# Partial data inverse problem for hyperbolic equation with time-dependent
damping coefficient and potential

Boya Liu  B. Liu, Department of Mathematics
North Carolina State University, Raleigh
NC 27695, USA<EMAIL_ADDRESS>,  Teemu Saksala  T. Saksala, Department of
Mathematics
North Carolina State University, Raleigh
NC 27695, USA<EMAIL_ADDRESS>and  Lili Yan  L. Yan, School of Mathematics,
University of Minnesota, Minneapolis, MN 55455, USA<EMAIL_ADDRESS>
###### Abstract.

We study an inverse problem of determining a time-dependent damping
coefficient and potential appearing in the wave equation in a compact
Riemannian manifold of dimension three or higher. More specifically, we are
concerned with the case of conformally transversally anisotropic manifolds, or
in other words, compact Riemannian manifolds with boundary conformally
embedded in a product of the Euclidean line and a transversal manifold. With
an additional assumption of the attenuated geodesic ray transform being
injective on the transversal manifold, we prove that the knowledge of a
certain partial Cauchy data set determines time-dependent damping coefficient
and potential uniquely.

##  1\. Introduction and Statement of Results

This paper is devoted to an inverse problem of a hyperbolic initial boundary
value problem, with the aim to determine lower order time-dependent
perturbations, namely, a scalar-valued damping coefficient and potential of a
Riemannian wave operator, from a set of partial Cauchy data. As introduced in
[38], from the physical point of view, this inverse problem is concerned with
determining properties such as the time-evolving damping force and the density
of an inhomogeneous medium by probing the medium with disturbances generated
on the lateral boundary and at the initial time, and by measuring the response
at the end of the experiment as well as on some part of the lateral boundary.

To state the inverse problem considered in this paper, let $(M,g)$ be a
smooth, compact, oriented Riemannian manifold of dimension $n\geq 3$ with
smooth boundary $\partial M$. We denote the spacetime $Q=(0,T)\times
M^{\mathrm{int}}$ with $0<T<\infty$, $\overline{Q}$ the closure of $Q$, and
$\Sigma=(0,T)\times\partial M$ the lateral boundary of $Q$. Recall that the
Laplace-Beltrami operator $\Delta_{g}$ of the metric $g$ acts on
$C^{2}$-smooth functions according to the following expression in local
coordinates $x_{1},\ldots,x_{n}$ of the manifold $M$

$\Delta_{g}v(x)=|g|^{-1/2}(x)\partial_{x^{j}}\left(g^{jk}(x)|g(x)|^{1/2}\partial_{x^{k}}v(x)\right),\quad
x\in M.$

Here $|g|$ and $g^{jk}$ denote the absolute value of the determinant and the
inverse of $g_{jk}$, respectively.

For a given smooth and strictly positive function $c(x)$ on $M$, we consider
the wave operator

$\Box_{c,g}=c(x)^{-1}\partial_{t}^{2}-\Delta_{g},$ (1.1)

whose coefficients are time-independent. In this paper we study an inverse
problem for the following linear hyperbolic partial differential operator

$\mathcal{L}_{c,g,a,q}=\Box_{c,g}+a(t,x)\partial_{t}+q(t,x),\quad(t,x)\in Q,$
(1.2)

with time-dependent lower order coefficients $a\in W^{1,\infty}(Q)$ (the
damping coefficient) and $q\in C(\overline{Q})$ (the potential). Our first
geometric assumption is the following:

######  Definition 1.1.

A Riemannian manifold $(M,g)$ of dimension $n\geq 3$ with boundary $\partial
M$ is called conformally transversally anisotropic (CTA) if $M$ is a compact
subset of a manifold $\mathbb{R}\times M_{0}^{\text{int}}$, with smooth
boundary and nonempty interior, and $g=c(e\oplus g_{0})$. Here
$(\mathbb{R},e)$ is the real line, $(M_{0},g_{0})$ is a smooth compact
$(n-1)$-dimensional Riemannian manifold with smooth boundary, called the
transversal manifold, and $c\in C^{\infty}(\mathbb{R}\times M_{0})$ is a
strictly positive function.

Examples of CTA manifolds include precompact smooth proper subsets of
Euclidean, spherical, and hyperbolic spaces. We refer readers to [25] for more
examples. Since the manifold $M$ is embedded into the product manifold
$\mathbb{R}\times M_{0}^{\mathrm{int}}$, we can write every point $x\in M$ in
the form $x=(x_{1},x^{\prime})$, where $x_{1}\in\mathbb{R}$ and $x^{\prime}\in
M_{0}$. In particular, the projection $\varphi(x)=x_{1}$ is a limiting
Carleman weight. It was established in [23, Theorem 1.2] that the existence of
a limiting Carleman weight implies that a conformal multiple of the metric $g$
admits a parallel unit vector field, and the converse holds for simply
connected manifolds. Locally, the latter condition is equivalent to the fact
that the manifold $(M,g)$ is conformal to the product of an interval and some
Riemannian manifold $(M_{0},g_{0})$ of one dimension less.

The limiting Carleman weight $\varphi$ gives us a canonical way to define the
front and back faces of $\partial M$ and $\partial Q$. Let $\nu$ be the
outward unit normal vector to ${\partial M}$ with respect to the metric $g$.
We denote $\partial M_{\pm}=\\{x\in\partial M:\pm\partial_{\nu}\varphi(x)\geq
0\\}$ and $\Sigma_{\pm}=(0,T)\times\partial M_{\pm}^{\text{int}}$. Then we
define $U=(0,T)\times U^{\prime}$ and $V=(0,T)\times V^{\prime}$, where
$U^{\prime},V^{\prime}\subset\partial M$ are open neighborhoods of $\partial
M_{+}$, $\partial M_{-}$, respectively.

The goal of this paper is to show that the time-dependent damping coefficient
$a(t,x)$ and potential $q(t,x)$, appearing in (1.2), can be uniquely
determined from the following set of partial Cauchy data

$\mathcal{C}_{g,a,q}=\\{(u|_{\Sigma},u|_{t=0},u|_{t=T},\partial_{t}u|_{t=0},\partial_{\nu}u|_{V}):\>u\in
H^{1}(0,T;L^{2}(M)),\>\mathcal{L}_{c,g,a,q}u=0\\}.$ (1.3)

We will define this data carefully in Section 2.

Notice that in addition to the data measured on the lateral boundary, the set
of Cauchy data $\mathcal{C}_{g,a,q}$ also includes measurements made at the
initial time $t=0$ and the end time $t=T$. Indeed, it was established in [32]
that the full lateral boundary data with vanishing initial conditions

$\mathcal{C}_{g,a,q}^{\mathrm{lat}}=\\{(u|_{\Sigma},\partial_{\nu}u|_{\Sigma}):u\in
H^{1}(0,T;L^{2}(M)),\>\mathcal{L}_{g,a,q}u=0,\>u|_{t=0}=\partial_{t}u|_{t=0}=0\\}$
(1.4)

determines time-independent damping coefficients and potentials uniquely for
$T>\mathrm{diam}(M)$, where $M$ is a bounded domain in $\mathbb{R}^{n}$.
However, due to domain of dependence arguments, as explained for instance in
[40, Subsection 1.1], it is only possible to recover a general time-dependent
coefficient in the optimal set

$\mathcal{D}=\\{(t,x)\in Q:\mathrm{dist}(x,\partial
M)<t<T-\mathrm{dist}(x,\partial M)\\}$

from $\mathcal{C}_{g,a,q}^{\mathrm{lat}}$. Thus, even for large measurement
time $T>0$, a global recovery of time-dependent lower order coefficients of
the hyperbolic operator (1.2) needs some additional information at the
beginning $\\{t=0\\}$ and at the end $\\{t=T\\}$ of the measurement.

Unfortunately, the product structure of the ambient space $\mathbb{R}\times
M_{0}$ of the manifold $(M,g)$ is not quite sufficient for the recovery method
presented in this paper. We need to also assume that certain geodesic ray
transforms on the transversal manifold $(M_{0},g_{0})$ are injective. Such
assumptions have been successfully implemented to solve many important inverse
problems on CTA manifolds, see for instance [21, 25, 46, 68] and the
references therein.

Let us now recall some definitions related to geodesic ray transforms on
Riemannian manifolds with boundary. Geodesics of $(M_{0},g_{0})$ can be
parametrized (non-uniquely) by points on the unit sphere bundle
$SM_{0}=\\{(x,\xi)\in TM_{0}:|\xi|=1\\}$. Moreover, we use the notation

$\partial_{\pm}SM_{0}=\\{(x,\xi)\in SM_{0}:x\in\partial
M_{0},\pm\langle\xi,\nu(x)\rangle>0\\}$

for the incoming (–) and outgoing (+) boundaries of $SM_{0}$. These sets
correspond to the geodesics touching the boundary, and
$\langle\cdot,\cdot\rangle$ is the Riemannian inner product of
$(M_{0},g_{0})$.

Let $(x,\xi)\in\partial_{-}SM_{0}$, and let $\gamma=\gamma_{x,\xi}$ be a
geodesic of $M_{0}$ with initial conditions $\gamma(0)=x$ and
$\dot{\gamma}(0)=\xi$. Then $\tau_{\mathrm{exit}}(x,\xi)>0$ stands for the
first time when $\gamma$ meets $\partial M_{0}$ with the convention that
$\tau_{\mathrm{exit}}(x,\xi)=+\infty$ if $\gamma$ stays in the interior of
$M_{0}$. We say that a unit speed geodesic segment
$\gamma:[0,\tau_{\mathrm{exit}}(x,\xi)]\to M_{0}$,
$0<\tau_{\mathrm{exit}}(x,\xi)<\infty$, is non-tangential if $\gamma(0)$,
$\gamma(\tau_{\mathrm{exit}}(x,\xi))\in{\partial M}_{0}$, $\dot{\gamma}(0)$
and $\dot{\gamma}(\tau_{\mathrm{exit}}(x,\xi))$ are non-tangential vectors to
$\partial M_{0}$, and $\gamma(\tau)\in M_{0}^{\mathrm{int}}$ for all
$0<\tau<\tau_{\mathrm{exit}}(x,\xi)$.

In this paper we shall reduce the determination of unknown time-dependent
coefficients $a(x,t)$ and $q(x,t)$ from the set of partial Cauchy data (1.3)
to the invertibility of the attenuated geodesic ray transform on the
transversal manifold $(M_{0},g_{0})$. Given a smooth function $\alpha$ on
$M_{0}$, the attenuated geodesic ray transform of a function $f\colon
M_{0}\to\mathbb{R}$ is given by

$I^{\alpha}(f)(x,\xi)=\int_{0}^{\tau_{\mathrm{exit}}(x,\xi)}\exp\bigg{[}\int_{0}^{t}\alpha(\gamma_{x,\xi}(s))ds\bigg{]}f(\gamma_{x,\xi}(t))dt,\quad(x,\xi)\in\partial_{-}SM_{0}\setminus\Gamma_{-},$
(1.5)

where
$\Gamma_{-}=\\{(x,\xi)\in\partial_{-}SM_{0}:\tau_{\mathrm{exit}}(x,\xi)=+\infty\\}$.
Our second geometric assumption is the following:

Assumption 1. There exists $\varepsilon>0$ such that for each smooth function
$\alpha$ on $M_{0}$ with $\|\alpha\|_{L^{\infty}(M_{0})}<\varepsilon$, the
respective attenuated geodesic ray transform $I^{\alpha}$ on $(M_{0},g_{0})$
is injective over continuous functions $f$ in the sense that if
$I^{\alpha}(f)(x,\xi)=0$ for all
$(x,\xi)\in\partial_{-}SM_{0}\setminus\Gamma_{-}$ such that $\gamma_{x,\xi}$
is a non-tangential geodesic, then $f=0$ in $M_{0}$.

It was verified in [23, Theorem 7.1] that simple manifolds always satisfy
Assumption 1. Traditionally, a compact, simply connected Riemannian manifold
with smooth boundary is called simple if its boundary is strictly convex, and
no geodesic has conjugate points. Also, the injectivity of the geodesic ray
transform ($\alpha=0$) on simple manifolds is well-known, see [52, 62].

In addition to simple manifolds, there are some other geometric conditions
under which the attenuated geodesic ray transform $I^{\alpha}$ is known to be
injective. For instance, if the manifold is radially symmetric and satisfies
the Herglotz condition, then $I^{\alpha}$ is injective whenever the
attenuation $\alpha$ is also radially symmetric and Lipschitz continuous [22,
Theorem 29]. For the purposes of the current paper it suffices to only
consider constant attenuations. The Herglotz condition is a special case of a
manifold satisfying a convex foliation condition, and in [53] the injectivity
of $I^{\alpha}$ is verified on this type of manifolds of dimension $n\geq 3$.
Some examples of manifolds satisfying the global foliation condition are the
punctured Euclidean space $\mathbb{R}^{n}\setminus\\{0\\}$ and the torus
$\mathbb{T}^{n}$. We refer readers to [53, Section 2] for more examples.
Finally, we would like to recall that a convex foliation condition does not
forbid the existence of conjugate points. Hence, there are many non-simple
Riemannian manifolds with invertible attenuated geodesic ray transform.

The main result of this paper is the following.

######  Theorem 1.2.

Let $T>0$. Suppose that $(M,g)$ is a CTA manifold of dimension $n\geq 3$ and
that Assumption 1.1 holds for the transversal manifold $(M_{0},g_{0})$. Let
$a_{i}\in W^{1,\infty}(Q)$ and $q_{i}\in C(\overline{Q})$, $i=1,2$. If
$a_{1}=a_{2}$ and $q_{1}=q_{2}$ on $\partial Q$, then
$\mathcal{C}_{g,a_{1},q_{1}}=\mathcal{C}_{g,a_{2},q_{2}}$ implies that
$a_{1}=a_{2}$ and $q_{1}=q_{2}$ in $Q$.

Theorem 1.2 can be viewed as an extension of [41, Theorem 1.4], where only the
potential was considered, to the case of recovering both damping coefficient
and potential from the set of partial Cauchy data $\mathcal{C}_{g,a,q}$. From
the perspective of geometric setting, this paper extends [38] from the
Euclidean space, as well as [41] from CTA manifolds with a simple transversal
manifold, to a larger class of CTA manifolds. As in [41], we attack the
problem by utilizing tools from the theory of inverse problems for elliptic
operators. However, in comparison with the earlier works, we would not be able
to relax the simplicity assumption on the transversal manifold without
significant modifications to the construction of CGO solutions.

Assumption 1.1 of this paper is different from the literature concerning
inverse problems for elliptic operators on CTA manifolds; see, for instance,
[25, 36, 46]. These works assume the invertibility of the geodesic ray
transform $I^{\alpha}$ for $\alpha=0$. In the case of elliptic operators,
where there is only one Euclidean direction $x_{1}$, the authors reduced the
problem first to the attenuated geodesic ray transform. Then the authors
recovered the geodesic ray transform for each coefficient in the Taylor
expansions of the unknown functions by differentiating an expression similar
to (5.21) in our manuscript with respect to the variable $\lambda$ at zero.
Unfortunately, this approach is not applicable in our case, as the mapping
$(\lambda,\beta)\mapsto-\lambda(\beta,1)$, appearing in (5.21), is a
diffeomorphism only if $\lambda\neq 0$. Thus, computing $\lambda$ and
$\beta$-derivatives of (5.21) at $\lambda=0$ will not give us the geodesic ray
transform of Taylor coefficients of the unknown functions at the origin in the
Fourier variables.

###  1.1. Previous literature

The recovery of coefficients appearing in hyperbolic equations from boundary
measurements has attracted lots of attention in recent years. Results in this
direction are generally divided into two categories with respect to time-
independent and time-dependent coefficients.

Starting with seminal works [11, 14], there has been extensive literature
related to the recovery of time-independent coefficients appearing in
hyperbolic equations. We refer readers to [8, 19, 30, 31, 48] and references
therein for some works in this direction. A powerful tool to prove uniqueness
results for time-independent coefficients of hyperbolic equations, including
the leading order coefficient, is the Boundary Control Method, which was
developed in [11, 14], as well as a time-sharp unique continuation theorem
proved in [66]. We refer readers to [42] for an introduction to the method and
[12, 34] for reviews. However, it was discovered in [5, 6] that the unique
continuation theorem analogous to [66] may fail when the dependence of
coefficients on time is not analytic, which means that the Boundary Control
Method is not well-suited to recover time-dependent coefficients in general.

Aside from the Boundary Control Method, the approach of geometric optic (GO)
solutions is also widely utilized to recover time-independent coefficients of
hyperbolic equations. Using this approach, the unique recovery of time-
independent potential $q$ (with $a=0$) from full lateral boundary Dirichlet-
to-Neumann map was established in [55], and [32] extended this result to
recovery of time-independent damping coefficients using the same boundary
data. A uniqueness result from partial boundary measurements was considered in
[26]. The GO solution approach has also been used to obtain stronger stability
results [16, 17, 37, 65] than the Boundary Control Method [7], but it gives
less sharp uniqueness results from the perspective of geometric assumptions
than the latter.

Turning the attention to the time-dependent category, most of the results in
this direction rely on the use of GO solutions. This approach was first
implemented in the context of determining time-dependent coefficients of
hyperbolic equations from the knowledge of scattering data by using properties
of the light-ray transform [64]. Recovery of time-dependent potential $q$ from
the full lateral boundary data $\mathcal{C}_{q}^{\mathrm{lat}}$, given by
(1.4), on the infinite cylinder $\mathbb{R}\times\Omega$, where $\Omega$ is a
domain in $\mathbb{R}^{n}$, was established in [58]. On a finite cylinder
$(0,T)\times\Omega$ with $T>\mathrm{diam}(\Omega)$, it was proved in [54] that
$\mathcal{C}_{q}^{\mathrm{lat}}$ determines $q$ uniquely in the optimal set
$\mathcal{D}$ of $(0,T)\times\Omega$. Uniqueness and stability results for
determining a general time-dependent potential $q$ from partial data were
established in [40] and [18, 39], respectively.

Going beyond the Euclidean space, uniqueness results for time-dependent
potential $q$ from both full and partial boundary measurements were
established in [41] on a CTA manifold $(M,g)$, with a simple transversal
manifold $M_{0}$, by using the GO solution approach. For more general
manifolds, recently it was proved in [3] that the set of full Cauchy data
uniquely determines the potential $q$ in Lorentzian manifolds satisfying
certain two-sided curvature bounds and some other geometric assumptions, and
this curvature bound was weakened in [4] near Minkowski geometry. In
particular, the proof of [3] is based on a new optimal unique continuation
theorem and can be viewed as a generalization of the Boundary Control Method
to the cases without real analyticity assumptions.

There is also some literature related to determining time-dependent first
order perturbations appearing in hyperbolic equations from boundary data
analogous to (1.3). In the Euclidean setting, [38] extended the result of [40]
to a unique determination of time-dependent damping coefficients and
potentials from $\mathcal{C}_{g,a,q}$. When the vector field perturbation
appears in the wave equation, similar to elliptic operators such as the
magnetic Schrödinger operator, one can only recover the first order
perturbation up to a gauge invariance, i.e., the differential of a test
function in $Q$, see [27] for a uniqueness result when the dependence of
coefficients on time is real-analytic, and this analyticity assumption was
removed later in [59]. Logarithmic type stability estimates for the vector
field perturbation as well as the potential were proved in [15]. A uniqueness
result analogous to [27, 59] from partial Dirichlet-to-Neumann map was
obtained in [44]. In the non-Euclidean setting, it is established in [28] that
the hyperbolic Dirichlet-to-Neumann map determines the first order and the
zeroth order perturbations up a gauge invariance on a certain non-optimal
subset of $Q$ by inverting the light-ray transform of the Lorentzian metric
$-dt^{2}+g(x)$ for one forms and functions.

To summarize, there are only two known methods to recover the coefficients
appearing in hyperbolic equations from boundary measurements. Since in the
current paper the unknown lower order coefficients of the hyperbolic operator
$\mathcal{L}_{c,g,a,q}$, as in (1.2), are time-dependent, we cannot apply the
Boundary Control Method. Thus, our proofs are based on GO solutions. We
believe that an introduction of any new method, which can be used to attack
the hyperbolic inverse problems, would be a major breakthrough. Obviously,
this is not in the scope of the current paper.

Finally, we would like to emphasize that to the best of our knowledge, the
global recovery of a full first order time-dependent perturbation (a one-form
and potential function in $Q$) of the Riemannian wave operator from a set of
partial Cauchy data, and the optimal recovery of these coefficients from the
respective hyperbolic Dirichlet-to-Neumann map, remain important open
problems.

###  1.2. Outline for the proof of Theorem 1.2

The first main ingredient of the proof is the construction of exponentially
growing and decaying complex geometric optic (CGO) solutions to the equation
$\mathcal{L}_{c,g,a,q}u=0$ of the form

$u(t,x)=e^{\pm s(\beta t+\varphi(x))}(v_{s}(t,x)+r_{s}(t,x)),\quad(t,x)\in Q.$

Here $s=\frac{1}{h}+i\lambda$ is a complex number, $h\in(0,1)$ is a
semiclassical parameter, $\lambda\in\mathbb{R}$ and
$\beta\in(\frac{1}{\sqrt{3}},1)$ are some fixed numbers, $v_{s}$ is a Gaussian
beam quasimode, and $r_{s}$ is a correction term that decays with respect to
the parameter $h$. The function $\varphi(x)=x_{1}$ is a limiting Carleman
weight on $M$.

We exploit the existence of the limiting Carleman weight $\beta t+x_{1}$ in
$Q$ and derive necessary boundary and interior Carleman estimates, see
Proposition 3.1 and Lemma 3.4, respectively. The boundary Carleman estimates
are used to control the solutions on the inaccessible part of the boundary,
while the interior Carleman estimates are needed to verify the existence of
the correction term $r_{s}$ in Proposition 3.5. In the current paper the
Dirichlet boundary values are given on the full lateral boundary, which is in
line with many earlier works involving a first order term, see for instance
[38, 44, 51, 61]. Meanwhile, in the absence of the damping term, but with
Dirichlet data measured only on a part of the lateral boundary, the authors of
[40, 41] constructed GO solutions to the respective hyperbolic equation that
vanish initially and on part of the lateral boundary. In this way the authors
were able to utilize the boundary Carleman estimate to control their GO
solutions on the inaccessible part of the boundary. Unfortunately, this method
only provides an $L^{2}$-estimate for the correction term $r_{s}$, and due to
the existence of the damping coefficient, we need an $H^{1}$-estimate for
$r_{s}$. This is provided in Proposition 3.5.

Since the transversal manifold $(M_{0},g_{0})$ is not necessarily simple in
this paper, the approach based on global GO solutions is not applicable to us.
To medicate this, in Theorem 4.2 we construct Gaussian beam quasimodes for
every non-tangential geodesic in the transversal manifold $M_{0}$ by using
techniques developed in solving inverse problems for elliptic operators, see
for instance [21, 25, 46], followed by a concentration property for the
quasimodes given in Theorem 4.4. The construction of CGO solutions is
finalized in Theorem 4.5. In this part we need the regularity conditions
imposed to the unknown time-dependent coefficients $a$ and $q$.

The second main component in the proof is the integral identity (5.6), whose
derivation needs the equivalence of the partial Cauchy data. When the obtained
CGO solutions are inserted in the integral identity, the boundary Carleman
estimate of Proposition 3.2 forces the right-hand side of (5.6) to vanish in
the limit $h\to 0$. On the other hand, the concentration property given in
Theorem 4.4 implies that the left-hand side of (5.6) converges to the
attenuated geodesic ray transform of the Fourier transform (in the two
Euclidean variables $(t,x_{1})$) of the unknown coefficients in the
transversal manifold $(M_{0},g_{0})$. To carry on this reduction step, we need
the regularity and boundary conditions imposed to the unknown time-dependent
coefficients $a$ and $q$. We need Assumption 1.1 to invert the attenuated
geodesic ray transform. We first provide a proof for the uniqueness result for
the damping coefficient $a(t,x)$, followed by verifying the uniqueness for the
potential $q(t,x)$.

This paper is organized as follows. We begin by carefully defining the set of
partial Cauchy data (1.3) in Section 2. In Section 3 we derive the boundary
and interior Carleman estimates. In Section 4 we construct the CGO solutions
to the hyperbolic equation $\mathcal{L}_{c,g,a,q}u=0$ based on Gaussian beam
quasimodes in $Q$. Finally, the proof of Theorem 1.2 is presented in Section
5.

### Acknowledgments

We would like to express our gratitude to Katya Krupchyk and Lauri Oksanen for
their valuable discussions and suggestions. T.S. is partially supported by the
National Science Foundation (DMS 2204997). L.Y. is partially supported by the
National Science Foundation (DMS 2109199). We are also very grateful to the
anonymous referees for their invaluable feedback, which led to significant
improvements of the paper.

##  2\. Definition of the Partial Cauchy Data

The goal of this short section is to recall some properties of the weak
solutions to the initial boundary value problem

$\begin{cases}\mathcal{L}_{c,g,a,q}u(t,x)=0\quad\text{in}\quad Q,\\\
u(0,x)=h_{0}(x),\>\partial_{t}u(0,x)=h_{1}(x)\quad\text{in}\quad M,\\\
u(t,x)=f(t,x)\quad\text{on}\quad\Sigma,\end{cases}$ (2.1)

as introduced in [38, Section 2].

We define the space

$H_{\Box_{c,g}}(Q)=\\{u\in
H^{1}(0,T;L^{2}(M)):\Box_{c,g}u=(c^{-1}\partial_{t}^{2}-\Delta_{g})u\in
L^{2}(Q)\\},$

equipped with the norm

$\|u\|^{2}_{H_{\Box_{c,g}}(Q)}=\|u\|^{2}_{H^{1}(0,T;L^{2}(M))}+\|\Box_{c,g}u\|^{2}_{L^{2}(Q)}.$

Our starting point is the following result, originally presented in [40,
Theorem A.1].

######  Lemma 2.1.

The space $H_{\Box_{c,g}}$ is continuously embedded into the closure of
$C^{\infty}(\overline{Q})$ in the space

$K_{\Box_{c,g}}(Q)=\\{u\in H^{-1}(0,T;L^{2}(M)):\Box_{c,g}u\in L^{2}(Q)\\}$

equipped with the norm

$\|u\|^{2}_{K_{\Box_{c,g}}(Q)}=\|u\|^{2}_{H^{-1}(0,T;L^{2}(M))}+\|\Box_{c,g}u\|^{2}_{L^{2}(Q)}.$

For each $w\in C^{\infty}(\overline{Q})$, we introduce two linear maps

$\iota_{0}w=(\iota_{0,1}w,\iota_{0,2}w,\iota_{0,3}w)=(w|_{\Sigma},w|_{t=0},\partial_{t}w|_{t=0})$

and

$\iota_{1}w=(\iota_{1,1}w,\iota_{1,2}w,\iota_{1,3}w)=(\partial_{\nu}w|_{\Sigma},w|_{t=T},\partial_{t}w|_{t=T}).$

######  Lemma 2.2.

The maps $\iota_{0}$ and $\iota_{1}$ defined above can be extended
continuously to

$\iota_{0}:H_{\Box_{c,g}}(Q)\to H^{-3}(0,T;H^{-1/2}(\partial M))\times
H^{-2}(M)\times H^{-4}(M)$

and

$\iota_{1}:H_{\Box_{c,g}}(Q)\to H^{-3}(0,T;H^{-3/2}(\partial M))\times
H^{-2}(M)\times H^{-4}(M),$

respectively.

###### Proof.

Since the conformal factor $c$ in $\Box_{c,g}$ is time-independent, the proof
is a straightforward modification of the proof for [40, Proposition A.1]. ∎

We note that by the same argument as in [38, Section 2], the set

$\mathcal{J}=\\{u\in H^{1}(0,T;L^{2}(M)):\Box_{c,g}u=0\\}$

is a closed vector subspace of $H^{1}(0,T;L^{2}(M))$, contained in
$H_{\Box_{c,g}}(Q)$. Finally, we record the range of the map $\iota_{0}$

$\mathcal{K}:=\\{\iota_{0}w:u\in H_{\Box_{c,g}}(Q)\\}\subset
H^{-3}(0,T;H^{-1/2}(\partial M))\times H^{-2}(M)\times H^{-4}(M).$

By an analogous argument to the proof for [40, Proposition 2.1], we get the
following result.

######  Lemma 2.3.

The linear map $\iota_{0}\colon\mathcal{J}\to\mathcal{K}$ is a bijection.

By Lemma 2.3, the inverse function $\iota_{0}^{-1}:\mathcal{K}\to\mathcal{J}$
exists, and we can use it to define a norm in $\mathcal{K}$ via the formula

$\|(f,h_{0},h_{1})\|_{\mathcal{K}}=\|\iota_{0}^{-1}(f,h_{0},h_{1})\|_{H^{1}(0,T;L^{2}(M))},\quad(f,h_{0},h_{1})\in\mathcal{K}.$

We would like to recall that we have defined $\partial M_{\pm}=\\{x\in\partial
M:\pm\partial_{\nu}\varphi(x)\geq 0\\}$ and $V=(0,T)\times V^{\prime}$, where
$V^{\prime}\subset\partial M$ is an open neighborhood of $\partial M_{-}$. We
are now ready to state and prove the existence and uniqueness of solutions to
the initial boundary value problem (2.1) with the datum
$(f,h_{0},h_{1})\in\mathcal{K}$.

######  Proposition 2.4.

Let $a\in W^{1,\infty}(Q)$ and $q\in C(\overline{Q})$. For each datum
$(f,h_{0},h_{1})\in\mathcal{K}$, the initial boundary value problem (2.1) has
a unique weak solution $u\in H^{1}(0,T;L^{2}(M))$ that satisfies

$\|u\|_{H^{1}(0,T;L^{2}(M))}\leq C\|(f,h_{0},h_{1})\|_{\mathcal{K}}.$ (2.2)

Furthermore, the boundary operator

$\mathcal{B}_{a,q}:\mathcal{K}\to H^{-3}(0,T;H^{-3/2}(V^{\prime}))\times
H^{-2}(M),\quad\mathcal{B}_{a,q}(f,h_{0},h_{1})=(\iota_{1,1}u|_{V},\iota_{1,2}u)$
(2.3)

is bounded, and the partial Cauchy data set $\mathcal{C}_{g,a,q}$, as in
(1.3), is the graph of the map $\mathcal{B}_{a,q}$.

###### Proof.

The proof is a straightforward modification of the proof of [38, Proposition
2.1]. ∎

##  3\. Carleman Estimates

Our goal of this section is to prove a boundary Carleman estimate as well as
an interior Carleman estimate for the operator $\mathcal{L}_{c,g,a,q}$
conjugated by an exponential weight corresponding to a linear function $\beta
t+x_{1}$, where $0<\beta<1$ is a constant. We shall utilize the boundary
Carleman estimate to control boundary terms over subsets of the boundary
$\partial Q$ where measurements are not accessible, and the interior Carleman
estimates will be used in Section 4 to construct the remainder term for both
exponentially decaying and growing CGO solutions.

Let $(M,g)$ be a CTA manifold as defined in Definition 1.1, and let
$\widetilde{g}=e\oplus g_{0}$. By the conformal properties of the Laplace-
Beltrami operator, we have

$c^{\frac{n+2}{4}}(-\Delta_{g})(c^{-\frac{n-2}{4}}u)=-\Delta_{\widetilde{g}}u-\big{(}c^{\frac{n+2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}})\big{)}u,$
(3.1)

see [25, Section 2]. Also, since $c$ is independent of $t$, we get

$c^{\frac{n+2}{4}}a\partial_{t}(c^{-\frac{n-2}{4}}u)=ca\partial_{t}u,\quad\text{and}\quad
c^{\frac{n+2}{4}}\partial_{t}^{2}(c^{-\frac{n-2}{4}}u)=c\partial_{t}^{2}u.$
(3.2)

Thus, it follows from (3.1) and (3.2) that for the hyperbolic operator
$\mathcal{L}_{c,g,a,q}$, we have

$c^{\frac{n+2}{4}}\circ\mathcal{L}_{c,g,a,q}\circ
c^{-\frac{n-2}{4}}=\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}},$
(3.3)

where

$\widetilde{a}=ca,\quad\widetilde{q}=c(q-c^{\frac{n-2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}})).$
(3.4)

Hence, by replacing the metric $g$ and coefficients $a,q$ with
$\widetilde{g},\widetilde{a},\widetilde{q}$, respectively, we can assume that
the conformal factor $c=1$. In this section we shall make use of this
assumption and consider the leading order wave operator $\Box_{e\oplus
g_{0}}=\partial_{t}^{2}-\Delta_{e\oplus g_{0}}$. Let us denote
$\mathcal{L}_{g,a,q}$ the hyperbolic partial differential operator
$\mathcal{L}_{c,g,a,q}$ when $c=1$.

###  3.1. Boundary Carleman estimate

Due to the damping coefficient, we need to use a convexification argument
similar to [38, 44] to establish the needed boundary Carleman estimate. To
elaborate, let us first introduce a new parameter $\varepsilon>0$, which is
independent of $h$ and to be determined later. For $0<h<\varepsilon<1$, we
consider the perturbed weight

$\varphi_{\pm h,\varepsilon}(t,x)=\pm\frac{1}{h}(\beta
t+x_{1})-\frac{t^{2}}{2\varepsilon}.$ (3.5)

Our first result in this section can be viewed as an extension of [38, Theorem
3.1] from the Euclidean setting to that of Riemannian manifolds with
dependence on a parameter $\beta$. Note that [38, Theorem 3.1] is not directly
applicable in our case since the parameter $\beta$ is strictly less than $1$.

######  Proposition 3.1.

Let $a,q\in L^{\infty}(Q,\mathbb{C})$ and $u\in C^{2}(\overline{Q})$. If $u$
satisfies the conditions

$u|_{\Sigma}=u|_{t=0}=\partial_{t}u|_{t=0}=0,$ (3.6)

then for all $0<h\ll\varepsilon\ll 1$ we have

$\displaystyle\|e^{-\varphi_{h,\varepsilon}}h^{2}\mathcal{L}_{g,a,q}(e^{\varphi_{h,\varepsilon}}u)\|^{2}_{L^{2}(Q)}+(\frac{4}{\beta}-\frac{\beta}{2})h^{3}\|\nabla_{g}u(T,\cdot)\|^{2}_{L^{2}(M)}+3\beta
h\|u(T,\cdot)\|^{2}_{L^{2}(M)}$ (3.7) $\displaystyle\geq$
$\displaystyle\frac{(3\beta^{2}-1)h^{2}}{4\varepsilon}\|u\|^{2}_{L^{2}(Q)}+\frac{\beta
h^{3}}{4}\|\partial_{t}u(T,\cdot)\|^{2}_{L^{2}(M)}+\frac{h^{4}}{2\varepsilon}(\|\partial_{t}u\|^{2}_{L^{2}(Q)}+\|\nabla_{g}u\|^{2}_{L^{2}(Q)})$
$\displaystyle+h^{3}\int_{\Sigma}{\color[rgb]{0,0,0}\nu_{1}}|\partial_{\nu}u|^{2}dS_{g}dt,$

and

$\displaystyle\|e^{-\varphi_{-h,\varepsilon}}h^{2}\mathcal{L}_{g,a,q}e^{\varphi_{-h,\varepsilon}}u\|^{2}_{L^{2}(Q)}+2(\beta+1)h^{3}(\|\nabla_{g}u(T,\cdot)\|^{2}_{L^{2}(M)}+\|\partial_{t}u(T,\cdot)\|^{2}_{L^{2}(M)})$
(3.8)
$\displaystyle\geq\frac{(3\beta^{2}-1)h^{2}}{4\varepsilon}\|u\|^{2}_{L^{2}(Q)}+\frac{h^{4}}{2\varepsilon}(\|\partial_{t}u\|_{L^{2}(Q)}+\|\nabla_{g}u\|^{2}_{L^{2}(Q)})-h^{3}\int_{\Sigma}\nu_{1}|\partial_{\nu}u|^{2}dS_{g}dt,$

where $\varphi_{\pm h,\varepsilon}$ is given by (3.5),
$\frac{1}{\sqrt{3}}\leq\beta<1$, and
$\nu_{1}:=\langle\nu,\partial_{x_{1}}\rangle_{g}$.

###### Proof.

We shall only provide a detailed proof for estimate (3.7). The derivation of
(3.8) is analogous and therefore omitted. To proceed, we omit the subscripts
$h,\varepsilon$ in $\varphi_{h,\varepsilon}$ to simplify the notation.

Step 1: The conjugated operator
$e^{-\varphi}h^{2}\mathcal{L}_{g,a,q}e^{\varphi}u$. By direct computations, we
have

$\displaystyle e^{-\varphi}h^{2}\mathcal{L}_{g,a,q}e^{\varphi}u=$
$\displaystyle
h^{2}\left[\partial_{t}^{2}u+2\partial_{t}\varphi\partial_{t}u+u\partial_{t}^{2}\varphi+u(\partial_{t}\varphi)^{2}-(\Delta_{g}u+2\langle\nabla_{g}\varphi,\nabla_{g}u\rangle_{g}+u\Delta_{g}\varphi\right.$
(3.9)
$\displaystyle\left.+u|\nabla_{g}\varphi|^{2})+a\partial_{t}u+au\partial_{t}\varphi+qu\right]$
$\displaystyle:=$ $\displaystyle P_{1}u+P_{2}u+P_{3}u,$

where

$\displaystyle
P_{1}u=h^{2}(\Box_{g}u+(\partial_{t}\varphi)^{2}u-|\nabla_{g}\varphi|^{2}u+(\Box_{g}\varphi)u),\quad
P_{2}u=h^{2}(2\partial_{t}\varphi\partial_{t}u-2\langle\nabla_{g}\varphi,\nabla_{g}u\rangle_{g}),$
(3.10) $\displaystyle
P_{3}u=h^{2}(a\partial_{t}u+(a\partial_{t}\varphi)u+qu).$

Thus, $P_{1}u$ includes the even order derivatives of $u$, $P_{2}u$ contains
the odd order derivatives of $u$, and $P_{3}u$ has all the terms involving $a$
or $q$. Due to (3.9), (3.10), and the triangle inequality, we have

$\displaystyle\|e^{-\varphi}h^{2}\mathcal{L}_{g,a,q}e^{\varphi}u\|_{L^{2}(Q)}^{2}\geq\frac{1}{2}\|P_{1}u+P_{2}u\|^{2}_{L^{2}(Q)}-\|P_{3}u\|^{2}_{L^{2}(Q)}.$
(3.11)

In the last two steps of this proof we shall bound
$\|P_{1}u+P_{2}u\|^{2}_{L^{2}(Q)}$ from below and $\|P_{3}u\|^{2}_{L^{2}(Q)}$
from above. We first use the Cauchy’s inequality to convert the first term on
the right hand side of (3.11) into a product. The choice of the operators
$P_{1}$ and $P_{2}$ simplifies the subsequent computations, which involve
several integration by parts. The estimate for the second term in the right
hand side of (3.11) is short and is based on the boundedness of $a$ and $q$.
Both estimations rely on the choice of the perturbed weight given in (3.5).
Finally, the inequality (3.7) is obtained by combining these two estimates.

Step 2: A lower bound of $\|P_{1}u+P_{2}u\|^{2}_{L^{2}(Q)}$. To start, we have
by the Cauchy’s inequality that

$\displaystyle\frac{1}{2}\|P_{1}u+P_{2}u\|^{2}_{L^{2}(Q)}\geq\int_{Q}\mathrm{Re}(P_{1}u\overline{P_{2}u})dV_{g}dt.$

Since $g(x_{1},x^{\prime})=(dx_{1})^{2}+g_{0}(x^{\prime})$, we get from direct
computations that

$\partial_{t}\varphi=\frac{1}{h}\beta-\frac{1}{\varepsilon}t,\quad\partial_{t}^{2}\varphi=-\frac{1}{\varepsilon},\quad\partial_{x_{1}}\varphi=\frac{1}{h},\quad\langle\nabla_{g}\varphi,\nabla_{g}u\rangle_{g}=\frac{1}{h}\partial_{x_{1}}u,$
(3.12)

which yield

$\displaystyle\int_{Q}\mathrm{Re}(P_{1}u\overline{P_{2}u})dV_{g}dt=\mathrm{Re}\int_{Q}2h^{4}\left[\partial_{t}^{2}u(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\overline{\partial_{t}u}-\frac{1}{h}\partial_{t}^{2}u\overline{\partial_{x_{1}}u}-\Delta_{g}u(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\overline{\partial_{t}u}\right.$
(3.13)
$\displaystyle\left.+\frac{1}{h}\Delta_{g}u\overline{\partial_{x_{1}}u}-\big{(}\frac{1}{h^{2}}(1-\beta^{2})+\frac{1}{\varepsilon}+\frac{2\beta}{h\varepsilon}t-\frac{1}{\varepsilon^{2}}t^{2}\big{)}u\big{(}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\overline{\partial_{t}u}-\frac{1}{h}\overline{\partial_{x_{1}}u}\big{)}\right]dV_{g}dt.$

Let us proceed to estimate each term on the right-hand side of (3.13). For the
first term, we integrate by parts and use the assumption
$\partial_{t}u|_{t=0}=0$ to deduce

$\displaystyle
2h^{4}\mathrm{Re}\int_{Q}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}^{2}u\overline{\partial_{t}u}dV_{g}dt$
$\displaystyle=h^{4}(\frac{1}{h}\beta-\frac{1}{\varepsilon}T)\|\partial_{t}u(T,\cdot)\|^{2}_{L^{2}(M)}+\frac{h^{4}}{\varepsilon}\|\partial_{t}u\|^{2}_{L^{2}(Q)}.$
(3.14)

Turning attention to the second term, we note that the Lie bracket
$[\partial_{t},\partial_{x_{1}}]$ vanishes. Thus, we integrate by parts and
apply $\partial_{t}u|_{t=0}=0$ to obtain

$2h^{4}\mathrm{Re}\int_{Q}(-\frac{1}{h})\partial_{t}^{2}u\overline{\partial_{x_{1}}u}dV_{g}dt=-2h^{3}\mathrm{Re}\int_{M}\partial_{t}u(T,x)\overline{\partial_{x_{1}}u(T,x)}dV_{g}+h^{3}\int_{Q}\partial_{x_{1}}|\partial_{t}u|^{2}dV_{g}dt.$

Since the vector field $\partial_{x_{1}}$ is divergence free, we get from the
assumption $u|_{\Sigma}=0$ and integration by parts that the last term in the
equation above vanishes. Hence, we have the following equality for the second
term

$2h^{4}\mathrm{Re}\int_{Q}(-\frac{1}{h})\partial_{t}^{2}u\overline{\partial_{x_{1}}u}dV_{g}dt=-2h^{3}\mathrm{Re}\int_{M}\partial_{t}u(T,x)\overline{\partial_{x_{1}}u(T,x)}dV_{g}.$
(3.15)

Before estimating the third term, we recall that in local coordinates
$(t,(x_{j})_{j=1}^{n})$ of $Q$ we have $[\partial_{t},\partial_{x_{j}}]=0$ for
every $j=1,\ldots,n$, and $\nabla_{g}u(t,x)=g^{ik}(x)\partial_{x^{k}}u(t,x)$.
Furthermore, since the metric $g$ is time-independent, we have

$\displaystyle\partial_{t}|\nabla_{g}u|^{2}=2\langle\nabla_{g}\partial_{t}u,\nabla_{g}u\rangle_{g}.$

Thus, by Green’s identities and $u|_{\Sigma}=0$, we obtain

$-2h^{4}\mathrm{Re}\int_{Q}\Delta_{g}u(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\overline{\partial_{t}u}dV_{g}dt=h^{4}\int_{Q}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}|\nabla_{g}u|^{2}dV_{g}dt.$

Since $u|_{t=0}=0$, it follows immediately that $\nabla_{g}u(0,\cdot)=0$. Then
we integrate by parts to get

$\int_{Q}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}|\nabla_{g}u|^{2}dV_{g}dt=\int_{M}(\frac{1}{h}\beta-\frac{1}{\varepsilon}T)|\nabla_{g}u(T,\cdot)|^{2}dV_{g}+\int_{Q}\frac{1}{\varepsilon}|\nabla_{g}u|^{2}dV_{g}dt.$

Therefore, we have verified that the third term on the right-hand side of
(3.13) satisfies

$-2h^{4}\mathrm{Re}\int_{Q}\Delta_{g}u(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}udV_{g}dt=h^{4}(\frac{1}{h}\beta-\frac{1}{\varepsilon}T)\|\nabla_{g}u(T,\cdot)\|^{2}_{L^{2}(M)}+\frac{h^{4}}{\varepsilon}\|\nabla_{g}u\|^{2}_{L^{2}(Q)}.$
(3.16)

We next follow the proof of [41, Lemma 4.2] to estimate the fourth term. Since
the metric $g$ is $x_{1}$-independent, it follows from the Leibniz rule and
the local representation of the divergence operator [50, Proposition 2.46]
that

$\displaystyle 2\overline{\partial_{x_{1}}u}\Delta_{g}u$
$\displaystyle=2\operatorname{div}_{g}(\overline{\partial_{x_{1}}u}\nabla_{g}u)-\operatorname{div}_{g}(|\nabla_{g}u|^{2}_{g}\partial_{x_{1}}).$

Thus, an application of the divergence theorem yields

$\displaystyle
2h^{4}\mathrm{Re}\int_{Q}\frac{1}{h}\overline{\partial_{x_{1}}u}\Delta_{g}udV_{g}dt$
$\displaystyle=h^{3}\mathrm{Re}\int_{\Sigma}2\partial_{\nu}u\overline{\partial_{x_{1}}u}-|\nabla_{g}u|^{2}\langle\nu,\partial_{x_{1}}\rangle_{g}dS_{g}dt.$

Since $u|_{\Sigma}=0$, we see that
$\nabla_{g}u|_{\Sigma}=(\partial_{\nu}u)\nu$ and
$\partial_{x_{1}}u|_{\Sigma}=\partial_{\nu}u\langle\nu,\partial_{x_{1}}\rangle_{g}:=\partial_{\nu}u\nu_{1}$.
Therefore, we have

$2h^{4}\mathrm{Re}\int_{Q}\frac{1}{h}\Delta_{g}u\overline{\partial_{x_{1}}u}dV_{g}dt=h^{3}\int_{\Sigma}{\nu_{1}}|\partial_{\nu}u|^{2}dS_{g}dt.$
(3.17)

We now turn our attention to the last term on the right-hand side of (3.13).
To that end, we integrate by parts, and use
$\operatorname{div}_{g}(\partial_{x_{1}})=0$ and $u|_{\Sigma}=0$ to write

$2\mathrm{Re}\int_{M}u(t,\cdot)\overline{\partial_{x_{1}}u(t,\cdot)}dV_{g}=\int_{M}\partial_{x_{1}}|u(t,\cdot)|^{2}dV_{g}=\int_{\partial
M}|u(t,\cdot)|^{2}\nu_{1}dS_{g}=0.$

Hence, by utilizing the condition $u|_{t=0}=0$, the last term on the right-
hand side of (3.13) can be written as

$\displaystyle
2h^{4}\mathrm{Re}\int_{Q}-\big{(}\frac{1}{h^{2}}(1-\beta^{2})+\frac{1}{\varepsilon}+\frac{2\beta}{h\varepsilon}t-\frac{1}{\varepsilon^{2}}t^{2}\big{)}u\big{(}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\overline{\partial_{t}u}-\frac{1}{h}\overline{\partial_{x_{1}}u}\big{)}dV_{g}dt$
$\displaystyle=$
$\displaystyle-h^{4}\int_{Q}\big{(}\frac{1}{h^{2}}(1-\beta^{2})+\frac{1}{\varepsilon}+\frac{2\beta}{h\varepsilon}t-\frac{1}{\varepsilon^{2}}t^{2}\big{)}(\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}|u|^{2}dV_{g}dt$
$\displaystyle=$
$\displaystyle-h^{4}\big{(}\frac{1-\beta^{2}}{h^{2}}+\frac{1}{\varepsilon}+\frac{2\beta}{\varepsilon
h}T-\frac{1}{\varepsilon^{2}}T^{2}\big{)}\big{(}\frac{1}{h}\beta-\frac{1}{\varepsilon}T\big{)}\|u(T,\cdot)\|_{L^{2}(M)}^{2}$
$\displaystyle+h^{4}\int_{Q}\big{(}\frac{3\beta^{2}-1}{\varepsilon
h^{2}}-\frac{1}{\varepsilon^{2}}-\frac{6\beta}{\varepsilon^{2}h}t+\frac{3}{\varepsilon^{3}}t^{2}\big{)}|u|^{2}dV_{g}dt.$

We now choose the numbers $\varepsilon,h>0$ such that

$0<\varepsilon<3T^{2},\quad\frac{1}{\sqrt{3}}<\beta<1,\quad\text{ and
}\quad\frac{1}{h}>\max\bigg{\\{}\frac{2T}{\varepsilon\beta},\frac{12\beta
T}{\varepsilon(3\beta^{2}-1)},\frac{2\beta
T}{\varepsilon},\frac{1}{\varepsilon}\bigg{\\}}.$ (3.18)

These choices yield $h<\varepsilon$,

$\frac{3\beta^{2}-1}{\varepsilon
h^{2}}-\frac{1}{\varepsilon^{2}}-\frac{6\beta}{\varepsilon^{2}h}t+\frac{3}{\varepsilon^{3}}t^{2}\geq\frac{3\beta^{2}-1}{2\varepsilon
h^{2}},$

and

$0<\left(\frac{1-\beta^{2}}{h^{2}}+\frac{1}{\varepsilon}+\frac{2\beta}{\varepsilon
h}T-\frac{1}{\varepsilon^{2}}T^{2}\right)\left(\frac{1}{h}\beta-\frac{1}{\varepsilon}T\right)\leq\frac{3\beta}{h^{3}}.$

The choices of $h$ and $\varepsilon$ in (3.18) allow the term
$\frac{1}{h^{2}}$ to absorb the lower order terms when $0<h\ll\varepsilon\ll
1$. Therefore, we get from these choices of $\varepsilon$ and $h$ that

$\displaystyle
2h^{4}\mathrm{Re}\int_{Q}-\big{(}\frac{1}{h^{2}}(1-\beta^{2})+\frac{1}{\varepsilon}+\frac{2\beta}{h\varepsilon}t-\frac{1}{\varepsilon^{2}}t^{2}\big{)}u((\frac{1}{h}\beta-\frac{1}{\varepsilon}t)\partial_{t}u-\frac{1}{h}\partial_{x_{1}}u)dV_{g}dt$
(3.19) $\displaystyle\geq$ $\displaystyle-3\beta
h\|u(T,\cdot)\|_{L^{2}(M)}^{2}+\frac{(3\beta^{2}-1)h^{2}}{2\varepsilon}\|u\|^{2}_{L^{2}(Q)}.$

By combining estimates (3.14)–(3.17) and (3.19), we obtain

$\displaystyle\frac{1}{2}\|P_{1}u+P_{2}u\|^{2}\geq$ $\displaystyle
h^{4}\big{(}\frac{1}{h}\beta-\frac{1}{\varepsilon}T\big{)}\|\partial_{t}u(T,\cdot)\|^{2}_{L^{2}(M)}+\frac{h^{4}}{\varepsilon}(\|\partial_{t}u\|^{2}_{L^{2}(Q)}+\|\nabla_{g}u\|^{2}_{L^{2}(Q)})$
$\displaystyle-2h^{3}\mathrm{Re}\int_{M}\partial_{t}u(T,x)\overline{\partial_{x_{1}}u}(T,x)dV_{g}+h^{4}\big{(}\frac{1}{h}\beta-\frac{1}{\varepsilon}T\big{)}\|\nabla_{g}u(T,\cdot)\|^{2}_{L^{2}(M)}$
$\displaystyle+h^{3}\int_{\Sigma}\nu_{1}|\partial_{\nu}u|^{2}dS_{g}dt+\frac{(3\beta^{2}-1)h^{2}}{2\varepsilon}\|u\|^{2}_{L^{2}(Q)}-3\beta
h\|u(T,\cdot)\|_{L^{2}(M)}^{2}.$

We sharpen the estimate above by implementing
$\frac{1}{h}>\frac{2T}{\varepsilon\beta}$ from (3.18) and utilizing the
following inequality

$\displaystyle\mathrm{Re}\int_{M}\partial_{t}u(T,x)\overline{\partial_{x_{1}}u}(T,x)dV_{g}\leq\frac{\beta}{8}\|\partial_{t}u(T,\cdot)\|_{L^{2}(M)}^{2}+\frac{8}{\beta}\|\nabla_{g}u(T,\cdot)\|_{L^{2}(M)}^{2}$

to obtain

$\displaystyle\frac{1}{2}\|P_{1}u+P_{2}u\|^{2}\geq$
$\displaystyle\frac{1}{4}\beta
h^{3}\|\partial_{t}u(T,\cdot)\|^{2}_{L^{2}(M)}+\frac{h^{4}}{\varepsilon}(\|\partial_{t}u\|^{2}_{L^{2}(Q)}+\|\nabla_{g}u\|^{2}_{L^{2}(Q)})$
(3.20)
$\displaystyle-h^{3}(\frac{16}{\beta}-\frac{\beta}{2})\|\nabla_{g}u(T,\cdot)\|_{L^{2}(M)}^{2}+\frac{(3\beta^{2}-1)h^{2}}{2\varepsilon}\|u\|^{2}_{L^{2}(Q)}$
$\displaystyle+h^{3}\int_{\Sigma}\nu_{1}|\partial_{\nu}u|^{2}dS_{g}dt-3\beta
h\|u(T,\cdot)\|_{L^{2}(M)}^{2}.$

Step 3: An upper bound of $\|P_{3}u\|_{L^{2}(Q)}$. We deduce from (3.10),
(3.12), as well as the triangle inequality that

$\displaystyle\|P_{3}u\|^{2}_{L^{2}(Q)}\leq
3h^{4}\big{(}\|a\|^{2}_{L^{\infty}(Q)}\|\partial_{t}u\|^{2}_{L^{2}(Q)}+\big{(}\frac{\beta^{2}}{h^{2}}\|a\|^{2}_{L^{\infty}(Q)}+\|q\|^{2}_{L^{\infty}(Q)}\big{)}\|u\|^{2}_{L^{2}(Q)}\big{)}.$
(3.21)

Here we have used the inequality $(x+y+z)^{2}\leq 3(x^{2}+y^{2}+z^{2})$ for
$x,y,z\in\mathbb{R}$.

In addition to the choice $0<\varepsilon<3T^{2}$ made in (3.18), we will
further require that

$\frac{1}{2\varepsilon}\geq 3\|a\|^{2}_{L^{\infty}(Q)}\quad\text{ and
}\quad\frac{3\beta^{2}-1}{4\varepsilon}\geq
3(\beta^{2}\|a\|^{2}_{L^{\infty}(Q)}+\|q\|^{2}_{L^{\infty}(Q)}).$

After combining estimates (3.11), (3.20), and (3.21), we obtain the claimed
estimate (3.7). This completes the proof of Proposition 3.1. ∎

We are now ready to state and prove the boundary Carleman estimate.

######  Proposition 3.2.

Let $a,q\in L^{\infty}(Q,\mathbb{C})$ and $v\in C^{2}(\overline{Q})$. If $v$
satisfies

$v|_{\Sigma}=v|_{t=0}=\partial_{t}v|_{t=0}=0,$ (3.22)

then for all $0<h\ll\varepsilon\ll 1$, we have

$\displaystyle\|e^{-\frac{1}{h}(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,a,q}v\|_{L^{2}(Q)}+\mathcal{O}(h^{3/2})\|e^{-\frac{1}{h}(\beta
T+x_{1})}\nabla_{g}v(T,\cdot)\|_{L^{2}(M)}$ (3.23)
$\displaystyle+\mathcal{O}(h^{1/2})\|e^{-\frac{1}{h}(\beta
T+x_{1})}v(T,\cdot)\|_{L^{2}(M)}+\mathcal{O}(h^{3/2})\bigg{(}\int_{\Sigma_{-}}|\partial_{\nu}\varphi||e^{-\frac{1}{h}(\beta
t+x_{1})}\partial_{\nu}v|^{2}dS_{g}dt\bigg{)}^{1/2}$
$\displaystyle\geq\mathcal{O}(h)\|e^{-\frac{1}{h}(\beta
t+x_{1})}v\|_{L^{2}(Q)}+\mathcal{O}(h^{3/2})\|e^{-\frac{1}{h}(\beta
T+x_{1})}\partial_{t}v(T,\cdot)\|_{L^{2}(M)}+\mathcal{O}(h^{2})(\|e^{-\frac{1}{h}(\beta
t+x_{1})}\partial_{t}v\|_{L^{2}(Q)}$ $\displaystyle+\|e^{-\frac{1}{h}(\beta
t+x_{1})}\nabla_{g}v\|_{L^{2}(Q)})+\mathcal{O}(h^{3/2})\bigg{(}\int_{\Sigma_{+}}\partial_{\nu}\varphi|e^{-\frac{1}{h}(\beta
t+x_{1})}\partial_{\nu}v|^{2}dS_{g}dt\bigg{)}^{1/2}.$

Here $\varphi(x)=x_{1}$, $\Sigma_{\pm}=(0,T)\times\partial
M_{\pm}^{\text{int}}$, and $\partial M_{\pm}=\\{x\in\partial
M:\pm\partial_{\nu}\varphi(x)\geq 0\\}$.

###### Proof.

By following the steps in the proof of [38, Theorem 3.1], we deduce the
claimed estimate (3.23) from estimate (3.7) by substituting
$u=\exp(-\varphi_{h,\varepsilon})v$, where $\varphi_{h,\varepsilon}$ given by
(3.5), as well as choosing $\varepsilon>0$ small but fixed. ∎

######  Remark 3.3.

The Carleman estimate (3.23) can be extended to any function
$v\in\mathcal{H}:=C^{1}([0,T];L^{2}(M))\cap C([0,T];H^{1}(M))$ satisfying
(3.22) and $\Box_{e\oplus g_{0}}v\in L^{2}(Q)$. Indeed, we may approximate
$f:=\Box_{e\oplus g_{0}}v\in L^{2}(Q)$ by a sequence $f_{j}\in
C_{0}^{\infty}(Q)$ such that $f_{j}\to f$ in $L^{2}(Q)$ as $j\to\infty$. If
$v_{j}$ solves $\Box_{e\oplus g_{0}}v_{j}=f_{j}$ and satisfies
$v_{j}|_{\Sigma}=v_{j}|_{t=0}=\partial_{t}v_{j}|_{t=0}=0$, then $v_{j}\in
C^{\infty}(\overline{Q})$ by [47, Remark 2.10]. In particular, the boundary
Carleman estimate (3.23) holds for $v_{j}$.

Furthermore, we have

$\|v_{j}-v\|_{\mathcal{H}}+\|\partial_{\nu}v_{j}-\partial_{\nu}v\|_{L^{2}(\Sigma)}\leq
C\|f_{j}-f\|_{L^{2}(Q)}\to 0,\quad j\to 0$

by the energy estimate [47, Theorem 2.1] together with [47, Remark 2.2]. Thus,
estimate (3.23) extends to $v$.

###  3.2. Semiclassical pseudodifferential operators

In this subsection we recall some fundamental concepts of semiclassical
pseudodifferential calculus on closed Riemannian manifolds by following the
expositions of [63] and [69, Chapter 14]. Let $(N,\widetilde{g})$ be a smooth
compact $n$-dimensional Riemannian manifold without boundary. For each
$m\in\mathbb{R}$, the Kohn-Nirenberg symbol class $S^{m}(T^{\ast}N)$ consists
of smooth functions on the cotangent bundle $T^{\ast}N$, which in local
coordinates of $N$ are given by

$S^{m}_{1,0}(T^{\ast}N)=S^{m}(T^{\ast}N)=\\{a(x,\xi)\in
C^{\infty}(T^{\ast}N):|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq
C_{\alpha\beta}\langle\xi\rangle^{m-|\beta|}\\},$ (3.24)

where $\langle\xi\rangle=(1+|\xi|^{2})^{1/2}$. For a parameter-dependent
symbol $a(x,\xi;h)$, we say that $a\in S^{m}(T^{\ast}N)$ if the estimate in
(3.24) holds uniformly for every $h\in(0,h_{0})$ and for some $h_{0}>0$. A
linear operator $B\colon C^{\infty}(N)\to C^{\infty}(N)$ is called negligible
if its Schwartz kernel $K_{B}\in C^{\infty}(N\times N)$ locally satisfies the
estimate
$\partial_{x}^{\alpha}\partial_{y}^{\beta}K_{B}(x,y)=\mathcal{O}(h^{\infty})$
for all $\alpha,\beta\in\mathbb{N}^{n}.$

A linear map $A\colon C^{\infty}(N)\to C^{\infty}(N)$ is a semiclassical
pseudodifferential operator of order $m\in\mathbb{R}$ if there exists $a\in
S^{m}(T^{\ast}N)$ such that in local coordinates, the operator $A$ is given by
the standard $h$-quantization

$Au(x)=\frac{1}{(2\pi h)^{n}}\int\int
e^{\frac{i}{h}(x-y)\cdot\xi}a(x,\xi;h)u(y)dyd\xi+Bu(x),$ (3.25)

where the operator $B$ is negligible, and the operator $\psi A\varphi$ is
negligible for each $\varphi,\psi\in C^{\infty}(N)$ with disjoint supports. We
denote $\Psi^{m}(N)$ the set of semiclassical pseudodifferential operators of
order $m$ on $(N,g)$.

We recall that the correspondence from an operator to a symbol is not globally
well-defined, but there exists a bijective map between the following
equivalence classes

$\Psi^{m}(N)/\Psi^{m-1}(N)\to S^{m}(T^{\ast}N)/S^{m-1}(T^{\ast}N).$

The image $\sigma_{A}(x,\xi;h)$ of $A\in\Psi^{m}(N)$ under this map is called
the principal symbol of $A$. These definitions allow us to compose the
operators $A_{j}\in\Psi^{m_{j}}(N)$, $j=1,2$, and we have
$A_{1}A_{2}\in\Psi^{m_{1}+m_{2}}(N)$ with principal symbol
$\sigma_{A_{1}A_{2}}=\sigma_{A_{1}}\sigma_{A_{2}}$.

An operator $A\in\Psi^{m}(N)$ is called elliptic if there is a constant $C>0$,
independent of $h$, such that the principal symbol satisfies

$|\sigma_{A}(x,\xi;h)|>\frac{1}{C}\langle\xi\rangle^{m}.$

An elliptic semiclassical operator $A\in\Psi^{m}(N)$ has an inverse
$R\in\Psi^{-m}(N)$ in the sense that there exists $h_{0}>0$ such that for all
$h\in(0,h_{0})$ we have $RA=AR=I$ as linear operators on $C^{\infty}(M)$ and

$\sigma_{R}(x,\xi;h)=\sigma_{A}(x,\xi;h)^{-1}\in
S^{-m}(T^{\ast}N)/S^{-m-1}(T^{\ast}N).$

By [63, Proposition 10.1], the operator

$J^{s}=(1-h^{2}\Delta_{\widetilde{g}})^{\frac{s}{2}},\quad s\in\mathbb{R},$
(3.26)

which is defined by the means of spectral theorem, is elliptic and belongs to
the class $\Psi^{s}(N)$ with principal symbol $\langle\xi\rangle^{s}$. We note
that for all $s_{1},s_{2}\in\mathbb{R}$, we have

$J^{s_{1}+s_{2}}=J^{s_{1}}J^{s_{2}},\quad(J^{s_{1}})^{-1}=J^{-s_{1}},\quad
J^{0}=I.$ (3.27)

We now define the semiclassical inner product of order $s\in\mathbb{R}$:

$(u,v)_{H^{s}_{\mathrm{scl}}(N)}:=(J^{s}u,J^{s}v)_{L^{2}(N)},\quad u,v\in
C^{\infty}(N).$

Then the semiclassical Sobolev space $H^{s}_{\mathrm{scl}}(N)$ is defined as
the completion of $C^{\infty}(N)$ with respect to a related norm. Furthermore,
every operator $A\in\Psi^{m}(N)$ yields a bounded map $A\colon
H^{s}_{\mathrm{scl}}(N)\to H^{s-m}_{\mathrm{scl}}(N)$. Also, we recall that if
$A$ is negligible, then the operator norm satisfies

$\|A\|_{H^{-s}_{\mathrm{scl}}(N)\to
H^{s}_{\mathrm{scl}}(N)}=\mathcal{O}(h^{\infty})\quad\text{for all
}s\in\mathbb{R}.$ (3.28)

Finally, we discuss the definition of semiclassical Sobolev spaces on an open
subset $U\subset N$. We recall that for $u\in C^{\infty}(N)$, the norms
$\|u\|_{H^{1}_{\mathrm{scl}}(N)}$ and

$\|u\|^{2}_{h}:=\|u\|^{2}_{L^{2}(N)}+\|h\nabla_{\widetilde{g}}u\|^{2}_{L^{2}(N)}$

are equivalent. We use the latter to define the semiclassical Sobolev space
$H^{1}_{\mathrm{scl}}(U)$ as a completion of $C^{\infty}(U)$ with respect to
the norm $\|\cdot\|_{h}$ restricted on $U$, and $H^{-1}_{\mathrm{scl}}(U)$ as
the topological dual of $H^{1}_{\mathrm{scl}}(U)$. We would like to recall the
following characterization [1, Section 3.13] of $H^{-1}_{\mathrm{scl}}(N)$ via
the completion of $L^{2}(N)$ with respect to the norm

$\|v\|_{H^{-1}_{\mathrm{scl}}(N)}=\sup_{0\neq\psi\in
H^{1}_{\mathrm{scl}}(N)}\frac{|\langle
v,\psi\rangle_{L^{2}(N)}|}{\|\psi\|_{H^{1}_{\mathrm{scl}}(N)}}.$ (3.29)

###  3.3. Interior Carleman estimate

In this subsection we assume that $(\overline{Q},e\oplus g)$ is isometrically
embedded into a closed Riemannian manifold $(N,g^{\prime})$ without boundary,
where $g^{\prime}=e\oplus g$ in some open neighborhood $U\subset N$ of
$\overline{Q}$. Here $U=(a,b)\times\widehat{M}$, where $[0,T]\subset(a,b)$ and
$\widehat{M}$ is an open manifold such that $M\subset\widehat{M}$. To prove
the existence of suitable solutions to $\mathcal{L}_{g,a,q}u=0$ in $Q$, we
need the following interior Calerman estimate for negative order Sobolev
spaces.

######  Lemma 3.4.

Let $a\in W^{1,\infty}(Q)$ and $q\in L^{\infty}(Q,\mathbb{C})$. Then for all
$0<h\ll\varepsilon\ll 1$ and $w\in C^{\infty}_{0}(Q)$, there exists a constant
$C>0$ such that

$h\|w\|_{L^{2}(N)}\leq C\|e^{\mp\frac{1}{h}(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,a,q}e^{\pm\frac{1}{h}(\beta
t+x_{1})}w\|_{H^{-1}_{\mathrm{scl}}(N)}.$ (3.30)

###### Proof.

We shall follow the arguments from [46, Section 2], see also [24, 44]. We
start by extending $a$ and $q$ from $W^{1,\infty}(Q)$ and $L^{\infty}(Q)$ to
$W^{1,\infty}(N)$ and $L^{\infty}(N)$, respectively, and denote the extensions
by the same letters. Throughout the proof of this lemma, we will use the
following shorthand notations for the convexified operators

${\Box_{\varphi_{\pm h,\varepsilon}}}=e^{-\varphi_{\pm
h,\varepsilon}}h^{2}{\Box_{g}}e^{\varphi_{\pm
h,\varepsilon}}\quad\text{and}\quad\mathcal{P}_{\varphi_{\pm
h,\varepsilon}}=e^{-\varphi_{\pm
h,\varepsilon}}h^{2}\mathcal{L}_{g,a,q}e^{\varphi_{\pm h,\varepsilon}}.$

Here the convexified weight $\varphi_{\pm h,\varepsilon}$ is defined as in
(3.5). Let us also define $u:=e^{\frac{t^{2}}{2\varepsilon}}w\in
C^{\infty}_{0}(Q)$.

We get from the triangle inequality that

$\displaystyle\|\mathcal{P}_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}\geq\|\Box_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}-\|e^{-\varphi_{\pm
h,\varepsilon}}h^{2}(a\partial_{t}+q)e^{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)},$ (3.31)

and begin by estimating the first term on the right-hand side of inequality
(3.31). This will be followed by a perturbation of the first term with the
second term, which contains the lower-order terms.

Let the open set $U\subset N$ be the same as at the beginning of this
subsection. We note that by taking $a=q=0$ and $v\in C^{\infty}_{0}(U)$,
Proposition 3.1 yields

$\frac{h}{\sqrt{\varepsilon}}\|v\|_{H^{1}_{\mathrm{scl}}(N)}\leq
C\|\Box_{\varphi_{\pm h,\varepsilon}}v\|_{L^{2}(N)},$ (3.32)

where $C=\frac{2}{\sqrt{3\beta^{2}-1}}$.

Let $\chi\in C^{\infty}_{0}(U)$ equal to $1$ in a neighborhood of
$\overline{Q}$. Since the operator $J^{-1}$, as defined in (3.26), is in
$\Psi^{-1}(N)$, and $\hbox{supp }(1-\chi)\cap\hbox{supp }u=\emptyset$, we have
by (3.28) that

$\|(1-\chi)J^{-1}u\|_{H^{1}_{\mathrm{scl}}(N)}=\mathcal{O}(h^{\infty})\|u\|_{H^{-1}_{\mathrm{scl}(N)}}\leq\mathcal{O}(h^{\infty})\|u\|_{L^{2}(N)}.$
(3.33)

Therefore, for all $0<h\ll\varepsilon\ll 1$ we have

$\displaystyle\|u\|_{L^{2}(N)}$ $\displaystyle\leq\|\chi
J^{-1}u\|_{H^{1}_{\mathrm{scl}}(N)}+\|(1-\chi)J^{-1}u\|_{H^{1}_{\mathrm{scl}}(N)}\leq\|\chi
J^{-1}u\|_{H^{1}_{\mathrm{scl}}(N)}+\mathcal{O}(h^{\infty})\|u\|_{L^{2}(N)}.$
(3.34)

To estimate the first term on the right-hand side of (3.34), we apply (3.32)
with $v=\chi J^{-1}u\in C^{\infty}_{0}(U)$ to get from the triangle inequality
and the commutator $[A,B]=AB-BA$ that

$\displaystyle\|\chi J^{-1}u\|_{H^{1}_{\mathrm{scl}}(N)}$ (3.35)
$\displaystyle\leq\mathcal{O}\left(\frac{\sqrt{\varepsilon}}{h}\right)\left(\|\chi
J^{-1}\Box_{\varphi_{\pm
h,\varepsilon}}u\|_{L^{2}(N)}+\|\chi[\Box_{\varphi_{\pm
h,\varepsilon}},J^{-1}]u\|_{L^{2}(N)}+\|[\Box_{\varphi_{\pm
h,\varepsilon}},\chi]J^{-1}u\|_{L^{2}(N)}\right).$

We now estimate the two terms containing the commutators on the right-hand
side of (3.35). Noting that $\Box_{\varphi_{\pm
h,\varepsilon}}\in\Psi^{2}(N)$, an application of [69, Theorem 9.5 (iii)]
yields that the commutator satisfies the equation $[\Box_{\varphi_{\pm
h,\varepsilon}},J^{-1}]=hR_{1}$, where $R_{1}\in\Psi^{0}(N)$. Thus, for the
second term on the right-hand side of (3.35) we have

$\|\chi[\Box_{\varphi_{\pm h,\varepsilon}},J^{-1}]u\|_{L^{2}(N)}\leq
h\|u\|_{L^{2}(N)}.$ (3.36)

For the third term on the right hand side of (3.35), we observe that

$[\Box_{\varphi_{\pm
h,\varepsilon}},\chi]=h^{2}\Box\chi+2h^{2}\left(\partial_{t}\chi\partial_{t}\varphi_{\pm
h,\varepsilon}-\langle\nabla_{g}\chi,\nabla_{g}\varphi_{\pm
h,\varepsilon}\rangle\right)+2h^{2}(\partial_{t}\chi\partial_{t}-\langle\nabla_{g}\chi,\nabla_{g}\cdot\rangle).$

Thus, $\hbox{supp }([\Box_{\varphi_{\pm
h,\varepsilon}},\chi])\subset\hbox{supp }(\partial_{t}\chi,\nabla_{g}\chi)$
and $\hbox{supp }(\partial_{t}\chi,\nabla_{g}\chi)\cap\hbox{supp
}u=\emptyset$. Therefore, we get from (3.28) that

$\|[\Box_{\varphi_{\pm
h,\varepsilon}},\chi]J^{-1}u\|_{L^{2}(N)}\leq\mathcal{O}(h^{\infty})\|u\|_{L^{2}(N)}.$
(3.37)

Since $\chi J^{-1}\in\Psi^{-1}(N)$, together with (3.35), (3.36) and (3.37),
we deduce from (3.34) that

$\displaystyle\|u\|_{L^{2}(N)}$
$\displaystyle\leq\mathcal{O}\left(\frac{\sqrt{\varepsilon}}{h}\right)\|\Box_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}+\mathcal{O}(\sqrt{\varepsilon})\|u\|_{L^{2}(N)}+\mathcal{O}(h^{\infty})\|u\|_{L^{2}(N)}.$

For later purposes, let us rewrite this inequality as

$\frac{h}{\sqrt{\varepsilon}}\|u\|_{L^{2}(N)}\leq\mathcal{O}(1)\|\Box_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}+\mathcal{O}(h)\|u\|_{L^{2}(N)}.$
(3.38)

To estimate the lower order terms in (3.31), taking $0\neq\psi\in
H^{1}_{\mathrm{scl}}(N)$ with $\|\psi\|_{H^{1}_{\mathrm{scl}}(N)}=1$, we
integrate by parts to obtain

$\displaystyle\langle e^{-\varphi_{\pm
h,\varepsilon}}h^{2}(a\partial_{t}+q)e^{\varphi_{\pm
h,\varepsilon}}u,\psi\rangle_{L^{2}(N)}$ $\displaystyle=-\langle
u,h^{2}\left((\partial_{t}a)\psi+a\partial_{t}\psi-(\partial_{t}\varphi_{\pm
h,\varepsilon})a\psi\right)\rangle_{L^{2}(N)}+\langle
h^{2}qu,\psi\rangle_{L^{2}(N)}.$

By recalling that $\partial_{t}\varphi_{\pm
h,\varepsilon}=\pm\frac{\beta}{h}-\frac{t}{\varepsilon}$,
$\|\psi\|_{H^{1}_{\mathrm{scl}}(N)}=1$, $h<\varepsilon$, and $\beta<1$, we get
from the Cauchy–Schwartz inequality that

$\displaystyle|\langle e^{-\varphi_{\pm
h,\varepsilon}}h^{2}(a\partial_{t}+q)e^{\varphi_{\pm
h,\varepsilon}}u,\psi\rangle_{L^{2}(N)}|\leq
h(3+T)(\|a\|_{W^{1,\infty}(N)}+\|q\|_{L^{\infty}(N)})\|u\|_{L^{2}(N)}.$ (3.39)

Therefore, the characterization (3.29) of the semiclassical $H^{-1}$-norm and
(3.39) implies that

$\displaystyle\|e^{-\varphi_{\pm
h,\varepsilon}}h^{2}(a\partial_{t}+q)e^{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}\leq\mathcal{O}(h)\|u\|_{L^{2}(N)}.$
(3.40)

Using (3.38) and (3.40), we derive from (3.31) that

$\displaystyle\mathcal{O}(1)\|\mathcal{P}_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}+\mathcal{O}(h)\|u\|_{L^{2}(N)}\geq\frac{h}{\sqrt{\varepsilon}}\|u\|_{L^{2}(N)},$

which can be rewritten as

$\displaystyle\mathcal{O}(\frac{\sqrt{\varepsilon}}{h})\|\mathcal{P}_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}+\mathcal{O}(\sqrt{\varepsilon})\|u\|_{L^{2}(N)}\geq\|u\|_{L^{2}(N)}.$

We now take $\varepsilon$ small enough but fixed to absorb the second term on
the left-hand side and get

$\displaystyle\mathcal{O}(\frac{1}{h})\|\mathcal{P}_{\varphi_{\pm
h,\varepsilon}}u\|_{H^{-1}_{\mathrm{scl}}(N)}\geq\|u\|_{L^{2}(N)}.$ (3.41)

Finally, we use $w=e^{-\frac{t^{2}}{2\varepsilon}}u$ for $0<t<T$ and apply
(3.41) to obtain

$\displaystyle\|w\|_{L^{2}(N)}\leq\mathcal{O}(\frac{1}{h})e^{\frac{T^{2}}{2\varepsilon}}\|e^{\mp\frac{1}{h}(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,a,q}e^{\pm\frac{1}{h}(\beta
t+x_{1})}w\|_{H^{-1}_{\mathrm{scl}}(N)}.$

Here the inequality relies on the assumption $u\in C_{0}^{\infty}(Q)$. This
completes the proof of Proposition 3.4. ∎

The following solvability result will be implemented in the next section to
construct CGO solutions for the operator $\mathcal{L}_{g,a,q}$.

######  Proposition 3.5.

Let $a\in W^{1,\infty}(Q)$, $q\in L^{\infty}(Q,\mathbb{C})$, and
$s=\frac{1}{h}+i\lambda$ with $\lambda\in\mathbb{R}$ fixed. If $h>0$ is small
enough, then for all $v\in L^{2}(Q)$ there exists a solution $u\in
H^{1}_{\mathrm{scl}}(Q)$ to the equation

$e^{\pm s(\beta t+x_{1})}h^{2}\mathcal{L}_{g,a,q}e^{\mp s(\beta
t+x_{1})}u=v\textit{ in }Q$ (3.42)

such that

$\|u\|_{H^{1}_{\mathrm{scl}}(Q)}\leq\mathcal{O}(h^{-1})\|v\|_{L^{2}(Q)}.$
(3.43)

###### Proof.

The proof uses standard functional analysis arguments, which have been
utilized to prove analogous results for elliptic operators, see for instance
[23, 45], as well as hyperbolic operators in [44]. ∎

##  4\. Construction of Complex Geometric Optic Solutions Based on Gaussian
Beam Quasimodes

Let $(M,g)$ be a CTA manifold, $a\in W^{1,\infty}(Q)$, and $q\in
C(\overline{Q})$. Let $0<h<1$, $\lambda\in\mathbb{R}$, and
$s=\frac{1}{h}+i\lambda\in\mathbb{C}$. In the first part of this section we
shall assume that the conformal factor $c=1$ and write $\mathcal{L}_{g,a,q}$
for $\mathcal{L}_{c,g,a,q}$. The goal of this section is to construct an
exponentially decaying, with respect to the real part of $s$, solution to the
equation $\mathcal{L}_{g,a,q}^{*}u_{1}=0$ in $Q$ of the form

$u_{1}=e^{-s(\beta t+x_{1})}(v_{s}+r_{1}),$ (4.1)

where
$\mathcal{L}_{g,a,q}^{\ast}=\mathcal{L}_{g,-\overline{a},\overline{q}-\partial_{t}\overline{a}}$
is the formal $L^{2}$-adjoint of the operator $\mathcal{L}_{g,a,q}$, as well
as an exponentially growing solution to the equation
$\mathcal{L}_{g,a,q}u_{2}=0$ in $Q$ of the form

$u_{2}=e^{s(\beta t+x_{1})}(w_{s}+r_{2}).$ (4.2)

Here $v_{s}$ and $w_{s}$ are smooth Gaussian beam quasimodes, and
$r_{1}=r_{1,s},r_{2}=r_{2,s}$ are correction terms that vanish in the limit
$h\to 0$. We construct the Gaussian beam quasimodes $v_{s}$ and $w_{s}$ for
each non-tangential geodesic of $(M_{0},g_{0})$ in Subsection 4.1. In
particular, the functions $v_{s}$ and $w_{s}$ satisfy the estimates given in
(4.4) and (4.5). In Subsection 4.2 we will establish a concentration property
of the quasimodes along the non-tangential geodesic in the limit $h\to 0$.
Finally, in Subsection 4.3 we will construct the remainder terms $r_{1}$ and
$r_{2}$, whose existence and decaying properties follow directly from the
estimates for the quasimode and the interior Carleman estimate Proposition
3.5. We emphasize that the CGO solutions (4.1) and (4.2) are constructed under
the assumption $c=1$. We shall incorporate general conformal factors $c$ and
modify our CGO solutions accordingly in Subsection 4.3.

Let us write $x=(x_{1},x^{\prime})$ for coordinates in $\mathbb{R}\times
M_{0}$, globally in $\mathbb{R}$ and locally in $M_{0}$. To justify our
construction we note that a function $u_{1}$ of the form (4.1) solves the
equation $\mathcal{L}^{\ast}_{g,a,q}u_{1}=0$ if

$e^{s(\beta t+x_{1})}\mathcal{L}^{\ast}_{g,a,q}e^{-s(\beta
t+x_{1})}r_{1}=-e^{s(\beta t+x_{1})}\mathcal{L}^{\ast}_{g,a,q}e^{-s(\beta
t+x_{1})}v_{s}.$

###  4.1. Construction of Gaussian beam quasimodes

In this subsection we focus on constructing Gaussian beam quasimodes.
Initially introduced in [10, 57], the construction of Gaussian beam quasimodes
has a very long tradition in spectral theory and in microlocal analysis, see
also [9, 56]. Gaussian beam quasimodes have also been used extensively to
solve inverse problems, starting with [13, 33]. Among the literature in this
direction, we refer readers to [21, 23, 25, 46, 68] for applications in
elliptic operators and [28, 34] in hyperbolic operators.

Let $(M,g)$ be a CTA manifold with the conformal factor $c=1$, and let $T>0$.
Replacing the transversal manifold $(M_{0},g_{0})$ by a slightly larger
manifold if necessary, we may assume without loss of generality that
$(M,g)\subset(\mathbb{R}\times M_{0}^{\mathrm{int}},e\oplus g_{0})$. The
Gaussian beam quasimodes will be constructed in $\mathbb{R}^{2}\times
M_{0}^{\mathrm{int}}$. To obtain $C^{\infty}$-smooth Gaussian beam quasimodes,
we shall regularize the damping coefficient $a$ and explain the necessity of
doing so in the proof of Theorem 4.2. We extend $a$ to
$W^{1,\infty}(\mathbb{R}^{2}\times M_{0}^{\mathrm{int}})$ with compact
support. Using a partition of unity argument combined with a regularization in
each coordinate patch, we have the following result, see [60, Lemma 2.1] for
details.

######  Proposition 4.1.

For any $a\in W_{0}^{1,\infty}(\mathbb{R}^{2}\times M_{0}^{int})$, there
exists an open and bounded set $W\subset\mathbb{R}^{2}\times M_{0}^{int}$ and
a family $a_{\zeta}\in C_{0}^{\infty}(W,\mathbb{C})$ such that

$\displaystyle\|a-a_{\zeta}\|_{L^{\infty}}=o(1),\quad\|a_{\zeta}\|_{L^{\infty}}=\mathcal{O}(1),\quad\|\nabla_{g}a_{\zeta}\|_{L^{\infty}}=o(\zeta^{-1}),$
(4.3)
$\displaystyle\|\partial_{t}a_{\zeta}\|_{L^{\infty}}=o(\zeta^{-1}),\quad\|\partial_{t}^{2}a_{\zeta}\|_{L^{\infty}}=o(\zeta^{-2}),\quad\|\Delta_{g}a_{\zeta}\|_{L^{\infty}}=o(\zeta^{-2}),\quad\zeta\to
0.$

Here the $L^{\infty}$-norms are taken over the set $\mathbb{R}^{2}\times
M_{0}^{int}$.

We are now ready to state and prove our first main result of this section.

######  Theorem 4.2.

Let $(M,g)$ be a smooth CTA manifold with boundary, $T>0$,
$\beta\in(\frac{1}{\sqrt{3}},1)$, and let $s=\frac{1}{h}+i\lambda$, $0<h\ll
1$, $\lambda\in\mathbb{R}$ fixed. Let $a\in W^{1,\infty}(Q)$ and $q\in
C(\overline{Q})$. Then for every unit speed non-tangential geodesic $\gamma$
of $(M_{0},g_{0})$ there exist one parameter families of Gaussian beam
quasimodes $v_{s},w_{s}\in C^{\infty}(\mathbb{R}^{2}\times M_{0})$ such that
the estimates

$\displaystyle\|v_{s}\|_{L^{2}(Q)}=\mathcal{O}(1),\quad\|\partial_{t}v_{s}\|_{L^{2}(Q)}=o(h^{-1/2}),\quad\|e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}^{\ast}_{g,a,q}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}(Q)}=o(h),$ (4.4)

and

$\displaystyle\|w_{s}\|_{L^{2}(Q)}=\mathcal{O}(1),\quad\|\partial_{t}w_{s}\|_{L^{2}(Q)}=o(h^{-1/2}),\quad\|e^{-s(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,a,q}e^{s(\beta t+x_{1})}w_{s}\|_{L^{2}(Q)}=o(h)$
(4.5)

are valid as $h\to 0$.

The proof of Theorem 4.2 is very long and given in the following subsections.

####  4.1.1. Preparations for the proof of Theorem 4.2

Let $\gamma=\gamma(\tau)$ be a non-tangential geodesic in the transversal
manifold $(M_{0},g_{0})$ of length $L>0$. By following [49, Example 9.32], we
embed $(M_{0},g_{0})$ into a closed manifold $(\widehat{M}_{0},g_{0})$ of the
same dimension. We also extend $\gamma$ as a unit speed geodesic in
$\widehat{M}_{0}$. Since $\gamma$ is non-tangential, we can choose
$\varepsilon>0$ so that $\gamma(\tau)\in\widehat{M}_{0}\setminus M_{0}$ and it
does not self-intersect for $\tau\in[-2\varepsilon,0)\cup(L,L+2\varepsilon]$.

We begin with the construction of a Gaussian beam quasimode for the conjugated
operator $e^{-s(\beta t+x_{1})}\mathcal{L}^{\ast}_{g,a,q}e^{s(\beta
t+x_{1})}$. We follow the main ideas from [21, 24, 46] and modify the argument
in accordance with the extra time variable $t$. Consider the CGO ansatz
$v_{s}$ of the form

$v_{s}(t,x_{1},x^{\prime};h)=e^{is\Theta(x^{\prime})}b(t,x_{1},x^{\prime};h),$

where $s=\frac{1}{h}+i\lambda$ is a complex number, and the amplitude $b$
depends implicitly on the semiclassical parameter $h$.

Since the phase function $\Theta$ is independent of $t$, we have

$e^{-is\Theta}a\partial_{t}(e^{is\Theta}b)=a\partial_{t}b\quad\text{and}\quad
e^{-is\Theta}\partial_{t}^{2}(e^{is\Theta}b)=\partial_{t}^{2}b.$ (4.6)

Also, as $\Theta$ is independent of $x_{1}$ and $g=e\oplus g_{0}$, we get

$e^{-is\Theta}(-\Delta_{g})e^{is\Theta}b=-\Delta_{g}b-is[2\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b(x_{1},\cdot)\rangle_{g_{0}}+(\Delta_{g_{0}}\Theta)b]+s^{2}\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}b.$
(4.7)

Using (4.6) and (4.7), we obtain

$\displaystyle e^{s(\beta
t+x_{1})}\mathcal{L}_{g,-\overline{a},\overline{q}-\partial_{t}\overline{a}}e^{-s(\beta
t+x_{1})}v_{s}=$ $\displaystyle
e^{is\Theta}[s^{2}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b$
(4.8)
$\displaystyle+s(2\partial_{x_{1}}b-2\beta\partial_{t}b-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b(x_{1},\cdot)\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b+\beta\overline{a}b)$
$\displaystyle+\mathcal{L}_{g,-\overline{a},\overline{q}-\partial_{t}\overline{a}}b].$

The computations in (4.8) suggest that in order to verify the estimates in
(4.4), we should construct the phase function $\Theta$ and the amplitude $b$
such that they approximately solve the eikonal and transport equations
appearing on right-hand side of (4.8) as multipliers of the terms $s^{2}$ and
$s$, respectively.

The construction of the Gaussian beam quasimode $v_{s}$ is divided into the
several steps, which are addressed in the following subsections.

  1. (1)

Fermi coordinates near $\gamma$: In these local coordinates
$x^{\prime}=(\tau,y)\in M_{0}$ is given by its closest point $\gamma(\tau)$ to
the fixed geodesic $\gamma$ and by its location $y$ in the respective geodesic
plane perpendicular to $\dot{\gamma}(\tau)$. Due to these choices, the metric
tensor $g_{0}$ on $\gamma$ is Euclidean up to the first order. This simplifies
many subsequent computations.

  2. (2)

Eikonal equation and phase function: To satisfy estimate (4.4) we solve the
eikonal equation, which is the multiplier of $s^{2}$ in (4.8), up to a certain
order. The solution is called the phase function $\Theta$.

  3. (3)

Transport equation and amplitude: To satisfy estimate (4.4) we solve the
transport equation, which is the multiplier of $s$ in (4.8), up to a certain
order, and call the obtained solution $b$ the amplitude. Since we want to
construct a smooth Gaussian beam quasimode, we shall solve the transport
equation with a regularized damping coefficient $a_{\zeta}$ and relate the
regularization parameter $\zeta$ to the semiclassical parameter $h$ with an
explicit expression. We solve the transport equation via a change of
variables, which converts the transport equation into a better understood
$\overline{\partial}$-equation.

  4. (4)

Local verification of estimate (4.4): In this step we provide a proof for
estimate (4.4) in a neighborhood of a fixed point $z_{0}=\gamma(\tau_{0})$.
The key of the proof is the fact that map $x\mapsto|x|^{k}e^{-d|x|^{2}}$ is in
$L^{2}(\mathbb{R}^{n})$ for all $d>0$ and $k\geq 0$.

  5. (5)

Global construction of Gaussian beam quasimodes: We provide the global
construction of $v_{s}$ by a gluing together quasimodes defined along the
small pieces of the geodesic.

####  4.1.2. Fermi coordinates near $\gamma$.

We fix a point $z_{0}=\gamma(\tau_{0})$ on
$\gamma([-\varepsilon,L+\varepsilon])$ and construct the quasimode locally
near $z_{0}$. Let
$(\tau,y)\in\Omega:=\\{(\tau,y)\in\mathbb{R}\times\mathbb{R}^{n-2}:|\tau-\tau_{0}|<\delta,|y|<\delta^{\prime}\\}$,
$\delta,\delta^{\prime}>0$, be Fermi coordinates near $z_{0}$. The detailed
construction of these coordinates is given in [36, Lemma 7.4]. Heuristically,
the idea is to first choose the number $\delta>0$ to be sufficiently small, so
that the geodesic segment $\gamma|_{[\tau_{0}-\delta,\tau_{0}+\delta]}$ is not
self-intersecting. Then one uses the parallel transport to choose some vector
fields $E_{1}(\tau),\ldots,E_{n-2}(\tau)$ along $\gamma$ such that the vector
fields $\dot{\gamma}(\tau),E_{1}(\tau),\ldots,E_{n-2}(\tau)$ span a parallel
orthonormal frame along $\gamma$. Due to the choice of $\delta$, as well as
the inverse function theorem, there exists $\delta^{\prime}>0$ such that the
map $F(\tau,y)=\exp_{\gamma(\tau)}\left(y^{\alpha}E_{\alpha}(\tau)\right)$ is
a diffeomorphism in the set $\Omega$. Here $\exp$ is the exponential map of
$(M_{0},g_{0})$ and $\alpha\in\\{1,\ldots,n-2\\}$.

We note that near $z_{0}=\gamma(\tau_{0})$ the trace of the geodesic $\gamma$
is given by the set $\Gamma=\\{(\tau,0):|\tau-\tau_{0}|<\delta\\}$. Due to
this construction, we get

$g_{0}^{jk}(\tau,0)=\delta^{jk}\quad\text{and}\quad\partial_{y_{l}}g_{0}^{jk}(\tau,0)=0.$
(4.9)

Hence, by Taylor’s theorem, for small $|y|$ we have

$g_{0}^{jk}(\tau,y)=\delta^{jk}+\mathcal{O}(|y|^{2}).$ (4.10)

In these coordinates the Gaussian beam ansatz takes the form

$v_{s}(t,x_{1},\tau,y)=e^{is\Theta(\tau,y)}b(t,x_{1},\tau,y;h),$ (4.11)

and our aim is to find the phase function $\Theta\in
C^{\infty}(\Omega,\mathbb{C})$ that satisfies

$\mathrm{Im}\Theta\geq
0,\quad\mathrm{Im}\Theta|_{\Gamma}=0,\quad\mathrm{Im}\Theta(\tau,y)\text{ is
bi-Lipschitz equivalent to }|y|^{2},$ (4.12)

and an amplitude $b\in
C^{\infty}(\mathbb{R}\times\mathbb{R}\times\Omega,\mathbb{C})$ such that
$\hbox{supp }(b(t,x_{1},\cdot))\subset\\{|y|<\delta^{\prime}/2\\}$. In
particular, the name “Gaussian beam” comes from (4.12).

####  4.1.3. Eikonal equation and phase function.

Our goal in this step is to find a phase function $\Theta$ by solving the
eikonal equation
$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}=1-\beta^{2}$
up to order $|y|^{3}$ on $\Gamma$ by arguing similarly as in [25, 34, 46, 56,
57]. That is, we find a function $\Theta(\tau,y)\in
C^{\infty}(\Omega,\mathbb{C})$ that satisfies

$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2})=\mathcal{O}(|y|^{3}),\quad
y\to 0,$ (4.13)

and

$\mathrm{Im}\Theta(\tau,y)\geq d|y|^{2}$ (4.14)

for some constant $d>0$ depending on $\beta$. The equation (4.13), combined
with a scaling in the semiclassical parameter $h$, will be used in Subsection
4.1.5 to prove estimate (4.4).

In order to utilize the Taylor expansion (4.10) for the metric $g_{0}$, we
look for a function $\Theta$ of the form
$\Theta=\Theta_{0}+\Theta_{1}+\Theta_{2}$, where

$\Theta_{j}(\tau,y)=\sum_{|\alpha|=j}\frac{\Theta_{j,\alpha}(\tau)}{\alpha!}y^{\alpha},\quad
j=0,1,2,$

are the homogeneous polynomials in the $y$-variable. We also write
$g^{jk}_{0}=g_{0,0}^{jk}+g_{0,1}^{jk}+g_{0,2}^{jk}+r^{jk}_{3}$, where

$g_{0,l}^{jk}(\tau,y)=\sum_{|\mu|=l}\frac{g_{0,l,\mu}^{jk}(\tau)}{\mu!}y^{\mu},\quad
l=0,1,2,$

and $r_{3}=\mathcal{O}(|y|^{3})$ is the remainder in Taylor’s theorem. By the
properties of Fermi coordinates (4.9), we have $g_{0,0}^{jk}=\delta^{jk}$ and
$g_{0,1}^{jk}=0$. We then choose accordingly that

$\Theta_{0}(\tau,y)=\sqrt{1-\beta^{2}}\tau\quad\text{ and
}\quad\Theta_{1}(\tau,y)=0.$ (4.15)

Let us next find $\Theta_{2}$. To this end, we write the metric
$g^{jk}_{0}=\delta^{jk}+g_{0,2}^{jk}+\mathcal{O}(|y|^{3})$. With the
understanding that $j,k$ run from 1 to $n-1$ and $\alpha,\mu$ run from 2 to
$n-1$, we see that

$\displaystyle\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2})=[2\sqrt{1-\beta^{2}}\partial_{\tau}\Theta_{2}+\nabla_{y}\Theta_{2}\cdot\nabla_{y}\Theta_{2}+(1-\beta^{2})g_{0,2}^{11}]+\mathcal{O}(|y|^{3}).$

Similar to [21, 25, 36], our aim is to find a function $\Theta_{2}$ such that

$2\sqrt{1-\beta^{2}}\partial_{\tau}\Theta_{2}+\nabla_{y}\Theta_{2}\cdot\nabla_{y}\Theta_{2}+(1-\beta^{2})g_{0,2}^{11}=0.$
(4.16)

Due to our previous choice for the form of $\Theta_{2}$, we have

$\Theta_{2}(\tau,y)=\frac{1}{2}\sqrt{1-\beta^{2}}H(\tau)y\cdot y,$

and in order to satisfy (4.12), we seek to obtain a smooth complex-valued
symmetric matrix $H(\tau)$ with a positive definite imaginary part. Since each
term of (4.16) is quadratic in $y$, it is sufficient for $H(\tau)$ to satisfy
the following matrix initial value problem, called the Ricatti equation

$\dot{H}(\tau)+H(\tau)^{2}=F(\tau),\quad H(\tau_{0})=H_{0},\quad\text{ for
}\tau\in\mathbb{R},$ (4.17)

where $F(\tau)$ is a symmetric matrix such that
$g_{0,2}^{11}(\tau,y)=-F(\tau)y\cdot y$ in
$(\tau_{0}-\delta,\tau_{0}+\delta)$. If we choose $H_{0}$ to be a symmetric
complex-valued matrix such that $\mathrm{Im}(H_{0})$ is positive definite,
then (4.17) has a unique smooth symmetric complex-valued solution $H(\tau)$
with $\mathrm{Im}H(\tau)$ positive definite, see [34, Lemma 2.56] for details.

Therefore, the phase function $\Theta$ reads

$\Theta(\tau,y)=\sqrt{1-\beta^{2}}(\tau+\frac{1}{2}H(\tau)y\cdot y).$ (4.18)

Due to the compactness and the positive definiteness of
$\mathrm{Im}(H(\tau))$, the function $\Theta$ satisfies the properties in
(4.12).

####  4.1.4. Transport equation and the amplitude.

We next seek an amplitude $b$ of the form

$b(t,x_{1},\tau,y;h,\zeta)=h^{-\frac{n-2}{4}}b_{0}(t,x_{1},\tau;\zeta)\chi(y/\delta^{\prime}),$
(4.19)

where $b_{0}\in
C^{\infty}(\mathbb{R}_{t}\times\mathbb{R}_{x_{1}}\times[\tau_{0}-\delta,\tau_{0}+\delta])$
is independent of $y$ and satisfies the approximate transport equation

$2\partial_{x_{1}}b_{0}-2\beta\partial_{t}b_{0}-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}(x_{1},\cdot)\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b_{0}+\beta\overline{a_{\zeta}}b_{0}=\mathcal{O}(|y|\zeta^{-1})$
(4.20)

as $y,\zeta\to 0$. The cut-off function $\chi\in
C^{\infty}_{0}(\mathbb{R}^{n-2})$ in (4.19) is chosen such that $\chi=1$ for
$|y|\leq 1/4$ and $\chi=0$ for $|y|\geq 1/2$. Here $\zeta$ is the
regularization parameter and $a_{\zeta}$ is the regularized damping
coefficient given in the Proposition 4.1. Instead of the transport equation
appearing the coefficient of $s$ in (4.8), we solve (4.20) and obtain a smooth
Gaussian beam quasimode. Eventually, we want the Gaussian beam to only depend
on the semiclassical parameter $h$. Therefore, at the end of this subsection
we will give an explicit relation between $h$ and $\zeta$. We shall make the
expression $|y|\zeta^{-1}$ rigorous in the next subsection when we prove
estimate (4.4). Also, we will address the effect of the change from $a$ to
$a_{\zeta}$ to the proof of (4.4). Equation (4.20), combined with a scaling in
the semiclassical parameter $h$, will be used in Subsection 4.1.5 to prove the
third estimate in (4.4).

In order to find a function $b_{0}$ such that (4.20) holds, we first compute
$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}(x_{1},\cdot)\rangle_{g_{0}}$.
It follows from (4.18) that

$\partial_{\tau}\Theta(\tau,y)=\sqrt{1-\beta^{2}}+\mathcal{O}(|y|^{2}).$
(4.21)

Therefore, we get from (4.10) that

$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}(x_{1},\cdot)\rangle_{g_{0}}=\sqrt{1-\beta^{2}}(\partial_{\tau}b_{0}+H(\tau)y\cdot\partial_{y}b_{0})+\mathcal{O}(|y|^{2})\partial_{\tau}b_{0}+\mathcal{O}(|y|^{2})\partial_{y}b_{0}.$
(4.22)

We next compute $\Delta_{g_{0}}\Theta$ near the geodesic $\gamma$. Using
(4.10) and (4.18), we have

$(\Delta_{g_{0}}\Theta)(\tau,0)=\sqrt{1-\beta^{2}}\delta^{jk}H_{jk}=\sqrt{1-\beta^{2}}\hbox{tr}\,H(\tau),$

which implies

$(\Delta_{g_{0}}\Theta)(\tau,y)=\sqrt{1-\beta^{2}}\hbox{tr}\,H(\tau)+\mathcal{O}(|y|).$
(4.23)

Finally, we Taylor expand the coefficients appearing on the left-hand side of
(4.20). Writing

$\displaystyle
a_{\zeta}(t,x_{1},\tau,y)=a_{\zeta}(t,x_{1},\tau,0)+\int_{0}^{1}(\nabla_{y}a_{\zeta}(t,x_{1},\tau,ys))yds,$

and utilizing (4.3), we get

$a_{\zeta}(t,x_{1},\tau,y)=a_{\zeta}(t,x_{1},\tau,0)+\mathcal{O}(|y|\zeta^{-1}).$
(4.24)

To achieve (4.20), we require that $b_{0}(t,x_{1},\tau;\zeta)$ satisfies

$(\beta\partial_{t}-\partial_{x_{1}}+i\sqrt{1-\beta^{2}}\partial_{\tau})b_{0}=\frac{1}{2}[-i\sqrt{1-\beta^{2}}\hbox{tr}\,H(\tau)+\beta\overline{a_{\zeta}}(t,x_{1},\tau,0)]b_{0}.$
(4.25)

To solve this equation we perform a change of variables and write the left-
hand side of (4.25) as a $\overline{\partial}$-equation. To that end, let
$S:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be an invertible linear function such that
for a fixed $\beta\in(\frac{1}{\sqrt{3}},1)$, its inverse function is

$S^{-1}(t,x_{1},\tau)=\left(\frac{1}{\beta}t,x_{1}+\frac{1}{\beta}t,\frac{1}{\sqrt{1-\beta^{2}}}\tau\right):=(\widetilde{t},p,r).$
(4.26)

Then we get from (4.26) that

$\partial_{t}=\frac{\partial}{\partial\widetilde{t}}\frac{\partial\widetilde{t}}{\partial
t}+\frac{\partial}{\partial p}\frac{\partial p}{\partial
t}+\frac{\partial}{\partial\tau}\frac{\partial\tau}{\partial
t}=\frac{1}{\beta}(\partial_{\widetilde{t}}+\partial_{p})\quad\text{ and
}\quad\partial_{x_{1}}=\frac{\partial}{\partial\widetilde{t}}\frac{\partial\widetilde{t}}{\partial
x_{1}}+\frac{\partial}{\partial p}\frac{\partial p}{\partial
x_{1}}+\frac{\partial}{\partial\tau}\frac{\partial\tau}{\partial
x_{1}}=\partial_{p}.$ (4.27)

By substituting (4.26) and (4.27) into (4.25), we obtain the equation

$(\partial_{\widetilde{t}}+i\partial_{r})b_{0}^{\prime}=\frac{1}{2}[-i\sqrt{1-\beta^{2}}\hbox{tr}\,H(\sqrt{1-\beta^{2}}r)+\beta\overline{a_{\zeta}}^{\prime}(\widetilde{t},p,r,0)]b_{0}^{\prime},$
(4.28)

where $a_{\zeta}^{\prime}=a_{\zeta}\circ S$ and $b_{0}^{\prime}=b_{0}\circ S$.

Writing
$\overline{\partial}=\frac{1}{2}(\partial_{\widetilde{t}}+i\partial_{r}),$ we
look for a solution to (4.28) of the form
$b_{0}^{\prime}(\widetilde{t},p,r;\zeta)=e^{\Phi_{1,\zeta}(\widetilde{t},p,r)+f_{1}(r)}$.
By a direct computation, we see that in order for such a function
$b_{0}^{\prime}$ to solve (4.28), the functions $\Phi_{1,\zeta}$ and $f_{1}$
need to satisfy

$\overline{\partial}\Phi_{1,\zeta}(\widetilde{t},p,r)=\frac{\beta}{4}\overline{a_{\zeta}}^{\prime}(\widetilde{t},p,r,0)=\frac{\beta}{4}\overline{a_{\zeta}}^{\prime}(\widetilde{t},p,\gamma(r))$
(4.29)

and

$\partial_{r}f_{1}=-\frac{\sqrt{1-\beta^{2}}}{2}\hbox{tr}\,H(\sqrt{1-\beta^{2}}r).$
(4.30)

Note that $f_{1}$ can be obtained by integrating the right-hand side of (4.30)
with respect to $r$.

In order to solve the $\overline{\partial}$-equation (4.29), we use the
fundamental solution $E(\widetilde{t},r)=\frac{1}{\pi(\widetilde{t}+ir)}$ of
the $\overline{\partial}$-operator [29, Section 5.4] to take

$\Phi_{1,\zeta}(\widetilde{t},p,r)=\frac{\beta}{4}(E\ast\overline{a_{\zeta}}^{\prime})(\widetilde{t},p,\gamma(r)).$
(4.31)

While forming the convolution over the complex variable $\widetilde{t}+ir$, we
note that by Proposition 4.1, the function $a_{\zeta}^{\prime}$ is compactly
supported in $\mathbb{R}^{2}\times M_{0}^{\mathrm{int}}$. Since $\gamma$ is a
non-tangential geodesic in $(M_{0},g_{0})$, we may assume without loss of
generality that the map
$(\widetilde{t},p,r)\mapsto\overline{a_{\zeta}}^{\prime}(\widetilde{t},p,\gamma(r))$
is smooth and compactly supported in the entire $(\widetilde{t},p,r)$-space so
that estimate (4.3) still holds. Therefore, we have obtained a
$C^{\infty}$-smooth solution
$b_{0}^{\prime}(\widetilde{t},p,r;\zeta)=e^{\Phi_{1,\zeta}(\widetilde{t},p,r)+f_{1}(r)}$
of (4.28) defined in the whole $(\widetilde{t},p,r)$-space.

To verify that $b_{0}$ satisfies (4.20), we need to estimate
$b_{0}(\cdot;\zeta)$, as well as its first and second order derivatives over
the set
$[0,\frac{T}{\beta}]\times\overline{J}_{p}\times[r_{0}-\delta,r_{0}+\delta]$,
where $J_{p}\subset\mathbb{R}$ is an open and bounded interval such that the
respective $p$-coordinate of each point in $Q$ is in $J_{p}$. Since the
function $\overline{a}_{\zeta}$ is supported in some open and bounded set of
$\mathbb{R}^{2}\times M_{0}^{\mathrm{int}}$, as given in Proposition 4.1,
there exists some compact set $K\subset\mathbb{R}^{2}$ such that the following
inequality holds for every
$(\widetilde{t},p,r)\in[0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}]$

$|\Phi_{1,\zeta}(\widetilde{t},p,r)|\leq\int_{K}|E(\widetilde{t}-t,r-s)||\overline{a}^{\prime}_{\zeta}(t,p,s)|dtds\leq\|E_{(\widetilde{t},r)}\|_{L^{1}(K)}\|\overline{a}^{\prime}_{\zeta}\|_{L^{\infty}},$
(4.32)

where $E_{\widetilde{t},r}(t,s)=E(\widetilde{t}-t,r-s)$. Due to the local
integrability of $E$, the term $\|E_{(\widetilde{t},r)}\|_{L^{1}(K)}$ has a
uniform bound for all
$(\widetilde{t},r)\in[0,\frac{T}{\beta}]\times[0,\frac{L}{\sqrt{1-\beta^{2}}}]$.
Then it follows from estimate (4.3) that
$\|\Phi_{1,\zeta}\|_{L^{\infty}}=\mathcal{O}(1)$. Furthermore, by replacing
the function $\overline{a}_{\zeta}$ with
$\partial_{\widetilde{t}}\overline{a}_{\zeta}$ in (4.32) and utilizing (4.3)
again, we get
$\|\partial_{\widetilde{t}}\Phi_{1,\zeta}\|_{L^{\infty}}=o(\zeta^{-1})$. We
also obtain the following estimates by using similar arguments:

$\displaystyle\|\nabla_{g_{0}}\Phi_{1,\zeta}\|_{L^{\infty}},\|\partial_{p}\Phi_{1,\zeta}\|_{L^{\infty}}=o(\zeta^{-1}),\quad\|\Delta_{g_{0}}\Phi_{1,\zeta}\|_{L^{\infty}},\|\partial_{\widetilde{t}}^{2}\Phi_{1,\zeta}\|_{L^{\infty}}=o(\zeta^{-2}),\quad\zeta\to
0.$

To complete the verification of (4.20), we connect the semiclassical parameter
$h$ and the regularization parameter $\zeta$ by setting $\zeta=h^{\alpha}$,
$0<\alpha<\frac{1}{2}$. Note that the change of coordinates function $S$,
given by (4.26), is independent of $\zeta$. Hence, this choice of $\zeta$, in
conjunction with estimates (4.3), (4.31), and
$\|f_{1}\|_{L^{\infty}}=\mathcal{O}(1)$, yields

$\displaystyle\|b_{0}(\cdot;h)\|_{L^{\infty}}=\mathcal{O}(1),\quad\|\nabla_{g_{0}}b_{0}(\cdot;h)\|_{L^{\infty}},\|\partial_{t}b_{0}(\cdot;h)\|_{L^{\infty}},\|\partial_{x_{1}}b_{0}(\cdot;h)\|_{L^{\infty}}=o(h^{-\alpha}),$
(4.33)
$\displaystyle\|\Delta_{g_{0}}b_{0}(\cdot;h)\|_{L^{\infty}},\|\partial_{t}^{2}b_{0}(\cdot;h)\|_{L^{\infty}}=o(h^{-2\alpha}),\quad
h\to 0.$

By substituting (4.21)–(4.25) into the left-hand side of (4.20), we get from
(4.33) that

$\displaystyle
2\partial_{x_{1}}b_{0}-2\beta\partial_{t}b_{0}-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}(x_{1},\cdot)\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b_{0}+\beta\overline{a_{\zeta}}b_{0}$
$\displaystyle=-2i\sqrt{1-\beta^{2}}H(\tau)y\cdot\partial_{y}b_{0}+\mathcal{O}(|y|^{2})\partial_{\tau}b_{0}+\mathcal{O}(|y|^{2})\partial_{y}b_{0}+\mathcal{O}(|y|)+\mathcal{O}(|y|h^{-\alpha}).$
$\displaystyle=\mathcal{O}(|y|h^{-\alpha}).$

Thus, equation (4.20) is verified.

####  4.1.5. Local verification of estimate $\eqref{eq:estimate_v}$.

Let us now verify that the estimates in (4.4) hold for the quasimode

$v_{s}(\widetilde{t},p,r,y;h)=e^{is\Theta(\sqrt{1-\beta^{2}}r,y)}b^{\prime}(\widetilde{t},p,r,y;h)=e^{is\Theta(\sqrt{1-\beta^{2}}r,y)}h^{-\frac{n-2}{4}}b_{0}^{\prime}(\widetilde{t},p,r;h)\chi(y/\delta^{\prime})$
(4.34)

in the open set $(0,\frac{T}{\beta})\times J_{p}\times\Omega$ of $Q$, where
$\Omega\subset M_{0}$ is the domain of Fermi coordinates near the point
$z_{0}=\gamma(\tau_{0})$. To establish this, we shall need the following
estimate for any $k\geq 0$:

$\begin{split}\|h^{-\frac{n-2}{4}}|y|^{k}e^{-\frac{\mathrm{Im}\Theta}{h}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}&\leq\|h^{-\frac{n-2}{4}}|y|^{k}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}\\\
&\leq\bigg{(}\int_{\mathbb{R}^{n-2}}h^{k}|z|^{2k}e^{-2d|z|^{2}}dz\bigg{)}^{1/2}\\\
&=\mathcal{O}(h^{k/2}),\quad h\to 0.\end{split}$ (4.35)

Here we applied estimate (4.14) and the change of variables $z=h^{-1/2}y$.

We are now ready to start verifying (4.4) locally. To that end, we use (4.14),
(4.33), and (4.35) with $k=0$ to get

$\displaystyle\|v_{s}\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle\leq\|b_{0}^{\prime}\|_{L^{\infty}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times[r_{0}-\delta,r_{0}+\delta])}\|e^{is\Theta}h^{-\frac{n-2}{4}}\chi(y/\delta^{\prime})\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
(4.36)
$\displaystyle\leq\mathcal{O}(1)\|h^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}=\mathcal{O}(1),\quad
h\to 0.$

Let us next estimate
$\|\partial_{t}v_{s}\|_{L^{2}([0,T]\times\overline{J}_{p}\times\Omega)}$. By
utilizing (4.33), (4.34), and $\zeta=h^{\alpha}$, $0<\alpha<\frac{1}{2}$, we
obtain

$\|\partial_{t}v_{s}\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}=o(h^{-1/2}),\quad
h\to 0.$ (4.37)

We now proceed to estimate $\|e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,-\overline{a},\overline{q}-\partial_{t}\overline{a}}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}([0,\frac{T}{\beta}]\times J_{p}\times\Omega)}$ by
estimating each term on the right-hand of (4.8) independently. Let us start
with the first term. By applying (4.13), (4.14), (4.33), and (4.35) with
$k=3$, we get

$\displaystyle
h^{2}\|e^{is\Theta}s^{2}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
(4.38)
$\displaystyle=h^{2}\|e^{is\Theta}s^{2}h^{-\frac{n-2}{4}}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b_{0}^{\prime}\chi(y/\delta^{\prime})\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle\leq\mathcal{O}(1)\|h^{-\frac{n-2}{4}}|y|^{3}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}=\mathcal{O}(h^{3/2}),\quad
h\to 0.$

We next consider the second term on the right-hand side of (4.8). From a
direct computation, we see that

$|e^{is\Theta}|=e^{-\frac{1}{h}\mathrm{Im}\Theta}e^{-\lambda\mathrm{Re}\Theta}=e^{-\frac{\sqrt{1-\beta^{2}}}{2h}\mathrm{Im}H(\tau)y\cdot
y}e^{-\lambda\sqrt{1-\beta^{2}}\tau}e^{-\lambda\mathcal{O}(|y|^{2})}.$

We observe that $e^{-\frac{1}{h}}=\mathcal{O}(h^{\infty})$. Therefore, on the
support of $\nabla_{g_{0}}\chi(y/\delta^{\prime})$ we deduce from (4.14) that

$|e^{is\Theta}|\leq e^{-\frac{\widetilde{d}}{h}}\quad\text{for some
}\widetilde{d}>0.$

Thus, using estimates (4.20) and (4.35) with $k=1$,
$\alpha\in(0,\frac{1}{2})$, and the triangle inequality, we have

$\displaystyle
h^{2}\|e^{is\Theta}s(2\partial_{x_{1}}b-2\beta\partial_{t}b-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b(x_{1},\cdot)\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b+\beta\overline{a_{\zeta}}b)\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
(4.39)
$\displaystyle\leq\mathcal{O}(h)\|e^{is\Theta}h^{-\frac{n-2}{4}}[|y|h^{-\alpha}\chi(y/\delta^{\prime})-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\chi(y/\delta^{\prime})\rangle_{g_{0}}]\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle\leq\mathcal{O}(h)\|h^{-\frac{n-2}{4}}|y|h^{-\alpha}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}+\mathcal{O}(e^{-\frac{\widetilde{d}}{h}})$
$\displaystyle=\mathcal{O}(h^{3/2-\alpha})=o(h),\quad h\to 0.$

We want to emphasize that in the second term on the right-hand side of (4.8)
we have the damping coefficient $\bar{a}$ instead of its smooth approximation
$\overline{a_{\zeta}}$, which appeared in (4.39). To medicate this
discrepancy, we use estimates (4.3) and (4.35) with $k=0$ to get

$\displaystyle
h^{2}\|e^{is\Theta}s\beta(\overline{a}-\overline{a_{\zeta}})b\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle=\mathcal{O}(h)\|e^{is\Theta}(\overline{a}-\overline{a_{\zeta}})h^{-\frac{n-2}{4}}b_{0}^{\prime}\chi(y/\delta^{\prime})\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
(4.40)
$\displaystyle\leq\mathcal{O}(h)\|\overline{a}-\overline{a_{\zeta}}\|_{L^{\infty}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}\|h^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}$
$\displaystyle=o(h),\quad h\to 0.$

Finally, we estimate the third term on the right-hand side of (4.8). To that
end, we utilize estimates (4.33) and (4.35) with $k=0$ to obtain

$h^{2}\|e^{is\Theta}\partial_{t}^{2}b\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}=o(h^{2(1-\alpha)})=o(h),\quad
h\to 0.$ (4.41)

To estimate the term involving the $\Delta_{g}$, we incorporate estimates
(4.19), (4.33), and (4.35) with $k=0$, as well as the triangle inequality, to
get

$\displaystyle h^{2}$
$\displaystyle\|e^{is\Theta}(-\Delta_{g}b)\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
(4.42) $\displaystyle\leq$
$\displaystyle\mathcal{O}(h^{2})\|h^{-\frac{n-2}{4}}e^{is\Theta}\chi(y/\delta^{\prime})\Delta_{g}b_{0}^{\prime}\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle+\mathcal{O}(h^{2})\|h^{-\frac{n-2}{4}}e^{is\Theta}[b_{0}^{\prime}\Delta_{g}\chi(y/\delta^{\prime})+2\langle\nabla_{g}b_{0}^{\prime},\nabla_{g}\chi(y/\delta^{\prime})\rangle_{g}]\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}$
$\displaystyle\leq$
$\displaystyle\mathcal{O}(h^{2})\left(\|h^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}h^{-2\alpha}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}+\mathcal{O}(e^{-\frac{\widetilde{d}}{h}})\right)$
$\displaystyle=$
$\displaystyle\mathcal{O}(h^{2(1-\alpha)})+\mathcal{O}(e^{-\frac{\widetilde{d}}{h}})=o(h),\quad
h\to 0.$

For the lower order terms, it follows from (4.33) that

$h^{2}\|e^{is\Theta}(-\overline{a}\partial_{t}b+(\overline{q}-\partial_{t}\overline{a})b)\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}=o(h^{2-\alpha})=o(h^{3/2}),\quad
h\to 0.$ (4.43)

Therefore, by combining estimates (4.38)–(4.43), we conclude from (4.8) that

$\|e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{g,-\overline{a},\overline{q}-\partial_{t}\overline{a}}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times\Omega)}=o(h),\quad
h\to 0.$ (4.44)

This completes the verification of estimate (4.4) locally in the set
$(0,\frac{T}{\beta})\times J_{p}\times\Omega$.

Before proceeding to the global construction, we need an estimate for
$\|v(t,x_{1},\cdot)\|_{L^{2}({\partial M}_{0})}$ for later purposes. If
$\Omega$ contains a boundary point $x_{0}=(\tau_{0},0)\in\partial M_{0}$, then
$\dot{\gamma}(\tau_{0})$ is transversal to $\partial M_{0}$. Let $\rho$ be a
boundary defining function for $M_{0}$ so that $\partial M_{0}$ is given by
the level set $\rho(r,y)=0$ near $x_{0}$, and $\nabla_{g_{0}}\rho$ is normal
to $\partial M_{0}$ at $x_{0}$. These imply $\partial_{\tau}\rho(x_{0})\neq
0$. By the implicit function theorem, there exists a smooth function $y\mapsto
r(y)$ near 0 such that $\partial M_{0}$ near $x_{0}$ is given by
$\\{(r(y),y):|y|<r_{0}\\}$ for some $r_{0}>0$ small, see the proof of [36,
Proposition 7.5].

Using (4.12), (4.14), and (4.33), we see that there exists a constant $C$ such
that

$|v_{s}(t,x_{1},\tau,y;h)|\leq
Ch^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}\chi(y/\delta^{\prime}).$ (4.45)

Thus, after shrinking the set $\Omega$ if necessary and using (4.35) with
$k=0$ along with (4.45), we get

$\displaystyle\|v(t,x_{1},\cdot)\|^{2}_{L^{2}({\partial M}_{0}\cap\Omega)}$
$\displaystyle=\int_{|y|_{e}<r_{0}}|v(t,x_{1},r(y),y)|^{2}dS_{g}(y)$ (4.46)
$\displaystyle\leq\mathcal{O}(1)\int_{\mathbb{R}^{n-2}}h^{-\frac{n-2}{2}}e^{-\frac{2d}{h}|y|^{2}}dy=\mathcal{O}(1),\quad
h\to 0.$

####  4.1.6. Global construction of the Gaussian beam quasimodes via gluing.

Finally, we glue together the quasimodes defined along small pieces of the
geodesic $\gamma$ to obtain the quasimode $v_{s}$ in $\mathbb{R}^{2}\times
M_{0}^{\mathrm{int}}$. Since $\widehat{M}_{0}$ is compact and
$\gamma(r):(-2\varepsilon,\frac{L}{\sqrt{1-\beta^{2}}}+2\varepsilon)\to\widehat{M}_{0}$
is a non-tangential geodesic that is not a loop, it follows from [36, Lemma
7.2] and the choice of $\varepsilon>0$ that the curve
$\gamma|_{(-2\varepsilon,\frac{L}{\sqrt{1-\beta^{2}}}+2\varepsilon)}$ has
finitely many self intersection times $r_{\ell}\geq 0$ with
$\ell\in\\{1,\ldots,R\\}$ and

$-\varepsilon=r_{0}<r_{1}<\dots<r_{R}<r_{R+1}=\frac{L}{\sqrt{1-\beta^{2}}}+\varepsilon.$

Due to [36, Lemma 7.4], there exists an open cover
$\\{(\Omega_{\ell},\kappa_{\ell})_{\ell=0}^{R+1}\\}$ of
$\gamma([-\varepsilon,\frac{L}{\sqrt{1-\beta^{2}}}+\varepsilon])$ consisting
of Fermi coordinate neighborhoods that have the following properties:

  1. (1)

$\kappa_{\ell}(\Omega_{\ell})=I_{\ell}\times B$, where $I_{\ell}$ are open
intervals and $B=B(0,\delta^{\prime})$ is an open ball in $\mathbb{R}^{n-2}$.
Here $\delta^{\prime}>0$ can be taken arbitrarily small and same for each
$\Omega_{\ell}$.

  2. (2)

$\kappa_{\ell}(\gamma(r))=(r,0)$ for $r\in I_{\ell}$.

  3. (3)

$r_{\ell}$ only belongs to $I_{\ell}$ and
$\overline{I_{\ell}}\cap\overline{I_{k}}=\emptyset$ unless $|\ell-k|\leq 1$.

  4. (4)

$\kappa_{\ell}=\kappa_{k}$ on $\kappa_{\ell}^{-1}((I_{\ell}\cap I_{k})\times
B)$.

In particular, the intervals $I_{0}$ and $I_{R+1}$ are chosen in such a way
that they do not contain any self-intersection times. In the case when
$\gamma$ does not self-intersect, there is a single coordinate neighborhood of
$\gamma|_{[-\varepsilon,\frac{L}{\sqrt{1-\beta^{2}}}+\varepsilon]}$ such that
(1) and (2) are satisfied.

We proceed as follows to construct the quasimode $v_{s}$. Suppose first that
$\gamma$ does not self-intersect at $r=0$. Using the procedure from the
earlier part of this proof, we find a quasimode

$v_{s}^{(0)}(\widetilde{t},p,r,y;h)=h^{-\frac{n-2}{4}}e^{is\Theta^{(0)}(\sqrt{1-\beta^{2}}r,y)}e^{\Phi_{1,h}(\widetilde{t},p,r)+f_{1}(r)}\chi(y/\delta^{\prime})$

in $\Omega_{0}$ with some fixed initial conditions at $r=-\varepsilon$ for the
Riccati equation (4.17) determining $\Theta^{(0)}$. We now choose some
$r_{0}^{\prime}$ such that $\gamma(r_{0}^{\prime})\in\Omega_{0}\cap\Omega_{1}$
and let

$v_{s}^{(1)}(\widetilde{t},p,r,y;h)=h^{-\frac{n-2}{4}}e^{is\Theta^{(1)}(\sqrt{1-\beta^{2}}r,y)}e^{\Phi_{1,h}(\widetilde{t},p,r)+f_{1}(r)}\chi(y/\delta^{\prime})$

be the quasimode in $\Omega_{1}$ by choosing the initial conditions for
$\Theta^{(1)}$ such that
$\Theta^{(1)}(r_{0}^{\prime})=\Theta^{(0)}(r_{0}^{\prime})$. Here we have used
the same functions $\Phi_{1,h}$ and $f_{1}$ in $v_{s}^{(0)}$ and $v_{s}^{(1)}$
since $\Phi_{1,h}$ and $f_{1}$ are both globally defined for all
$r\in(-2\varepsilon,\frac{L}{\sqrt{1-\beta^{2}}}+2\varepsilon)$, and neither
of the functions depends on $y$. On the other hand, since the equations
determining the phase functions $\Theta^{(0)}$ and $\Theta^{(1)}$ have the
same initial data in $\Omega_{0}$ and in $\Omega_{1}$, and the local
coordinates $\kappa_{0}$ and $\kappa_{1}$ coincide on
$\kappa_{0}^{-1}((I_{0}\cap I_{1})\times B)$, we have
$\Theta^{(1)}=\Theta^{(0)}$ in $\Omega_{0}\cap\Omega_{1}$. Therefore, we
conclude that $v_{s}^{(0)}=v_{s}^{(1)}$ in the overlapped region
$\Omega_{0}\cap\Omega_{1}$. Continuing in this way, we obtain quasimodes
$v_{s}^{(2)},\dots,v_{s}^{(R+1)}$ such that

$v_{s}^{(\ell)}(\widetilde{t},p,\cdot)=v_{s}^{(\ell+1)}(\widetilde{t},p,\cdot)\quad\text{in}\quad\Omega_{\ell}\cap\Omega_{\ell+1}$
(4.47)

for all $\widetilde{t}$ and $p$. If $\gamma$ self-intersects at $r=0$, we
start the construction from $v^{(1)}$ by fixing initial conditions for (4.17)
at $r=0$ and find $v^{(0)}$ by going backwards.

Let $\chi_{j}(r)$ be a partition of unity subordinate to the intervals
$(I_{\ell})_{\ell=0}^{R+1}$. We denote
$\widetilde{\chi}_{\ell}(\widetilde{t},p,r,y)=\chi_{\ell}(r)$ and define a
smooth function

$v_{s}=\sum_{\ell=0}^{R+1}\widetilde{\chi}_{\ell}v_{s}^{(\ell)},\quad\text{in
$\mathbb{R}^{2}\times\widehat{M}_{0}$.}$

Let $z_{1},\dots,z_{R^{\prime}}\in M_{0}$, $R^{\prime}<R$, be the distinct
self-intersection points of $\gamma$, corresponding to the self intersection
times $0\leq r_{1}<\cdots<r_{R}$. Let $V_{j}$ be a small neighborhood in
$\widehat{M}_{0}$ centered at $z_{j}$, for $j\in\\{1,\dots,R^{\prime}\\}$. We
proceed as in [36, Proposition 7.5], and use the definition of the intervals
$I_{0},\ldots,I_{R+1}$ and (4.47) to pick a finite cover $W_{0},\ldots W_{S}$
of the remaining points on the geodesic $\gamma$ such that for each
$k\in\\{1,\ldots,S\\}$ we have $W_{k}\subset\Omega_{\ell(k)}$ for some
$\ell(k)\in\\{0,\ldots,R+1\\}$. This gives us an open cover for $\hbox{supp
}(v_{s}(\widetilde{t},p,\cdot))\cap M_{0}$

$\displaystyle\hbox{supp }(v_{s}(\widetilde{t},p,\cdot))\cap
M_{0}\subset\left(\bigcup_{j=1}^{R^{\prime}}V_{j}\right)\cup\left(\bigcup_{k=0}^{S}W_{k}\right),$

and the quasimode restricted to $V_{j}$ and $W_{k}$ is of the form

$v_{s}(\widetilde{t},p,\cdot)|_{V_{j}}=\sum_{\ell:\gamma(r_{\ell})=z_{j}}v_{s}^{(\ell)}(\widetilde{t},p,\cdot)\quad\text{
and }\quad
v_{s}(\widetilde{t},p,\cdot)|_{W_{k}}=v_{s}^{(\ell(k))}(\widetilde{t},p,\cdot),$
(4.48)

respectively. Here (4.48) follows from the construction of the intervals
$(I_{\ell})_{\ell=0}^{R+1}$, the partition of unity subordinate to these
intervals, and choosing the set $V_{j}$ small enough.

Since in the both cases of (4.48) the function $v_{s}$ is a finite sum of
$v^{(\ell)}$, the estimate $\|v_{s}(\widetilde{t},p,\cdot)\|_{L^{2}(\partial
M_{0})}=\mathcal{O}(1)$ and those in (4.4) follow from the corresponding local
considerations (4.37), (4.44), and (4.46) for each of $v_{s}^{(\ell)}$,
respectively. This completes the construction of the Gaussian beam quasimode
$v_{s}$.

####  4.1.7. Construction of a Gaussian beam quasimode for the operator
$e^{-s(\beta t+x_{1})}\mathcal{L}_{g,a,q}e^{s(\beta t+x_{1})}$.

We next seek a Gaussian beam quasimode for the operator $e^{-s(\beta
t+x_{1})}\mathcal{L}_{g,a,q}e^{s(\beta t+x_{1})}$ of the form

$w_{s}(t,x_{1},\tau,y;h,\zeta)=e^{is\Theta(\tau,y)}B(t,x_{1},\tau,y;h,\zeta)$

with the phase function $\Theta\in C^{\infty}(\Omega,\mathbb{C})$ satisfying
(4.12) and $B(t,x_{1},\tau,y)\in
C^{\infty}(\mathbb{R}\times\mathbb{R}\times\Omega)$ supported near $\Gamma$.
By replacing $s$ in (4.6) and (4.7) by $-s$, and recalling that $\Theta$ is
independent of $x_{1}$, we obtain

$\displaystyle e^{-s(\beta t+x_{1})}\mathcal{L}_{g,a,q}e^{s(\beta
t+x_{1})}w_{s}=$
$\displaystyle+s(-2\partial_{x_{1}}B+2\beta\partial_{t}B-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}B(x_{1},\cdot)\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)B+\beta
aB)$ (4.49) $\displaystyle+\mathcal{L}_{g,a,q}B].$

Since the eikonal equation above is identical to the one in (4.8), we see from
Subsection 4.1.3 that the phase function $\Theta$ is given by (4.18). We next
find the amplitude $B$ in the form of

$B(t,x_{1},\tau,y;h)=h^{-\frac{n-2}{4}}B_{0}(t,x_{1},\tau;h)\chi(y/\delta^{\prime}),$
(4.50)

where $B_{0}\in
C^{\infty}([\mathbb{R}\times\mathbb{R}\times\\{\tau:|\tau-\tau_{0}|<\delta\\})$.
To that end, by proceeding similarly as the construction of $b_{0}$ in
Subsection 4.1.4, we require that $B_{0}$ solves

$(\beta\partial_{t}-\partial_{x_{1}}-i\sqrt{1-\beta^{2}}\partial_{\tau})B_{0}=\frac{1}{2}[(i\sqrt{1-\beta^{2}}\hbox{tr}\,H(\tau)-\beta
a(t,x_{1},\tau,0)]B_{0}$ (4.51)

Using change of coordinates (4.26) again, we get

$(\partial_{\widetilde{t}}-i\partial_{r})B_{0}^{\prime}=\frac{1}{2}[i\sqrt{1-\beta^{2}}\hbox{tr}\,H(\sqrt{1-\beta^{2}}r)-\beta
a_{\zeta}^{\prime}(\widetilde{t},p,r,0)]B_{0}^{\prime},$ (4.52)

where $B_{0}^{\prime}=B_{0}\circ S$ and $a_{\zeta}^{\prime}=a_{\zeta}\circ S$.

By writing $\partial=\frac{1}{2}(\partial_{\widetilde{t}}-i\partial_{r})$ and
looking for a solution of the form
$B_{0}=e^{\Phi_{2}(\widetilde{t},p,r)+f_{2}(r)}\eta(\widetilde{t},p,r)$ with
$\partial\eta=0$, we see that the functions $\Phi_{2,\zeta}$ and $f_{2}$ must
satisfy

$\partial\Phi_{2,\zeta}=-\frac{1}{4}\beta
a_{\zeta}^{\prime}(\widetilde{t},p,\gamma(r))$ (4.53)

and

$\partial_{r}f_{2}=-\frac{\sqrt{1-\beta^{2}}}{2}\hbox{tr}\,H(\sqrt{1-\beta^{2}}r).$
(4.54)

Using similar arguments as in the construction of $v_{s}$, we obtain a
Guassian beam quasimode $w_{s}\in C^{\infty}(Q)$ such that the estimates in
(4.5) holds.

This completes the proof of Theorem 4.2.

###  4.2. Concentration property of the Gaussian beam quasimodes

By the proof of Theorem 4.2, for each non-tangential geodesic
$\gamma\colon[0,\frac{L}{\sqrt{1-\beta^{2}}}]\to M_{0}$ and $h>0$ there exist
smooth functions $\Phi_{1,h},\Phi_{2,h}$ in
$[0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}]$
satisfying

$(\partial_{\widetilde{t}}+i\partial_{r})\Phi_{1,h}(\widetilde{t},p,r)=\frac{1}{2}\beta\overline{a}_{h}^{\prime}(\widetilde{t},p,\gamma(r))\text{
and
}(\partial_{\widetilde{t}}-i\partial_{r})\Phi_{2,h}(\widetilde{t},p,r)=-\frac{1}{2}\beta
a^{\prime}_{h}(\widetilde{t},p,\gamma(r)),$

where $a_{h}^{\prime}=a_{h}\circ S$, and the change of coordinates $S$ is
given by (4.26). Here $J_{p}\subset\mathbb{R}$ is an open and bounded interval
such that for each point in $Q$ the respective $p$-coordinate is in $J_{p}$.
In the next lemma we study the behavior for these functions as $h\to 0$.

######  Lemma 4.3.

Let $\beta\in(\frac{1}{\sqrt{3}},1)$, and let
$\gamma\colon[0,\frac{L}{\sqrt{1-\beta^{2}}}]\to M_{0}$ be a non-tangential
geodesic in $(M_{0},g_{0})$ as in Proposition 4.2. Then there exist continuous
functions $\Phi_{1}$ and $\Phi_{2}$ in
$[0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}]$
that satisfy

$(\partial_{\widetilde{t}}+i\partial_{r})\Phi_{1}(\widetilde{t},p,r)=\frac{1}{2}\beta\overline{a}^{\prime}(\widetilde{t},p,\gamma(r))\text{
and
}(\partial_{\widetilde{t}}-i\partial_{r})\Phi_{2}(\widetilde{t},p,r)=-\frac{1}{2}\beta
a^{\prime}(\widetilde{t},p,\gamma(r)),$ (4.55)

respectively. Furthermore, the following estimate holds

$\|\Phi_{j,h}-\Phi_{j}\|_{L^{\infty}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}])}=o(1),\quad
j=1,2,\quad h\to 0.$ (4.56)

###### Proof.

With a slight abuse of notation, we consider the compactly supported function
$\bar{a}^{\prime}(\widetilde{t},p,r)=\overline{a}^{\prime}(\widetilde{t},p,\gamma(r))$
in $\mathbb{R}^{3}$ and define a continuous function
$\Phi_{1}(\widetilde{t},p,r)=\frac{\beta}{4}(E\ast\overline{a}^{\prime})(\widetilde{t},p,r).$
Here the convolution is taken over the complex variable $\widetilde{t}+ir$.
Since $E:=\frac{1}{\pi(\widetilde{t}+ir)}$ is the fundamental solution for the
$\bar{\partial}$-operator, we see that
$\overline{\partial}\Phi_{1}=\frac{1}{4}\beta\overline{a}^{\prime}.$ Lastly,
estimate (4.56) follows from the local integrability of $E$, estimate (4.3),
and an inequality analogous to (4.32). The analogous claims for $j=2$ follow
by the same arguments. This completes the proof of Lemma 4.3. ∎

In the following theorem we show that a Gaussian beam quasimode concentrates
along the geodesic in the limit $h\to 0$.

######  Theorem 4.4.

Let $s=\frac{1}{h}+i\lambda$, $0<h\ll 1$, $\lambda\in\mathbb{R}$ fixed, and
$\beta\in(\frac{1}{\sqrt{3}},1)$. Let
$\gamma\colon[0,\frac{L}{\sqrt{1-\beta^{2}}}]\to M_{0}$ be a non-tangential
geodesic in $(M_{0},g_{0})$ as in Proposition 4.2. Let $J_{p}$ be as above.
Let $v_{s}$ and $w_{s}$ be the quasimodes from Proposition 4.2. Then for each
$\psi\in C(M_{0})$ and
$(\widetilde{t}^{\prime},p^{\prime})\in[0,\frac{T}{\beta}]\times\overline{J}_{p}$
we have

$\displaystyle\lim_{h\to
0}\int_{M_{0}}\overline{v_{s}}(\widetilde{t}^{\prime},p^{\prime},\cdot)w_{s}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi
dV_{g_{0}}=(1-\beta^{2})^{-\frac{n-6}{4}}$
$\displaystyle\int_{0}^{\frac{L}{\sqrt{1-\beta^{2}}}}e^{-2(1-\beta^{2})\lambda
r}e^{\overline{\Phi_{1}}(\widetilde{t}^{\prime},p^{\prime},r)+\Phi_{2}(\widetilde{t}^{\prime},p^{\prime},r)}$
(4.57)
$\displaystyle\quad\quad\quad\times\eta(\widetilde{t}^{\prime},p^{\prime},r)\psi(\gamma(r))dr.$

Here the functions $\Phi_{1},\Phi_{2}\in
C([0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}])$
are as in Lemma 4.3, and $\eta\in
C^{\infty}([0,\frac{T}{\beta}]\times\overline{J}_{p}\times[0,\frac{L}{\sqrt{1-\beta^{2}}}])$
with $(\partial_{\widetilde{t}}-i\partial_{r})\eta=0$.

###### Proof.

By a partition of unity, it suffices to verify (4.57) for $\psi\in
C_{0}(V_{j}\cap M_{0})$ and $\psi\in C_{0}(W_{k}\cap M_{0})$, where $V_{j}$
and $W_{k}$ are the same as in the subsection 4.1.6 in the proof of Theorem
4.2.

Case 1: $\psi\in C_{0}(W_{k}\cap M_{0})$. We first consider the easier case
that $\psi\in C_{0}(W_{k}\cap M_{0})$ for some $k$. Here $\hbox{supp }\psi$
may extend to $\partial M_{0}$, and we extend $\psi$ by zero outside
$W_{k}\cap M_{0}$. Using the quasimodes $v_{s}$ and $w_{k}$ are given in
(4.48), by arguing similarly as in the proof of Theorem 4.2, we obtain
Gaussian beam quasimodes

$\displaystyle
v_{s}(\widetilde{t},p,r,y)=e^{is\Theta(\sqrt{1-\beta^{2}}r,y)}h^{-\frac{n-2}{4}}e^{\Phi_{1,h}(\widetilde{t},p,r)+f_{1}(r)}\chi(y/\delta^{\prime}),$
(4.58) $\displaystyle
w_{s}(\widetilde{t},p,r,y)=e^{is\Theta(\sqrt{1-\beta^{2}}r,y)}h^{-\frac{n-2}{4}}e^{\Phi_{2,h}(\widetilde{t},p,r)+f_{2}(r)}\eta(\widetilde{t},p,r)\chi(y/\delta^{\prime}).$

In order to establish (4.57), we substitute the quasimodes given in (4.58)
directly into the left-hand side of (4.57). After that, we apply the dominated
convergence theorem to simplify our computation since the term
$e^{-\frac{1}{h}\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot y}$
dominates the other exponentials in the phase function. Then we utilize
properties of the solution $H(r)$ to the Riccati equation (4.17), as well as
the definitions of functions $f_{1},f_{2}$ given by (4.30) and (4.54),
respectively.

Let us now provide the detailed proof. Using (4.10), we see that the
determinant satisfies

$|g_{0}(r,y)|^{1/2}=\sqrt{1-\beta^{2}}+\mathcal{O}(|y|^{2}).$ (4.59)

It then follows from (4.18), (4.26), (4.58), and (4.59) that

$\displaystyle\int_{M_{0}}\overline{v_{s}}(\widetilde{t}^{\prime},p^{\prime},\cdot)w_{s}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi
dV_{g_{0}}$ (4.60) $\displaystyle=$
$\displaystyle\sqrt{1-\beta^{2}}\int_{0}^{\frac{L}{\sqrt{1-\beta^{2}}}}\int_{\mathbb{R}^{n-2}}e^{-\frac{2}{h}\mathrm{Im}\Theta}e^{-2\lambda\mathrm{Re}\Theta}h^{-\frac{n-2}{2}}e^{\overline{\Phi_{1,h}}(\widetilde{t}^{\prime},p^{\prime},r)+\Phi_{2,h}(\widetilde{t}^{\prime},p^{\prime},r)+\overline{f_{1}}(r)+f_{2}(r)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\chi^{2}(y/\delta^{\prime})\eta(\widetilde{t},p,r)\psi(r,y)|g_{0}|^{1/2}dydr$
$\displaystyle=$
$\displaystyle\sqrt{1-\beta^{2}}\int_{0}^{\frac{L}{\sqrt{1-\beta^{2}}}}\int_{\mathbb{R}^{n-2}}e^{-\frac{1}{h}\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}e^{-2\lambda(1-\beta^{2})r}e^{\lambda\mathcal{O}(|y|^{2})}h^{-\frac{n-2}{2}}\chi^{2}(y/\delta^{\prime})\eta(\widetilde{t}^{\prime},p^{\prime},r)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times
e^{\overline{\Phi_{1,h}}(\widetilde{t}^{\prime},p^{\prime},r)+\Phi_{2,h}(\widetilde{t}^{\prime},p^{\prime},r)+\overline{f_{1}}(r)+f_{2}(r)}\psi(r,y)(\sqrt{1-\beta^{2}}+\mathcal{O}(|y|^{2}))dydr$
$\displaystyle=$
$\displaystyle(1-\beta^{2})\int_{0}^{\frac{L}{\sqrt{1-\beta^{2}}}}\int_{\mathbb{R}^{n-2}}e^{-\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}e^{-2(1-\beta^{2})\lambda
r}e^{h\lambda\mathcal{O}(|y|^{2})}\chi^{2}(h^{1/2}y/\delta^{\prime})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\times
e^{\overline{\Phi_{1,h}}(\widetilde{t}^{\prime},p^{\prime},r)+\Phi_{2,h}(\widetilde{t}^{\prime},p^{\prime},r)+\overline{f_{1}}(r)+f_{2}(r)}\psi(r,h^{\frac{1}{2}}y)(1+h\mathcal{O}(|y|^{2}))\eta(\widetilde{t}^{\prime},p^{\prime},r)dydr,$

where we have performed a change of variables $y\mapsto h^{\frac{1}{2}}y$ in
the last step.

Passing to the limit $h\to 0$ in (4.60), we get the following pointwise limits

$e^{h\lambda\mathcal{O}(|y|)^{2}}\to
1,\quad\chi^{2}(h^{1/2}y/\delta^{\prime})\to
1,\quad\psi(r,h^{\frac{1}{2}}y)\to\psi(r,0)=\psi(\gamma(r)),\quad\Phi_{i,h}\to\Phi_{i},$

where we used (4.56) in the verification of the last limit. We recall that

$\mathrm{Im}\Theta=\frac{1}{2}\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y\geq d|y|^{2}\quad\text{ and
}\quad\int_{\mathbb{R}^{n-2}}e^{-d|y|^{2}}dy<\infty.$

Hence, by the dominated convergence theorem, we get

$\displaystyle\lim_{h\to
0}\int_{M_{0}}\overline{v_{s}}(\widetilde{t}^{\prime},p^{\prime},\cdot)w_{s}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi
dV_{g_{0}}$ (4.61)
$\displaystyle=(1-\beta^{2})\int_{0}^{\frac{L}{\sqrt{1-\beta^{2}}}}e^{\overline{f_{1}}(r)+f_{2}(r)}\bigg{(}\int_{\mathbb{R}^{n-2}}e^{-\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}dy\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\eta(\widetilde{t}^{\prime},p^{\prime},r)e^{-2(1-\beta^{2})\lambda
r}e^{\overline{\Phi_{1}}(\widetilde{t}^{\prime},p^{\prime},r)+\Phi_{2}(\widetilde{t}^{\prime},p^{\prime},r)}\psi(\gamma(r))dr.$

To simplify the expression on the right-hand side of (4.61), we perform a
change of variable $y\mapsto(1-\beta^{2})^{-\frac{1}{4}}y$ to obtain

$\displaystyle\int_{\mathbb{R}^{n-2}}e^{-\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}dy$
$\displaystyle=\int_{\mathbb{R}^{n-2}}(1-\beta^{2})^{-\frac{n-2}{4}}e^{-\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}dy$ (4.62)
$\displaystyle=\frac{\pi^{\frac{n-2}{2}}(1-\beta^{2})^{-\frac{n-2}{4}}}{\sqrt{\hbox{det}(\mathrm{Im}H(\sqrt{1-\beta^{2}}r))}}.$

We set $r_{0}=\sqrt{1-\beta^{2}}\tau_{0}$ and recall from [34, Lemma 2.58]
that

$\hbox{det}(\mathrm{Im}H(\sqrt{1-\beta^{2}}r))=\hbox{det}(\mathrm{Im}H(\sqrt{1-\beta^{2}}r_{0}))e^{-2\int_{r_{0}}^{r}\sqrt{1-\beta^{2}}\mathrm{tr}\mathrm{Re}H(\sqrt{1-\beta^{2}}w)dw}.$
(4.63)

This implies

$\int_{\mathbb{R}^{n-2}}e^{-\sqrt{1-\beta^{2}}\mathrm{Im}H(\sqrt{1-\beta^{2}}r)y\cdot
y}dy=\frac{\pi^{\frac{n-2}{2}}(1-\beta^{2})^{-\frac{n-2}{4}}e^{\int_{r_{0}}^{r}\sqrt{1-\beta^{2}}\mathrm{tr}\mathrm{Re}(H(\sqrt{1-\beta^{2}}w))dw}}{\sqrt{\hbox{det}(\mathrm{Im}H(\sqrt{1-\beta^{2}}r_{0}))}}.$
(4.64)

Furthermore, since

$\partial_{r}f_{j}(r)=-\frac{1}{2}\sqrt{1-\beta^{2}}\hbox{tr}\,H(\sqrt{1-\beta^{2}}r),\quad
j=1,2,$ (4.65)

we get

$\partial_{r}(\overline{f_{1}}(r)+f_{2}(r))=-\sqrt{1-\beta^{2}}\hbox{tr}\,\mathrm{Re}H(\sqrt{1-\beta^{2}}r).$

Thus, by the fundamental theorem of calculus, we have

$\overline{f_{1}}(r)+f_{2}(r)=\overline{f_{1}}(r_{0})+f_{2}(r_{0})-\int_{r_{0}}^{r}\sqrt{1-\beta^{2}}\mathrm{tr}\mathrm{Re}(H(\sqrt{1-\beta^{2}}w))dw.$
(4.66)

We next choose $f_{1}(r_{0})$ and $f_{2}(r_{0})$ so that

$\frac{e^{\overline{f_{1}}(r_{0})+f_{2}(r_{0})}\pi^{\frac{n-2}{2}}}{\sqrt{\hbox{det}(\mathrm{Im}H(\sqrt{1-\beta^{2}}r_{0}))}}=1.$
(4.67)

Thus, it follows from (4.64), (4.66), and (4.67) that

$\displaystyle
e^{\overline{f_{1}}(r)+f_{2}(r)}\bigg{(}\int_{\mathbb{R}^{n-2}}e^{-\mathrm{Im}H(\sqrt{1-\beta^{2}}r)x\cdot
x}dx\bigg{)}=(1-\beta^{2})^{-\frac{n-2}{4}}.$ (4.68)

Finally, we obtain (4.57) for $\psi\in C_{0}(W_{k}\cap M_{0})$ by substituting
(4.68) into (4.61).

Case 2: $\psi\in C_{0}(V_{j}\cap M_{0})$. Let us now verify (4.57) when
$\psi\in C_{0}(V_{j}\cap M_{0})$ for some $j\in\\{1,\ldots,R^{\prime}\\}$,
where $R^{\prime}$ is the number of self-intersection points of $\gamma$. In
this case, we obtain from (4.48) that the quasimodes are of the form

$v_{s}=\sum_{l:\gamma(r_{l})=z_{j}}v_{s}^{(l)},\quad
w_{s}=\sum_{l:\gamma(r_{l})=z_{j}}w_{s}^{(l)}$

on $\hbox{supp }(\psi)$, thus

$\overline{v_{s}}w_{s}=\sum_{l:\gamma(r_{l})=z_{j}}\overline{v_{s}^{(l)}}w_{s}^{(l)}+\sum_{l\neq
l^{\prime}:\gamma(r_{l})=\gamma(r_{l^{\prime}})=z_{j}}\overline{v_{s}^{(l)}}w_{s}^{(l^{\prime})}.$
(4.69)

We aim to prove that the contribution of the cross terms vanish in the limit
$h\to 0$. More precisely,

$\lim_{h\to
0}\int_{M_{0}}\overline{v_{s}^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)w_{s}^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi
dV_{g_{0}}=0,\quad l\neq l^{\prime}.$ (4.70)

If so, then the limit (4.57) follows from the first part of this proof.

In order to verify (4.70), we split $\psi$ into a smooth and a sufficiently
small part. In the latter case, the limit (4.70) follows from Proposition 4.1.
For the smooth part, we integrate by parts, which requires us to differentiate
$\psi$. In addition, we need the estimate (4.46) to show that the boundary
term vanishes. This estimate was not needed to prove Theorem 4.2.

To start the proof, following similar arguments as in the proof of [25,
Proposition 3.1], we write

$v_{s}^{(l)}=e^{\frac{i}{h}\mathrm{Re}\Theta^{(l)}}\phi^{(l)},\quad\phi^{(l)}=e^{-\lambda\mathrm{Re}\Theta^{(l)}}e^{-s\mathrm{Im}\Theta^{(l)}}b^{(l)},$

and

$w_{s}^{(l^{\prime})}=e^{\frac{i}{h}\mathrm{Re}\Theta^{(l^{\prime})}}\omega^{(l^{\prime})},\quad\omega^{(l^{\prime})}=e^{-\lambda\mathrm{Re}\Theta^{(l^{\prime})}}e^{-s\mathrm{Im}\Theta^{(l^{\prime})}}B^{(l^{\prime})},$

which imply that

$\overline{v_{s}^{(l)}}w_{s}^{(l^{\prime})}=e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}\omega^{(l^{\prime})},\quad\sigma=\mathrm{Re}\Theta^{(l^{\prime})}-\mathrm{Re}\Theta^{(l)}.$
(4.71)

Hence, in view of (4.71), we need to show that

$\lim_{h\to
0}\int_{M_{0}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi
dV_{g_{0}}=0,\quad l\neq l^{\prime}.$ (4.72)

To prove (4.72), let us write $\psi=\psi_{h}+(\psi-\psi_{h})$, where the
regularization $\psi_{h}\in C^{\infty}_{0}(V_{j}\cap M_{0})$, but its support
can meet $\partial M_{0}$, and $\psi-\psi_{h}$ is continuous. We recall that,
due to the estimates in (4.4) and (4.5), we have
$\|\phi^{(l)}\|_{L^{2}(V_{j}\cap
M_{0})},\|\omega^{(l^{\prime})}\|_{L^{2}(V_{j}\cap M_{0})}=\mathcal{O}(1)$.
Thus, by Hölder’s inequality and estimate (4.3), we obtain

$\bigg{|}\int_{M_{0}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)(\psi-\psi_{h})dV_{g_{0}}\bigg{|}\leq\|v_{s}^{(l)}\|_{L^{2}}\|w_{s}^{(l^{\prime})}\|_{L^{2}}\|\psi-\psi_{h}\|_{L^{\infty}}=o(1),\quad
h\to 0,$ (4.73)

where the $L^{p}$-norms are taken over the set $V_{j}\cap M_{0}$.

To analyze the term involving the smooth part $\psi_{h}$, we note that by
(4.10) and (4.18), the gradients of $\mathrm{Re}\Theta^{(l)}$ and
$\mathrm{Re}\Theta^{(l^{\prime})}$ at $z_{j}$ are parallel to
$\dot{\gamma}(r_{l})$ and $\dot{\gamma}(r_{l^{\prime}})$, respectively. Since
the geodesic $\gamma$ intersects itself transversally at $z_{j}$, we have
$\nabla_{g_{0}}\sigma(z_{j})\neq 0$. Therefore, by shrinking the set $V_{j}$
if necessary, we may assume that $\sigma$ has no critical points in $V_{j}$.
Thus, the vector field
$L=\frac{\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}}{|\nabla_{g_{0}}\sigma|^{2}}\nabla_{g_{0}}\sigma$
is well-defined and satisfies
$e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}=-ihL(e^{\frac{i}{h}\sigma}).$
Therefore, we integrate by parts in (4.72) to obtain

$\displaystyle\int_{M_{0}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi_{h}dV_{g_{0}}=$
$\displaystyle-ih\int_{V_{j}\cap\partial
M_{0}}\frac{\partial_{\nu}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi_{h}dS_{g_{0}}$
(4.74)
$\displaystyle+ih\int_{M_{0}}e^{\frac{i}{h}\sigma}\operatorname{div}(L)(\widetilde{t}^{\prime},p^{\prime},\cdot)dV_{g_{0}},$

To show that the boundary term on the right-hand side of (4.74) vanishes as
$h\to 0$, we use estimate (4.46) to observe that
$\|\phi^{(l)}\|_{L^{2}(\partial M_{0})}$,
$\|\omega^{(l^{\prime})}\|_{L^{2}(\partial M_{0})}=\mathcal{O}(1)$.
Furthermore, $\sigma$ is real-valued and independent of $h$. Then by estimate
(4.3) and Hölder’s inequality, we have

$\displaystyle-ih\int_{V_{j}\cap\partial M_{0}}$
$\displaystyle\frac{\partial_{\nu}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi_{h}dS_{g_{0}}=\mathcal{O}(h),\quad
h\to 0.$

To prove that the second term on the right-hand side of (4.74) vanishes as
$h\to 0$, we first compute that

$\operatorname{div}L=\operatorname{div}\left(\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}\frac{\nabla_{g_{0}}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}\right)=\left\langle\nabla_{g_{0}}(\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}),\frac{\nabla_{g_{0}}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}\right\rangle_{g_{0}}+\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}\operatorname{div}\left(\frac{\nabla_{g_{0}}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}\right).$

Using estimates (4.3),
$\|\phi^{(l)}\|_{L^{2}(M_{0})},\|\omega^{(l^{\prime})}\|_{L^{2}(M_{0})}=\mathcal{O}(1)$,
and Hölder’s inequality, we get

$\displaystyle
ih\int_{M_{0}}e^{\frac{i}{h}\sigma}\overline{\phi^{(l)}}(\widetilde{t}^{\prime},p^{\prime},\cdot)\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\psi_{h}\operatorname{div}\left(\frac{\nabla_{g_{0}}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}\right)dV_{g_{0}}=\mathcal{O}(h),\quad
h\to 0.$

Next, we write

$\displaystyle\overline{\phi^{(l)}}\omega^{(l^{\prime})}\psi_{h}$
$\displaystyle=[e^{-\lambda(\mathrm{Re}\Theta^{(l)}+\mathrm{Re}\Theta^{(l^{\prime})})}e^{-i\lambda(\mathrm{Im}\Theta^{(l^{\prime})}-\mathrm{Im}\Theta^{(l)})}][e^{-\frac{1}{h}(\mathrm{Im}\Theta^{(l)}+\mathrm{Im}\Theta^{(l^{\prime})})}h^{-\frac{n-2}{2}}][b_{0}^{(l)}\overline{B_{0}^{(l^{\prime})}}\chi^{2}(y/\delta^{\prime})]\psi_{h}$
$\displaystyle=f_{1}f_{2}f_{3}\psi_{h}.$

To streamline the proof, we only provide the estimate for the worst case
scenario, which occurs when $\nabla_{g_{0}}$ acts on
$f_{2}=e^{-\frac{1}{h}(\mathrm{Im}\Theta^{(l)}+\mathrm{Im}\Theta^{(l^{\prime})})}h^{-\frac{n-2}{2}}$.
To that end, due to (4.18), there exists $C>0$ such that by Cauchy-Schwarz
inequality, as well as estimates (4.3) and (4.35) with $k=\frac{1}{2}$, we
have

$\displaystyle
h\int_{M_{0}}|e^{\frac{i}{h}\sigma}f_{1}f_{2}\psi_{h}\langle\nabla_{g_{0}}f_{2},\frac{\nabla_{g_{0}}\sigma}{|\nabla_{g_{0}}\sigma|^{2}}\rangle|dV_{g_{0}}$
$\displaystyle\leq
C\|\psi_{h}\|_{L^{\infty}}\|\phi^{(l)}(\widetilde{t}^{\prime},p^{\prime},\cdot)\|_{L^{2}}\|\omega^{(l^{\prime})}(\widetilde{t}^{\prime},p^{\prime},\cdot)\|_{L^{2}}\int_{M_{0}}h^{-\frac{n-2}{2}}|y|e^{-\frac{d}{h}|y|^{2}}dV_{g_{0}}=\mathcal{O}(h^{1/2}),\quad
h\to 0.$

Hence, we conclude that the second term in the right-hand side of (4.74) is of
order $\mathcal{O}(h^{1/2})$, thus the limit (4.72) is verified. This
completes the proof of Proposition 4.4. ∎

###  4.3. Construction of CGO solutions

We now proceed to construct CGO solutions of the forms (4.1) and (4.2). Thanks
to the interior Carleman estimate established in Proposition 3.5, we can put
the ingredients together in a simple way. Let $(M,g)$ be a CTA manifold given
by Definition 1.1. We already computed in Section 3 that

$c^{\frac{n+2}{4}}\circ\mathcal{L}_{c,g,a,q}\circ
c^{-\frac{n-2}{4}}=\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}},$

where $\widetilde{g}=e\oplus g_{0}$, $\widetilde{a}=ca$, and
$\widetilde{q}=c(q-c^{\frac{n-2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}}))$. This
implies that $u=c^{-\frac{n-2}{4}}\widetilde{u}$ satisfies
$\mathcal{L}_{c,g,a,q}u=0$ if $\widetilde{u}$ solves
$\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}}\widetilde{u}=0$ in
$Q$.

Let us write $(t,x)=(t,x_{1},x^{\prime})$ for local coordinates in $Q$ and
recall that $s=\frac{1}{h}+i\lambda$, $0<h\ll 1$, where $\lambda\in\mathbb{R}$
is fixed. We are interested in finding CGO solutions to the equation

$\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}}\widetilde{u}=0\quad\text{in}\quad
Q$ (4.75)

of the form

$\widetilde{u}=e^{-s(\beta t+x_{1})}(v_{s}+r),$

where $v_{s}$ is the Gaussian beam quasimode given in Proposition 4.2, and
$r=r_{s}$ is a correction term that vanishes in the limit $h\to 0$. Indeed,
$\widetilde{u}$ is a solution to (4.75) if

$e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}}e^{-s(\beta
t+x_{1})}r=-e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}}e^{-s(\beta
t+x_{1})}v_{s}.$ (4.76)

Then we apply Proposition 3.5 with $v=-e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{a},\widetilde{q}}e^{-s(\beta
t+x_{1})}v_{s}$ and estimate (4.4) to conclude that there exists $r\in
H^{1}(Q^{\mathrm{int}})$ such that (4.76) holds and
$\|r\|_{H^{1}_{\mathrm{scl}}(Q)}=o(1)$ as $h\to 0$.

We summarize our discussion above in the following theorem. In particular, we
have general conformal factor $c$ in this result instead of $c=1$, which we
assumed in all of the earlier results in this paper.

######  Theorem 4.5.

Let $a\in W^{1,\infty}(Q)$ and $q\in C(\overline{Q})$. Let
$s=\frac{1}{h}+i\lambda$ with $\lambda\in\mathbb{R}$ fixed. For all $h>0$
small enough, there exists a solution $u_{1}\in H^{1}(Q)$ to
$\mathcal{L}_{c,g,a,q}^{*}u_{1}=0$ of the form

$u_{1}=e^{-s(\beta t+x_{1})}c^{-\frac{n-2}{4}}(v_{s}+r_{1}),$ (4.77)

where $v_{s}\in C^{\infty}(Q)$ is the Gaussian beam quasimode given in
Proposition 4.2, and $r_{1}\in H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})$ is such
that $\|r_{1}\|_{H^{1}_{\mathrm{\mathrm{scl}}}(Q^{\mathrm{int}})}=o(1)$ as
$h\to 0$.

There also exists a solution $u_{2}\in H^{1}(Q)$ to
$\mathcal{L}_{c,g,a,q}u_{2}=0$ that has the form

$u_{2}=e^{s(\beta t+x_{1})}c^{-\frac{n-2}{4}}(w_{s}+r_{2}),$ (4.78)

where $w_{s}\in C^{\infty}(Q)$ is the Gaussian beam quasimode given in
Proposition 4.2, and $r_{2}\in H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})$ is such
that $\|r_{2}\|_{H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})}=o(1)$ as $h\to 0$.

##  5\. Proof of Theorem 1.2

Let $u_{1}\in H^{1}(Q)$ be an exponentially decaying CGO solution given by
(4.77) to the equation $\mathcal{L}_{c,g,a_{1},q_{1}}^{*}u_{1}=0$ in $Q$, and
let $u_{2}\in H^{1}(Q)$ be an exponentially growing CGO solution given by
(4.78) satisfying $\mathcal{L}_{c,g,a_{2},q_{2}}u_{2}=0$ in $Q$. Due to the
main assumption $\mathcal{C}_{g,a_{1},q_{1}}=\mathcal{C}_{g,a_{2},q_{2}}$, it
follows from Proposition 2.4 that there exists a function $v\in
H_{\Box_{c,g}}(Q)$ such that $\mathcal{L}_{c,g,a_{1},q_{1}}v=0$ and

$(u_{2}-v)|_{\Sigma}=(u_{2}-v)|_{t=0}=(u_{2}-v)|_{t=T}=\partial_{t}(u_{2}-v)|_{t=0}=\partial_{\nu}(u_{2}-v)|_{V}=0.$

Then $u:=u_{2}-v\in H_{\Box_{c,g}}(Q)$ solves the equation

$\mathcal{L}_{c,g,a_{1},q_{1}}u=a\partial_{t}u_{2}+qu_{2}\quad\text{in}\quad
Q,\quad
u|_{\Sigma}=u|_{t=0}=u|_{t=T}=\partial_{t}u|_{t=0}=\partial_{\nu}u|_{V}=0.$
(5.1)

Here and in what follows we denote $a:=a_{1}-a_{2}$ and $q:=q_{1}-q_{2}$.
Since $a\partial_{t}u_{2}+qu_{2}\in L^{2}(Q)$, it follows from [47, Theorem
2.1] that

$u\in C^{1}([0,T];L^{2}(M))\cap C([0,T];H^{1}_{0}(M))\subset H^{1}(Q)\text{
with }\partial_{\nu}u\in L^{2}(\Sigma).$

Since $u_{1}\in H^{1}(Q)$ and $\Box_{c,g}u_{1}\in L^{2}(Q)$, we see that

$(c^{-1}\partial_{t}u_{1},-\nabla_{g}u_{1})\in
H_{\operatorname{div}}(Q):=\\{F\in L^{2}(Q,TQ):\operatorname{div}_{(t,x)}F\in
L^{2}(Q)\\}.$

We view $\overline{Q}$ as a compact Riemannian manifold of dimension $n+1$
with the metric $\overline{g}=dt^{2}\oplus g$, and let $\bar{\nu}$ be the
outward unit normal vector to $\partial Q$. In view of [35, Lemma 2.2], for
$F\in H_{\operatorname{div}}(Q)$, $F\cdot\bar{\nu}|_{\partial Q}$ can be
defined as an element of $H^{-1/2}(\partial Q)$, and for $\psi\in H^{1}(Q)$ we
have

$\langle\psi,{F}\cdot\bar{\nu}\rangle_{H^{1/2}(\partial Q),H^{-1/2}(\partial
Q)}=\langle\psi,\operatorname{div}_{\overline{g}}({F})\rangle_{L^{2}(Q)}+\langle\nabla_{\overline{g}}\psi,{F}\rangle_{L^{2}(Q)}.$

By taking $\psi=u$ and
$F=\overline{(c^{-1}\partial_{t}u_{1},-\nabla_{g}u_{1})}$, we deduce

$\displaystyle\langle
u,\overline{(c^{-1}\partial_{t}u_{1},-\nabla_{g}u_{1})}\cdot\bar{\nu}\rangle_{H^{1/2}(\partial
Q),H^{-1/2}(\partial Q)}$ (5.2) $\displaystyle=\langle
u,\Box_{c,g}\overline{u_{1}}\rangle_{L^{2}(Q)}+\langle(\partial_{t}u,\nabla_{g}u),(c^{-1}\partial_{t}\overline{u_{1}},-\nabla_{g}\overline{u_{1}})\rangle_{L^{2}(Q)}.$

Arguing similarly and using $\partial_{t}u|_{t=0},\>\partial_{t}u|_{t=T}\in
L^{2}(M)$, $\partial_{\nu}u\in L^{2}(\Sigma)$, and $u_{1}\in H^{1}(Q)$, we get

$\displaystyle\langle(c^{-1}\partial_{t}u,-\nabla_{g}u)\cdot\bar{\nu},\overline{u_{1}}\rangle_{L^{2}(\partial
Q)}=\langle\Box_{c,g}u,\overline{u_{1}}\rangle_{L^{2}(Q)}+\langle(c^{-1}\partial_{t}u,-\nabla_{g}u),(\partial_{t}\overline{u_{1}},\nabla_{g}\overline{u_{1}})\rangle_{L^{2}(Q)}.$
(5.3)

Since $a\in W^{1,\infty}(Q)$ and $u,\overline{u_{1}}\in H^{1}(Q)$, it follows
from the proof [20, Proposition 9.4] that $a\overline{u_{1}}\in H^{1}(Q)$ and
$\partial_{t}(a\overline{u_{1}})=\partial_{t}a\overline{u_{1}}+a\partial_{t}\overline{u_{1}}$
in the weak sense. Therefore, the following integration by parts is justified: