1 Introduction
Let
$\overline {X}$
be a compact manifold with boundary of dimension
$n+1$
. A smooth metric g in the interior X of
$\overline {X}$
is conformally compact (CC) if
$\overline {g}=\rho ^{2} g$
extends smoothly to a Riemannian metric in
$\overline {X}$
, where
$\rho \in C^{\infty }(\overline {X},[0,\infty ))$
is a boundary defining function for
$\partial \overline {X}$
(that is,
$d\rho |_{\partial \overline {X}} \neq 0$
and
$\{\rho =0\}=\partial \overline {X}$
). The boundary
$\partial \overline {X}$
equipped with the conformal class of
$\overline {g}|_{T \partial \overline {X}}$
is called the conformal boundary or conformal infinity of
$(\overline {X}, g)$
. If
$|d\rho |_{\overline {g}}=1$
at the boundary, g is called asymptotically hyperbolic (AH). A more restrictive condition is the Einstein condition
$\operatorname {\mathrm {Ric}}_{g}=-ng$
. In this case, g is called conformally compact Einstein (CCE) or Poincaré–Einstein.
In these particular ‘edge’ geometries (in the sense of [Reference Mazzeo28]), we study the inverse scattering problem for the generalised eigenvalue problem
where
$s=(n+1)/2$
. If
$(\overline {X},g)$
is an AH manifold with constant scalar curvature
${R_{g}\equiv -n(n+1)}$
, given
$f \in C^{\infty }(\partial \overline {X})$
, the problem (1.1) has a unique solution u with
$u|_{\partial \overline {X}}=f$
and the expansion
(see [Reference Graham and Zworski7–Reference Guillarmou and Guillopé9] and Lemma 2.1). Here, S (sometimes written as
$S_{g}$
to make clear the dependence on g) is the scattering matrix at energy
$(n+1)/2$
, and it depends on the choice of the boundary defining function (or, equivalently, the choice of the metric on the conformal infinity), although it can be defined without this dependence (see [Reference Graham and Zworski7, Section 3]). Hereafter, we denote by
$P_{s}$
the operator appearing in (1.1).
As is mentioned in [Reference Graham and Zworski7, Reference Mazzeo and Melrose30], the scattering matrix can be considered as a version of the Dirichlet-to-Neumann (DN) map for these geometries. It then makes sense to consider the analogue of the Calderón problem in this context. In inverse problems, this means determining the metric (or even the differential or topological structure) of a compact Riemannian manifold with boundary from the DN map associated to the Laplace–Beltrami operator. On compact Riemannian manifolds, this problem is solved on surfaces and in the analytic category [Reference Lassas and Uhlmann23], and in conformally anisotropic geometries [Reference Dos Santos Ferreira, Kenig, Salo and Uhlmann4]. The Calderón problem in our context, known as the inverse scattering problem, asks if it is possible to recover information of the manifold
$(\overline {X},g)$
from the knowledge of the scattering matrix
$S_{g}(s)$
for certain fixed s. In this direction, our main result is as follows.
Theorem 1.1. Let
$(\overline {X}_{i},g_{i})$
be AH manifolds of dimension
$n+1$
, with constant scalar curvature
$R_{g_{i}} \equiv -n(n+1)$
. Suppose that the
$\partial \overline {X}_{i}$
contain a common open set
$\Gamma $
such that the identity map
$\mathrm {id} \colon \Gamma \subset \partial \overline {X}_{i} \to \Gamma \subset \overline {X}_{2}$
is a diffeomorphism. Assume that
$\ker _{L^{2}}P_{{(n+1)}/{2}}^{i}=\emptyset $
for
$i=1,2$
. If for some boundary defining functions
$\rho _{i}$
the scattering matrices at energy
$(n+1)/2$
agree for the two metrics, that is,
$S_{g_{1}}((n+1)/2)=S_{g_{2}}((n+1)/2)$
, then there exists
$c\in C^{\infty }(\overline {X}_{1})$
such that:
-
(1) when $n=1$
, there exists a diffeomorphism
$\psi \colon \overline {X}_{1} \to \overline {X}_{2}$
with
$\psi |_{\Gamma }=\mathrm {id}_{\Gamma }$
and such that
$c\psi ^* g_{2}=g_{1}$
; -
(2) when $n\geq 2$
, there exists a diffeomorphism
$\psi \colon U_{1} \to U_{2}$
between neighbourhoods
$U_{i}$
of
$\Gamma $
in
$\overline {X}_{i}$
, with
$\psi |_{\Gamma }=\mathrm {id}_{\Gamma }$
, and such that
$c\psi ^* g_{2}$
and
$g_{1}$
coincide up to infinite order, that is,
$\partial _{\rho _{1}}^{k}g|_{\rho _{1}=0}=\partial _{\rho _{1}}^{k}(c\psi ^{*}g_{2})|_{\rho _{1}=0}$
for all
$k \geq 0$
.
The result is inspired by the classical result for compact Riemannian manifolds with boundary obtained in [Reference Lee and Uhlmann24]. In contrast to the proof given in [Reference Lee and Uhlmann24], which is based on a factorisation of the Laplace–Beltrami operator, our proof follows an indirect approach (although we also have a ‘formal’ factorisation of the Laplacian in AH geometries (see Lemma 2.9)). We use the relation between the Dirichlet problem for (1.1) at
$s=(n+1)/2$
and the Dirichlet problem for the conformal Laplacian (also known as the Yamabe operator), described first in [Reference Guillarmou and Guillopé9] for AH manifolds with
$R_{g} \equiv -n(n+1)$
and then in [Reference Chang and del Mar González3] for more general equations in CCE manifolds. This means we have conformal invariance for the scattering operator at this energy, so the conformal factors appearing in the result are natural. They show that, for manifolds as in Theorem 1.1, the DN map of
$\rho ^{2}g$
is the scattering of g at energy
$(n+1)/2$
. According to the results of [Reference Lassas, Liimatainen and Salo22], the DN map for the conformal Laplacian determines the jet of a metric at the boundary, which is conformal to
$\rho ^{2}g$
. We remark, however, that the determination of the jet on the boundary in [Reference Lassas, Liimatainen and Salo22] relies on a factorisation of the magnetic Schrödinger operator carried out in [Reference Dos Santos Ferreira, Kenig, Salo and Uhlmann4].
Although the theorem has a short proof, the result is new since there are no inverse scattering results at this energy level and, furthermore, we impose no restrictions on the dimension. To the authors’ knowledge, the first inverse scattering result in the AH manifold context was given in [Reference Joshi and Sá Barreto21], where it was shown that the scattering matrix on AH manifolds associated with (1.1) with a potential determines the metric up to diffeomorphism at infinite order, assuming the same potential, and also the potential up to infinite order if one fixes the metric. These results do not apply in our case due to their restrictions on the energy levels. Using radiation fields, it was proved in [Reference Sá Barreto32] that the scattering matrix at all energies determines the metric up to a diffeomorphism on AH manifolds. In the same spirit, inverse scattering results for asymptotically complex hyperbolic manifolds were obtained in [Reference Guillarmou and Sá Barreto10]. On even dimensional CCE manifolds (of dimension
$n+1$
), it is known [Reference Guillarmou and Sá Barreto11] that the scattering matrix at energy n determines the metric up to a diffeomorphism. On CC manifolds, it was shown in [Reference Marazzi25] that the scattering matrix at two energy levels, associated with (1.1) with potential, determines the potential and the metric (and even the curvature at the boundary) up to the usual gauge. In [Reference Guillarmou and Tzou12], an inverse scattering result for the Schrödinger equation on AH manifolds was obtained, showing that the scattering matrix determines the metric, assuming that the potentials vanish up to order 2 in the boundary. Later, we mention an updated version of this result, which also recovers the metric (see Theorem 2.11). The results from [Reference Guillarmou and Sá Barreto11] were extended in [Reference Marazzi26], where it is shown that on CCE manifolds, the scattering matrix at energy n determines the metric at certain orders involving
$\log $
terms, up to diffeomorphism, and a modified scattering operator determines the metric up to diffeomorphism. In [Reference Isozaki15], inverse scattering problems in hyperbolic space were studied and related, similarly to this work, to inverse problems for the Schrödinger operator in Euclidean domains. In [Reference Isozaki, Kurylev and Lassas17], inverse scattering was considered for manifolds with cylindrical ends, while in [Reference Isozaki, Kurylev and Lassas20], surfaces with conical singularities, cusps and regular ends were considered (see also the monographs [Reference Isozaki and Kurylev16, Reference Isozaki, Kurylev, Lassas, Stefanov, Vasy and Zworski18, Reference Isozaki, Kurylev, Lassas, Eskin, Friedlander and Garnett19]). The results in [Reference Sá Barreto32] were extended to partial data in [Reference Hora and Sá Barreto13], and for scattering with disjoint source and observation sets in [Reference Hora and Sá Barreto14]. We also highlight the work [Reference Marx-Kuo27] on AH manifolds, where, assuming some evenness conditions on the metric, it is shown that one can determine it from the knowledge of minimal areas.
2 Proof of the main result
Before dealing with the inverse problem, we show that the direct problem has a solution with asymptotics as mentioned in Section 1. The most fundamental work in this direction is [Reference Graham and Zworski7]. The following is mentioned in [Reference Guillarmou and Guillopé9] as a consequence of [Reference Guillarmou8, Lemma 4.1], but we provide a short proof for completeness.
Lemma 2.1. Let
$(\overline {X},g)$
be an AH manifold with
$R_{g} \equiv -n(n+1)$
. Assume that
$\ker _{L^{2}}P_{({n+1)}/{2}}=\emptyset $
. Given
$f \in C^{\infty }(\partial \overline {X})$
, there exists a unique solution u to
having the expansion
The term of order
$(n+1)/2$
, that is,
$S((n+1)/2)$
, is defined as the scattering matrix of energy
$(n+1)/2$
.
Proof. We follow the strategy from [Reference Graham and Zworski7, Reference Guillarmou8, Reference Guillarmou and Sá Barreto11]. First, we construct a solution modulo infinite order. Then, the hypothesis on the kernel and the metric allow us to apply [Reference Guillarmou8, Lemma 4.1] to ensure that the resolvent has no poles at
$(n+1)/2$
. Therefore, we can modify the solution just constructed to obtain a full solution having the required asymptotics.
The first step is to construct
$F_{\infty } \in C^{\infty }(\overline {X})$
such that
where
$\rho $
is a Graham–Lee geodesic boundary defining function (see [Reference Graham and Lee6, Lemma 5.2 and the subsequent paragraph], and also [Reference Graham5, Lemma 2.1]). As mentioned in [Reference Guillarmou8, page 24], it is enough to work in a neighbourhood of the boundary. Using the flow
$\phi _t(y)$
of the gradient
$\nabla ^{\rho ^2 g} \rho $
of the Graham–Lee boundary defining function, we obtain a diffeomorphism
$\phi \colon [0, \varepsilon )_{t} \times \partial \overline {X} \to \phi ([0, \varepsilon ) \times \partial \overline {X}) \subset \overline {X}$
defined by
$\phi (t, y):=\phi _t(y)$
, and the metric pulls back to
for some smooth one-parameter family of metrics
$h(t)$
on the boundary
$\partial \overline {X}$
, where
$\phi ^* \rho =t$
. It can be shown that
where
Observe that for
$j \in \mathbb {N}_{0}=\{0,1,\ldots \}$
,
where
Suppose now that
$\tilde {F} \in C^{\infty }(\overline {X})$
satisfies
$\mathscr {D}\tilde {F}=O(t^{j})$
for
$j \geq 1$
. Since
$G(z)f=O(t)$
(a priori, we analyse this further below), using (2.2) with
$f=(t^{-j}\mathscr {D}\tilde {F})|_{t=0}/(j(1-j))$
,
for
$j \neq 1$
. Indeed, on the left-hand side of (2.3), we are subtracting the term of order
$O(t^{j})$
from
$\mathscr {D}\tilde {F}$
. This procedure suggests the following recursive definition of
$f_{j}$
and
$F_{j}$
:
To construct
$f_{1}$
,
$f_{2}$
and
$f_{3}$
, we proceed as follows. To construct
$f_{1}$
and, hence,
$F_{1}$
, we look for terms of order
$1$
in
$\mathscr {D}F_{0}$
. First, recall from [Reference Guillarmou and Guillopé9, Section 4] that
$R_{g} \equiv -n(n+1)$
implies
$\operatorname {\mathrm {tr}}_{h_{0}}h_{1}=0$
. Indeed, since g is conformal to
$t^{2}\overline {g}=dt^{2}+h(t)$
,
Since
$R_{g} \equiv -n(n+1)$
, after simplification, we obtain, by first dividing by t and then letting
$t \to 0$
, that
$\operatorname {\mathrm {tr}}_{h_{0}}h_{1}=0$
, where
$h_{0}=h(0)$
and
$h_{1}=\partial _{t}h(0)$
. This proves the claim. Now, we can use [Reference Guillarmou8, (4.8) and (4.9)] and, since
$\operatorname {\mathrm {tr}}_{h_{0}}h_{1}=0$
, we have
where
$D^{j}$
are differential operators of order j on
$\partial \overline {X}$
. So,
$\mathscr {D}f$
only contains terms of order
$3$
or more. Hence,
$f_{1}:=0$
and
$F_{1}:=F_{0}$
. Similarly, for
$f_{2}$
, we look for terms of order
$2$
in
$\mathscr {D}F_{1}$
. By the previous argument, there are none. Thus, we set
$f_{2}:=0$
and
$F_{2}:=F_{0}$
. Finally, for
$f_{3}$
, we see that
$t^{3}(\Delta _{h_{0}}+D^{2})f$
is the term of order 3 in
$\mathscr {D}F_{2}=\mathscr {D}f$
, and hence, we define
$f_{3}:=(\Delta _{h_{0}}+D^{2})f$
and
$F_{3}=t^{3}f_{3}+f_{0}$
. We can now continue using (2.4) inductively. By Borel’s lemma, we can find
$F_{\infty } \in C^{\infty }(\overline {X})$
satisfying (2.1) and such that
$F_{\infty }-F_{j}=O(t^{j+1})$
.
To complete the construction of the full solution to the eigenvalue problem, we now use our assumption on g and the
$L^{2}$
kernel to invoke [Reference Guillarmou8, Lemma 4.1], which shows that the resolvent of
$P_{s}$
has no poles at
$s=(n+1)/2$
. Hence, it is an analytic family by [Reference Graham and Zworski7, Proposition 3.1] (and the results in [Reference Mazzeo and Melrose30]) and the resolvent of
$P_{{(n+1)}/{2}}$
satisfies
where
$\dot {C}^{\infty }(\overline {X}):=\{u \in C^{\infty }(\overline {X}):u=O(\kern0.5pt\rho ^{\infty })\}$
. Therefore, we can set
which solves the eigenvalue problem.
To see the uniqueness, note that if
$u_{1}$
and
$u_{2}$
are two solutions with the same term of order
$\rho ^{{(n-1)}/{2}}$
, then
$u_{1}-u_{2} \in \rho ^{{(n+1)}/{2}}C^{\infty }(\overline {X})$
. This implies that
$u_{1}-u_{2} \in L^{2}(\overline {X})$
, which contradicts our hypothesis. This finishes the proof.
Remark 2.2. Alternatively, one can use a more direct method in the following sense. One can obtain a solution to the problem
$P_{s}u^{\prime }_{s}=O(\kern0.5pt\rho ^{\infty })$
with
$u^{\prime }_{s}|_{\partial \overline {X}}=f$
for s close to
$(n+1)/2$
. Then, a full solution can be obtained by using the resolvent at energy s. Since there are no poles,
$R(s)$
extends to
$R((n+1)/2)$
, and one then obtains a solution u to
$P_{{(n+1)}/{2}}u=0$
and
$u_{|\partial \overline {X}}=f$
.
Remark 2.3. We only used the curvature condition to ensure that
$\operatorname {\mathrm {tr}}_{h_{0}}h_{1}=0$
. The vanishing of the trace means that
$\partial \overline {X}$
is minimal in
$(\overline {X},\rho ^{2}g)$
. In particular, the result is also true if we change the curvature assumption to evenness of the metric g modulo
$O(\kern0.5pt\rho ^{3})$
. We recall that
$(\overline{X},g)$
is even modulo
$O(\kern0.5pt\rho ^{2k+1})$
if there is a boundary defining function
$\rho $
with
$|d\rho |_{\rho ^{2}g}=1$
, and such that in a collar neighbourhood, the pullback of g by the gradient flow of
$\nabla ^{\rho ^{2}g}\rho $
takes the form
where the
$h_{j}$
are symmetric tensors on
$\partial \overline {X}$
. It was shown in [Reference Guillarmou8, Lemma 2.1] that this is a well-defined notion, meaning that it is independent of the boundary defining function. Examples of even metrics can be found in [Reference Guillarmou8, remark after Theorem 1.4]. For instance, hyperbolic metrics perturbed on a compact set are even modulo
$O(\kern0.5pt\rho ^{\infty })$
, and CCE manifolds of dimension
$n+1$
are even modulo
$O(\kern0.5pt\rho ^{n})$
.
The key relation between inverse scattering on AH manifolds and the Calderón problem for the conformal Laplacian is given by the following result of Guillarmou and Guillopé.
Lemma 2.4 [Reference Guillarmou and Guillopé9, Section 4]
Let
$(\overline {X},g)$
be an AH manifold, with
$R_{g} \equiv -n(n+1)$
. Then,
$u \in C^{\infty }(\overline {X})$
is a solution to
if and only if
$U:=\rho ^{-{(n-1)}/{2}}u$
is a solution to the conformal Laplacian problem
where
$R_{\overline {g}}$
is the scalar curvature associated to
$\overline {g}$
. Furthermore, the scattering operator for the problem (2.5) and the DN map for (2.6) coincide, that is,
as operators from
$C^{\infty }(\partial \overline {X})$
to itself, where
$\Lambda _{\overline {g}}$
is the DN map with respect to the metric
$\overline {g}$
.
On CCE manifolds, this was extended to any energy level in [Reference Chang and del Mar González3, Lemma 4.1 and Theorem 4.3].
To reach our goal, we need the following result concerning the boundary jet recovery in the inverse problem for the conformal Laplacian.
Lemma 2.5 [Reference Lassas, Liimatainen and Salo22, Lemmas 2.1 and 2.2]
Let
$(\overline {X}_{1},\overline {g}_{1})$
,
$(\overline {X}_{2},\overline {g}_{2})$
be compact Riemannian manifolds with boundary, with a common boundary portion
$\Gamma \subset \partial \overline {X}_{1}, \partial \overline {X}_{2}$
. Assume that DN maps for the conformal Laplacian agree, that is,
$\Lambda _{\overline {g}_{1}}=\Lambda _{\overline {g}_{2}}$
. Then, there exist positive functions
$c_{i} \in C^\infty(\overline{X}_i)$
with
$c_{i}|_{\Gamma }=1$
such that, if
$\tilde {g}_{i}=c_{i}\overline {g}_{i}$
, then the jets of
$\tilde {g}_{1}$
and
$\tilde {g}_{2}$
on the respective
$\tilde {g}_{i}$
-normal coordinates agree.
Proof of Theorem 1.1
Let
$\rho _{i}$
be geodesic boundary defining functions for
$g_{i}$
such that the scattering matrices at energy
$(n+1)/2$
coincide. From Lemma 2.4, the same holds for the DN maps associated to the conformal Laplacian of the metrics
$\rho _{1}^{2}g_{1}$
and
$\rho _{2}^{2}g_{2}$
. If
$n \geq 2$
, Lemma 2.5 shows that there exist
$c_{i} \in C^{\infty }(\overline {X}_{i})$
with
$c_{i}|_{\Gamma }=1$
(
$i=1,2$
) such that given coordinates y in
$\Gamma $
, if
$\psi _1$
and
$\psi _2$
are
$c_{1}\rho _{1}^{2}g_1$
- and
$c_{2}\rho _{2}^{2}g_2$
-boundary normal coordinates constructed by using the same y-coordinates, then
$(\psi _{1}^{-1})^{*}(c_{1}\rho _{1}^{2}g_{1})=(\psi _{2}^{-1})^{*}(c_{2}\rho _{2}^{2}g_{2})+O(t^{\infty })$
. Hence,
$g_{1}=(c_{1}\rho _{1}^{2})^{-1}(\psi _{1}\circ \psi _{2}^{-1})^{*}(c_{2}\rho _{2}^{2}g_{2})+O(\kern0.5pt\rho _{1}^{\infty })$
. To deal with surfaces, one applies the result from [Reference Lassas and Uhlmann23] instead of [Reference Lassas, Liimatainen and Salo22] to conclude the proof.
Remark 2.6. Note that for surfaces, the hypothesis on the kernel is superfluous by the results in [Reference Graham and Zworski7, Reference Mazzeo and Melrose30].
Remark 2.7. By the results in [Reference Guillarmou and Guillopé9], one actually obtains equivalence between the inverse scattering problem at energy
$1$
and the anisotropic Calderón problem for surfaces. However, in higher dimensions, one also needs information about the mean curvature of the boundary with respect to the compactified metric.
Remark 2.8. When
$n+1 \geq 2$
, if all the components are analytic, that is, the metrics, the conformal factors and the diffeomorphisms, then one can extend the diffeomorphism as in [Reference Lee and Uhlmann24] to obtain a global one. Furthermore, if one assumes that the compactified manifolds
$(\overline {X}_{i},c_{i}g_{i})$
are locally conformally real analytic in the sense of [Reference Lassas, Liimatainen and Salo22], then a global result can be achieved as in the case of surfaces (Theorem 2.11). We point out that this condition seems to be stronger than being analytic in the interior, so it is not immediate that CCE are locally conformally real analytic up to the boundary. Since we recover metrics conformal to the initial ones, even if the initial ones are CCE, their conformal versions are, in general, not Einstein. Hence, we cannot apply the methods of [Reference Guillarmou and Sá Barreto10] in our case to obtain a global result.
One needs the conformal factors to put the metric in a certain ‘conformal gauge’ (which does not change the DN map) to obtain the vanishing of derivatives of
$\partial _{x_{n+1}}\log (cg^{\alpha \beta })=0$
and recover the jet of the metric at the boundary. The recovery of the jet is obtained by the following procedure (see [Reference Dos Santos Ferreira, Kenig, Salo and Uhlmann4, (8.2)], [Reference Lassas, Liimatainen and Salo22, page 1136]). (We point out that there is a little typo there; the condition should be
$\partial _{\tilde {x}_n}^{\kern1.2pt j} (\tilde {g}_{\alpha \beta } \partial _{\tilde {x}_n} \tilde {g}^{\alpha \beta }) (x', 0)=0$
for
$j \geq 1$
.) First, one factorises the conformal Laplacian and then one shows that the DN map is equal to a pseudodifferential operator of order 1 appearing in the factorisation, modulo a smoothing operator. Hence, to recover the jet of the metric, one shows that the DN map determines this
$\Psi $
DO and, finally, one shows that from the full symbol, one can recover the jet of the metric. In the context of inverse problems, this technique was first used in [Reference Sylvester and Uhlmann33] for the conductivity equation, in [Reference Lee and Uhlmann24] for the Laplace–Beltrami operator and then in [Reference Nakamura, Sun and Uhlmann31] for the magnetic Schrödinger operator with Euclidean metric. In [Reference Dos Santos Ferreira, Kenig, Salo and Uhlmann4], the procedure was generalised for the Schrödinger operator in the presence of a magnetic field on Riemannian manifolds. In particular, if we have no magnetic field, this procedure gives a (local) factorisation of
$P_{s}$
. Indeed, in the coordinates given by the geodesic boundary defining function
$\rho $
(that is, on
$\partial \overline {X} \times [0,\varepsilon )$
, we use coordinates
$(y,t)$
so that
$\overline {g}=\rho ^{2}g=dt^{2}+h_{t}(y)$
, with dual variables
$(\xi ,\eta )$
), we have for
$L_{\overline {g}}$
as in (2.6):
where
and
$B \in \Psi ^{1}$
has symbol
$b\sim \sum _{j \leq 1}b_{j}$
, where each term in the expansion is given by
Here,
$q_{2}$
and
$q_{1}$
are the symbols of
respectively. Here, the left quantisation is used. Hence, using the invariance of the conformal Laplacian together with (2.7), we obtain
We summarise this discussion as follows.
Lemma 2.9. Formally, we have the factorisation
We refer to this as a formal factorisation since it is merely an algebraic manipulation, but not a proper factorisation in the standard algebra of
$\Psi $
DO. Since
$\Delta _{g}$
is a 0-differential operator ([Reference Mazzeo28, Proposition 2.6]), one should expect to have a factorisation in the 0-calculus of Mazzeo and Melrose. Lemma 2.9 suggests, in general, a factorisation of the form
Remark 2.10. It is shown in [Reference Chang and del Mar González3] that there are ‘special’ boundary defining functions
$\rho _{1}$
,
$\rho _{2}$
so that, in a collar neighbourhood of the boundary,
$P_{s}$
becomes:
-
(1) the Laplace–Beltrami operator associated to $\rho _{1}^{2}g$
, when
$s=(n+1)/2$
; -
(2) the Bakry–Émery Laplacian (also known as the Ornstein–Uhlenbeck operator, Witten Laplacian or weighted Laplacian) $-\operatorname {\mathrm {div}} ((\kern0.5pt\rho _{2})^{a} \nabla \bullet )$
. Inverse problems for this operator have been studied recently in [Reference Borthwick and Kamran1]. The equation can be thought as applying the operator
$ L=e^{V}\operatorname {\mathrm {div}} (e^{-V} \nabla \bullet ), $
where
$V=-\log \rho _{2}^{a}$
.
Finally, we state the following stronger version of Theorem 1.1 in dimension 2.
Theorem 2.11. Let
$(\overline {X},g_{i})$
be two AH surfaces with common boundary
$\partial \overline {X}$
and let
$V_{i}\in \rho _{i}^2 C^{\infty }(\overline {X})$
be two potentials. Assume that
and let
$S_{i}(1)$
(associated to
$\rho _{i}$
) be the scattering map for the operator
$\Delta _{g_{i}}+V_i$
for
$i=~1, 2$
. If
$S_{1}(1)=S_{2}(1)$
, then there exists a diffeomorphism
$\psi \colon \overline {X} \to \overline {X}$
fixing the boundary pointwise, and a smooth function c such that
$ g_{1}=c\psi ^{*}g_{2}$
and
$V_{1}=c^{-1}\psi ^{*}V_{2}. $
The proof of [Reference Guillarmou and Tzou12, Corollary 6.1] applies
mutatis mutandis
: it is not hard to see that the DN maps for the operators
$\Delta _{\overline {g}_{i}}+\overline {V}_i$
are the same. Then, [Reference Cârstea, Liimatainen and Tzou2, Theorem 1.1] allows one to recover both the metrics and the potentials with the gauge mentioned in the result. The difference with [Reference Guillarmou and Tzou12, Corollary 6.1] lies in the fact that, previously, only the potential could be recovered as an application of the main theorem of [Reference Guillarmou and Tzou12].
Acknowledgements
The author would like to thank Colin Guillarmou, Rafe Mazzeo, Mikko Salo, Gunther Uhlmann and András Vasy for helpful discussions. Additionally, the author expresses gratitude to Stanford University and the University of Washington for their hospitality during his visit, while this work was in progress. The author was partly supported by the Ross–Lynn Scholar Grant from Purdue University’s College of Science.
