In this case, $A$ generates a so-called cosine family.

  2. (b)

For all $t\in[0,T]$ let $A(t)$ generate a cosine family. Then by [43],
w.l.o.g. we may assume that there exists $(B(t))_{t\in[0,T]}$ such that
$B(t)^{2}=A(t)$ for all $t\in[0,T]$. Let us assume that $(B(t))_{t\in[0,T]}$
satisfies Assumption 2.1 and Assumption 3.1. Then, under some further
assumptions, the space $Z$ can be chosen to be $Z:=\mathrm{D}(B(t))$ for
some/all $t\in[0,T]$ (equipped with the graph norm); cf. [18, 19, 20].

As we have seen in the above remark, in the autonomous case we have an
explicit description of the space $Z$.

######  Conjecture 3.7.

Let $(S(t,s))_{(t,s)\in\Delta}$ be an evolutionary fundamental solution on $X$
of (nACP2) associated to $(A(t),\mathrm{D}(A(t)))_{t\in[0,T]}$. We conjecture
that $Z$ has the form

$Z=\\{x\in X:\ S(\cdot,\cdot)x\in\mathrm{C}(\Delta;D)\\}$

equipped with the norm $\|\cdot\|_{Z}$ given by

$\|x\|_{Z}:=\|x\|_{X}+\sup_{(t,s)\in\Delta}\|A(t)S(t,s)x\|_{X}.$

##  4\. A Bounded Perturbation Type Result

Let $(B(t))_{t\in\left[0,T\right]}$ be a family of bounded operators on $X$.
We first prove a first-order non-autonomous bounded perturbation type result
similar to [6, Chapter VI, Cor. 9.20] which fits into our framework of
fundamental solutions.

######  Proposition 4.1.

Let $(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ be a family of densely
defined closed operators on a Banach space $X$ satisfying Assumption 2.1,
Assumption 3.1 and Assumption 3.2. Let $(S(t,s))_{(t,s)\in\Delta}$ be an
evolutionary fundamental solution of the corresponding (nACP) on $X$ and let
$Z$ be the space occuring in Theorem 3.4 such that for all $(t,s)\in\Delta$ we
have

  * •

$S(t,s)X\subseteq Z$, $S(t,s)Z\subseteq D$, $(t,s)\mapsto S(t,s)x\in Z$ is
continuous for all $x\in X$,

  * •

$\mathop{\frac{\partial}{\partial{t}}}S(t,s)Z\subseteq Z$,
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x$ exists for all $x\in Z$ and
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x=A(t)S(t,s)x$,

  * •

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ exists for all $x\in Z$,
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)Z\subseteq Z$ and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in Z$ is continuous
for all $x\in Z$,

  * •

$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)D\subseteq
Z$,
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$
exists for all $x\in Z$ and there exists $C\geq 0$ such that
$\|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\|_{X}\leq
C\|x\|_{Z}$ for all $x\in Z$ and $(t,s)\in\Delta$.

Let $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ be a fundamental solution of (nACP)
on $\mathcal{Z}:=Z\times X$ corresponding to
$(\mathcal{A}(t),\mathrm{D}(\mathcal{A}(t)))_{t\in\left[0,T\right]}$. Let
$B(\cdot)\in\mathrm{C}\left(\left[0,T\right];\mathscr{L}_{\mathrm{s}}(Z)\right)$.
Then there exists a fundamental solution $(\mathcal{V}(t,s))_{(t,s)\in\Delta}$
of (nACP) on $\mathcal{Z}$ corresponding to the family of operators
$(\mathcal{A}(t)+\mathcal{B}(t),\mathrm{D}(\mathcal{A}(t)))_{t\in\left[0,T\right]}$,
where

$\mathcal{B}(t):=\begin{pmatrix}0&0\\\ B(t)&0\end{pmatrix},\quad
t\in\left[0,T\right].$

###### Proof.

Firstly, by Theorem 3.4 we know that $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is
indeed a fundamental solution of (nACP) on $\mathcal{Z}=Z\times X$. By [6,
Chapter VI, Cor. 9.20] there exists a family of operators
$(\mathcal{V}(t,s))_{(t,s)\in\Delta}$ satisfying (U1) and (U2). Moreover, we
know that the variation of constants formula holds, i.e., one has that

(4.1)
$\displaystyle\begin{split}\mathcal{V}(t,s)\binom{z}{x}&=\mathcal{U}(t,s)\binom{z}{x}+\int_{s}^{t}{\mathcal{U}(t,r)\mathcal{B}(r)\mathcal{V}(r,s)\binom{z}{x}\
\mathrm{d}{r}}\\\
&=\mathcal{U}(t,s)\binom{z}{x}+\int_{s}^{t}{\mathcal{V}(t,r)\mathcal{B}(r)\mathcal{U}(r,s)\binom{z}{x}\
\mathrm{d}{r}},\end{split}$

for all $z\in Z$ and $x\in X$. In order to show that
$(\mathcal{V}(t,s))_{(t,s)\in\Delta}$ is a fundamental solution according to
Definition 2.3 we have to show that also (U3) and (U4) hold. To do so, let
$\binom{x}{y}\in\mathcal{D}:=D\times Z$ be arbitrary. Then
$\mathcal{U}(t,s)\binom{x}{y}\in\mathcal{D}$ by the assumption that
$(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ is a fundamental solution, cf.
Definition 2.3. As $(\mathcal{V}(t,s))_{(t,s)\in\Delta}$ is a family of
operators in $Z\times X$ we obviously have $\mathcal{V}(t,s)(D\times
X)\subseteq Z\times X$. Now, we observe that by the assumption that
$B(\cdot)\in\mathrm{C}\left(\left[0,T\right];\mathscr{L}_{\mathrm{s}}(Z)\right)$
and the explicit representation of the operators we have
$\mathcal{B}(t)(Z\times X)\subseteq\left\\{0\right\\}\times Z\subseteq D\times
Z$ so that the integral term appearing in the variation of constant formula
(4.1) is in $D\times Z$ as well. This shows that (U3) holds. For (U4), let
$\binom{x}{y}\in\mathcal{D}$. Then we have

$\displaystyle\frac{\partial}{\partial{t}}\mathcal{V}(t,s)\binom{x}{y}$
$\displaystyle=\frac{\partial}{\partial{t}}\mathcal{U}(t,s)\binom{x}{y}+\frac{\partial}{\partial{t}}\int_{s}^{t}{\mathcal{U}(t,r)\mathcal{B}(r)\mathcal{V}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=\mathcal{A}(t)\mathcal{U}(t,s)\binom{x}{y}+\mathcal{U}(t,t)\mathcal{B}(t)\mathcal{V}(t,s)\binom{x}{y}+\int_{s}^{t}{\frac{\partial}{\partial{t}}\mathcal{U}(t,r)\mathcal{B}(r)\mathcal{V}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=\mathcal{A}(t)\mathcal{U}(t,s)\binom{x}{y}+\mathcal{B}(t)\mathcal{V}(t,s)\binom{x}{y}+\int_{s}^{t}{\mathcal{A}(t)\mathcal{U}(t,r)\mathcal{B}(r)\mathcal{V}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=\mathcal{A}(t)\mathcal{U}(t,s)\binom{x}{y}+\mathcal{B}(t)\mathcal{V}(t,s)\binom{x}{y}+\mathcal{A}(t)\int_{s}^{t}{\mathcal{U}(t,r)\mathcal{B}(r)\mathcal{V}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=(\mathcal{A}(t)+\mathcal{B}(t))\mathcal{V}(t,s)\binom{x}{y},$

where we have used Hille’s theorem. Moreover,

$\displaystyle\frac{\partial}{\partial{s}}\mathcal{V}(t,s)\binom{x}{y}$
$\displaystyle=\frac{\partial}{\partial{s}}\mathcal{U}(t,s)\binom{x}{y}+\frac{\partial}{\partial{s}}\int_{s}^{t}{\mathcal{V}(t,r)\mathcal{B}(r)\mathcal{U}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=-\mathcal{U}(t,s)\mathcal{A}(s)\binom{x}{y}-\mathcal{V}(t,s)\mathcal{B}(s)\mathcal{U}(s,s)\binom{x}{y}+\int_{s}^{t}{\frac{\partial}{\partial{s}}\mathcal{V}(t,r)\mathcal{B}(r)\mathcal{U}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=-\mathcal{U}(t,s)\mathcal{A}(s)\binom{x}{y}-\mathcal{V}(t,s)\mathcal{B}(s)\binom{x}{y}+\int_{s}^{t}{\mathcal{V}(t,r)\mathcal{B}(r)\frac{\partial}{\partial{t}}\mathcal{U}(r,s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=-\mathcal{U}(t,s)\mathcal{A}(s)\binom{x}{y}-\mathcal{V}(t,s)\mathcal{B}(s)\binom{x}{y}-\int_{s}^{t}{\mathcal{V}(t,r)\mathcal{B}(r)\mathcal{U}(r,s)\mathcal{A}(s)\binom{x}{y}\
\mathrm{d}{r}}$
$\displaystyle=-\mathcal{V}(t,s)(\mathcal{A}(s)+\mathcal{B}(s))\binom{x}{y}.$

Hence, we conclude that there exists a fundamental solution of (nACP)
corresponding to the family of operators
$(\mathcal{A}(t)+\mathcal{B}(t),\mathrm{D}(\mathcal{A}(t)))_{t\in\left[0,T\right]}$
according to Definition 2.3. ∎

Now, by using Theorem 3.4 in combination with Proposition 4.1 we obtain the
following result.

######  Theorem 4.2.

Let $(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ be a family of densely
defined closed operators on a Banach space $X$ satisfying Assumption 2.1,
Assumption 3.1 and Assumption 3.2. Let $(S(t,s))_{(t,s)\in\Delta}$ be an
evolutionary fundamental solution of (nACP2) such that for all
$(t,s)\in\Delta$ we have

  * •

$S(t,s)X\subseteq Z$, $S(t,s)Z\subseteq D$, $(t,s)\mapsto S(t,s)x\in Z$ is
continuous for all $x\in X$,

  * •

$\mathop{\frac{\partial}{\partial{t}}}S(t,s)Z\subseteq Z$,
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x$ exists for all $x\in Z$ and
$\frac{\partial^{2}}{\partial t^{2}}S(t,s)x=A(t)S(t,s)x$,

  * •

$\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$ exists for all $x\in Z$,
$\mathop{\frac{\partial}{\partial{s}}}S(t,s)Z\subseteq Z$ and
$(t,s)\mapsto\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\in Z$ is continuous
for all $x\in Z$,

  * •

$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)D\subseteq
Z$,
$\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x$
exists for all $x\in Z$ and there exists $C\geq 0$ such that
$\|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}S(t,s)x\|_{X}\leq
C\|x\|_{Z}$ for all $x\in Z$ and $(t,s)\in\Delta$.

Let
$B(\cdot)\in\mathrm{C}\left(\left[0,T\right];\mathscr{L}_{\mathrm{s}}(X)\right)\cap\mathrm{C}\left(\left[0,T\right];\mathscr{L}_{\mathrm{s}}(Z)\right)$.
Then there exists an evolutionary fundamental solution of (nACP2) on $X$
associated to a family of operator
$(A(t)+B(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$.

###### Proof.

By the assumptions on the evolutionary fundamental solution
$(S(t,s))_{(t,s)\in\Delta}$ and Theorem 3.4, there exists a fundamental
solution $(\mathcal{U}(t,s))_{(t,s)\in\Delta}$ of (nACP) on
$\mathcal{Z}:=Z\times X$. For $t\in\left[0,T\right]$ we define a family of
operators on $\mathcal{Z}=Z\times X$ by

$\displaystyle\mathcal{B}(t):=\begin{pmatrix}0&0\\\ B(t)&0\end{pmatrix}.$

By Proposition 4.1 we obtain a fundamental solution
$(\mathcal{V}(t,s))_{(t,s)\in\Delta}$ of (nACP) on $\mathcal{Z}$ corresponding
to the family of operators
$(\mathcal{A}(t)+\mathcal{B}(t),\mathrm{D}(\mathcal{A}(t)))_{t\in\left[0,T\right]}$.
By using Theorem 3.4 again we conclude the result. ∎

######  Remark 4.3.

We observe that we do not need any additional assumption on
$(S(t,s))_{(t,s)\in\Delta}$ for Theorem 4.2 besides the ones already appearing
in Theorem 3.4. Hence, perturbation just relies on the continuity assumption
on the perturbing operators.

##  5\. Example: Non-Autonomous Wave Equation

Motivated by [12, Sec. 5], we consider the following perturbed non-autonomous
wave equation on $\mathrm{L}^{2}\left(0,\pi\right)$ given by

(5.1) $\displaystyle\begin{cases}\frac{\partial^{2}}{\partial
t^{2}}w(t,\xi)=\alpha(t)\frac{\partial^{2}}{\partial\xi^{2}}w(t,\xi)+\beta(t,\xi)w(t,\xi),&\quad
t\in\left(0,T\right],\ \xi\in\left(0,\pi\right),\\\ w(t,0)=w(t,\pi)=0,&\quad
t\in\left(0,T\right],\\\ w(0,\xi)=\varphi(\xi),&\\\ \frac{\partial}{\partial
t}w(0,\xi)=\psi(\xi),&\quad\xi\in\left[0,\pi\right],\end{cases}$

where $T>0$, $\varphi,\psi\in\mathrm{L}^{2}\left(0,\pi\right)$ and
$\alpha\colon\left[0,T\right]\to\mathbb{R}$ is continuously differentiable
such that $\alpha(t)\geq 1$ for all $t\in\left[0,T\right]$. Moreover, we
assume that $\beta\in\mathrm{C}^{2}([0,T]\times\left[0,\pi\right])$. For
$t\in\left[0,T\right]$, we define a family of operators
$(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$ on
$X:=\mathrm{L}^{2}\left(0,\pi\right)$ by $A(t)=\alpha(t)A_{0}$ with dense
domain $\mathrm{D}(A(t))=\mathrm{D}(A_{0})=:D$, $t\in\left[0,T\right]$, where

$A_{0}f:=f^{\prime\prime},\quad\mathrm{D}(A_{0}):=\left\\{f\in\mathrm{H}^{2}\left(0,\pi\right):\
f(0)=f(\pi)=0\right\\}$

is the Dirichlet Laplacian. In particular, Assumption 2.1 is satisfied. Since
$\alpha$ is continuous, there exists $C>0$ such that $\alpha(t)\leq C$ for all
$t\in[0,T]$, and therefore by the assumption $\alpha\geq 1$, we observe

$\|\cdot\|_{A_{0}}\leq\|\cdot\|_{A(t)}\leq C\|\cdot\|_{A_{0}}$

for all $t\in[0,T]$, which implies that Assumption 3.1 holds. Moreover, as
$\alpha$ is continuously differentiable, Assumption 3.2 is satisfied as well.

Furthermore, we introduce a family of bounded operators
$(B(t))_{t\in\left[0,T\right]}$ by

$B(t)f:=\beta(t,\cdot)f,\quad t\in\left[0,T\right]$

Then (5.1) has an abstract form of a non-autonomous second-order abstract
Cauchy problem

(5.2) $\begin{cases}\ddot{u}(t)=(A(t)+B(t))u(t),&\quad
t\in\left(0,T\right],\\\ u(0)=\varphi,&\\\ \dot{u}(0)=\psi.&\end{cases}$

As elaborated by Henríquez and Pozo in [12], for $n\in\mathbb{N}$ let
$z_{n}\colon[0,\pi]\to\mathbb{R}$,
$z_{n}(\xi):=\sqrt{\frac{2}{\pi}}\sin(n\xi)$. Then $(z_{n})_{n\in\mathbb{N}}$
is an orthonormal basis of $\mathrm{L}^{2}(0,\pi)$ of eigenfunctions of
$A_{0}$ corresponding to the sequence of eigenvalues
$(-n^{2})_{n\in\mathbb{N}}$ and the family of bounded linear operators
$(S(t,s))_{(t,s)\in\Delta}$ on $\mathrm{L}^{2}\left(0,\pi\right)$ defined by

$S(t,s)x:=\sum_{n=1}^{\infty}{r_{n}(t,s)\left\langle x,z_{n}\right\rangle
z_{n}},$

provides a fundamental solution to the second-order non-autonomous abstract
Cauchy problem associated to $(A(t),\mathrm{D}(A(t)))_{t\in\left[0,T\right]}$,
where the functions $r_{n}$ denote the solution of the initial value problem

(5.3)
$\displaystyle\begin{cases}r^{\prime\prime}(t)+n^{2}\alpha(t)r(t)=0,&\quad
0\leq s\leq t\leq T,\\\ r(s)=0,&\\\ r^{\prime}(s)=1.&\end{cases}$

Note that we have
$|r_{n}(t,s)|\leq\frac{1}{\sqrt{\alpha(s)}n}\leq\frac{1}{n}$,
$|\mathop{\frac{\partial}{\partial{t}}}r_{n}(t,s)|\leq 1$,
$|\mathop{\frac{\partial}{\partial{s}}}r_{n}(t,s)|\leq 1$ and
$|\mathop{\frac{\partial}{\partial{t}}}\mathop{\frac{\partial}{\partial{s}}}r_{n}(t,s)|\leq
n$ for all $(t,s)\in\Delta$ and $n\in\mathbb{N}$; cf. [12, (5.12)] as well as
[29]. By spectral theory,

$D=\\{x\in\mathrm{L}^{2}(0,\pi):\;\sum_{n=1}^{\infty}n^{4}\langle
x,z_{n}\rangle^{2}<\infty\\}.$

Let

$Z:=\mathrm{H}_{0}^{1}\left(0,\pi\right)=\left\\{f\in\mathrm{H}^{1}\left(0,\pi\right):\
f(0)=f(\pi)=0\right\\}=\\{x\in\mathrm{L}^{2}(0,\pi):\
\sum_{n=1}^{\infty}n^{2}\langle x,z_{n}\rangle^{2}<\infty\\}.$

The estimates on the $r_{n}$ imply that the hypotheses on
$(S(t,s))_{(t,s)\in\Delta}$ in Theorem 4.2 are satisfied, due to uniform
convergence of the series representations. Note that these hypotheses are the
same as those in Theorem 3.4(a).

Since $\beta\in\mathrm{C}^{2}([0,T]\times[0,\pi])$ we easily obtain
$B(\cdot)\in\mathrm{C}([0,T];\mathscr{L}_{\mathrm{s}}(X))\cap\mathrm{C}([0,T];\mathscr{L}_{\mathrm{s}}(Z))$.
Thus, Theorem 4.2 yields a fundamental solution to (5.2).

## Acknowledgment

This work is based on the research supported by the National Research
Foundation (Grant number: 150417). It is acknowledged that opinions, findings
and conclusions or recommendations expressed in any publication generated by
this supported research is that of the author(s). The National Research
Foundation accepts no liability whatsoever in this regard.

C.S. thanks for a very pleasent stay at the University of the Free State,
South Africa, where this work was done.

## References

  * [1] P. Acquistapace and B. Terreni.  A unified approach to abstract linear nonautonomous parabolic equations.  Rend. Sem. Mat. Univ. Padova, 78:47–107, 1987.
  * [2] H. Amann.  Parabolic evolution equations in interpolation and extrapolation spaces.  J. Funct. Anal., 78(2):233–270, 1988.
  * [3] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander.  Vector-valued Laplace transforms and Cauchy problems, volume 96 of Monographs in Mathematics.  Birkhäuser/Springer Basel AG, Basel, second edition, 2011.
  * [4] J. Bochenek.  Existence of the fundamental solution of a second order evolution equation.  Ann. Polon. Math., 66:15–35, 1997.  Volume dedicated to the memory of Włodzimierz Mlak.
  * [5] G. Da Prato and E. Giusti.  Una caratterizzazione dei generatori di funzioni coseno astratte.  Boll. Un. Mat. Ital. (3), 22:357–362, 1967.
  * [6] K.-J. Engel and R. Nagel.  One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2000.  With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
  * [7] D. E. Evans.  Time dependent perturbations and scattering of strongly continuous groups on Banach spaces.  Math. Ann., 221(3):275–290, 1976.
  * [8] H. O. Fattorini.  Second order linear differential equations in Banach spaces, volume 108 of North-Holland Mathematics Studies.  North-Holland Publishing Co., Amsterdam, 1985.
  * [9] J. A. Goldstein.  Semigroups of linear operators & applications.  Dover Publications, Inc., Mineola, NY, 2017.  Second edition, Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany.
  * [10] H. R. Henríquez.  Existence of solutions of non-autonomous second order functional differential equations with infinite delay.  Nonlinear Anal., 74(10):3333–3352, 2011.
  * [11] H. R. Henríquez.  Existence of solutions of the nonautonomous abstract Cauchy problem of second order.  Semigroup Forum, 87(2):277–297, 2013.
  * [12] H. R. Henríquez and J. C. Pozo.  Existence of solutions of abstract non-autonomous second order integro-differential equations.  Bound. Value Probl., pages Paper No. 168, 24, 2016.
  * [13] J. S. Howland.  Stationary scattering theory for time-dependent Hamiltonians.  Math. Ann., 207:315–335, 1974.
  * [14] T. Kato.  Integration of the equation of evolution in a Banach space.  J. Math. Soc. Japan, 5:208–234, 1953.
  * [15] T. Kato.  Abstract evolution equations of parabolic type in Banach and Hilbert spaces.  Nagoya Math. J., 19:93–125, 1961.
  * [16] T. Kato.  Linear evolution equations of “hyperbolic” type.  J. Fac. Sci. Univ. Tokyo Sect. I, 17:241–258, 1970.
  * [17] J. Kisyński.  On cosine operator functions and one-parameter groups of operators.  Studia Math., 44:93–105, 1972.
  * [18] M. Kozak.  An abstract linear second-order temporally inhomogeneous differential equation. I.  Univ. Iagel. Acta Math., 1137(31):21–30, 1994.
  * [19] M. Kozak.  An abstract second-order temporally inhomogeneous linear differential equation. II.  Univ. Iagel. Acta Math., 1169(32):263–274, 1995.
  * [20] M. Kozak.  A fundamental solution of a second-order differential equation in a Banach space.  Univ. Iagel. Acta Math., 1169(32):275–289, 1995.
  * [21] N. T. Lan.  On nonautonomous second-order differential equations on Banach space.  Int. J. Math. Math. Sci., 26(1):45–54, 2001.
  * [22] Y. Lin.  Time-dependent perturbation theory for abstract evolution equations of second order.  Studia Math., 130(3):263–274, 1998.
  * [23] D. Lutz.  On bounded time-dependent perturbations of operator cosine functions.  Aequationes Math., 23(2-3):197–203, 1981.
  * [24] I. V. Melnikova and A. Filinkov.  Abstract Cauchy problems: three approaches, volume 120 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics.  Chapman & Hall/CRC, Boca Raton, FL, 2001.
  * [25] R. Nagel.  Semigroup methods for nonautonomous Cauchy problems.  In Evolution equations (Baton Rouge, LA, 1992), volume 168 of Lecture Notes in Pure and Appl. Math., pages 301–316. Dekker, New York, 1995.
  * [26] R. Nagel and A. Rhandi.  A characterization of Lipschitz continuous evolution families on Banach spaces.  In Operator theory in function spaces and Banach lattices, volume 75 of Oper. Theory Adv. Appl., pages 275–288. Birkhäuser, Basel, 1995.
  * [27] F. Neubrander.  Well-posedness of higher order abstract Cauchy problems.  Trans. Amer. Math. Soc., 295(1):257–290, 1986.
  * [28] G. Nickel.  Evolution semigroups for nonautonomous Cauchy problems.  Abstr. Appl. Anal., 2(1-2):73–95, 1997.
  * [29] S. Noyo, R. Obaya, and J. Rojo.  Ecuaciones y Sistemas Diferenciales.  McGraw-Hill, Madrid, 1995.
  * [30] A. Pazy.  Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences.  Springer-Verlag, New York, 1983.
  * [31] S. Piskarev and S.-Y. Shaw.  Perturbation and comparison of cosine operator functions.  Semigroup Forum, 51(2):225–246, 1995.
  * [32] S. Piskarev and S.-Y. Shaw.  On certain operator families related to cosine operator functions.  Taiwanese J. Math., 1(4):527–546, 1997.
  * [33] F. Räbiger, A. Rhandi, and R. Schnaubelt.  Perturbation and an abstract characterization of evolution semigroups.  J. Math. Anal. Appl., 198(2):516–533, 1996.
  * [34] F. Räbiger and R. Schnaubelt.  The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions.  Semigroup Forum, 52(2):225–239, 1996.
  * [35] F. Räbiger, R. Schnaubelt, A. Rhandi, and J. Voigt.  Non-autonomous Miyadera perturbations.  Differential Integral Equations, 13(1-3):341–368, 2000.
  * [36] R. Schnaubelt.  Asymptotic behaviour of parabolic nonautonomous evolution equations.  In Functional analytic methods for evolution equations, volume 1855 of Lecture Notes in Math., pages 401–472. Springer, Berlin, 2004.
  * [37] H. Serizawa and M. Watanabe.  Time-dependent perturbation for cosine families in Banach spaces.  Houston J. Math., 12(4):579–586, 1986.
  * [38] M. Shimizu and I. Miyadera.  Perturbation theory for cosine families on Banach spaces.  Tokyo J. Math., 1(2):333–343, 1978.
  * [39] M. Sova.  Cosine operator functions.  Rozprawy Mat., 49:47, 1966.
  * [40] T. Takenaka and N. Okazawa.  A Phillips-Miyadera type perturbation theorem for cosine functions of operators.  Tohoku Math. J. (2), 30(1):107–115, 1978.
  * [41] H. Tanabe.  On the equations of evolution in a Banach space.  Osaka Math. J., 12:363–376, 1960.
  * [42] H. Tanabe.  Equations of evolution, volume 6 of Monographs and Studies in Mathematics.  Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.  Translated from the Japanese by N. Mugibayashi and H. Haneda.
  * [43] C. C. Travis and G. F. Webb.  Perturbation of strongly continuous cosine family generators.  Colloq. Math., 45(2):277–285, 1981.
  * [44] V. V. Vasilev and S. I. Piskarev.  Differential equations in Banach spaces. II. Theory of cosine operator functions.  J. Math. Sci. (N.Y.), 122(2):3055–3174, 2004.
  * [45] M. Veraar and J. Zimmerschied.  Non-autonomous stochastic Cauchy problems in Banach spaces.  Studia Math., 185(1):1–34, 2008.
  * [46] T. Winiarska.  Second order Cauchy problem with a damping operator.  Univ. Iagel. Acta Math., 1285(43):11–20, 2005.
  * [47] T. Winiarska.  Second order evolution problem with dependent on $t$ and not densely defined operators.  Demonstratio Math., 42(3):513–520, 2009.
  * [48] T.-J. Xiao and J. Liang.  Higher order abstract Cauchy problems: their existence and uniqueness families.  J. London Math. Soc. (2), 67(1):149–164, 2003.
  * [49] T. Xio and L. Jin.  On complete second order linear differential equations in Banach spaces.  Pacific J. Math., 142(1):175–195, 1990.


MnLargeSymbols’164 MnLargeSymbols’171

#  Dressing in AdS and a Conformal Bethe–Salpeter Equation

Sylvain Fichet<EMAIL_ADDRESS>ICTP-SAIFR & IFT-UNESP, R. Dr. Bento
Teobaldo Ferraz 271, São Paulo, Brazil

###### Abstract

We initiate the study of Dyson equations of perturbative QFT in AdS and their
consequences for large-$N$ CFT. We show that the dressed one-particle AdS
propagator features wavefunction renormalization and operator mixing, giving
rise to finite corrections to OPE data. We show how the resummation of $1/N$
effects in the CFT emerges from the dressing in AdS. When a boundary-to-bulk
propagator is dressed by propagators whose sum of conformal dimensions is
lower than the main dimension, it cannot map onto a CFT source following the
standard prescription; we relate this to an AdS/CFT version of particle
instability. We investigate the dressing of the two-particle propagator and
obtain a conformal Bethe–Salpeter equation for the conformal partial wave of a
“bound state” operator. In case of a ladder kernel, we briefly show that a
bound state with conformal dimension equal to the sum of its constituents plus
a $1/N$-suppressed “binding energy” emerges. Resummation of the Dyson
equations is essential to deriving these results.

## Introduction

The AdS/CFT correspondence establishes a profound connection between two
fields of physics: gravity and strongly-coupled gauge theories. In its most
studied form, the duality implies that the boundary amplitudes of weakly-
coupled theories of gravity in $d$+$1$-dimensional Anti-de Sitter (AdSd+1)
spacetime correspond to correlators of a strongly-coupled $d$-dimensional
conformal field theory (CFTd) with many degrees of freedom (large-$N$). The
$1/N$ expansion of the CFT correlators maps onto the perturbative expansion of
the AdS QFT amplitudes Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj ;
Freedman:1998bj ; Liu:1998ty ; Freedman:1998tz ; DHoker:1999mqo ;
DHoker:1999kzh ; Aharony:1999ti ; Zaffaroni:2000vh ; Nastase:2007kj ;
Kap:lecture ,

The concepts and tools of flat space perturbative QFT have been gradually
identified/generalized in the context of AdS/CFT. For example, the structure
of the AdS boundary “S-matrix” has been identified in Balasubramanian:1999ri ;
Giddings:1999qu . More recently, the CFT Froissard-Gribov formula for analytic
continuation in spin was found in Caron-Huot:2017vep . AdS/CFT is now studied
at loop-level Cornalba:2007zb ; Penedones:2010ue ; Fitzpatrick:2011hu ;
Alday:2017xua ; Alday:2017vkk ; Alday:2018pdi ; Alday:2018kkw ;
Meltzer:2018tnm ; Ponomarev:2019ltz ; Shyani:2019wed ; Alday:2019qrf ;
Alday:2019nin ; Meltzer:2019pyl ; Aprile:2017bgs ; Aprile:2017xsp ;
Aprile:2017qoy ; Giombi:2017hpr ; Cardona:2017tsw ; Aharony:2016dwx ;
Yuan:2017vgp ; Yuan:2018qva ; Bertan:2018afl ; Bertan:2018khc ; Liu:2018jhs ;
Carmi:2018qzm ; Aprile:2018efk ; Ghosh:2018bgd ; Mazac:2018ycv ;
Beccaria:2019stp ; Chester:2019pvm ; Beccaria:2019dju ; Carmi:2019ocp ;
Aprile:2019rep ; Fichet:2019hkg ; Meltzer:2019nbs ; Drummond:2019hel ;
Albayrak:2020isk ; Albayrak:2020bso ; Meltzer:2020qbr ; Costantino:2020vdu ;
Carmi:2021dsn , and AdS/CFT unitarity methods have been identified and
explored in Fitzpatrick:2011dm ; Meltzer:2019nbs ; Meltzer:2020qbr , both in
the space of conformal dimensions and in momentum space.

One item of the pertubative QFT toolkit has not been deeply explored yet: the
quantum dressing of AdS amplitudes induced by interactions—as described by the
Dyson equations. Its CFT counterpart corresponds to the resummation of $1/N$
effects. The aim of this note is to initiate a study of these dressing
equations and its consequences in AdS/CFT.

Some aspects related to dressing and resummation have been addressed in the
literature. A bubble resummation in the $O(N)$ and Gross-Neveu models has been
made in Carmi:2018qzm . Effects of 1PI insertions on 2pt functions have been
discussed in Meltzer:2019nbs  in the context of unitarity methods. A
resummation in AdS has been done in Costantino:2020vdu , with a focus on
propagation in timelike regime. Aspects of ladder diagrams have also been
discussed very recently in Carmi:2021dsn .

Our aim here is to initiate a systematic understanding of dressing in AdS/CFT.
The focus is more conceptual than technical—detailed calculations will be
shown elsewhere. We will work with the one- and two-particle propagators.

## Preliminary Observations

At large but finite $N$, the conformal algebra can be viewed as the
$N$=$\infty$ conformal algebra with small, $1/N$-suppressed conformal
deformations to the generators Fitzpatrick:2010zm ; Fitzpatrick:2011dm .
Denoting the free CFT generators as
$D_{0},K^{\mu}_{0},P^{\mu}_{0},M^{\mu\nu}$, the $N$= finite generators are
$D=D_{0}+D_{I}$, $K^{\mu}=K^{\mu}_{0}+K^{\mu}_{I}$,
$P^{\mu}=P^{\mu}_{0}+P^{\mu}_{I}$, with $D_{I},K^{\mu}_{I},P^{\mu}_{I}\propto
1/N$. 111$M^{\mu\nu}$ is assumed to remain unaffected by this continuous
deformation.

Using radial quantization, the CFT correlators can in principle be computed
perturbatively in $1/N$, in a way structurally similar to the flat space
$S$-matrix. The free “multiparticle” states are CFT states built from single-
trace operators of the $N$=$\infty$ algebra. These map onto free bulk fields
in AdS. In the interaction picture, the interaction piece of the Hamiltonian
(namely $D_{I}$) is exponentiated and generates the correlators in a
perturbative expansion around $N$=$\infty$. The perturbative picture is
explicitly realized in AdS, where one effectively has a weakly-coupled QFT in
the bulk, and the free multiparticle states are formed by the free bulk fields
ending on the boundary.

The irreducible representations of the $N$=$\infty$ and $N$= finite algebras
are related by $1/N$-suppressed deformations. For example, the conformal
dimensions of operators in the $N=\,$finite algebra take the form
$\Delta=\Delta_{0}+\gamma$ with the anomalous dimension
$\gamma\propto{1}/{N^{2}}$. Similarly, since the operators in $N$=$\infty$ and
$N$= finite theories differ, their respective OPE coefficients should differ
and be related by a $1/N$-suppressed deformation $c=c_{0}+\delta c$ with
$\delta c\propto{1}/{N^{2}}$.

We may also expect a notion of “operator mixing” induced by $1/N$ corrections.
How may such a mixing appear? Consider flat space QFT. At the level of
propagators, mixing between states appears from the resummation of 1PI
insertions dressing the propagator (i.e. the Born series generated by the 2pt
Dyson equation). Hence we should investigate the analogous dressing in AdS.

As another simple “invitation” to the question of dressing, consider a 2pt
correlator of a CFT with $N$= finite. Using $\Delta=\Delta_{0}+\gamma$ we can
always write

$\frac{1}{x^{2\Delta}}=\frac{1}{x^{2\Delta_{0}}}\left(1-\gamma\log
x^{2}+\frac{1}{2}\gamma^{2}\log^{2}x^{2}+\ldots\right)$ (1)

where the $1/N$-suppressed terms explicitly appear as corrections to the
correlator from the $N$=$\infty$ CFT. The form of the series is totally fixed
by the dilatation symmetry of the $N$=finite conformal algebra. We can then
wonder: How does this series emerge from the AdS dual? Since conceptually the
series Eq. (1) is generated by repeated application of the dilatation operator
of the $N=\,$finite CFT, on the AdS side the corresponding series should be
generated by repeated insertions of the interaction Hamiltonian. We can thus
expect that the above exponential series should emerge from the dressed AdS
propagator—we will show how this happens in the following. 222While a study at
the leading order of the expansion is enough to determine $\gamma$, such a
truncation is not conformally invariant—only the full exponential series is.
Hence one can expect that the resummed series will teach us more than its
leading term.

Having set the stage and motivations, we proceed to the AdS calculations.

## The Dressed One-Particle Propagator

We consider scalar fields in the Poincaré patch of $AdS_{d+1}$, with metric
$ds^{2}=(kz)^{-2}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2})$, and Fourier
transform along the constant $z$ slices to work in “Poincaré momentum space”
$(p^{\mu},z)$. The free propagator between arbitrary points is denoted
$G^{(0)}_{p}(z,z^{\prime})$, with
$G^{(0)}(X,X^{\prime})=\int\frac{d^{d}p}{(2\pi)^{d}}G^{(0)}_{p}(z,z^{\prime})$.

We use the “harmonic” (or conformal spectral) representation, which takes the
form

$G^{(0)}_{p,\alpha}(z,z^{\prime})=\int^{i\infty}_{-i\infty}d\hat{\alpha}\,P(\hat{\alpha},\alpha)\,\Omega_{\hat{\alpha}}(z,z^{\prime})\,,$
(2)

where

$P(\hat{\alpha},\alpha)=\frac{1}{\hat{\alpha}^{2}-\alpha^{2}}\,,\quad\Omega_{\hat{\alpha}}(z,z^{\prime})\propto{\cal
K}^{-}_{\hat{\alpha}}(z){\cal K}^{+}_{\hat{\alpha}}(z^{\prime})$ (3)

with ${\cal K}^{\pm}(z)$ the boundary-to-bulk propagators
($\Omega_{\hat{\alpha}}$ is the kernel of the harmonic transform). Here the
overall constants are not needed, see e.g. Costa:2014kfa ; Liu:2018jhs ;
Carmi:2018qzm ; Meltzer:2019nbs ; Costantino:2020vdu  for detailed formalism.
The ${\cal K}_{\alpha}^{\pm}(z)$ map onto CFT operators with dimension
$\Delta_{\pm}=d/2\pm\alpha$.

Let us consider the propagator dressed by generic 1PI insertions
$i\Pi(z,z^{\prime})$, including coupling constants, as pictured in Fig. 1. We
first establish its general form, using the harmonic representation for the
1PI blobs

$i\Pi(z,z^{\prime})=\int^{i\infty}_{-i\infty}d\hat{\alpha}{\cal
B}(\hat{\alpha})\,\Omega_{\hat{\alpha}}(z,z^{\prime})\,.$ (4)

We have

$\int dzdz^{\prime}{\cal K}_{\alpha}^{-}(z)i\Pi(z,z^{\prime}){\cal
K}_{\alpha^{\prime}}^{-}(z^{\prime})\propto\delta(\alpha-\alpha^{\prime})i{\cal
B}(\alpha)+s.t.$ (5)

because the l.h.s. amounts to a conformal 2pt function (where $s.t.$ stands
for shadow transform, see e.g. Karateev:2018oml ). This allows us to resum the
Born series, giving the dressed propagator

$G_{p,{\alpha}}(z,z^{\prime})=\int_{-i\infty}^{i\infty}d\hat{\alpha}\,\frac{1}{P(\hat{\alpha},\alpha)^{-1}+{\cal
B}(\hat{\alpha})}\Omega^{(0)}_{\hat{\alpha}}\,(z,z^{\prime})\,.$ (6)

The loop insertion is, in general, challenging to evaluate, see Giombi:2017hpr
; Carmi:2018qzm ; Costantino:2020vdu  for the bubble case. However, we can
establish its general form by using the AdS Kallen-Lehmann (KL)
representation. In the AdS viewpoint one inserts a complete set of
multiparticle states Dusedau:1985ue , expressing the $i\Pi(z,z^{\prime})$ as
an infinite sum over propagators with dimension of multitrace operators
(primaries and descendants). Focusing for simplicity on nonderivative
interactions, the exchanged operators are scalar. A generic $m$-tuple trace
operator takes the form ${\cal O}_{1}\partial^{2n_{1}}{\cal
O}_{2}\ldots\partial^{2n_{m-1}}{\cal O}_{m}$ with total dimension
$\sum^{m}_{i=1}\Delta_{i}+2n$ where $n$ counts the derivatives. Introducing
$\alpha_{m}=\sum^{m}_{i=1}\Delta_{i}-d/2$, the KL form of the 1PI insertion
reads

$i\Pi_{p}(z,z^{\prime})=\sum_{m\geq 2,n\geq
0}g_{m}^{2}a_{m,n}G^{(0)}_{p,\alpha_{m}+2n}(z,z^{\prime})$ (7)

where $a_{m,n}$ is the spectral function. The $m$=$1$ case is excluded from
the sum by 1PI requirement. A bubble diagram, for example, gives a series of
double-trace propagators $G^{(0)}_{\Delta_{1}+\Delta_{2}+2n-d/2}$. The
$a_{2,n}$ coefficients for the bubble were found in various ways in
Fitzpatrick:2011hu ; Costantino:2020vdu . Finally, the $g_{m}$ contain the
coupling constants from bulk vertices, scaling as $g_{m}\sim 1/N^{m-1}$ as
dictated by the mapping onto CFT correlators.

In the harmonic representation this general form of the 1PI insertion becomes

${\cal B}(\hat{\alpha})=\sum_{m\geq 2,n\geq
0}g_{m}^{2}a_{m,n}P\left(\hat{\alpha},\alpha_{m}+2n\right)\,.$ (8)

Plugging Eq. (8) into Eq. (6) gives the general form of the AdS propagator
dressed by arbitrary 1PI insertions.

We now extract information about the CFT from the dressed propagator
$G_{p,\alpha}$.

Figure 1:  The dressed one-particle propagator (blue line). Top: The 2pt Dyson
equation with 1PI insertion $i\Pi$. Bottom: The dressed propagator as a
combination of free propagators (dark lines) with different conformal
dimensions.

### Anomalous Dimension

The first bit of information we extract is the anomalous dimension at the
$\hat{\alpha}\sim\pm\alpha$ pole. Since ${\cal B}(\alpha)$ is a perturbative
correction, we obtain the correction to $\alpha$ by expanding ${\cal
B}(\alpha)$ near the pole, ${\cal B}(\hat{\alpha})={\cal B}(\alpha)+\ldots$
The corresponding CFT operator given by that pole aquires an anomalous
dimension $\gamma$, with

$\gamma=-\frac{{\cal
B}(\alpha)}{2\alpha}\,,\quad\Delta_{\pm}=\Delta^{0}_{\pm}\pm\gamma=\frac{d}{2}\pm\alpha\pm\gamma$
(9)

where $\Delta^{0}_{\pm}$ is the conformal dimension in the $N$=$\infty$
theory. This method is explicitly verified for the bubble via Giombi:2017hpr ;
Costantino:2020vdu .

### Wavefunction/OPE renormalization

We consider the next-to-leading term in the expansion of $\cal
B(\hat{\alpha})$,

${\cal B}(\hat{\alpha})={\cal
B}(\alpha)+\frac{\partial}{\partial\hat{\alpha}^{2}}{\cal
B}|_{\alpha=\hat{\alpha}}\,(\hat{\alpha}^{2}-\alpha^{2})+\ldots$ (10)

Introducing $\delta Z=\frac{\partial}{\partial\hat{\alpha}^{2}}{\cal
B}|_{\alpha=\hat{\alpha}}$, we get that the residue of the pole is corrected
by $1-\delta Z$. This is the spectral equivalent of “wavefunction
renormalization”. From the CFT viewpoint, upon unit-normalizing the 2pt
functions, the factor becomes a correction to the OPE coefficients of the
$N$=$\infty$ theory. For e.g. a 3pt function, the correction takes the form
$\delta c_{123}=-\frac{1}{2}(\delta Z_{1}+\delta Z_{2}+\delta Z_{3})c_{123}$.

One can notice a general relation between $\gamma$ and $\delta c$,

$\frac{\delta
c}{c}=\frac{1}{2}\left(\frac{\gamma}{\alpha}+\frac{\partial\gamma}{\partial\alpha}\right)\,.$
(11)

Hence for a known $\gamma$ we can readily deduce the associated correction to
the OPE coefficient.

### Operator Mixing

There is also an infinite sequence of new simple poles arising in
$G_{p,\alpha}$ because of the poles in ${\cal B}(\alpha)$, see Eq. (8). For
simplicity, consider poles far from the main one,
$|\hat{\alpha}|\sim|\alpha_{m}+2n|\gg|\alpha|$. The new poles are

$\hat{\alpha}\approx\pm\left(\alpha_{m}+2n-\frac{g^{2}_{m}a^{2}_{m,n}}{2(\alpha_{m}+2n)^{3}}\right)$
(12)

where the last term is $1/N$-suppressed. The associated residues in this limit
are

$r_{m,n}\approx\mp\frac{g^{2}_{m}a^{2}_{m,n}}{2(\alpha_{m}+2n)^{5}}$ (13)

and are thus $1/N$-suppressed. Using the definition Eq. (2) these poles
contribute to the dressed propagator as

$G_{p,\alpha}(z,z^{\prime})\supset-\sum_{m\geq 2,n\geq
0}\frac{g_{m}^{2}a^{2}_{m,n}}{(\alpha_{m}+2n)^{5}}G^{(0)}_{p,\alpha_{m}+2n}(z,z^{\prime})$
(14)

That is, an infinite sequence of multitrace propagators arises in the dressed
propagator with $1/N$-suppressed coefficients. The complete form of
$G_{p,\alpha}(z,z^{\prime})$ is obtained similarly and is summarized in Fig.
1.

These contributions to $G_{p,\alpha}(z,z^{\prime})$ introduce a notion of
mixing between the original CFT operator and the sequence of multitrace
operators in the following sense: If the main operator appears in a given OPE,
then the whole sequence of multitrace operators also appear in the OPE—with
$1/N$-suppressed coefficients. Our AdS result Eq. (14) dictates what are
precisely the coefficients.

This can be seen explicitly at the level of a 4pt exchange diagram. When
cutting on $G_{p,\alpha}(z,z^{\prime})$—in either $\alpha$-space
Meltzer:2019nbs  or momentum space Meltzer:2020qbr , one obtains a sum of
squared 3pt correlators (or transition amplitudes) over the sequence of
multitrace operators weighted by the $r_{i,n}$ residues.

### Boundary Propagators and (In)Stability in AdS/CFT

Let us consider the dressed boundary-to-bulk propagator (i.e. ${\cal K}$),
obtained from the dressed $G_{p,\alpha}(z,z^{\prime})$ by sending e.g. $z$ to
the boundary and rescaling by $z^{d/2-\alpha-\gamma}$. If the dimensions in
Eq. (14) satisfy $\alpha_{m}>\alpha$, all multitrace contributions drop faster
than the main, single-trace term when approaching the boundary, leaving only
this main operator—with dimension $\Delta_{0}+\gamma$ and wavefunction
corrected by $\delta Z$. Hence in spite of dressing, the dressed bulk field
still maps onto a source for the single-trace operator in the CFT, and thus
the standard AdS/CFT prescription remains unaffected.

In contrast, if $\alpha_{m}<\alpha$, the multitrace contribution grows faster
than the single-trace one near the boundary. Thus the usual rescaling from the
standard AdS/CFT prescription does not give a finite result. We understand
this apparent failure of the AdS/CFT prescription for $\alpha_{m}<\alpha$ as
follows. When $\alpha_{m}<\alpha$, rather than using ${\cal K}_{\alpha}$ as
external leg, one should use the ${\cal K}_{i}$ from the fields inside the 1PI
insertion—defined at some given order in perturbation theory. The ${\cal
K}_{i}$ have dimension lower than the main operator, and a subset of them
satisfies the AdS/CFT prescription at the given order in perturbation theory.
These ${\cal K}_{i}$ can thus be used as external legs.

This resolution matches the notion of particle instability in flat space QFT
Veltman:1963th ; Denner:2014zga ; Rody : if a heavy particle with mass $M$ can
decay into particles of mass $\sum^{n}_{i=1}m_{i}<M$ at some order, then the
space of final states is built from these lighter offspring particles rather
than the original particle—which is seen as unstable at this order of
perturbation theory. We have essentially obtained the analogous picture for
bulk fields/sources in AdS/CFT, where the analog of the “kinematic threshold”
is given by

$\sum^{m}_{i=1}\Delta_{i}<\Delta\,.$ (15)

This condition was also found in Giddings:1999qu . This notion of instability
in AdS/CFT deserves further study.

### Emergence of the $1/N$ Resummation in CFT

Finally, we show how the conformal series Eq. (1) appears from the AdS side.
We write the dressed propagator as a geometric series in $\alpha$ space,
$G_{p,\alpha}=\sum_{n}G_{p,\alpha}^{(n)}$, with

$G^{(n)}_{p,\alpha}(z,z^{\prime})=\int_{-i\infty}^{i\infty}d\hat{\alpha}\,P(\hat{\alpha},\alpha)\,\left[-P(\hat{\alpha},\alpha){\cal
B}(\hat{\alpha})\right]^{n}\Omega^{(0)}_{\hat{\alpha}}\,(z,z^{\prime})$ (16)

Closing the contour, there is a $n+1$-tuple pole at $\hat{\alpha}=\pm\alpha$.
Ignoring the other poles—which correspond to the operator mixing discussed
above, we have

$G^{(n)}_{p,\alpha}(z,z^{\prime})\supset\frac{(-1)^{n}}{n!}\frac{\partial^{n}}{\partial\hat{\alpha}^{n}}\left[\left(\frac{{\cal
B}(\hat{\alpha})}{2\hat{\alpha}}\right)^{n}G^{(0)}_{p,\hat{\alpha}}(z,z^{\prime})\right]_{\hat{\alpha}=\alpha}\,.$
(17)

We then focus only on the derivatives acting on $G$, because we already know
that derivatives on $\cal B$ lead to wavefunction renormalization or to
effects neglected here wip . The anomalous dimension defined in Eq. (9)
appears. The sequence of derivatives exponentiates, giving

$G_{p,\alpha}(z,z^{\prime})\supset
e^{\gamma\partial_{\alpha}}G^{(0)}_{p,\alpha}(z,z^{\prime})\,.$ (18)

Finally, we take the boundary-to-bulk limit and trade $\alpha$ for the
conformal dimension using $\Delta^{0}_{\pm}=d/2\pm\alpha$. We obtain an
exponentiated operator acting on the $N$=$\infty$ 2pt CFT correlator,

$e^{\pm\gamma\partial_{\Delta_{0}}}\frac{1}{x^{2\Delta_{0}}}\,$ (19)

whose action is to shift the conformal dimension $\Delta^{0}_{\pm}$ by
$\pm\gamma$. We have therefore reproduced the conformal series of Eq. (1) from
the AdS side, both for dimension larger and lower than $d/2$.

Reproducing the above steps by acting with one derivative on the bubble
function in Eq. (16), and using the definition Eq. (10), we similarly obtain
the $1-\delta Z$ correction to the normalization of the 2pt correlator.

## The Dressed Two-Particle Propagator

Figure 2:  The dressed two-particle propagator (blue). Top: The 4pt Dyson
equation with 2PI insertion $K$. Bottom: The boundary Bethe–Salpeter equation
with a ladder kernel.

In flat space the 4pt Dyson equation and associated Bethe–Salpeter equation
(BSE) for a weakly-coupled bound state give rise to rich physics and
challenging problems BSE_review ; Gross:1993zj . Here we present a conformal
version of the BSE and show the existence of a “bound state” in spectral
space.

The general AdS 4pt Dyson equation is shown in Fig. 2 (top). One could study
it for arbitrary bulk points, but our focus is on sending the endpoints to the
boundary. Rescaling appropriately the legs, one obtains the Witten diagram
version of the Dyson equation. The diagrams map onto 4pt CFT correlators, and
can be generically described using a decomposition over conformal partial
waves (CPW) Meltzer:2019nbs

${\cal A}^{1234}(x_{i})=\int^{i\infty}_{-i\infty}d\hat{\alpha}_{\cal
O}\rho(\hat{\alpha}_{\cal O})\Psi^{1234}_{\cal O}(x_{i})\,$ (20)

where $\rho(\alpha)$ is the OPE function. In momentum space the CPW
$\Psi^{1234}_{\cal O}$ is simply the product of 3pt functions,

$\Psi_{\cal O}^{1234}(p_{i})=\Gamma^{12{\cal
O}}(p_{1,2})\,\Gamma^{34\tilde{\cal
O}}(p_{3,4})(2\pi)^{d}\delta^{(d)}\left(\sum_{i}p_{i}\right)$ (21)

with

$\Gamma^{12{\cal O}}(p_{1,2})=\llangle{{\cal O}}_{1}(p_{1}){\cal
O}_{2}(p_{2}){\cal O}(-p_{1}-p_{2})\rrangle$ (22)

and dimensions $[{\cal O}_{i}]=d/2+\alpha_{i}$, $[\tilde{\cal
O}_{i}]=d/2-\alpha_{i}$.

Then, in analogy with the flat space BSE approach, we assume the existence of
a simple pole with dimension $\Delta_{B}$ in the OPE function of $G_{2}$,

$\rho_{G_{2}}(\hat{\alpha})\propto P(\hat{\alpha},\alpha_{B})$ (23)

for $\hat{\alpha}$ near $\alpha_{B}$, with $\Delta_{B}=d/2+\alpha_{B}$.

Finally we project the Dyson equation to focus on the exchange of operator
with dimension near $\Delta_{B}$. This can be done using e.g. a CPW, or more
directly by contracting the $3,4$ legs with a $\Gamma^{\tilde{3}\tilde{4}{\cal
O}^{\prime}}$, giving a relation between 3pt correlators in the
$\alpha\sim\alpha_{B}$ region. Because of the nearby pole in the connected
diagrams, the diagram with free propagators is negligible. We end up with a
self-consistent equation between 3pt Witten diagrams, which amounts to the
AdS/CFT version of the BSE, as shown in Fig. 2 (bottom).

We see that the role of the “vertex function” of flat space BSE is here played
by a 3pt diagram (i.e. a 3pt CFT correlator), which is fully constrained by
conformal symmetry. Thus the only unknown of our BSE is the dimension of the
“bound state” operator $\Delta_{B}$.

For a given interaction kernel, the value(s) of $\Delta_{B}$ can be extracted
from the BSE at large $N$ as follows. The vertices from the kernel are
$1/N$-suppressed. Thus each side of the BSE can match only at values of
$\Delta_{B}$ for which an $N$-enhancement cancels the $1/N$-suppression from
the vertices.

We show briefly how this happens for the simplest case of a ladder kernel
(shown in Fig. 2). Using the harmonic representation (see Eqs. (2), (3)) we
split the internal lines of the triangle loop. We then decompose the
$X$-exchange $t$-channel subdiagram onto $s$-channels using the 6j symbol
Liu:2018jhs

${\cal J}^{1,2}_{X,B}=\begin{Bmatrix}1&2&X\\\ 2&1&B\end{Bmatrix}$ (24)

where for simplicity we already restrict to legs with pairwise equal
dimension. This step is, up to normalization constants,

(25)

The conformal bubble integral gives a
$\delta(\alpha_{B}-\hat{\alpha}_{S})+s.t.$ Karateev:2018oml ; Meltzer:2020qbr
, which eliminates the $\hat{\Delta}_{S}$ integral and gives ${\cal
J}^{1,2}_{\hat{X},B}$.

The non-trivial information is inside the 6j symbol. For legs with pairwise
equal dimension, $\cal J$ contains double poles of the form
$1/(\Delta_{B}-\Delta_{1}-\Delta_{2}-2n)^{2}$ Liu:2018jhs . Thus, to cancel
the $1/N^{2}$ suppression from the two cubic vertices, $\Delta_{B}$ must take
the form $\Delta_{B}=\Delta_{1}+\Delta_{2}+2n+\delta_{B}$, where the “binding
energy” $\delta_{B}\propto 1/N$ is determined by the BSE and depends on
$\Delta_{1,2,X}$ and $n$. The picture is confirmed by explicit calculation in
$d=4$, which will be presented elsewhere. This method provides
$\delta^{2}_{B}$ but extra input seems needed to fix its sign.

The BSE is an important perturbative tool whose implications for AdS/CFT
remain to be analyzed in detail, including the $c_{12B}$ OPE coefficient, the
spectator equation (i.e. large $\Delta_{1}$), and cross-ladder diagrams. More
broadly, it is about the resummation of $1/N$ corrections in 4pt CFT
correlators, which would also be interesting to study directly from CFT
methods such as the conformal bootstrap.

## Acknowledgments

We thank David Meltzer, Alexandria Costantino and Flip Tanedo for useful
discussions and comments. The author is supported by the São Paulo Research
Foundation (FAPESP) under grants #2011/11973, #2014/21477-2 and #2018/11721-4,
by CAPES under grant #88887.194785, and by the University of California,
Riverside.

## References

  * (1) J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133, [hep-th/9711200]. [Adv. Theor. Math. Phys.2,231(1998)].
  * (2) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105–114, [hep-th/9802109].
  * (3) E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150].
  * (4) D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Comments on 4 point functions in the CFT / AdS correspondence, Phys. Lett. B 452 (1999) 61–68, [hep-th/9808006].
  * (5) H. Liu and A. A. Tseytlin, On four point functions in the CFT / AdS correspondence, Phys. Rev. D 59 (1999) 086002, [hep-th/9807097].
  * (6) D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Correlation functions in the CFT(d) / AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96–118, [hep-th/9804058].
  * (7) E. D’Hoker, D. Z. Freedman, and L. Rastelli, AdS / CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys. B 562 (1999) 395–411, [hep-th/9905049].
  * (8) E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton exchange and complete four point functions in the AdS / CFT correspondence, Nucl. Phys. B 562 (1999) 353–394, [hep-th/9903196].
  * (9) O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386, [hep-th/9905111].
  * (10) A. Zaffaroni, Introduction to the AdS-CFT correspondence, Class. Quant. Grav. 17 (2000) 3571–3597.
  * (11) H. Nastase, Introduction to AdS-CFT, arXiv:0712.0689.
  * (12) J. Kaplan, “Lectures on AdS/CFT from the Bottom Up.” https://sites.krieger.jhu.edu/jared-kaplan/files/2016/05/AdSCFTCourseNotesCurrentPublic.pdf.
  * (13) V. Balasubramanian, S. B. Giddings, and A. E. Lawrence, What do CFTs tell us about Anti-de Sitter space-times?, JHEP 03 (1999) 001, [hep-th/9902052].
  * (14) S. B. Giddings, The Boundary S matrix and the AdS to CFT dictionary, Phys. Rev. Lett. 83 (1999) 2707–2710, [hep-th/9903048].
  * (15) S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078, [arXiv:1703.0027].
  * (16) L. Cornalba, M. S. Costa, and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP 09 (2007) 037, [arXiv:0707.0120].
  * (17) J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025, [arXiv:1011.1485].
  * (18) A. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-Matrix, JHEP 10 (2012) 127, [arXiv:1111.6972].
  * (19) L. F. Alday and A. Bissi, Loop Corrections to Supergravity on $AdS_{5}\times S^{5}$, Phys. Rev. Lett. 119 (2017), no. 17 171601, [arXiv:1706.0238].
  * (20) L. F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017, [arXiv:1711.0203].
  * (21) L. F. Alday, A. Bissi, and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP 06 (2019) 010, [arXiv:1809.1067].
  * (22) L. F. Alday, On Genus-one String Amplitudes on $AdS_{5}\times S^{5}$, arXiv:1812.1178.
  * (23) D. Meltzer, Higher Spin ANEC and the Space of CFTs, JHEP 07 (2019) 001, [arXiv:1811.0191].
  * (24) D. Ponomarev, E. Sezgin, and E. Skvortsov, On one loop corrections in higher spin gravity, JHEP 11 (2019) 138, [arXiv:1904.0104].
  * (25) M. Shyani, Lorentzian inversion and anomalous dimensions in Mellin space, arXiv:1908.0001.
  * (26) L. F. Alday and E. Perlmutter, Growing Extra Dimensions in AdS/CFT, JHEP 08 (2019) 084, [arXiv:1906.0147].
  * (27) L. F. Alday and X. Zhou, Simplicity of AdS Supergravity at One Loop, arXiv:1912.0266.
  * (28) D. Meltzer, AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering, JHEP 05 (2020) 133, [arXiv:1912.0558].
  * (29) F. Aprile, J. Drummond, P. Heslop, and H. Paul, Quantum Gravity from Conformal Field Theory, JHEP 01 (2018) 035, [arXiv:1706.0282].
  * (30) F. Aprile, J. Drummond, P. Heslop, and H. Paul, Unmixing Supergravity, JHEP 02 (2018) 133, [arXiv:1706.0845].
  * (31) F. Aprile, J. Drummond, P. Heslop, and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056, [arXiv:1711.0390].
  * (32) S. Giombi, C. Sleight, and M. Taronna, Spinning AdS Loop Diagrams: Two Point Functions, JHEP 06 (2018) 030, [arXiv:1708.0840].
  * (33) C. Cardona, Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS, arXiv:1708.0633.
  * (34) O. Aharony, L. F. Alday, A. Bissi, and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036, [arXiv:1612.0389].
  * (35) E. Y. Yuan, Loops in the Bulk, arXiv:1710.0136.
  * (36) E. Y. Yuan, Simplicity in AdS Perturbative Dynamics, arXiv:1801.0728.
  * (37) I. Bertan, I. Sachs, and E. D. Skvortsov, Quantum $\phi^{4}$ Theory in AdS4 and its CFT Dual, JHEP 02 (2019) 099, [arXiv:1810.0090].
  * (38) I. Bertan and I. Sachs, Loops in Anti–de Sitter Space, Phys. Rev. Lett. 121 (2018), no. 10 101601, [arXiv:1804.0188].
  * (39) J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, $d$-dimensional SYK, AdS Loops, and $6j$ Symbols, JHEP 03 (2019) 052, [arXiv:1808.0061].
  * (40) D. Carmi, L. Di Pietro, and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200, [arXiv:1810.0418].
  * (41) F. Aprile, J. Drummond, P. Heslop, and H. Paul, Double-trace spectrum of $N=4$ supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. D 98 (2018), no. 12 126008, [arXiv:1802.0688].
  * (42) K. Ghosh, Polyakov-Mellin Bootstrap for AdS loops, JHEP 02 (2020) 006, [arXiv:1811.0050].
  * (43) D. Mazac and M. F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163, [arXiv:1811.1064].
  * (44) M. Beccaria and A. A. Tseytlin, On boundary correlators in Liouville theory on AdS2, JHEP 07 (2019) 008, [arXiv:1904.1275].
  * (45) S. M. Chester, Genus-2 holographic correlator on AdS${}_{5}\times$ S5 from localization, JHEP 04 (2020) 193, [arXiv:1908.0524].
  * (46) M. Beccaria, H. Jiang, and A. A. Tseytlin, Supersymmetric Liouville theory in AdS2 and AdS/CFT, JHEP 11 (2019) 051, [arXiv:1909.1025].
  * (47) D. Carmi, Loops in AdS: From the Spectral Representation to Position Space, arXiv:1910.1434.
  * (48) F. Aprile, J. Drummond, P. Heslop, and H. Paul, One-loop amplitudes in AdS${}_{5}\times$ S5 supergravity from $\mathcal{N}$ = 4 SYM at strong coupling, JHEP 03 (2020) 190, [arXiv:1912.0104].
  * (49) S. Fichet, Opacity and effective field theory in anti–de Sitter backgrounds, Phys. Rev. D100 (2019), no. 9 095002, [arXiv:1905.0577].
  * (50) D. Meltzer, E. Perlmutter, and A. Sivaramakrishnan, Unitarity Methods in AdS/CFT, JHEP 03 (2020) 061, [arXiv:1912.0952].
  * (51) J. Drummond and H. Paul, One-loop string corrections to AdS amplitudes from CFT, arXiv:1912.0763.
  * (52) S. Albayrak, C. Chowdhury, and S. Kharel, An étude of momentum space scalar amplitudes in AdS, arXiv:2001.0677.
  * (53) S. Albayrak and S. Kharel, On spinning loop amplitudes in Anti-de Sitter space, arXiv:2006.1254.
  * (54) D. Meltzer and A. Sivaramakrishnan, CFT Unitarity and the AdS Cutkosky Rules, arXiv:2008.1173.
  * (55) A. Costantino and S. Fichet, Opacity from Loops in AdS, JHEP 02 (2021) 089, [arXiv:2011.0660].
  * (56) D. Carmi, Loops in AdS: From the Spectral Representation to Position Space II, arXiv:2104.1050.
  * (57) A. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-Matrix, JHEP 10 (2012) 032, [arXiv:1112.4845].
  * (58) A. L. Fitzpatrick, E. Katz, D. Poland, and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP 07 (2011) 023, [arXiv:1007.2412].
  * (59) $M^{\mu\nu}$ is assumed to remain unaffected by this continuous deformation.
  * (60) While a study at the leading order of the expansion is enough to determine $\gamma$, such a truncation is not conformally invariant—only the full exponential series is. Hence one can expect that the resummed series will teach us more than its leading term.
  * (61) M. S. Costa, V. Gonçalves, and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064, [arXiv:1404.5625].
  * (62) D. Karateev, P. Kravchuk, and D. Simmons-Duffin, Harmonic Analysis and Mean Field Theory, JHEP 10 (2019) 217, [arXiv:1809.0511].
  * (63) D. W. Dusedau and D. Z. Freedman, Lehmann Spectral Representation for Anti-de Sitter Quantum Field Theory, Phys. Rev. D 33 (1986) 389\.
  * (64) M. J. G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186–207.
  * (65) A. Denner and J.-N. Lang, The Complex-Mass Scheme and Unitarity in perturbative Quantum Field Theory, Eur. Phys. J. C 75 (2015), no. 8 377, [arXiv:1406.6280].
  * (66) J. Rodenburg, “Unstable Particles and Resonances.”
  * (67) A. Costantino and S. Fichet, “Work in progress.”
  * (68) N. Nakanishi, A General Survey of the Theory of the Bethe-Salpeter Equation, Progress of Theoretical Physics Supplement 43 (01, 1969) 1–81.
  * (69) F. Gross, Relativistic quantum mechanics and field theory.  1993\.