# Recovery of a time-dependent potential in hyperbolic equations on
conformally transversally anisotropic manifolds

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 potential
appearing in the wave equation in conformally transversally anisotropic
manifolds of dimension three or higher. These are compact Riemannian manifolds
with boundary that are conformally embedded in a product of the real line and
a transversal manifold. Under the assumption of the attenuated geodesic ray
transform being injective on the transversal manifold, we prove the unique
determination of time-dependent potentials from the knowledge of a certain
partial Cauchy data set.

##  1\. Introduction and Statement of Results

Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold of dimension
$n\geq 3$ with smooth boundary $\partial M$. Throughout this paper we denote
$Q=(0,T)\times M^{int}$ with $0<T<\infty$, $\overline{Q}$ being the closure of
$Q$, and $\Sigma=(0,T)\times\partial M$ the lateral boundary of $Q$. We
introduce the Laplace-Beltrami operator $\Delta_{g}$ of the metric $g$, and
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)

with time-independent coefficients.

In this paper we study an inverse problem for the linear hyperbolic partial
differential operator

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

with a time-dependent coefficient $q\in C(\overline{Q})$ called the potential.

We shall make two geometric assumptions, of which the first one 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 $\mathbb{R}\times M_{0}^{\text{int}}$ 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, see [10] for some more examples
of CTA manifolds. The global product structure of $M$ allows us to write every
point $x\in M$ as $x=(x_{1},x^{\prime})$, 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. 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, see [9, Theorem
1.2]. The latter condition holds if and only if the manifold $(M,g)$ is
locally isometric to the product of an interval and some $(n-1)$-dimensional
Riemannian manifold $(M_{0},g_{0})$.

In addition to the product structure of the ambient space $\mathbb{R}\times
M_{0}$ of the manifold $(M,g)$, we need to also assume the injectivity of
certain geodesic ray transforms on the transversal manifold $(M_{0},g_{0})$.
This type of assumption has been implemented to solve many important inverse
problems on CTA manifolds, see for instance [7, 10, 20, 23, 34] 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
$SM_{0}=\\{(x,\xi)\in TM_{0}:|\xi|=1\\}$. We denote

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

the incoming (–) and outgoing (+) boundaries of $SM_{0}$ corresponding to the
geodesics touching the boundary. Here $\langle\cdot,\cdot\rangle$ is the
Riemannian inner product of $(M_{0},g_{0})$, and $\nu$ is the outward unit
normal vector to $\partial M_{0}$ with respect to the metric $g_{0}$.

For any $(x,\xi)\in\partial_{-}SM_{0}$, we let $\gamma=\gamma_{x,\xi}$ be a
geodesic of $M_{0}$ with initial conditions
$(\gamma(0),\dot{\gamma}(0))=(x,\xi)$. 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 $\gamma(\tau)\in
M_{0}^{\text{int}}$ for all $\tau>0$. 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 $\dot{\gamma}(0)$
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
$0<\tau<\tau_{\mathrm{exit}}(x,\xi)$.

Given a continuous function $\alpha$ on $M_{0}$, the attenuated geodesic ray
transform of a function $f\colon M_{0}\to\mathbb{R}$ is given by

$I^{\alpha}(f)(x,\xi)=\int_{0}^{\tau_{\mathrm{exit}}(x,\xi)}\exp\bigg{[}\int_{0}^{t}\alpha(\gamma_{x,\xi}(s))ds\bigg{]}f(\gamma_{x,\xi}(t))dt,\quad(x,\xi)\in\partial_{-}SM_{0}\setminus\Gamma_{-},$
(1.3)

where
$\Gamma_{-}=\\{(x,\xi)\in\partial_{-}SM_{0}:\tau_{\mathrm{exit}}(x,\xi)=+\infty\\}$.
The attenuated geodesic ray transform is the mathematical basis for the
medical imaging method SPECT (single-photon emission computed tomography),
which is commonly used to diagnose and monitor heart problems as well as bone
and brain disorders. Inversion of an attenuated geodesic ray transform is a
crucial part of solving the Calderón problem on CTA manifolds [9].

The second geometric assumption we make in this paper is as follows.

Assumption 1. There exists $\varepsilon>0$ such that for any smooth
attenuation $\alpha$ 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 $M_{0}$.

Injectivity of the attenuated geodesic ray transform on simple manifolds for
small attenuations $\alpha$ was established in [9, Theorem 7.1]. A compact,
simply connected Riemannian manifold with smooth boundary is said to be simple
if its boundary is strictly convex, and no geodesic has conjugate points. When
$\alpha=0$, injectivity of the geodesic ray transform on simple manifolds is
well-known, see [24, 31].

The attenuated geodesic ray transform $I^{\alpha}$ is also known to be
injective when some other geometric conditions are imposed. For instance, it
was established in [8, Theorem 29] that $I^{\alpha}$ is injective on
spherically symmetric manifolds satisfying the Herglotz condition when the
attenuation $\alpha$ is radially symmetric and Lipschitz continuous. The
attenuation is a constant in this paper. The Herglotz condition is a special
case of a manifold satisfying a convex foliation condition, and in [25] 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 [25, Section 2] for more
examples. The convex foliation condition does not forbid the existence of
conjugate points in general.

Finally, we discuss the “measurements” made in this paper before presenting
our main result. We observe that the limiting Carleman weight $\varphi(x)$
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 prove the unique determination of the time-
dependent potential $q(t,x)$, which appears in (1.2), from the following set
of partial Cauchy data

$\mathcal{C}_{g,q}=\\{(u|_{U},u|_{t=T},\partial_{t}u|_{t=0},\partial_{\nu}u|_{V}):u\in
L^{2}(Q),\>\mathcal{L}_{c,g,q}u=0,u|_{t=0}=0,\>\hbox{supp }u|_{\Sigma}\subset
U\\}.$ (1.4)

The wellposedness of this set has been established in [18, Section 3]. From a
physical perspective, as introduced in [17], the inverse problem considered in
this paper can be interpreted as the determination of physical properties such
as the time-evolving density of an inhomogeneous medium by probing it with
disturbances generated on some parts of the boundary and at initial time, and
by measuring the response on some parts of the boundary and at the end of the
experiment.

We highlight that in $\mathcal{C}_{g,q}$ the Dirichlet value is measured and
supported only on roughly half of the lateral boundary $U$, and the Neumann
data is measured on approximately the other half of the lateral boundary $V$.
Measurements are also made at the initial time $t=0$ and the end time $t=T$.
It follows from the domain of dependence arguments given in [17, Subsection
1.1] that we can only hope to recover general time-dependent coefficients in
the optimal set

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

when only the lateral boundary data

$\mathcal{C}_{g,q}^{\mathrm{lat}}=\\{(u|_{\Sigma},\partial_{\nu}u|_{\Sigma}):u\in
L^{2}(Q),\>\mathcal{L}_{g,q}u=0,\>u|_{t=0}=\partial_{t}u|_{t=0}=0\\}$ (1.5)

is given. Hence, even for a large measurement time $T>0$, global unique
recovery of general time-dependent coefficients of the hyperbolic operator
(1.2) requires information at the beginning $\\{t=0\\}$ and at the end
$\\{t=T\\}$ of the measurement.

The main result of this paper is as follows.

######  Theorem 1.2.

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 $T>0$
and $q_{i}\in C(\overline{Q})$, $i=1,2$. If $q_{1}=q_{2}$ on $\partial Q$,
then $\mathcal{C}_{g,q_{1}}=\mathcal{C}_{g,q_{2}}$ implies that $q_{1}=q_{2}$
in $Q$.

######  Remark 1.3.

Theorem 1.2 can be viewed as an extension of [17] from the Euclidean space, as
well as [18] from CTA manifolds with a simple transversal manifold $M_{0}$, to
more general CTA manifolds. Due to the partial Dirichlet data assumption in
$\mathcal{C}_{g,q}$, Theorem 1.2 does not follow from our recent work [23].

######  Remark 1.4.

Theorem 1.2 states the unique determination of continuous potentials from the
set of partial Cauchy data $\mathcal{C}_{g,q}$. This is attributed to the
technique presented in this paper since the concentration property of Gaussian
beam quasimodes (Proposition 2.2) requires continuity.

######  Remark 1.5.

Assumption 1.1 of this paper is different from the literature concerning
inverse problems for elliptic operators on CTA manifolds, see for instance
[10, 20]. These works assume the invertibility of the geodesic ray transform.
In the case of elliptic operators, where there is only one Euclidean direction
$x_{1}$, the authors reduced the problem to the geodesic ray transform and
recovered the Taylor expansion of the unknown function by differentiating an
expression similar to (4.8) with respect to the variable $\lambda$ at zero.
However, this approach is not applicable in our case as the mapping
$(\lambda,\beta)\mapsto-\lambda(\beta,1)$, appearing in (4.8), is a
diffeomorphism only if $\lambda\neq 0$. Thus, computing $\lambda$ and
$\beta$-derivatives of (4.8) at $\lambda=0$ will not give us the Taylor
expansion of the unknown potential at the origin.

###  1.1. Previous literature

In this section we only review some literature concerning the recovery of
time-dependent coefficients appearing in hyperbolic equations from boundary
measurements. There is also a vast amount of literature about the time-
independent case, which has been discussed for instance in [23].

Most of the time-dependent results rely on the use of geometric optics (GO)
solutions to the hyperbolic equation. This approach was first implemented in
[32] to determine time-dependent coefficients of hyperbolic equations from the
knowledge of scattering data by using properties of the light-ray transform.
In the Euclidean setting, recovery of a time-dependent potential $q$ from the
full lateral boundary data $\mathcal{C}_{q}^{\mathrm{lat}}$ on the infinite
cylinder $\mathbb{R}\times\Omega$, where $\Omega$ is a bounded domain, was
established in [29]. On a finite cylinder $(0,T)\times\Omega$ with
$T>\mathrm{diam}(\Omega)$, it was proved in [26] that
$\mathcal{C}_{q}^{\mathrm{lat}}$ determines $q$ uniquely in the optimal subset
$\mathcal{D}$ of $(0,T)\times\Omega$. A uniqueness result for determining a
general time-dependent potential $q$ from the set of partial Cauchy data
$\mathcal{C}_{g,q}$ was established in [17].

Going beyond the Euclidean setting, global unique determination of a time-
dependent potential $q$ from both full and partial boundary measurements was
proved in [18] on a CTA manifold $(M,g)$ with a simple transversal manifold
$M_{0}$. In other classes of manifolds, it was recently established in [1]
that a set of full Cauchy data determines $q$ uniquely in Lorentzian manifolds
that satisfy certain two-sided curvature bounds and some other geometric
assumptions. This curvature bound was weakened in [2] near Minkowski geometry.
The proof of [1] is based on a new optimal unique continuation theorem and a
generalization of the Boundary Control Method, originally developed in [5], to
the cases when the dependence of coefficients on time is not analytic. Indeed,
the Boundary Control Method, which is a powerful tool to prove uniqueness
results for time-independent coefficients appearing in hyperbolic equations
[5, 6, 13, 14, 21], is not applicable to recover time-dependent coefficients
in general since it relies on an application of the unique continuation
theorem analogous to [33], which may fail without the aforementioned real
analyticity assumption, see [3, 4].

Aside from uniqueness results concerning only the potential, there is also
some literature about determining time-dependent first order perturbations
appearing in hyperbolic equations from boundary measurements as well. It was
established in [16] that boundary data $\mathcal{C}_{g,q}$, with $U=\Sigma$,
determines time-dependent damping coefficients and potentials uniquely in the
Euclidean setting. Very recently the authors extended this result to the
setting of CTA manifolds in [23].

If a full time-dependent vector field perturbation appears in the hyperbolic
equation, similar to the magnetic Schrödinger operator, it is only possible to
recover the vector field up to a differential of a test function in $Q$. A
global uniqueness result was proved in [11] when the dependence of
coefficients on the time variable is real-analytic. This analyticity
assumption was removed in [30], which proved a uniqueness result on an
infinite cylinder $\mathbb{R}\times\Omega$, where $\Omega$ is a bounded domain
in $\mathbb{R}^{n}$. We refer readers to [19] for a global uniqueness result
from partial Dirichlet-to-Neumann map on a finite cylinder $[0,T]\times\Omega$
with $T>\hbox{diam }(\Omega)$. In the Riemannian setting, it was established
in [12] that the lateral boundary data $\mathcal{C}_{g,q}^{\mathrm{lat}}$
determines the first order and the zeroth order perturbations up to the
described gauge invariance on a certain non-optimal subset of $Q$. This result
was obtained by reducing the problem to the inversion of the light-ray
transform of the Lorentzian metric $-dt^{2}+g(x)$. The authors of [12] also
showed that the light-ray transform is invertible whenever the respective
geodesic ray transform on the spatial manifold is invertible. To the best of
our knowledge, the global (optimal) recovery of a one-form and a potential
function, appearing in a hyperbolic operator, from a set of partial Cauchy
data $\mathcal{C}_{g,q}$ (1.4) (lateral boundary data
$\mathcal{C}_{g,q}^{\mathrm{lat}}$ (1.5)) is still an open problem.

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

The two main ingredients of the proof are the integral identity (4.2), which
was derived in [17, 18] from the set of partial Cauchy data
$\mathcal{C}_{g,q}$, and the construction of complex geometric optics (CGO)
solutions. Specifically, we shall construct a family of exponentially decaying
solutions $u_{1}$ to the equation $\mathcal{L}_{c,g,q}^{\ast}u_{1}=0$ of the
form

$u_{1}(t,x)=e^{-s(\beta t+\varphi(x))}(v_{s}(t,x)+r_{1}(t,x)),\quad(t,x)\in
Q.$

On the other hand, due to the restrictions $\hbox{supp }u|_{\Sigma}\subset U$
and $u|_{t=0}=0$ in $\mathcal{C}_{g,q}$, we need to construct a family of
exponentially growing solutions $u_{2}$ to the equation
$\mathcal{L}_{c,g,q}u_{2}=0$, which look like

$u_{2}(t,x)=e^{s(\beta t+\varphi(x))}(w_{s}(t,x)+r_{2}(t,x)),\quad(t,x)\in Q,$

and satisfy these two boundary conditions. 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, $v_{s}$ and $w_{s}$ are Gaussian beam quasimodes, $r_{1}$ and $r_{2}$
are correction terms that vanish in the limit $h\to 0$, and the function
$\varphi(x)=x_{1}$ is a limiting Carleman weight on $M$. We choose the values
of $\beta$ as above because the construction of $r_{1}$ relies on an
application of an interior Carleman estimate [23, Proposition 3.6]. This is
derived from a boundary Carleman estimate [23, Proposition 3.1], which is
valid for $\beta\in(\frac{1}{\sqrt{3}},1)$. For the construction of $r_{2}$,
we may take $\beta\in[\frac{1}{2},1]$, see [18, Theorem 4.1].

Since the transversal manifold $(M_{0},g_{0})$ is not necessarily simple, the
approach based on global GO solutions is not applicable. In Proposition 2.1 we
construct Gaussian beam quasimodes for every non-tangential geodesic in the
transversal manifold $M_{0}$ by using techniques originally developed in
solving inverse problems for elliptic operators, see for instance [7, 10, 20,
34], as well as [23] for hyperbolic operators. The quasimodes concentrate on
the geodesic in the semiclassical limit $h\to 0$, as we shall explain in
Proposition 2.2. The construction of the remainder terms $r_{1}$ and $r_{2}$
are given in Section 3. Here $r_{1}$ needs to have a stronger decay property,
namely, being $\mathcal{O}(h^{1/2})$ with respect to the semiclassical
$H^{1}$-norm. This is achieved with an interior Carleman estimate [23,
Proposition 3.6]. In order to find a remainder $r_{2}$ such that $u_{2}$
satisfies the required boundary conditions, we follow a different approach,
which was developed in [17, 18].

To complete the proof of Theorem 1.2, we shall substitute the CGO solutions
(3.1) and (3.5) into the integral identity (4.2) and pass to the limit $h\to
0$. Lemma 4.1 implies that the right-hand side of (4.2) vanishes in the limit
$h\to 0$. The proof of this lemma requires a decay in $H^{1}$-norm (3.2) for
$r_{1}$. Meanwhile, estimates (2.5), (3.2), and (3.6), in conjunction with the
concentration property of Gaussian beam quasimodes Proposition 2.2, yield that
the left-hand side of (4.2) converges to the attenuated geodesic ray transform
involving the function $q_{1}-q_{2}$ in the limit $h\to 0$. This is the reason
why we need Assumption 1.1 to complete the proof.

The paper is organized as follows. We begin with the construction of Gaussian
beam quasimodes in Section 2. In Section 3 we construct both exponentially
decaying and growing CGO solutions. Finally, we present the proof of Theorem
1.2 in Section 4.

### Acknowledgments

We would like to express our gratitude to Katya Krupchyk, Joonas Ilmavirta,
and Hanming Zhou for 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).

##  2\. Construction of Gaussian Beam Quasimodes

Let $(M,g)$ be a CTA manifold given by Definition 1.1 and $T>0$. The goal of
this section is to construct Gaussian beam quasimodes with desirable
concentration properties. Gaussian beam quasimodes have been utilized
extensively to solve inverse problems in Riemannian manifolds. We refer
readers to [7, 9, 10, 20, 34] for applications in elliptic operators and [12,
14, 23] in hyperbolic operators.

To streamline the construction, we first note that due to the conformal
properties of the Laplace-Beltrami operator explained in [10], we have

$c^{\frac{n+2}{4}}(-\Delta_{g})(c^{-\frac{n-2}{4}}u)=-(\Delta_{\widetilde{g}}-\big{(}c^{\frac{n+2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}})\big{)})u.$
(2.1)

Also, since $c$ is independent of the time variable $t$, we get

$c^{\frac{n+2}{4}}\partial_{t}^{2}(c^{-\frac{n-2}{4}}u)=c\partial_{t}^{2}u.$
(2.2)

Thus, equations (2.1) and (2.2) yield the following identity for the operator
$\mathcal{L}_{c,g,q}$:

$c^{\frac{n+2}{4}}\circ\mathcal{L}_{c,g,q}\circ
c^{-\frac{n-2}{4}}=\mathcal{L}_{\widetilde{g},\widetilde{q}},$ (2.3)

where

$\widetilde{g}=e\oplus
g_{0}\quad\text{and}\quad\widetilde{q}=c(q-c^{\frac{n-2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}})).$
(2.4)

Hence, by replacing the metric $g$ and coefficient $q$ with $\widetilde{g}$
and $\widetilde{q}$, respectively, we may assume that the conformal factor
$c=1$. In this section we shall use this assumption and consider the leading
order wave operator $\Box_{e\oplus g_{0}}=\partial_{t}^{2}-\Delta_{e\oplus
g_{0}}$. For simplicity, let us write $\mathcal{L}_{g,q}$ for
$\mathcal{L}_{c,g,q}$ with $c=1$. Also, throughout the rest of this paper we
shall denote $\mathcal{L}_{g,q}^{\ast}=\mathcal{L}_{g,\overline{q}}$ the
formal $L^{2}$-adjoint of the operator $\mathcal{L}_{g,q}$.

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

######  Proposition 2.1.

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

$\displaystyle\|v_{s}\|_{L^{2}(M_{0})}=\mathcal{O}(1),\quad\|e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}^{\ast}_{g,q}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}(Q)}=\mathcal{O}(h^{3/2}),$ (2.5)
$\displaystyle\|e^{-s(\beta t+x_{1})}h^{2}\mathcal{L}_{g,q}e^{s(\beta
t+x_{1})}v_{s}\|_{L^{2}(Q)}=\mathcal{O}(h^{3/2})$

hold as $h\to 0$.

###### Proof.

Let $L>0$ be the length of the geodesic $\gamma=\gamma(\tau)$. By [22, Example
9.32], we may embed $(M_{0},g_{0})$ into a larger closed manifold
$(\widehat{M}_{0},g_{0})$ of the same dimension. Also, we extend $\gamma$ as a
unit speed geodesic in $\widehat{M}_{0}$. Since $\gamma$ is non-tangential, we
can choose $\varepsilon>0$ such that $\gamma(\tau)\in\widehat{M}_{0}\setminus
M_{0}$ and does not self-intersect for
$\tau\in[-2\varepsilon,0)\cup(L,L+2\varepsilon]$.

Our goal is to construct Gaussian beam quasimodes near
$\gamma([-\varepsilon,L+\varepsilon])$. We start by fixing 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 $z_{0}$, see [15, Lemma
7.4]. We may assume that the coordinates $(\tau,y)$ extend smoothly to a
neighborhood of $\overline{\Omega}$.

We observe that near $z_{0}=\gamma(\tau_{0})$ the trace of the geodesic
$\gamma$ is given by the set $\Gamma=\\{(\tau,0):|\tau-\tau_{0}|<\delta\\}$,
and in these Fermi coordinates we have

$g_{0}^{jk}(\tau,0)=\delta^{jk}\quad\text{and}\quad\partial_{y_{l}}g_{0}^{jk}(\tau,0)=0.$
(2.6)

Hence, it follows from Taylor’s theorem that for small $|y|$ we can write

$g_{0}^{jk}(\tau,y)=\delta^{jk}+\mathcal{O}(|y|^{2}).$ (2.7)

We first construct quasimodes $v_{s}$ for the conjugated operator $e^{s(\beta
t+x_{1})}\mathcal{L}_{g,q}^{\ast}e^{-s(\beta t+x_{1})}$. To that end, let us
consider a Gaussian beam ansatz

$v_{s}(\tau,y;h)=e^{is\Theta(\tau,y)}b(\tau,y;h).$ (2.8)

Compared to [23], the quasimode $v_{s}$ constructed in this paper is
independent of the Euclidean variables $(t,x_{1})$.

Our goal is to find a phase function $\Theta\in C^{\infty}(\Omega,\mathbb{C})$
such that

$\mathrm{Im}\Theta\geq
0,\quad\mathrm{Im}\Theta|_{\Gamma}=0,\quad\mathrm{Im}\Theta(\tau,y)\sim|y|^{2},$
(2.9)

as well as an amplitude $b\in C^{\infty}(\Omega,\mathbb{C})$ such that
$\hbox{supp }(b(\tau,\cdot))\subset\\{|y|<\delta^{\prime}/2\\}$. We shall
follow the ideas originally presented in [14, 28].

Since $\Theta$ is independent of the Euclidean variables ($t,x_{1})$ and
$g=e\oplus g_{0}$, we have

$e^{-is\Theta}\partial_{t}^{2}(e^{is\Theta}b)=0$ (2.10)

and

$\quad
e^{-is\Theta}(-\Delta_{g})e^{is\Theta}b=-\Delta_{g_{0}}b-is[2\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b\rangle_{g_{0}}+(\Delta_{g_{0}}\Theta)b]+s^{2}\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}b.$
(2.11)

Therefore, we obtain from (2.10) and (2.11) that

$\displaystyle e^{s(\beta t+x_{1})}h^{2}\mathcal{L}^{\ast}_{g,q}e^{-s(\beta
t+x_{1})}v_{s}=$ $\displaystyle
h^{2}e^{is\Theta}[s^{2}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b$
(2.12)
$\displaystyle+s(-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b)$
$\displaystyle+(-\Delta_{g_{0}}+\overline{q})b].$

From the computation above, we see that in order to verify the estimates in
(2.5), we need to find a phase function $\Theta$ and an amplitude $b$ such
that they approximately solve the eikonal and transport equations appearing on
right-hand side of (2.12) as multipliers of the terms $s^{2}$ and $s$,
respectively.

Following similar arguments in [10, 14, 20, 27, 28], we aim to find
$\Theta(\tau,y)\in C^{\infty}(\Omega,\mathbb{C})$ such that

$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2})=\mathcal{O}(|y|^{3}),\quad
y\to 0,$ (2.13)

and

$\mathrm{Im}\Theta\geq d|y|^{2}$ (2.14)

for some constant $d>0$ that depends on $\beta$. As in [10, 27, 28], we choose

$\Theta(\tau,y)=\sqrt{1-\beta^{2}}(\tau+\frac{1}{2}H(\tau)y\cdot y),$ (2.15)

where the smooth complex-valued symmetric matrix $H(\tau)$ is the unique
solution of the initial value problem for the matrix Riccati equation

$\dot{H}(\tau)+H(\tau)^{2}=F(\tau),\quad H(\tau_{0})=H_{0},\quad\text{ for
}\tau\in\mathbb{R}.$ (2.16)

Here $\mathrm{Im}H(\tau)$ positive definite, $H_{0}$ is a complex symmetric
matrix such that $\mathrm{Im}(H_{0})$ is positive definite, and $F(\tau)$ is a
suitable symmetric matrix. We refer readers to [14, Lemma 2.56] for details.

We next look for an amplitude $b$ of the form

$b(\tau,y;h)=h^{-\frac{n-2}{4}}b_{0}(\tau)\chi(y/\delta^{\prime}),$ (2.17)

where $b_{0}\in C^{\infty}([\tau_{0}-\delta,\tau_{0}+\delta])$ depends on only
the travel time $\tau$ and satisfies the approximate transport equation

$-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b_{0}=\mathcal{O}(|y|),$
(2.18)

and the cut-off function $\chi\in C^{\infty}_{0}(\mathbb{R}^{n-2})$ is such
that $\chi=1$ for $|y|\leq 1/4$ and $\chi=0$ for $|y|\geq 1/2$.

In order to find $b_{0}$ that satisfies (2.18), we first compute
$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}\rangle_{g_{0}}$. To this end,
we deduce from (2.15) that

$\partial_{\tau}\Theta(\tau,y)=\sqrt{1-\beta^{2}}+\mathcal{O}(|y|^{2}).$
(2.19)

Therefore, we get from (2.7) that

$\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b_{0}\rangle_{g_{0}}=\sqrt{1-\beta^{2}}\partial_{\tau}b_{0}+\mathcal{O}(|y|^{2})\partial_{\tau}b_{0}.$
(2.20)

We next compute $\Delta_{g_{0}}\Theta$ near the geodesic $\gamma$. To that
end, we deduce from (2.7) and (2.15) that

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

This implies that

$(\Delta_{g_{0}}\Theta)(\tau,y)=\sqrt{1-\beta^{2}}\hbox{tr}\,H(\tau)+\mathcal{O}(|y|).$
(2.21)

To achieve (2.18), we require that $b_{0}(\tau)$ satisfies

$\partial_{\tau}b_{0}=-\frac{1}{2}\hbox{tr}\,H(\tau)b_{0}.$ (2.22)

Hence, we have

$b_{0}(\tau)=e^{f_{1}(\tau)},\quad\text{where}\quad\partial_{\tau}f_{1}(\tau)=-\frac{1}{2}\hbox{tr}\,H(\tau).$

Finally, we get (2.18) from (2.19)–(2.22) due to the $y$-independence of
$b_{0}$.

We next prove the estimates in (2.5) for the quasimode

$v_{s}(\tau,y;h)=e^{is\Theta(\tau,y)}b(\tau,y;h)=e^{is\Theta(\tau,y)}h^{-\frac{n-2}{4}}b_{0}(\tau)\chi(y/\delta^{\prime})$
(2.23)

locally in $\Omega$, where $\Omega\subset M_{0}$ is the domain of Fermi
coordinates near the point $z_{0}=\gamma(\tau_{0})$. To proceed, we shall need
the following estimate for any $k\in\mathbb{R}$:

$\begin{split}\|h^{-\frac{n-2}{4}}|y|^{k}e^{-\frac{\mathrm{Im}\Theta}{h}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}&\leq\|h^{-\frac{n-2}{4}}|y|^{k}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}\\\
&\leq\bigg{(}\int_{\mathbb{R}^{n-2}}h^{k}|z|^{2k}e^{-2d|z|^{2}}dz\bigg{)}^{1/2}=\mathcal{O}(h^{k/2}),\quad
h\to 0.\end{split}$ (2.24)

Here we applied estimate (2.14) and the change of variable $z=h^{-1/2}y$. Then
it follows from (2.14) and (2.24) with $k=0$ that

$\displaystyle\|v_{s}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|b_{0}\|_{L^{\infty}([\tau_{0}-\delta,\tau_{0}+\delta])}\|e^{is\Theta}h^{-\frac{n-2}{4}}\chi(y/\delta^{\prime})\|_{L^{2}(\Omega)}$
(2.25)
$\displaystyle\leq\mathcal{O}(1)\|h^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}=\mathcal{O}(1),\quad
h\to 0.$

We now proceed to estimate $\|e^{s(\beta
t+x_{1})}\mathcal{L}_{g,\overline{q}}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}(\Omega)}$, which requires estimating each term on the
right-hand side of (2.12). For the first term, by utilizing (2.13), (2.14),
and (2.24) with $k=3$, we obtain

$\displaystyle
h^{2}\|e^{is\Theta}s^{2}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b\|_{L^{2}(\Omega)}$
(2.26)
$\displaystyle=h^{2}\|e^{is\Theta}s^{2}h^{-\frac{n-2}{4}}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))b_{0}\chi(y/\delta^{\prime})\|_{L^{2}(\Omega)}$
$\displaystyle\leq\mathcal{O}(1)\|h^{-\frac{n-2}{4}}|y|^{3}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}=\mathcal{O}(h^{3/2}),\quad
h\to 0.$

We next estimate the second term on the right-hand side of (2.12). From a
direct computation, we get

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

We observe that $e^{-\frac{1}{h}}=\mathcal{O}(h^{\infty})$. Here we say that
$f=\mathcal{O}(h^{\infty})$ if $f=\mathcal{O}(h^{n})$ for every
$n\in\mathbb{N}$. Therefore, it follows from (2.14) that on the support of
$\nabla_{g_{0}}\chi(y/\delta^{\prime})$ we have

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

Thus, using equation (2.18), estimate (2.24) with $k=1$, along with the
triangle inequality, we have

$\displaystyle
h^{2}\|e^{is\Theta}s(-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}b\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)b)\|_{L^{2}(\Omega)}$
(2.27)
$\displaystyle\leq\mathcal{O}(h)\|e^{is\Theta}h^{-\frac{n-2}{4}}[|y|\chi(y/\delta^{\prime})-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\chi(y/\delta^{\prime})\rangle_{g_{0}}]\|_{L^{2}(\Omega)}$
$\displaystyle\leq\mathcal{O}(h)\|h^{-\frac{n-2}{4}}|y|e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}+\mathcal{O}(e^{-\frac{\widetilde{d}}{h}})$
$\displaystyle=\mathcal{O}(h^{3/2}),\quad h\to 0.$

Lastly, we estimate the third term on the right-hand side of (2.12). Since the
amplitude $b$ is independent of $t$, it suffices to estimate the term
involving $\Delta_{g}$ and the lower order term. To that end, we apply
estimate (2.24) with $k=0$ to get

$h^{2}\|e^{is\Theta}(-\Delta_{g}b)\|_{L^{2}(\Omega)}\leq\mathcal{O}(h^{2})\|h^{-\frac{n-2}{4}}e^{-\frac{d}{h}|y|^{2}}\|_{L^{2}(|y|\leq\delta^{\prime}/2)}=\mathcal{O}(h^{2}),\quad
h\to 0.$ (2.28)

For the lower order term, it follows from (2.24) with $k=0$ that

$h^{2}\|e^{is\Theta}\overline{q}b\|_{L^{2}(\Omega)}=\mathcal{O}(h^{2}),\quad
h\to 0.$ (2.29)

Therefore, by combining estimates (2.26)–(2.29), we conclude from (2.12) that

$\|e^{s(\beta t+x_{1})}h^{2}\mathcal{L}^{\ast}_{g,q}e^{-s(\beta
t+x_{1})}v_{s}\|_{L^{2}(\Omega)}=\mathcal{O}(h^{3/2}),\quad h\to 0.$ (2.30)

This completes the verification of (2.5) locally in the set $\Omega$.

To complete the construction of the quasimode $v_{s}$ on the transversal
manifold $M_{0}$, we glue together the quasimodes defined along small pieces
of the geodesic $\gamma$. Since $\widehat{M}_{0}$ is compact and
$\gamma(\tau):(-2\varepsilon,L+2\varepsilon)\to\widehat{M}_{0}$ is a unit
speed non-tangential geodesic that is not a loop, we get from [15, Lemma 7.2]
that $\gamma|_{(-2\varepsilon,L+2\varepsilon)}$ self-intersects at times
$\tau_{j}$, where $j\in\\{1,\ldots,N\\}$, and

$-\varepsilon=\tau_{0}<\tau_{1}<\dots<\tau_{N}<\tau_{N+1}=L+\varepsilon.$

By [15, Lemma 7.4], there exists an open cover
$\\{(\Omega_{j},\kappa_{j})_{j=0}^{N+1}\\}$ of
$\gamma([-\varepsilon,L+\varepsilon])$ consisting of coordinate neighborhoods
that have the following properties:

  1. (1)

$\kappa_{j}(\Omega_{j})=I_{j}\times B$, where $I_{j}$ 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 the same for each
$\Omega_{j}$.

  2. (2)

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

  3. (3)

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

  4. (4)

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

As explained in [15, Lemma 7.4], the intervals $I_{j}$ can be chosen as

$I_{0}=(-2\varepsilon,\tau_{1}-\widetilde{\delta}),\quad
I_{j}=(\tau_{j}-2\widetilde{\delta},\tau_{j+1}-\widetilde{\delta}),\>j=1,\dots,N,\quad
I_{N+1}=(\tau_{N+1}-2\widetilde{\delta},L+2\varepsilon)$

for some $\widetilde{\delta}>0$ small enough. When $\gamma$ does not self-
intersect, there is a single coordinate neighborhood of
$\gamma|_{[-\varepsilon,L+\varepsilon]}$ 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 $\tau=0$. By following the arguments from
the earlier part of this proof, we find a quasimode

$v_{s}^{(0)}(\tau,y;h)=h^{-\frac{n-2}{4}}e^{is\Theta^{(0)}(\tau,y)}e^{f_{1}(\tau)}\chi(y/\delta^{\prime})$

in $\Omega_{0}$ with some fixed initial conditions at $\tau=-\varepsilon$ for
the Riccati equation (2.16) determining $\Theta^{(0)}$. Next we choose some
$\tau_{0}^{\prime}$ such that
$\gamma(\tau_{0}^{\prime})\in\Omega_{0}\cap\Omega_{1}$ and let

$v_{s}^{(1)}(\tau,y;h)=h^{-\frac{n-2}{4}}e^{is\Theta^{(1)}(\tau,y)}e^{f_{1}(\tau)}\chi(y/\delta^{\prime})$

be a quasimode in $\Omega_{1}$ by choosing the initial conditions for (2.16)
such that $\Theta^{(1)}(\tau_{0}^{\prime})=\Theta^{(0)}(\tau_{0}^{\prime})$.
Here we have used the same function $f_{1}$ in both $v_{s}^{(0)}$ and
$v_{s}^{(1)}$ since $f_{1}$ is globally defined for all
$\tau\in(-2\varepsilon,L+2\varepsilon)$ and does not depend 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 get
$\Theta^{(1)}=\Theta^{(0)}$ in $I_{0}\cap I_{1}$. 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 $v_{s}^{(2)},\dots,v_{s}^{(N+1)}$
such that

$v_{s}^{(j)}=v_{s}^{(j+1)}\quad\text{in}\quad\Omega_{j}\cap\Omega_{j+1}$
(2.31)

If $\gamma$ self-intersects at $\tau=0$, we start the construction from
$v^{(1)}$ by fixing initial conditions for (2.16) at
$\tau=\tau_{0}^{\prime}\in I_{1}$ and find $v^{(0)}$ by going backwards.

Let $\chi_{j}(\tau)$ be a partition of unity subordinate to
$\\{I_{j}\\}_{j=1}^{N+1}$. We denote
$\widetilde{\chi_{j}}(\tau,y)=\chi_{j}(\tau)$ and define

$v_{s}=\sum_{j=0}^{N+1}\widetilde{\chi}_{j}v_{s}^{(j)}.$

Then we get $v_{s}\in C^{\infty}(M_{0})$.

Let $z_{1},\dots,z_{R}\in M_{0}$ be distinct self-intersection points of
$\gamma$, and let $0\leq\tau_{1}<\cdots<\tau_{N}$, $R\leq N$, be the times of
self-intersections. Let $V_{j}$ be a small neighborhood in $\widehat{M}_{0}$
centered at $z_{j},j=1,\dots,R$. Following the steps in [15], for
$\delta^{\prime}$ sufficiently small we can pick a finite cover
$W_{1},\dots,W_{S}$ of remaining points on the geodesic such that
$W_{k}\subset\Omega_{l(k)}$ for some index $l(k)$ and

$\displaystyle\hbox{supp }v_{s}\cap
M_{0}\subset(\cup_{j=1}^{R}V_{j})\cup(\cup_{k=1}^{S}W_{k}).$

Moreover, the quasimode restricted on $V_{j}$ and $W_{k}$ is of the form

$v_{s}|_{V_{j}}=\sum_{l:\gamma(\tau_{l})=z_{j}}v_{s}^{(l)}$ (2.32)

and

$v_{s}|_{W_{k}}=v_{s}^{l(k)},$ (2.33)

respectively. Since $v_{s}$ is a finite sum of $v_{s}^{(l)}$ in each case, the
first and second estimate in (2.5) follow from corresponding local
considerations (2.25) and (2.30) for each of $v_{s}^{(l)}$, respectively.

We next construct a Gaussian beam quasimode for the operator $e^{-s(\beta
t+x_{1})}\mathcal{L}_{g,q}e^{s(\beta t+x_{1})}$ of the form
$w_{s}(\tau,y;h)=e^{is\Theta(\tau,y)}B(\tau,y;h)$ with the same phase function
$\Theta\in C^{\infty}(\Omega,\mathbb{C})$ that satisfies (2.9), and
$B(\tau,y)\in C^{\infty}(\Omega)$ is supported near $\Gamma$.

By similar computations as in (2.12), we have

$\displaystyle e^{-s(\beta t+x_{1})}\mathcal{L}_{g,q}e^{s(\beta
t+x_{1})}w_{s}=$ $\displaystyle
e^{is\Theta}[s^{2}(\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}\Theta\rangle_{g_{0}}-(1-\beta^{2}))B$
(2.34)
$\displaystyle+s(-2i\langle\nabla_{g_{0}}\Theta,\nabla_{g_{0}}B\rangle_{g_{0}}-i(\Delta_{g_{0}}\Theta)B)$
$\displaystyle+(-\Delta_{g_{0}}+q)B].$

Notice that the eikonal equation and transport equation for $B$ coincide with
the corresponding equation for $b$. Therefore, we get
$B(\tau,y;h)=b(\tau,y;h)$. Furthermore, since the phase function $\Theta$ is
the same for $v_{s}$ and $w_{s}$, we have $w_{s}=v_{s}$. Finally, we obtain
the third estimate in (2.5) by arguing similarly as in the verification of the
second estimate in (2.5). This completes the proof of Proposition 2.1. ∎

We want the Gaussian beam quasimodes to concentrate along the geodesic as
$h\to 0$. By following the same arguments as in the proof of [10, Proposition
3.1], we have the following result.

######  Proposition 2.2.

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 $\gamma\colon[0,L]\to M_{0}$ be a non-
tangential geodesic in $(M_{0},g_{0})$ as in Proposition 2.1. Let $v_{s}$ be
the quasimode from Proposition 2.1. Then for each $\psi\in C(M_{0})$ and
$(t^{\prime},x_{1}^{\prime})\in[0,T]\times\mathbb{R}$ we have

$\lim_{h\to 0}\int_{\\{t^{\prime}\\}\times\\{x_{1}^{\prime}\\}\times
M_{0}}|v_{s}|^{2}\psi
dV_{g_{0}}=\int_{0}^{L}e^{-2\sqrt{1-\beta^{2}}\lambda\tau}\psi(\gamma(\tau))d\tau.$
(2.35)

##  3\. Construction of Complex Geometric Optics Solutions

In this section we construct a family of exponentially decaying solutions
$u_{1}\in H^{1}(Q)$ of the form

$u_{1}(t,x)=e^{-s(\beta
t+\varphi(x))}(v_{s}(x^{\prime})+r_{1}(t,x)),\quad(t,x)\in Q,$

as well as a family of exponentially growing solutions $u_{2}\in
H_{\Box_{c,g}}(Q):=\\{u\in L^{2}(Q):\Box_{c,g}u\in L^{2}(Q)\\}$ given by

$u_{2}(t,x)=e^{s(\beta
t+\varphi(x))}(v_{s}(x^{\prime})+r_{2}(t,x)),\quad(t,x)\in Q,$

satisfying $\hbox{supp }u_{2}|_{\Sigma}\subset U$ and $u_{2}|_{t=0}=0$. Here
$s=\frac{1}{h}+i\lambda$ with $\lambda\in\mathbb{R}$ fixed, $\varphi(x)=x_{1}$
is a limiting Carleman weight, $v_{s}$ is the Gaussian beam quasimodes given
in Proposition 2.1, and $r_{j}$, $j=1,2$, are correction terms that vanish as
$h\to 0$. These two types of solutions will play different roles in the proof
of Theorem 1.2, and the proofs for their existence are also somewhat
different. The construction of the exponentially decaying solution $u_{1}$
follows from an interior Carleman estimate [23, Proposition 3.6]. For the
exponentially growing solution $u_{2}$, we shall follow the approach
introduced in [17, 18].

In the following proposition, which shows the existence of exponentially
decaying solutions $u_{1}$, we equip $Q$ with a semiclassical Sobolev norm

$\|u\|^{2}_{H^{1}_{\mathrm{scl}}(Q)}=\|u\|^{2}_{L^{2}(Q)}+\|h\partial_{t}u\|^{2}_{L^{2}(Q)}+\|h\nabla_{g}u\|^{2}_{L^{2}(Q)}.$

######  Proposition 3.1.

Let $q\in C(\overline{Q})$, $\beta\in(\frac{1}{\sqrt{3}},1)$, and let
$s=\frac{1}{h}+i\lambda$ with $\lambda\in\mathbb{R}$ fixed. For all $h>0$
sufficiently small, there exists a solution $u_{1}\in H^{1}(Q)$ to
$\mathcal{L}_{c,g,q}^{*}u_{1}=0$ of the form

$u_{1}=e^{-s(\beta t+x_{1})}c^{-\frac{n-2}{4}}(v_{s}+r_{1}),$ (3.1)

where $v_{s}\in C^{\infty}(M_{0})$ is the Gaussian beam quasimode given in
Proposition 2.1, and $r_{1}\in H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})$
satisfies the estimate

$\|r_{1}\|_{H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})}=\mathcal{O}(h^{1/2}),\quad
h\to 0.$ (3.2)

###### Proof.

Since $(M,g)$ is a CTA manifold, the computations in Section 2 yield

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

where $\widetilde{g}=e\oplus g_{0}$ and
$\widetilde{q}=c(q-c^{\frac{n-2}{4}}\Delta_{g}(c^{-\frac{n-2}{4}}))$. Hence,
we see that if $\widetilde{u}$ is a solution to the equation
$\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}\widetilde{u}=0$, then the
function $u=c^{-\frac{n-2}{4}}\widetilde{u}$ solves
$\mathcal{L}^{\ast}_{c,g,q}u=0$. Thus, it suffices to look for solutions to
the equation $\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}\widetilde{u}=0$
of the form $\widetilde{u}=e^{-s(\beta t+x_{1})}(v_{s}+r_{1}).$ This is
equivalent to finding a function $r_{1}$ that solves the equation

$e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{q}}^{\ast}e^{-s(\beta
t+x_{1})}r_{1}=-e^{s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{q}}^{\ast}e^{-s(\beta
t+x_{1})}v_{s}.$ (3.3)

From here we use estimate (2.5) and apply an interior Carleman estimate [23,
Proposition 3.6] to deduce that there exists $r_{1}\in
H^{1}_{\mathrm{scl}}(Q^{\mathrm{int}})$ that solves (3.3) and satisfies
estimate (3.2). Lastly, we would like to recall that the interior Carleman
estimate we utilized in this proof needs the required assumption for the
parameter $\beta$. This completes the proof of Proposition 3.1. ∎

We now turn to the construction of exponentially growing solutions $u_{2}$
vanishing on part of $\partial Q$. We emphasize that the earlier construction
of $u_{1}$ requires an extension of the domain due to the interior Carleman
estimate. Therefore, we have no control over the traces of the solutions to
the wave equation on $\partial Q$ if we consider solutions on the extended
domain. Thus, we need to have a different approach.

For every $\varepsilon>0$ we set

$\partial M_{\varepsilon,-}=\\{x\in\partial
M:\partial_{\nu}\varphi(x)<-\varepsilon\\},\quad\partial
M_{\varepsilon,+}=\\{x\in\partial
M:\partial_{\nu}\varphi(x)\geq-\varepsilon\\},$

and $\Sigma_{\varepsilon,\pm}=(0,T)\times\partial M_{\varepsilon,\pm}$. To
find $u_{2}$, we use the following result, which was originally proved in [18,
Theorem 5.4]. For the convenience of the reader, we re-prove this result.

######  Proposition 3.2.

Let $q\in C(\overline{Q})$, $\beta\in[\frac{1}{2},1]$, and let
$s=\frac{1}{h}+i\lambda$ with $\lambda\in\mathbb{R}$ fixed. For all $h>0$
small enough, there exists a solution $u_{2}\in H_{\Box_{c,g}}(Q)$ to the
initial boundary value problem

$\begin{cases}\mathcal{L}_{c,g,q}u_{2}=0\quad\text{in}\quad Q,\\\
u_{2}(0,x)=0\quad\text{in}\quad M,\\\
u_{2}=0\quad\text{on}\quad\Sigma_{\varepsilon,-},\end{cases}$ (3.4)

of the form

$u_{2}=e^{s(\beta t+x_{1})}c^{-\frac{n-2}{4}}(v_{s}+r_{2}).$ (3.5)

Here $v_{s}\in C^{\infty}(M_{0})$ is the Gaussian beam quasimode given in
Proposition 2.1, and $r_{2}\in L^{2}(Q)$ satisfies the estimate

$\|r_{2}\|_{L^{2}(Q)}=\mathcal{O}(h^{1/2}),\quad h\to 0.$ (3.6)

Since $U^{\prime}\subset\partial M$ is an open neighborhood of ${\partial
M}_{+}=\\{x\in\partial M:\partial_{\nu}\varphi(x)\geq 0\\}$, we can choose
$\varepsilon>0$ sufficiently small such that $\Sigma_{\varepsilon,+}\subset
U=(0,T)\times U^{\prime}$. Thus, the claims $\hbox{supp
}u_{2}|_{\Sigma}\subset U$ and $u_{2}|_{t=0}=0$ follow from Proposition 3.2.

###### Proof of Propostion 3.2.

Arguing similarly as in the proof of Proposition 3.1, we may assume that the
conformal factor $c=1$ and look for solutions to the equation
$\mathcal{L}_{\widetilde{g},\widetilde{q}}\widetilde{u}=0$ of the form
$\widetilde{u}=e^{s(\beta t+x_{1})}(v_{s}+r_{2})$ such that
$\widetilde{u}(0,x)=0$ in $M$ and $\widetilde{u}=0$ on
$\Sigma_{\varepsilon,-}$. Here $\widetilde{g}$ and $\widetilde{q}$ are given
by (2.4). This is equivalent to finding a function $r_{2}$ that satisfies

$\begin{cases}e^{-s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{s(\beta
t+x_{1})}r_{2}=-e^{-s(\beta
t+x_{1})}h^{2}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{s(\beta
t+x_{1})}v_{s}=:f,\quad\text{in}\quad Q,\\\
r_{2}(0,x)=-v_{s}(x^{\prime})=:r_{0}\quad\text{in}\quad M,\\\
r_{2}(t,x)=-v_{s}(x^{\prime})=:r_{-}\quad\text{on}\quad\Sigma_{\varepsilon,-}.\end{cases}$
(3.7)

This can be achieved by solving the following initial boundary value problem

$\displaystyle e^{-\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{\frac{1}{h}(\beta
t+x_{1})}\widetilde{r}=-e^{i\lambda(\beta t+x_{1})}e^{-s(\beta
t+x_{1})}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{s(\beta
t+x_{1})}v_{s}=:\widetilde{f}\quad\text{in}\quad Q,$ (3.8a)
$\displaystyle\widetilde{r}(0,x)=-e^{i\lambda
x_{1}}v_{s}(x^{\prime})=:\widetilde{r}_{0}\quad\text{in}\quad M,$ (3.8b)
$\displaystyle\widetilde{r}(t,x)=-e^{i\lambda(\beta
t+x_{1})}v_{s}(x^{\prime})\psi(x)=:\widetilde{r}_{-}\quad\text{on}\quad\Sigma_{-},$
(3.8c)

and setting $r_{2}:=e^{-i\lambda(\beta t+x_{1})}\widetilde{r}$. Here $\psi\in
C^{\infty}_{0}(M)$ is a cut-off function such that $0\leq\psi\leq 1$,
$\hbox{supp }\psi\cap\partial M\subset\partial M_{\varepsilon/2,-}$, and
$\psi=1$ on $\partial M_{\varepsilon,-}$.

In order to verify the existence of $\widetilde{r}$, we need to derive a
boundary Carleman estimate for the operator
$\mathcal{L}_{\widetilde{g},\widetilde{q}}$. To that end, let us introduce the
space

$\mathcal{D}:=\left\\{u\in
C^{\infty}(\overline{Q}):u|_{\Sigma}=u|_{t=T}=\partial_{t}u|_{t=T}=u|_{t=0}=0\right\\}.$

By replacing $t$ by $T-t$ and $x_{1}$ by $-x_{1}$ in the boundary Carleman
estimate of [18, Lemma 4.2], we see that following estimate for the wave
operator $\Box_{\widetilde{g}}$ is valid for any $u\in\mathcal{D}$

$\displaystyle
h^{1/2}\|\partial_{t}u(0,\cdot)\|_{L^{2}(M)}+h^{1/2}\|\sqrt{\left|\partial_{\nu}\varphi\right|}\partial_{\nu}u\|_{L^{2}(\Sigma_{-})}+\|u\|_{L^{2}(Q)}$
(3.9) $\displaystyle\leq\mathcal{O}(h)\|e^{-\frac{1}{h}(\beta
t+x_{1})}\Box_{\widetilde{g}}e^{\frac{1}{h}(\beta
t+x_{1})}u\|_{L^{2}(Q)}+\mathcal{O}(h^{1/2})\|\sqrt{\partial_{\nu}\varphi}\partial_{\nu}u\|_{L^{2}(\Sigma_{+})}.$

To establish a boundary Carleman estimate for the oprtator
$\mathcal{L}_{\widetilde{g},\widetilde{q}}$, we first apply the triangle
inequality to obtain

$\|e^{-\frac{1}{h}(\beta t+x_{1})}\Box_{\widetilde{g}}e^{\frac{1}{h}(\beta
t+x_{1})}u\|_{L^{2}(Q)}\leq\|e^{-\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{\frac{1}{h}(\beta
t+x_{1})}u\|_{L^{2}(Q)}+\|\widetilde{q}u\|_{L^{2}(Q)}.$

Furthermore, we have

$\|\widetilde{q}u\|_{L^{2}(Q)}\leq\|\widetilde{q}\|_{L^{\infty}(Q)}\|u\|_{L^{2}(Q)}.$

Therefore, by absorbing the term
$h\|\widetilde{q}\|_{L^{\infty}(Q)}\|u\|_{L^{2}(Q)}$ into the left-hand side
of (3.9), we obtain the following boundary Carleman estimate for
$\mathcal{L}_{\widetilde{g},\widetilde{q}}$

$\displaystyle
h^{1/2}\|\partial_{t}u(0,\cdot)\|_{L^{2}(M)}+h^{1/2}\|\sqrt{\left|\partial_{\nu}\varphi\right|}\partial_{\nu}u\|_{L^{2}(\Sigma_{-})}+\|u\|_{L^{2}(Q)}$
(3.10) $\displaystyle\leq\mathcal{O}(h)\|e^{-\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}_{\widetilde{g},\widetilde{q}}e^{\frac{1}{h}(\beta
t+x_{1})}u\|_{L^{2}(Q)}+\mathcal{O}(h^{1/2})\|\sqrt{\partial_{\nu}\varphi}\partial_{\nu}u\|_{L^{2}(\Sigma_{+})}.$

Let us now recall the following estimates, which follow immediately from
Proposition 2.1:

$\|\widetilde{f}\|_{L^{2}(Q)}=\mathcal{O}(h^{-1/2}),\quad\|\widetilde{r}_{0}\|_{L^{2}(M)}=\mathcal{O}(1).$
(3.11)

Also, by utilizing the same arguments as in the proof of [23, Lemma 5.1], we
obtain the estimate
$\|v_{s}\|_{L^{2}(\Sigma_{\varepsilon/2,-})}=\mathcal{O}(1)$, which implies
that

$\||\partial_{\nu}\varphi|^{-1/2}\widetilde{r}_{-}\|_{L^{2}{(\Sigma_{-})}}\leq\||\partial_{\nu}\varphi|^{-1/2}e^{i\lambda(\beta
t+x_{1})}v_{s}\|_{L^{2}{(\Sigma_{\varepsilon/2,-})}}=\mathcal{O}(\varepsilon^{-1/2}).$
(3.12)

We shall now closely follow the proof of [17, Lemma 5.1] to verify the
existence of $\widetilde{r}\in H_{\Box_{\widetilde{g}}}(Q)$ satisfying
(3.8a)–(3.8c). We introduce the space

$\mathcal{M}:=\left\\{(e^{\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}e^{-\frac{1}{h}(\beta
t+x_{1})}u,\partial_{\nu}u|_{\Sigma_{+}}):u\in\mathcal{D}\right\\}$

and equip it with the norm

$\|(g_{1},g_{2})\|_{\mathcal{M}}=\|g_{1}\|_{L^{2}(Q)}+\|h^{-1/2}|\partial_{\nu}\varphi|^{1/2}g_{2}\|_{L^{2}(\Sigma_{+})}.$

Then $\mathcal{M}$ can be viewed as a subspace of $L^{2}(Q)\times
L_{\varphi,h,+}^{2}(\Sigma_{+})$, where

$L_{\varphi,h,+}^{2}(\Sigma_{+}):=\\{u:\|h^{-1/2}|\partial_{\nu}\varphi|^{1/2}u\|_{L^{2}(\Sigma_{+})}=\mathcal{O}(1)\\}.$

If $u\in\mathcal{D}$, by the boundary Carleman estimate (3.10) and the Cauchy-
Schwartz inequality, we have

$\displaystyle\left|\left<u,f\right>_{L^{2}(Q)}-\left<\partial_{t}u(0,\cdot),\widetilde{r}_{0}\right>_{L^{2}(M)}-\left<\partial_{\nu}u,\widetilde{r}_{-}\right>_{L^{2}(\Sigma_{-})}\right|$
$\displaystyle\leq\|u\|_{L^{2}(Q)}\|f\|_{L^{2}(Q)}+\|\partial_{t}u(0,\cdot)\|_{L^{2}(M)}\|\widetilde{r}_{0}\|_{L^{2}(M)}+\||\partial_{\nu}\varphi|^{1/2}\partial_{\nu}u\|_{L^{2}(\Sigma_{-})}\||\partial_{\nu}\varphi|^{-1/2}\widetilde{r}_{-}\|_{L^{2}{(\Sigma_{-})}}$
$\displaystyle\leq\mathcal{O}(1)\|(e^{\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}^{*}_{\widetilde{g},\widetilde{q}}e^{-\frac{1}{h}(\beta
t+x_{1})}u,\partial_{\nu}u|_{\Sigma_{+}})\|_{\mathcal{M}}$
$\displaystyle\qquad\quad\times\left(h\|f\|_{L^{2}(Q)}+h^{1/2}\|\widetilde{r}_{0}\|_{L^{2}(M)}+h^{1/2}\||\partial_{\nu}\varphi|^{-1/2}\widetilde{r}_{-}\|_{L^{2}{(\Sigma_{-})}}\right).$

Hence, we may define a linear functional $\mathcal{S}$ on $\mathcal{M}$ by
setting

$\mathcal{S}(e^{\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}e^{-\frac{1}{h}(\beta
t+x_{1})}u,\partial_{\nu}u|_{\Sigma_{+}}):=\left<u,f\right>_{L^{2}(Q)}-\left<\partial_{t}u(0,\cdot),\widetilde{r}_{0}\right>_{L^{2}(M)}-\left<\partial_{\nu}u,\widetilde{r}_{-}\right>_{L^{2}(\Sigma_{-})},\>u\in\mathcal{D}.$

By the Hahn-Banach theorem, we can extend the operator $\mathcal{S}$ to a
continuous linear form $\widetilde{\mathcal{S}}$ on $L^{2}(Q)\times
L^{2}_{\varphi,h}(\Sigma_{+})$ without increasing the norm. Hence, it follows
from estimates (3.11) and (3.12) that

$\|\widetilde{\mathcal{S}}\|=\|\mathcal{S}\|\leq\mathcal{O}(1)\left(h\|f\|_{L^{2}(Q)}+h^{1/2}\|\widetilde{r}_{0}\|_{L^{2}(M)}+h^{1/2}\||\partial_{\nu}\varphi|^{-1/2}\widetilde{r}_{-}\|_{L^{2}{(\Sigma_{-})}}\right)=\mathcal{O}(h^{1/2}).$
(3.13)

By the Riesz representation theorem, there exists
$(\widetilde{r},\widetilde{r}_{+})\in L^{2}(Q)\times
L_{\varphi,h,-}^{2}(\Sigma_{+})$, where
$L_{\varphi,h,-}^{2}(\Sigma_{+}):=\\{u:\|h^{1/2}|\partial_{\nu}\varphi|^{-1/2}u\|_{L^{2}(\Sigma_{+})}=\mathcal{O}(1)\\}$,
such that

$\|\widetilde{r}\|_{L^{2}(Q)}+\|h^{1/2}|\partial_{\nu}\varphi|^{-1/2}\widetilde{r}_{+}\|_{L^{2}(\Sigma_{+})}=\|\mathcal{S}\|=\mathcal{O}(h^{1/2}),$

and

$\widetilde{\mathcal{S}}(g_{1},g_{2})=\left<g_{1},\widetilde{r}\right>_{L^{2}(Q)}+\left<g_{2},\widetilde{r}_{+}\right>_{L^{2}(\Sigma_{+})},\quad(g_{1},g_{2})\in
L^{2}(Q)\times L_{\varphi,h,+}^{2}(\Sigma_{+}).$

Therefore, for all $u\in\mathcal{D}$, we get

$\displaystyle\left<e^{\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}e^{-\frac{1}{h}(\beta
t+x_{1})}u,\widetilde{r}\right>_{L^{2}(Q)}+\left<\partial_{\nu}u|_{\Sigma_{+}},\widetilde{r}_{+}\right>_{L^{2}(\Sigma_{+})}$
(3.14)
$\displaystyle=\left<u,f\right>_{L^{2}(Q)}-\left<\partial_{t}u(0,\cdot),\widetilde{r}_{0}\right>_{L^{2}(M)}-\left<\partial_{\nu}u|_{\Sigma_{-}},\widetilde{r}_{-}\right>_{L^{2}(\Sigma_{-})}.$

We now verify that $\widetilde{r}$ satisfies equations (3.8a)–(3.8c). By
taking $u\in C^{\infty}_{0}(Q)$ in (3.14) and using the fact that
$C^{\infty}_{0}(Q)$ is dense in $L^{2}(Q)$, we see that equation (3.8a) holds.
Furthermore, (3.8a) implies $\widetilde{r}\in H_{\Box_{\widetilde{g}}}(Q)$.
Thus, by [17, Proposition A.1], we can define the trace
$\widetilde{r}|_{\Sigma}\in H^{-3}(0,T;H^{-1/2}(\partial M))$ and
$\widetilde{r}|_{t=0}\in H^{-2}(M)$. Furthermore, the density result [17,
Theorem A.1], in conjunction with integration by parts, implies that for all
$u\in\mathcal{D}$, we have

$\displaystyle\left<e^{\frac{1}{h}(\beta
t+x_{1})}\mathcal{L}^{\ast}_{\widetilde{g},\widetilde{q}}e^{-\frac{1}{h}(\beta
t+x_{1})}u,\widetilde{r}\right>_{L^{2}(Q)}+\left<\partial_{\nu}u|_{\Sigma_{+}},\widetilde{r}|_{\Sigma_{+}}\right>_{H^{3}(0,T;H^{1/2}(\Sigma_{+})),H^{-3}(0,T;H^{-1/2}(\Sigma_{+}))}$
$\displaystyle=\left<u,f\right>_{L^{2}(Q)}-\left<\partial_{t}u(0,\cdot),\widetilde{r}|_{t=0}\right>_{H^{2}(M),H^{-2}(M)}-\left<\partial_{\nu}u,\widetilde{r}|_{\Sigma_{-}}\right>_{H^{3}(0,T;H^{1/2}(\Sigma_{-})),H^{-3}(0,T;H^{-1/2}(\Sigma_{-}))}.$

Finally, we compare the equality above with (3.14) and take $u\in\mathcal{D}$
to be arbitrary to conclude that $\widetilde{r}=\widetilde{r}_{0}$ and
$\widetilde{r}|_{\Sigma_{-}}=\widetilde{r}_{-}$. Therefore, we have verified
(3.8b) and (3.8c). This completes the proof of Proposition 3.2. ∎

##  4\. Proof of Theorem 1.2

Let $u_{1}\in H^{1}(Q)$ be an exponentially decaying CGO solution of the form
(3.1) satisfying $\mathcal{L}_{c,g,q_{1}}^{*}u_{1}=0$ in $Q$, and let
$u_{2}\in H_{\Box_{c,g}}(Q)$ be an exponentially growing CGO solution of
$\mathcal{L}_{c,g,q_{2}}u_{2}=0$ in $Q$ given by (3.5) such that
$u_{2}|_{t=0}=0$ and $\hbox{supp }u_{2}|_{\Sigma}\subset U$. Due to the
assumption $\mathcal{C}_{g,q_{1}}=\mathcal{C}_{g,q_{2}}$, by [18, Proposition
3.1], there exists a function $v\in H_{\Box_{c,g}}(Q)$ that satisfies the
equations $\mathcal{L}_{c,g,q_{1}}v=0$ and

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

Then the function $u:=u_{2}-v\in H_{\Box_{c,g}}(Q)$ is a solution to the
equation

$\mathcal{L}_{c,g,q_{1}}u=qu_{2}\quad\text{in}\quad Q,\quad
u|_{\Sigma}=u|_{t=0}=u|_{t=T}=\partial_{t}u|_{t=0}=\partial_{\nu}u|_{V}=0.$
(4.1)

Here, and in what follows, we used the notation $q:=q_{1}-q_{2}$.

After arguing similarly as in [18, Section 6], we obtain the following
integral identity

$\int_{Q}qu_{2}\overline{u_{1}}dV_{g}dt=\int_{M}c^{-1}\partial_{t}u(T,x)\overline{u_{1}(T,x)}dV_{g}-\int_{\Sigma\setminus
V}\partial_{\nu}u\overline{u_{1}}dS_{g}dt.$ (4.2)

We next substitute the CGO solutions (3.1) and (3.5) into (4.2) and pass to
the limit $h\to 0$.

To analyze the limit of the terms on right-hand side of (4.2), we have the
following lemma, which states that both terms on the right-hand side of (4.2)
vanish as $h\to 0$. Its proof is same as the proof of [23, Lemma 5.1], which
mainly relies on an application of a boundary Carleman estimate [18, Theorem
4.1], as well as estimates (2.5) and (3.2). This Lemma is the reason why we
need the $H^{1}$-norm decay (3.2) for $r_{1}$.

######  Lemma 4.1.

Let $u_{1}$ and $u$ be the functions described above. Then the following
estimates hold as $h\to 0$:

$\int_{M}c^{-1}\partial_{t}u(T,x)\overline{u_{1}(T,x)}dV_{g}=\mathcal{O}(h^{1/2})$
(4.3)

and

$\int_{\Sigma\setminus
V}\partial_{\nu}u\overline{u_{1}}dS_{g}dt=\mathcal{O}(h^{1/2}).$ (4.4)

To investigate the left-hand side of the integral identity (4.2), we compute
from the respective CGO solution (3.1) and (3.5) for $u_{1}$ and $u_{2}$ that

$u_{2}\overline{u_{1}}=e^{2i\lambda(\beta
t+x_{1})}c^{-\frac{n-2}{2}}(|v_{s}|^{2}+\overline{v_{s}}r_{2}+\overline{r_{1}}v_{s}+\overline{r_{1}}r_{2}).$

By estimates (2.5), (3.2), (3.6), and the Cauchy-Schwartz inequality, we
obtain the estimate

$\bigg{|}\int_{Q}e^{2i\lambda(\beta
t+x_{1})}c^{-\frac{n-2}{2}}(\overline{v_{s}}r_{2}+\overline{r_{1}}v_{s}+\overline{r_{1}}r_{2})dV_{g}dt\bigg{|}=\mathcal{O}(h^{1/2}),\quad
h\to 0.$ (4.5)

On the other hand, since $q_{1},q_{2}\in C(\overline{Q})$ and $q_{1}=q_{2}$ on
the boundary $\partial Q$, we may continuously extend $q$ on
$(\mathbb{R}^{2}\times M_{0})\setminus Q$ by zero and denote the extension by
the same letter. Then from the equality
$dV_{g}=c^{\frac{n}{2}}dV_{g_{0}}dx_{1}$, Fubini’s theorem, the dominated
convergence theorem, and the concentration property (2.35), we obtain the
following limit as $h\to 0$:

$\displaystyle\int_{Q}e^{2i\lambda(\beta
t+x_{1})}c^{-\frac{n-2}{2}}q|v_{s}|^{2}dV_{g}dt$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{M_{0}}e^{2i\lambda(\beta
t+x_{1})}cq|v_{s}|^{2}dV_{g_{0}}dx_{1}dt$ (4.6)
$\displaystyle\to\int_{0}^{L}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{2i\lambda(\beta
t+x_{1})-2\sqrt{1-\beta^{2}}\lambda\tau}(cq)(t,x_{1},\gamma(\tau))dx_{1}dtd\tau.$

Hence, by replacing $2\lambda$ with $\lambda$, we deduce from (4.2), (4.5),
(4.6), and Lemma 4.1 that the identity

$\int_{0}^{L}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\lambda(\beta
t+x_{1})-\sqrt{1-\beta^{2}}\lambda\tau}(cq)(t,x_{1},\gamma(\tau))dx_{1}dtd\tau=0$
(4.7)

holds for every non-tangential geodesic $\gamma$ in the transversal manifold
$(M_{0},g_{0})$.

We are ready to utilize Assumption 1.1, the invertibility of the attenuated
geodesic ray transform on $(M_{0},g_{0})$. To that end, we denote
$\mathcal{F}_{(t,x_{1})\to(\xi_{1},\xi_{2})}$ the Fourier transform in the two
Euclidean variables $(t,x_{1})$ and define

$\displaystyle f(x^{\prime},\beta,\lambda)$
$\displaystyle:=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\lambda(\beta
t+x_{1})}(cq)(t,x_{1},x^{\prime})dx_{1}dt=\mathcal{F}_{(t,x_{1})\to(\xi_{1},\xi_{2})}(cq)|_{(\xi_{1},\xi_{2})=-\lambda(\beta,1)},$

where $x^{\prime}\in M_{0}$, $\beta\in(\frac{1}{\sqrt{3}},1)$, and
$\lambda\in\mathbb{R}$. Since $q\in C(\overline{Q})$, the function
$f(\cdot,\beta,\lambda)$ is continuous on $M_{0}$. Since $\gamma$ is an
arbitrary non-tangential geodesic, it follows from (4.7) that the following
attenuated geodesic ray transform

$I^{-\sqrt{1-\beta^{2}}\lambda}(f(\cdot,\beta,\lambda))(x,\xi)=\int_{0}^{\tau_{\mathrm{exit}}(x,\xi)}e^{-\sqrt{1-\beta^{2}}\lambda\tau}f(\gamma_{x,\xi}(\tau),\beta,\lambda)d\tau,\quad(x,\xi)\in\partial_{-}SM_{0}\setminus\Gamma_{-},$
(4.8)

vanishes.

By Assumption 1.1, there exists $\varepsilon>0$ such that
$f(\gamma(\tau),\beta,\lambda)=0$ when
$\sqrt{1-\beta^{2}}|\lambda|<\varepsilon$. Hence, there exist
$\beta_{0}\in(\frac{1}{\sqrt{3}},1)$, $\lambda_{0}>0$, and $\delta>0$ such
that for every $(\lambda,\beta)\in\mathbb{R}^{2}$ satisfying
$|\beta-\beta_{0}|$, $|\lambda-\lambda_{0}|<\delta$, and $\lambda\neq 0$, we
have $\sqrt{1-\beta^{2}}|\lambda|<\varepsilon$. Thus, we see that
$\mathcal{F}_{(t,x_{1})\to(\xi_{1},\xi_{2})}(cq)=0$ in an open set of
$\mathbb{R}^{2}$. Furthermore, since $q$ is compactly supported, we get from
the Paley-Wiener theorem that $\mathcal{F}(cq)$ is real analytic. Therefore,
we conclude that $cq=0$ in $Q$. Since $c$ is a positive function, we have
$q=q_{1}-q_{2}=0$. This completes the proof of Theorem 1.2.

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