Inverse scattering on conformally compact manifolds

We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(\lambda_1,$ $\lambda_2)$ in a suitable subset of $\mc$, determines $\alf,$ and the Taylor series of both the potential and the metric at the boundary.


Introduction
In this note we study inverse scattering on conformally compact manifolds with non-constant asymptotic sectional curvatures. We work with ∆ the negative Laplacian. We prove that the scattering matrix of ∆ g +V , g conformally compact, V ∈ C ∞ , at fixed energies (λ 1 , λ 2 ) in a suitable subset of C, determines the curvature α, the Taylor series of the potential V at the boundary and the Taylor series of g near the boundary. This is a generalization and improvement of some results of [10]. It is a generalization to the case where the potential V does not vanish at infinity and the curvature at infinity is not constant. It is an improvement in the sense if we know a priori that α 1 = α 2 we determine both Taylor series from the scattering matrix at just one energy.
The potentials considered here can be thought of as symbols of order zero. In the Euclidean setting scattering for such potentials has been studied by Hassel-Melrose-Vasy [7,8], Saito [17], Herbst [9], and Agmon-Cruz-Herbst [1] among others. In this paper we get a first result on inverse scattering in the setting of conformally compact manifold with variable curvature at infinity. There does not seem to be any inverse result for the symbols which are potentials of order zero in the Euclidean setting.
Let X be a C ∞ compact manifold of dimension n+1 with boundary ∂X. We recall that x is a boundary defining function of ∂X if x ≥ 0, ∂X = {x = 0}, and dx = 0 on ∂X. We assume X is equipped with a Riemannian metric g such that for any defining function x of ∂X the metric x 2 g = g is a C ∞ non degenerate Riemannian metric up to ∂X. The manifold (X, g) is called a conformally compact manifold.
It is shown in [12] that if ν is the unit normal with respect to g, −(∂x/∂ν) 2 (y) = −α 2 (y) are the sectional curvatures at the boundary. When (∂x/∂ν) 2 (y) is constant, the manifold is called asymptotically hyperbolic. The scattering theory in the setting of variable sectional curvature α at infinity has been studied by Borthwick [2].
Following the proof of Lemma 2.1 of [3] one can show that once fixedg| ∂X there exists a unique C ∞ defining function x of ∂X, in a collar neighborhood [0, ǫ) × ∂X of ∂X, such that |dx|g = α. In this case we can write Mazzeo and Melrose [13] studied the resolvent of the Laplacian for asymptotically hyperbolic manifolds. They proved that the resolvent has a meromorphic continuation to C\{(1/2)(n − N 0 )}. Guillarmou [4] proved that in general the resolvent may have essential singularities at {(1/2)(n−N 0 )}. The generalization to a variable curvature at the boundary α(y) was carried out by Borthwick in [2]; he proved the existence of the Poisson operator, and meromorphic continuation of the resolvent, and the scattering matrix.
By the spectral theorem the resolvent is well defined for ℑλ << 0, and one would like to understand if it can meromorphically continued to a larger region of the complex plane. The proof of Proposition 5.3 of [2] can be modified to our case to obtain with Ω defined in (2.7) below, and D the discrete set of λ ∈ C such that the resolvent has a pole. Let x be such that (1.1) is satisfied. Given f ∈ C ∞ (∂X), there exists a unique u ∈ C ∞ (X), such that . We outline the proof of 1.1 in section 2. The Poisson operator is the map and the scattering matrix S λ is defined by In [10], Joshi and Sá Barreto deal with the asymptotically hyperbolic case and show that for a fixed λ ∈ C\Q, where Q is a discrete set, the scattering matrix S(λ) determines the Taylor series of the metric g or the potential V , with the assumption that the potential vanishes at the boundary. We carry out the natural extention of this approach to the conformally compact case and for potentials not vanishing at the boundary.
Let P 1 and P 2 be the operators and we fix a product structure in which (1.5) We introduce some notation necessary to state our Theorem. We denote by has a pole, i = 1, 2}, (1.6) we also denote by where α m = min ∂X α, α M = max ∂X α, and We denote by S 1 and S 2 the scattering matrices associated to P 1 and P 2 respectively. Our main Theorem is Theorem 1.2. Let g 1 , g 2 and V 1 , V 2 be as in (1.5), p ∈ ∂X, and σ = max{σ 1 , σ 2 }. Assume that near p, Theorem 1.2 can be restated invariantly as Theorem 1.3. Let g 1 , g 2 and V 1 , V 2 be as in (1.5), p ∈ ∂X, and σ = max{σ 1 , σ 2 }. Assume that near p, In Section 2, we recall the definitions of the spaces of polyhomogeneous distributions of [2] which are needed to carry out the analysis for the conformally compact geometry with variable curvature at infinity. The reason for the introduction of these spaces comes from the appearance of an indicial root which will depend on the space variable y, through the boundary curvature function α(y) and the potential V (0, y). In Section 3, we prove our main Theorem.
The author would like to thank his advisor Antônio Sá Barreto for his guidance and help on this paper, and anonymous referees for many helpful comments.

The Poisson Operator and the Scattering Matrix
In this section we prove Theorem 1.1.

Boundary asymptotics.
In this subsection we recall the spaces of functions used in [2].
Let M be a smooth manifold with corners, as defined in [15], and let ρ = (ρ 1 , ..., ρ p ) be the defining functions for the finitely many boundary faces Y 1 , ..., Y P of M . Let V b (M ) be the set of smooth vector fields tangent to the boundary. We recall the space of conormal distributions where m ∈ R p and ρ m = ρ m1 1 · · · ρ mp p . And call the set With this space defined, we recall for β ∈ C ∞ (M ; R p ) the space of polyhomogeneous distributions Lastly we recall the space of truncated expansion We refer the reader to [2] for a more detailed description of the later spaces and for a proof of the last equality. An important lemma which was proven in [2], tells us that these spaces only depend on the restriction to the boundary of β. For our case β will be the indicial root σ that will be discussed next; it appears in the asymptotic expansion that leads to the definition of the scattering matrix (1.3).
The space A β is independent of the choice of T j and depends on β only through the restrictions β| Yi .

The indicial operator.
We adapt the parametrix construction of [2]. For g as in (1.1), we consider the Schrödinger operator We consider the indicial roots when restricted to the boundary x = 0. The indicial root σ satisfy the equation ). (2.6) We denote σ = σ + , and therefore σ − = n − σ. Observe that σ is holomorphic in λ when where α m and α M are the minimum and maximum of α at ∂X respectively. Let and then let we have, just as in [2] Lemma 3.2: This is the first ingredient of the parametrix construction in [13]. The following corollary follows from the same arguments in [2], We also recall the construction of the stretched product, which is the manifold (with corners) obtained after blowing up the product X × X along ∂∆ι, where ∂∆ι = (∂X × ∂X) ∩ ∆ι ∼ = ∂X, and ∆ι is the set of fixed points of the involution I that exchanges the two projections, Where π L (X × X) is the projection onto the first component X × ∂X, and π r (X × X) the projection onto the second component ∂X × X.
We use the usual notation for the stretched product X × 0 X and denote the blow-down map by: The process of blowing-up just described, amounts to the introduction of singular coordinates near the corner, they are given near left face, in local projective coordinates, by (with Y = y − y ′ ) near the front face by near the right face by the left, right, and front faces are characterized by ρ = 0, ρ ′ = 0, and R = 0 respectively.

Pseudodifferential operators.
We recall the class of pseudodifferential operators that we need.
We are going to work on the space of half densities of the form h(x, y) α(y) The C ∞ multiples of such a density are sections of the bundle of singular half densities Γ 1/2 0 (X). Similarly, and we refer to [13,10] for the details, we can define the bundles Γ , whose lift to X × 0 X has a conormal singularity of order m at the lifted diagonal.

2.5.
The resolvent, the Poisson operator and the scattering matrix. We can now apply Proposition 4.2 of [2] to use the parametrix construction of [13] section 7 to get Proposition 2.1. Let λ ∈ C\(Ω ∪ D), then there exists M λ and F λ holomorphic, such that To apply analytic Fredholm theory we need the invertibility of I − F λ for at least one value of λ. To do that one modifies the parametrix M λ by adding a smoothing operator of finite rank which guaranties that (I − F λ ) −1 exists for λ such that ℜλ = 0, and ℑλ << 0. For the details of this construction we refer the reader to the second paragraph in the proof of Theorem 7.1 on page 301 of [13].
We decompose the resolvent as the pull-back using the blow-down map b (that is 0 Ψ m , 0 Ψ σ l ,σr ), and its residual class (Ψ σ l ,σr ) and state this as a Proposition, has a meromorphic continuation to λ ∈ C\(Ω ∪ D ′ ), and structure The proof of the existence of the Poisson operator and the scattering matrix follow the same as in [2], the Poisson operator is equal to The following Proposition, which is proven in [2], is the final ingredient needed to prove Theorem 1.1,

Proposition 2.3. For the Schwartz kernel of the Poisson operator
The proof of Theorem 1.1 follows; for the reader interested in the details we refer to [2]. The principal symbol of the scattering matrix is Remark 1. Notice that S λ is a pseudodifferential operator in Ψ m ǫ,0 for every ǫ > 0, and m = 2 max ∂X ℜσ − n.

Proof of Theorem 1.2
We proceed to analyze the relationship between scattering matrices associated to two distinct operators as in (1.4). Let's consider first the case where the scattering matrices agree at the principal symbol level. In this case where for j = 1, 2 and i − 1, 2 We use that |tξ| hi0 = t|ξ| hi0 for every t ∈ R to obtain 2 2σ2−2σ1 Γ(n/2−σ1) This implies that σ 1 and σ 2 are identical. Hence |ξ| h20 = |ξ| h10 for every ξ = 0 and thus h 10 and h 20 are also equal.
Going back to the metrics, we have obtained h 1 | ∂X = h 2 | ∂X , which means that there exists a tensor L(y, dy) such that Next we obtain the higher order Taylor coefficients of the metric and potential from the lower order symbols of the scattering matrix. We denote by Just as in [10] we have Tr(h 1 (0, y) −1 L(0, y)) + O(x 2 )).
For the rest of the proof we only need to use that the scattering matrices agree at one energy, so we take λ = λ 1 and drop the subindex. Also for the rest of the section. The fixed point y c will appear naturally after applying the normal operator.
Remark 2. Notice that if we assume that α 1 = α 2 , one only need one energy to obtain V 1 (0, y) = V 2 (0, y) from the previous argument. Now assume S 1λ − S 2λ ∈ Ψ 2 max ℜσ−n−2 (∂X), and we want to show that h 1 (x, y, dy) − h 2 (x, y, dy) = O(x 2 ), and V 1 − V 2 = O(x 2 ). To do so we go further and get information on the derivatives of V and the metric h. Let P 1 and P 2 be as defined in (1.4). First we compute P 2 − P 1 as in [10] 1 . The difference in the metric between our case and that of [10] is that g 00 = 1 α 2 x 2 , and δ i = det |g i | = det|h1| (αi(y)x n+1 ) 2 . So the only term that will change module higher order terms in the computation of is the i = j = 0 term. This term is equal to where T = Tr(h 1 (0, y) −1 L(0, y)), and δ = δ2 δ1 . Since α 1 = α 2 and V 1 (0, y) − V 2 (0, y) we have To find the expansion on the difference of the scattering matrices we can proceed as in [10]. Let R 1 and R 2 be the resolvents of P 1 and P 2 respectively, then where E is the right hand side of (3.3) after factoring out an x. To obtain R 2 as a perturbation of R 1 we need to find F so that P 2 (F ) = xER 1 .
We set x = x ′ s and F = x ′ F 1 , and as x ′ commutes with P 2 we obtain At this step we use a fundamental tool developed in [13], which is the normal operator. For the details of its construction and further properties we refer to [13] sections 2 and 5. We recall that the normal operator N P of ∆ g + V at a point y c ∈ ∂X is given by α 2 (y c )∆ 0 + V (0, y c ), where ∆ 0 is the Laplacian on H n+1 . Here we assume that the metric α 2 (y c )h 0 (y c ) is transformed by a linear change of variables to the identity. We apply the normal operator to (3.4) to get The right hand side of (3.5) is in A σ+1,σ−1 . We can now apply Proposition 6.19 of [13] to deduce that N P F 1 ∈ A σ,σ−1 , thus by the mapping properties of N P , we can write F = x ′ (F 1 ) = Rρ σ ρ ′σ γ(λ); with γ(λ) ∈ C ∞ (X × 0 X\∆ 0 , Γ 1/2 0 (X × 0 X)). Next we follow the construction of the expansion of γ which applies just as in [10]: 1 There is a little correction to the computation in [10], pointed out in [5].
We recall Proposition 4.4 of [10], which holds for our case and states that the kernel of S λ satisfies where b ∂ is the blow-up of the manifold ∂X × ∂X along the diagonal ∆ ∈ ∂X × ∂X (we refer the reader to [10] for the details of this blow-up). Thus we can write In the coordinates r = |y − y ′ |, w = (y − y ′ )/r, Taking the r −n factor out of the half-density we get that γ(σ, r, ω, y, 0, 0)|dωdy ′ | is the restriction of to the intersection of the left, right and front face, which is ρ = ρ ′ = R = 0. Next we explicitly calculate what this is.
To do that we recall the lemma from [10] Lemma 3.1.