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Scattering and Inverse Scattering on ACH Manifolds

We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.


Introduction
Scattering theory and inverse problems for real asymptotically hyperbolic manifold have been extensively studied, see for example [18,17,14,24,29,30,35] and references cited there. Their complex analogue, the asymptotically complex hyperbolic manifolds, ACH in short, have not been studied as much. They were introduced by Epstein, Melrose and Mendoza [7], and more recently have also been considered by Biquard [4] and Biquard-Herzlich [5]. This class of manifolds contains certain quotients of the complex hyperbolic space by discrete groups, as well as smooth pseudo-convex domains in C n+1 equipped with a Kähler metric of Bergman type. The purpose of this work is to extend to the complex case several results which are known for real asymptotically hyperbolic manifolds.
Before discussing asymptotically complex hyperbolic manifolds, we recall certain facts about real asymptotically hyperbolic manifolds. An (n + 1)-dimensional non compact manifold X equipped with a C ∞ Riemannian metric g is called asymptotically hyperbolic if it compactifies into a C ∞ manifoldX with boundary ∂X, and if ρ is a defining function of the boundary ∂X, ρ 2 g is a C ∞ metric which is non-degenerate up to ∂X, and moreover if |dρ| ρ 2 g = 1 at ∂X. It can be shown, see [12], that (X, g) is asymptotically hyperbolic if and only if there exists a diffeomorphism ψ : [0, ǫ) t × ∂X → U ⊂X with ψ({0} × ∂X) = ∂X such that (1.1) ψ * g = dt 2 + h(t) t 2 where h(t), t ∈ [0, ǫ) is a C ∞ 1-parameter family of C ∞ metrics on ∂X. The function ρ := ψ * t is a boundary defining function inX near ∂X, which can be extended smoothly toX. Note that the boundary represents the geometric infinity of X, as does the sphere S n for the hyperbolic space H n+1 .
The spectrum of ∆ g , the Laplacian of (X, g) was studied in [27]; it consists a finite pure point spectrum σ pp (∆), which is the set of L 2 (X) eigenvalues, and an absolutely continuous spectrum σ ac (∆) satisfying which is a bounded operator in L 2 (X), for ℜ(λ) > n 2 , has a finite meromorphic extension to C\ 1 2 (n−N 0 ), where N 0 is the set of non-zero natural numbers, as a map from C ∞ 0 (X) → C ∞ (X), [27]. The poles of R(λ) are called resonances. Here, and throughout the paper, we call a family of operators finite-meromorphic if it is meromorphic, i.e. it has a finite Laurent expansion at each point, and the rank of the polar part at a pole has finite rank. The finite meromorphic continuation of R(λ) to the entire complex plane exists if and only if h(t) has an even Taylor expansion at t = 0. If h(t) is not even, R(λ) might have essential singularities at the points 1 2 (n − N 0 ), [16].
It has been shown in [14,24] that for ℜ(λ) = n/2, ℑ(λ) = 0, and any f ∈ C ∞ (∂X), there exists a unique u λ ∈ C ∞ (X) satisfying (∆ g − λ(n − λ))u λ = 0 such that near ∂X One can use this to define the scattering operator S(λ), for ℜ(λ) = n/2, ℑ(λ) = 0, as a generalized Dirichlet-to-Neumann map Like the Dirichlet-to-Neumann map, S(λ) is an elliptic pseudo-differential operator, but of order 2λ − n. It extends meromorphically to C. The first author recently studied Krein theory in even dimension manifolds by introducing a generalized determinant of S(λ) and applied it to analyze Selberg zeta function for certain quotient of hyperbolic space by discrete groups of isometries, in continuation of work by Patterson-Perry [32]. The second author studied inverse problems using S(λ). He first proved with Joshi [24] that S(λ) for λ fixed determines the Taylor expansion of h(t) in (1.1), then more recently he proves in [35] that the map λ → S(λ) determines the whole manifold up to global isometry.
We give a precise definition of what we call an asymptotically complex hyperbolic metric in Section 3, but we will briefly explain this notion before stating our results. We consider a non-compact Riemannian manifold (X, g) that compactifies intoX smooth with boundary ∂X. We assume that the boundary admits a contact form Θ 0 and an almost complex structure J : ker Θ 0 → ker Θ 0 such that dΘ 0 (., J.) is symmetric positive definite on ker Θ 0 . The associated Reeb vector field T 0 is the one which satisfies Θ 0 (T 0 ) = 1 and dΘ 0 (T 0 , JZ) = 0 for any Z ∈ ker Θ 0 .
The structure of the operator S(λ) is established in Theorem 1.1. Let (X, g) be an ACH manifold even at order 2k, then the scattering operator S(λ) extends to C \ (−N 0 ∪ P k ) as a meromorphic family of conformally covariant operators in the class of Heisenberg pseudodifferential operators Ψ 4λ−2(n+1) Θ0 (∂X), which is unitary on L 2 (∂X, dvol h0 ) when ℜ(λ) = n+1 2 and ℑ(λ) = 0. The principal symbol of S(λ) is where c n ∈ C depends only on n and F denotes Fourier transform from T ∂X to T * ∂X. Moreover, S(λ) is finite-meromorphic in C \ (−N 0 ∪ P k ∪ (n + 1 − P k )) and has at most poles of order 1 at each λ k := n+1 2 + 1 4 k with k ∈ N, the residue of which is a Heisenberg differential operator in Ψ k Θ0 (∂X) plus a finite rank projector appearing if and only if λ k (n−1−λ k ) ∈ σ pp (∆ g ). Moreover at λ 2k , we have where ∆ b , T 0 are the horizontal sublaplacian and the Reeb vector field of (∂X, Θ 0 , J).
We also deduce from [17] an explicit formula between finite-multiplicity poles of S(λ) (scattering poles) and finite multiplicity poles of R(λ) (resonances) in Proposition 6.5 and show that essential singularities for S(λ) and R(λ) can occur at (n + 1/2 − N 0 )/2 if the metric has no evenness property, see Proposition 6.7.
The proof that S(λ), ℜ(λ) = n+1 2 , ℑ(λ) = 0, is a pseudodifferential operator in the Heisenberg calculus is sketched by Melrose in [31]. The novelties in this theorem are the computation of the principal symbol of S(λ), its meromorphic continuation, and the analysis of the poles. In the case where the manifoldX is a strictly pseudoconvex domain of C n+1 equipped with an approximate Einstein Kähler metric, the relationship between the residues Res λ 2k S(λ) and the Gover-Graham operators of [13] is announced in [21].
Then we study the scattering theory from a dynamical view point as in the Lax-Phillips theory. We define the radiation fields, show that they give unitary translation representations of the wave group which can be used to define the scattering matrix (6.1) from the wave equation.
The Cauchy problem for the wave equation has smooth solutions u ∈ C ∞ (R + × X), we consider the behavior of u at infinity along some bicharacteristics and prove Theorem 1.2. Let (ρ, z) ∈ [0, ǫ) × ∂X be some coordinates given by product decomposition φ as above. Let u(t, z) be the solution of (1.
Let E ac := Π ac (H 1 (X) × L 2 (X)) where Π ac is the orthogonal projection from L 2 (X) onto the space of absolute continuity of ∆ g , and let H 1 (X) be the first Sobolev space H 1 (X) = {f ∈ L 2 (X); |df | g ∈ L 2 (X)}. The space E ac is a Hilbert space when equipped with the norm Then we show Theorem 1.3. The forward and backward radiation fields R ± extend to isometric isomorphisms from E ac to L 2 (R × ∂X, drdvol h0 ). Moreover, the map defined by ) is unitary and is a convolution operator in s, and conjugating it with Fourier transform in s we have The operator S in (1.5) is the dynamical definition of the scattering operator. Next, using these tools and after proving a localization result for the support of functions f ∈ L 2 ac (X) for which R + (0, f ) = 0 in s ∈ (−∞, s 0 ) (a "support theorem" in the sense of Helgason, Lax-Phillips), we are able to prove the following result on inverse scattering: Theorem 1.4. Let (X 1 , g 1 ), (X 2 , g 2 ) two ACH manifolds with the same boundary M := ∂X 1 = ∂X 2 , and equipped with the same conformal class of contact forms [Θ 0,1 ] = [Θ 0,2 ]. Let S 1 (λ), S 2 (λ) be the corresponding scattering operators associated to a conformal representative Θ 0 ∈ [Θ 0,1 ]. If S 1 (λ) = S 2 (λ) on {ℜ(λ) = n+1 2 , ℑλ = 0}, then there exists a diffeomorphism Φ :X 1 →X 2 such that Φ = Id on M and Φ * g 2 = g 1 .
The method we use is very close to that introduced by the second author [35] in the asymptotically hyperbolic case, which was inspired by the boundary control theory of Belishev [3].
The paper is organized as follows: In Section 1, we consider the model case of the complex hyperbolic space H n+1 C , then we discuss the geometry of ACH manifolds near infinity in Section 2. We define the Θ-calculus on X and the Heisenberg calculus on ∂X (these are the "natural" class of pseudo-differential operators associated to the geometric structure) in Section 4 and we analyze the Poisson and the scattering operators in Section 6. The next sections consist in defining raidiation fields (Section 7), prove their relation with scattering operator (Section 8), the support Theorem (Section 10) and the inverse problem (Section 11). We conclude with a technical appendix.
and the boundary (the sphere S 2n+1 ) is a compactification of the Heisenberg group The variable u := ℜ(z 1 ) is the one lying in R and we have a contact form on H n given by Φ := du + y.dx − x.dy, where ω = x + iy ∈ C n = R n + iR n . The functions give coordinates on (0, ∞) × H n ≃ H n+1 C and the Bergman metric with holomorphic curvature −1 is given in this model by The Heisenberg group H n is a Lie group with the law the origin is 0 and the inverse (u, ω) −1 = (−u, −ω). A basis for the Lie algebra h of H n is given by the left invariant vector fields The map (u, ω) ∈ R × C n → uT + j ℜ(ω j )X j + ℑ(ω j )Y j identifies H n with h, and the group law becomes where π ker Φ is the projection on ker Φ parallel to T . The complex hyperbolic space has a Lie group structure, this is actually a semi-direct product of the multiplicative group ((0, ∞), ×) with (H n , . Hn ). We introduce the parabolic dilation M δ (ρ 0 , u, ω) := (δρ 0 , δ 2 u, δω) on (0, ∞) × H n (here δ > 0), then the group law on H n+1 and we have for this law It is easy to check that the corresponding Lie algebra has a basis (2.5) ρ 0 ∂ ρ0 , ρ 2 0 ∂ u , ρ 0 X 1 , . . . , ρ 0 X n , ρ 0 Y 1 , . . . , ρ 0 Y n , which is orthonormal with respect to the metric g 0 . This algebra will be denoted by Φ T 0 H n+1 C , to agree with the notation used in the next sections. Observe also that these vectors and the metric g 0 are homogeneous of degree 0 under the parabolic dilation M δ . , ∞), this leads to study the modified resolvent The Schwartz kernel of R(s) has been computed by Epstein-Melrose-Mendoza [7] and admits a meromorphic continuation to C, with poles at −N 0 of finite multiplicity (contrary to what is written in [7]). By symmetry arguments, this kernel R(s; z; z ′ ) is expressed as a function of the Bergman distance of d g0 (z; z ′ ). We have where Q is defined in (2.1). Using a polar decomposition around the diagonal, the kernel R(s; z, z ′ ) is obtained as a solution of an hypergeometric ODE, exactly like in the real case, and is given by with c n constant depending on n and 2 F 1 is a hypergeometric function (see [2]), we also used the formula A change of variables shows that if an operator K has a distributional Schwartz kernel which is of the form k(r(z, z ′ )), in other words, it depends only on d g0 (z, z ′ ), then K is a convolution operator with respect to the group law on H n+1 C : where µ −1 dµdtdz is a right invariant mesure. The resolvent kernel (2.6) is of this form (s is a parameter), so the action of the operator R(s) on a function is given by (2.7).
Remark: We see that the poles at −m ∈ −N 0 have residue for some a m,k ∈ C. But clearly P m has finite rank since it is a polynomial times Q(z, z) −m Q(z ′ , z ′ ) −m . So the poles are of finite multiplicity.
3. Asymptotically complex hyperbolic manifolds 3.1. Θ metrics. We start by describing the Θ structures of Epstein-Melrose-Mendoza [7], which generalize Bergman type metrics on pseudoconvex domains, as well as quotient Γ\H n+1 C of H n+1 C by convex co-compact groups of isometries. LetX = X ∪ ∂X be a smooth 2n + 2-dimensional compact manifold with boundary ∂X and let Θ ∈ C ∞ (∂X, T * X ) be a smooth 1-form on ∂X such that if i : ∂X →X is the inclusion, then Θ 0 := i * Θ does not vanish on ∂X. According to the terminology of [34], the boundary ∂X has the sturcture of a Heisenberg manifold equipped with the subbundle ker Θ 0 .
We first recall a few definitions introduced in [7]. If ρ is a boundary defining function of ∂X, we define the Lie subalgebra V Θ of C ∞ (X, TX) by the condition where Θ ∈ C ∞ (X, T * X ) is any smooth extension of Θ. It is shown in [7] that V Θ only depends on the conformal class of Θ. Let T, N, Y 1 , . . . , Y 2n be a smooth local frame inX near a point p ∈ ∂X such that Then any V ∈ V Θ can be written near p as and (3.2) ρN, ρ 2 T, ρY 1 , . . . , ρY 2n form a basis of V Θ over C ∞ (X) near p. The Lie algebra V Θ is the set of smooth sections of a vector bundle over TX, we denote by Θ TX this bundle. Let F p be the set of vector fields vanishing at p if p ∈ X or the set of vector fields of the form (3.1) satisfying a(p) = b(p) = c i (p) = 0 if p ∈ ∂X. The fibre Θ T pX at p ∈X can be defined by Θ T pX := V Θ /F p . If p ∈ ∂X, Θ T pX is a Lie algebra, and any vector v ∈ Θ T pX can be represented as The dual bundle Θ T * X of Θ TX has for local basis near p ∈ ∂X the dual basis to (3.2) A Θ-metric is a smooth positive symmetric 2-tensor on Θ T * X g ∈ C ∞ (X, S 2 + ( Θ T * X )). We are interested in the special cases of Θ-metrics for which Epstein, Melrose and Mendoza [7] proved the meromorphic extension of the resolvent. We begin by the first assumption, which allows to find particular boundary defining functions.

3.2.
Model boundary defining functions. Let g be a Θ-metric, it thus restricts to a standard metric in the interior X. If ρ is a boundary defining function, we can define the vector field X ρ ∈ V Θ as the dual of dρ/ρ via the metric g, i.e. g(X ρ , v) = ρ −1 dρ(v) for any v ∈ Θ TX, this is a smooth non-vanishing section of Θ TX. It is clear that in X, we have which extends to a non-vanishing vector in C ∞ (X, TX) transverse to ∂X since dρ(X ρ /ρ) = |dρ/ρ| 2 g = 0 on ∂X (here ∇ means gradient). We first assume that and it is easy to check that this condition does not depend on ρ. The restriction ρ 4 g| T ∂X is conformal to the tensor Θ 2 0 , which leads to the definition of the conformal class [Θ 0 ]. Lemma 3.1. Let e 2ω0 Θ 0 ∈ [Θ 0 ] with ω 0 ∈ C ∞ (∂X), then there exists a unique, up to C ∞ 0 (X), boundary defining function ρ of ∂X such that |X ρ | g = 1/2 in a neighbourhood of ∂X and ρ 4 g| T ∂X = e 4ω0 Θ 2 0 . Proof : If x is a boundary defining function we search for a function ω ∈ C ∞ (X) such that ρ := e ω x satisfies |X ρ | 2 g = |dρ/ρ| 2 g = 1/4 near ∂X, this can be rewritten under the form This a first order non-linear PDE with smooth coefficients since |dω| 2 g = O(x 2 ), |X x | g = 1/2 at x = 0 and since X x /x is transverse to ∂X , it is easy to check that this equation is noncharacteristic. By prescribing the value ω| x=0 = ω 0 , we obtain a unique solution in a neighbourhood of ∂X.
Such a boundary defining function will be called a model boundary defining function. Let φ t be the flow of the vector field 4X ρ /ρ, we consider the diffeomorphism φ : Then ρ(φ t (y)) = t and for any Z ∈ T ∂X φ * g(∂ t , ∂ t ) = 4 t 2 , φ * g(∂ t , Z) = dt(Z) t 2 = 0. We will write t = ρ and X ρ = ρ∂ ρ for what follows and we call this diffeomorphism a product decomposition near ∂X. Note also that Θ(∂ ρ ) = 0 since ρ∂ ρ ∈ V Θ . With this product decomposition, the metric g has the form a family of metrics on T ∂X for ρ = 0 and such that h(ρ)/ρ 2 ∈ C ∞ (∂X, S 2 ( Θ TX)) depending smoothly on ρ ∈ [0, ǫ). We will say that the metric is even is h(ρ) −1 , as a metric on T * ∂X has an even Taylor expansion at ρ = 0 in the product decomposition. It is straightforward to see that this condition is invariant with respect to the choice of model boundary defining function ρ (i.e. of product decomposition), for instance from the proof of Lemma 2.1 in [16] where the PDE is replaced in our case by (3.3). Indeed, if x is a model boundary defining function and ρ = e ω x another one, ω has to satisfy (3.3), that is h(x) = 0 and the evenness of the Taylor expansion of |dω| 2 h(x) at x = 0 was all that we needed in [16]. Note that evenness at order 2k can also be defined invariantly by requiring ∂ 2j+1 ρ h −1 (0) = 0 for all j < k (see again [16] for similar definition in the real case).
In view of this discussion, we have that ρ −2 h(ρ) ∈ C ∞ ([0, ǫ) × ∂X, S 2 + ( K * 1 )) and we can write near a point p ∈ ∂X . Note also that a| ρ=0 is globally defined and can be taken to be 1 by changing the conformal representative of [Θ 0 ]. Let us denote by g p the metric on Θ T pX , in terms of (3.4) and (3.5), this is The assumptions of [7] on the metric correspond to the minimal assumptions for which g p is isometric to the complex hyperbolic metric. The second assumption made in [7] is that (H2) Θ 0 is a contact form on ∂X which means that dΘ 0 is non-degenerate on ker Θ 0 . The next hypothesis is that for the orthogonal decomposition K 1,p = K 2,p ⊕ L p for g p , the map µ : is the identity. Since L p is spanned by some Z i = j l ij ρY j + k i ρ 2 T mod F p the assumption clearly reduces to k i = 0 since ρ 2 T mod F p commutes with any elements of K 1,p , therefore L p = (ker(Θ 0 /ρ 2 )) p . Then by orthogonality of the decomposition, this means that b i (p) = 0, i.e.
The last assumption of [7] is which, using the bundle isomorphism ψ, is actually equivalent to the following where the restriction ρ 2 g| ker Θ0 is the metric on the bundle ker Θ 0 ⊂ T ∂X whose value on fiber (ker Θ 0 ) p is the limit lim t→0 t 2 (φ * g) (t,p) .

3.4.
Asymptotically complex hyperbolic manifolds. An aymptotically complex hyperbolic manifold, or ACH manifold, is a non-compact Riemannian manifold (X, g) such that there exists a smooth compact manifold with boundaryX which compactifies X, equipped with a Θ-structure, such that g is a Θ-metric satisfying assumptions (H1) to (H4).
In view of the assumptions on Θ 0 for an ACH manifold, there exists a smooth global vector field, denoted T 0 , tangent to ∂X such that Θ 0 (T 0 ) = 1 and dΘ 0 (T 0 , Y ) = 0 for every Y ∈ ker Θ 0 ; this is Reeb's vector field. With the notation of (3.7), we can define Remark: An ACH manifold in the sense of Biquard-Herzlich [5] is quite similar to our setting, the difference lies in the term g of (3.8): for them, g = O(ρ δ ) for some δ > 0 and g does not have a polyhomogeneous expansion at the boundary, whereas in our case the metric is polyhomogeneous but we can allow terms of order O(ρ −3 ) in the Θ 0 direction, for instance. 4. Heisenberg pseudo-differential operators 4.1. Θ calculus. We denote by Diff m Θ (X) the set of differential operators of order m which are locally polynomial functions (with coefficients in C ∞ (X)) of vector fields of V Θ , i.e the envelopping algebra of V Θ . The Laplacian of a Θ-metric is an operator in Diff 2 Θ (X). In [7], the authors construct a class of pseudo-differential operators Ψ * Θ (X) onX which is large enough to capture the resolvent (∆ g − z) −1 . It is defined in term of distributional kernel lifted on a parabolic blown-up version ofX ×X We define the blow-up (or stretched product)X 2 Θ following Epstein-Melrose-Mendoza [7] we refer the reader to sections 5,6,7 of [7] for more details. Let π R , π L be the right and left projections ofX ×X ontoX, and for a choice of boundary defining function ρ inX, we denote Let ∆ be the diagonal inX ×X, ∂∆ its boundary, and let S ⊂ N * (∂∆) be the line subbundle of the conormal bundle of ∂∆ in T * (X ×X) spanned by π * L Θ − π * R Θ. We denote byX 2 Θ := [X ×X; ∂∆, S] the S-parabolic blow-up ofX ×X around ∂∆. This means that we construct a larger manifold thanX ×X by replacing ∂∆ by the S-parabolically spherical normal interior pointing bundle inX ×X: where SN S,+ means the bundle over ∂∆ such that each fiber is the quotient of the interior pointing normal bundle (without the 0 section) N + ∂∆\ {∂∆} of ∂∆ ⊂X ×X by the equivalence law after decomposing N + ∂∆ = S 0 ⊕ S ′ for S 0 annihilator of S and S ′ a complementary space. The complementary space has to be taken a certain way so that the total space is invariant under the dilation M δ of (4.1), but this can be done (see [7,Sec. 6]). We call SN S,+ ∂∆ the front face and denote it ff, we define the blow-down map β :X 2 Θ →X ×X to be the identity outside ff and the projection on the base on ff (recall ff is a bundle). We can put a topology and smooth structure onX 2 Θ by taking those ofX ×X far from ∂∆ via β and near ff, one can use a choice of normal fibration where U (resp. V ) is a small neighbourhood of the submanifold ∂∆ in N + ∂∆ viewed as zero section (resp.X ×X), and transport the topology and smooth structure of [N + ∂∆; ∂∆, S] through f after we have defined them on this space as follows: homogeneous functions of nonnegative integer order in N + ∂∆\∂∆ with respect to dilation M δ lift under β * f −1 * to well defined functions on β −1 (V ), the topology is the weakest such that these are continuous functions, the smooth structure is that generated by these homogeneous functions. This is proved in [7] to be invariant with respect to choices of f, S ′ as a smooth manifold with corners. The manifoldX 2 Θ has three boundary hypersurfaces, the front face ff, the right boundary rb and the left boundary lb (cl means closure) we denote by ρ ff , ρ lb , ρ rb some corresponding boundary defining function, for instance one can take ρ lb := β * (ρ)/ρ ff , ρ rb := β * (ρ ′ )/ρ ff if ρ ff is a chosen boundary defining function for ff. If one has a coordinate system (ρ, u, z) with (u, z) local coordinates on ∂X such that Θ 0 = du + y.dx − x.dy is in Darboux form and z = (x, y) ∈ R n × R n , denoting the same with prime for left factor onX ×X, then Next we define the full class of Θ-pseudodifferential operator Ψ * Θ (X) as in Section 12 of [7].
and vanishing at all order at all other boundary faces than ff. This is well-defined since ∆ ι meets the boundary ofX 2 Θ only at ff (in the interior of ff). We also define Ψ E,E ′ (X) to be the set of

The Normal operator.
If we look at the model case of H n+1 C , it is clear that the Lie algebra Φ T 0X (the boundary point is 0 ∈ H n ) is canonically isomorphic to the Lie algebra of the group H n+1 C with the law (2.4), which is the reason of the notation for this last Lie algebra in Section 1.
For p ∈ ∂X, consider X p := {dρ ≥ 0} ⊂ T pX the inward pointing part of T pX , then Θ T pX acts on C ∞ (X p ) as follows. On a neighbourhood U p ⊂X of p, we first define the dilation which is a well defined smooth vector field on X p , homogeneous of degree 0 with respect to M δ and depending only on V (p). Then the map realizes Θ T pX as a Lie algebra of smooth vector fields on X p , left invariant for the group action on X p generated by Θ T pX , this action on X p being The value of the left-invariant vectors on X p at e p identifies T ep X p with the Lie algebra Θ T pX of X p as usual. Using with assumption (H4 ′ ), there exists an orthonormal basis (X ′ 1 , . . . , is clearly identified as a group with H n+1 C and the group action on X p is isomorphic to the law using notations of Section 2. This gives for each p ∈ ∂X an isometric Lie algebra isomorphism acts by convolution as a left invariant differential operator on X p by (4.2). The normal operator can be considered as an operator on H n+1 The metric g p on Θ T pX ≃ T ep X p induces a left-invariant metric on X p and it is easy to check that this is the metric obtained by lim δ→0 M * δ g. A computation leads to where ∆ b (p) is the horizontal sublaplacian on ∂X p equipped with the Reeb field T 0 (p) and the contact distribution (ker Θ 0 ) p . Using conjugation with A p , this is the complex hyperbolic Laplacian on H n+1 C .
If we look at the action of vector fields in V Θ lifted through β * π * L onX 2 Θ , this action restricts smoothly on each fiber of the front face ff p to be the left invariant action of Θ T pX on X p with an identification between ff p and X p . First setX p := {dρ ≥ 0} ⊂ T pX , then . Setting |z| 2 := dΘ 0 (z, Jz), the identification of ff p with the quarter of sphere ρ ff := ((s 2 + s ′2 + |z| 2 /2) 2 + t 2 ) 1 4 = 1 gives the smooth structure of manifold with corners and the functions ρ lb := s/ρ ff , ρ rb := s ′ /ρ ff are boundary defining functions for left and right faces. If we fix s ′ = 1, we have an identification between X p and the interior of ff p given by "stereographic projection" with pole ρ rb = 1 for the parabolic dilation, these are projective coordinates of ff p . We then have in this model of ff p A vector in V p ∈ Θ T pX , value at p of a vector field V ∈ V Θ , acts as a homogeneous (for M δ ) left invariant vector field on X p , thus on ff p , by the dilated limit (4.2). This left invariant field acting on X p is N p (V ) but this is also the restriction of β * π * R V to the interior of ff p . It is proved in [7] that a vector field V ∈ V Θ actually lifts smoothly toX 2 Θ under β * π * R , tangent to any boundary hypersurface.

4.3.
Heisenberg pseudo-differential operators on Heisenberg manifolds. The space of Heisenberg pseudo-differential operators is defined in [6] (see also the monograph of Ponge [34]) but the approach we will use is that of Epstein-Melrose-Mendoza [8] since it is more naturally adapted to our case. In [8] they define the class Ψ m Θ0 (∂X) of classical Heisenberg pseudodifferential operators of order m by the structure of their Schwartz kernel.
Let M be a compact manifold of dimension n equipped with a one-dimensional subbundle L = RΘ 0 ⊂ T * M . one defines LT * M to be the parabolic compactification of T * M with respect to L, this is a smooth manifold with boundary and let q be a boundary defining function. A Heisenberg classical symbol of order m is a function in q −m C ∞ ( LT * M ). If ∆ is the diagonal of M × M , we denote by N ∆ the normal bundle of ∆ in M × M and there is a canonical Fourier transform of compactly supported distributions in each fiber and a neihgbouhood of the diagonal in M × M can be identified to a neighbourhood of the zero section of N ∆. Note also that I : where U is a neighbourhood of the zero section, we define by (m, τ ) = W −1 and ψ : its Schwartz kernel k is smooth outside the diagonal and can be written near the diagonal under the form The principal symbol is the class [8] proves that both definitions do not depend on the choice of h.
We can now define S to be the span of π * Let ρ ff(Y ) , ρ ff be boundary defining function for respectively ff ⊂ Y and ff ⊂ M 2 Θ0 , then for any s ∈ C, functions in ρ s ff C ∞ (M 2 Θ0 ) defined near ff lift under ψ to functions in ρ s ff(Y ) C ∞ (Y ) and the spaces , ρ ff and are in one to one correspondance (these are conormal densities to ff) which does not depend on the choice of Riemann-Weyl fibration in view of (4.6). Now if K ∈ ρ s ff(Y ) C ∞ (Y ) for s ∈ C\(−n−1−N), then k := β Y * K has an expansion near 0 in each fibre in k ∼ i k i with k i homogeneous of order s+i with respect to the parabolic dilation in the fiber and can thus be written like (4.5) for σ ∈ q −s−n−1 C ∞ ( LT * M ) the inverse Fourier transform of β Y * K in the fibres (Fourier transform keeps parabolic homogeneity). Choosing ρ ff(Y ) homogeneous of order 1 in the fibre, a term K ∈ ρ s ff(Y ) C ∞ (Y )/ρ s+1 ff(Y ) C ∞ (Y ) can be uniquely represented by a homogeneous distribution of order s with respect to the parabolic dilation in the fibre by taking  .7) is also equal to ρ s ff(Y ) (ρ −s ff K)| ff (understood in normal coordinates ff × [0, ∞) ρ ff(Y ) ) if K := ψ * K and ρ ff := ψ * ρ ff(Y ) , then the principal symbol of the operator is Notice that the Fourier transform on tangent and cotangent bundle needs to be given a density to be well defined, this is for instance the case if one has a metric on M . Note also that the definitions of Ψ * Θ0 (M ) being local, the metric h for the Riemann-Weyl fibration can be locally taken to be flat. Finally we remark that the contact form Θ 0 and the almost-complex structure J on its kernel induce a metric on M by (3.9 and a Reeb vector field T 0 fixing a representative for the parabolic blow-ups.

The Laplacian, its resolvent and the Poisson operator
We use the form (3.7) of the metric in the product decomposition [0, ǫ) ρ × ∂X of ∂X and the dilated metric k(ρ) defined in (3.10). We first write the Laplacian near infinity using this model form of the metric where ∆ b is the horizontal sub-laplacian on (∂X, Θ 0 , h 0 ).
Remark: If the metric is even at order 2k as defined in Subsection 3.2, then ∇ h(ρ) f is even in ρ at order 2k as well as the divergence div h(ρ) X = div k(ρ) X for any X ∈ T ∂X. Finally we have | det k(ρ)| = ρ 2 | det h(ρ)| which is then even in ρ at order 2k + 2, this clearly proves that (5.1) is (in local coordinates near ∂X) an operator of the form Q(ρ, y; ρ∂ρ, ∂ y ) where Q(x, y; D) is a family of polynomial in D with an even Taylor expansion at x = 0 at order 2k. ). Let us set for k ∈ N 0 then Epstein, Melrose and Mendoza [7] proved the Theorem 5.1. On an asymptotically complex hyperbolic manifold whose metric is even at order 2k, the modified resolvent of the Laplacian as a finite-meromorphic family of operators in the Θ-calculus: for λ not a pole, we have The poles are called resonances and we denote by R the set of resonances.
Remark: in [7], the property of metrics even at order 2k is only discussed for k = ∞, but this can be checked in general from their construction (see also [15] for a similar analysis in the real asymptotically hyperbolic case).
We define the multiplicity of a resonance λ 0 by which in turn is equivalent to the rank of the polar part of R(λ) at λ 0 .

The Poisson
Operator. Using the form (5.1) of the Laplacian and the structure of the resolvent, we are able to define the Poisson operator: Proposition 5.2. Suppose the metric is even at order 2k and let λ satisfy ℜ(λ) ≥ n+1 2 , λ / ∈ ( n+1 2 + k + 1 4 N) and λ(n + 1 − λ) / ∈ σ pp (∆ g ). Then for each f 0 ∈ C ∞ (∂X) there exists a unique function F (λ, f 0 ), linear in f 0 , solving the problem having an even Taylor expansion at ρ = 0 at order 2k in a product decomposition near ∂X. The Poisson operator defined by Proof : The proof is similar to the proof of Graham-Zworski [14] in the asymptotically hyperbolic case. We first deduce from (5.2) the indicial equation for f 0 ∈ C ∞ (M ) and j ∈ N 0 Then we can construct a meromorphic function Φ(λ) ∈ ρ 2(n+1−λ) C ∞ (X) with at most first order poles at n+1 We thus define which satisfies the required conditions. However we have to prove uniqueness of an operator satisfying (5.4), this is obtained easily using the indicial equation when ℜ(λ) > n+1 2 and using a Green formula at infinity if ℜ(λ) = n+1 2 (see [14] for details) : Using dvol h(ρ) = ρ −1 dvol h0 + O(1) and the asymptotics of u i gives which implies the Lemma by taking the limit as ǫ → 0.
Then the first part of the proposition is deduced classically from this Lemma and formula (5.6) extends clearly meromorphically to C \ (−N 0 ∪ P k ∪ (n + 1 − P 0 )) using Theorem 5.1, which proves the meromorphic extension of P (λ). The points (n + 1/2)/2 + j/2 are not singularities if j < 2k since by evenness of the metric at order 2k and the remark following (5.2), the function ρ −2(n+1−λ) Φ(λ, f 0 ) can be constructed to have even Taylor expansion at order 2k at ρ = 0, see [16] where it is done in greater details for the real asymptotically hyperbolic case.
¿From the proof of proposition 3.5 in [14], we also deduce that P (λ) : C ∞ (∂X) → C ∞ (X), is analytic at each , and P(λ k )f 0 can be defined as the unique solution of (5.8) and we are done.
Actually the Poisson operator was constructed in Section 15 of [7] by its Schwartz kernel as a weighted restriction of the resolvent kernel at the boundary face lb(X 2 Θ ). The Schwartz kernel of P (λ) is for some smooth positive function c(λ) ∈ C ∞ (∂X ×X). Actually one can use Lemma 5.3 as in [14,Prop. 3.9] to check the same result, which gives the value c(λ) = 2(2λ − n − 1). Using Theorem 5.1, we deduce that A property that we will use later is the relation between spectral measure and Poisson operator, the proof of which is an easy appalication of Green formula, and essentially similar to the real asymptotically hyperbolic case (see [33,19,16]): and λ ∈ (R ∪ n+1 2 + 1 4 N) by if the metric is even at order 2k by Proposition (5.2), we deduce that S(λ) continues meromor- The rank at a pole λ = s is finite if s ∈ −N 0 ∪ P k ∪ ( n+1 2 + 1 4 N) and in {ℜ(λ) > n+1 2 }, the same arguments than [14,Prop. 3.6] show that S(λ) has at most first order poles with residue with Π s a finite rank operator with Schwartz kernel and p k is a differential operator on ∂X defined by It is straightforward to see from the construction of Φ(λ) and (5.2) that p j (λ) is a differential operator on ∂X, actually a Heisenberg differential operator, the Heisenberg principal symbol σ pr (p 2k (λ)) of p 2k (λ) can be obtained by the induction formula Since [∆ b , T 0 ] ∈ Ψ 3 Θ0 (∂X) and we are interested only in the principal symbol, it suffices to deal as if ∆ b , T 0 were complex numbers. Observing that p 2k (λ) is holomorphic at (n + 2 + k + N 0 )/2, one can consider . But this last term q 2k (λ 2k ) is computed by Graham [11,Sec. 1] and is equal to We have thus proved Proposition 6.1. The scattering operator S(λ) is a finite meromorphic family of operators on C ∞ (∂X) in C \ (−N 0 ∪ P k ∪ (n + 1 − P k ) ∪ ( n+1 2 + 1 2 N)) and meromorphic in C \ (−N 0 ∪ P k ). In {ℜ(λ) > n+1 2 }, S(λ) has at most first order poles with residue with Π s is the finite rank operator defined in (6.2) and p k is the Heisenberg differential operator on ∂X defined in (6.3) and such that Note that S(λ) depends on the boundary defining function ρ and a different choiceρ = e ω ρ (i.e. a different conformal representative contact form e 2ω Θ 0 with ω 0 = ω| ∂X ) gives a scattering operatorŜ(λ) and residuesp k (when λ(n + 1 − λ) / ∈ σ pp(∆g) ) conformally related to S(λ), p k The definition of S(λ) and Lemma 5.3 directly imply (see [14,Sec. 3]) that where the adjoint is taken with respect to measure dvol h0 on ∂X and the left identity extends meromorphically in C \ ( 1 2 Z ∪ P k ∪ (n + 1 − P k )).
We have a direct relation between resolvent and scattering operator: For λ such that S(λ) exists, its Schwartz kernel is The right-hand side is a meromorphic family of distributions on ∂X in C \ −N 0 ∪ P k with first order poles at (n + 1 + N)/2 ∪ (n + 1 − P k ) if the metric is even at order 2k.
Proof : For ℜ(λ) < 0, we clearly have that Using (5.9) and (5.10) for ℜ(λ) < 0 gives thus by taking the limit we deduce that the kernel of S(λ) is β ∂ * (β ′ * (ρ −2λ P (λ))| lb ) for ℜ(λ) < 0, this gives the result in a half plane by using again (5.9). To extend it in the complex plane, we need to check that a distribution of the form ρ −4λ ff F λ with F λ ∈ C ∞ (∂X × Θ ∂X) meromorphic extends meromorphically to C. But this is a classical fact which can be proved as in the normal blow-up case (classical pseudo-differential operators) by Taylor expansion at the front face of the parabolic blow-up, we lead the reader to the Appendix where we describe it. This leads to first order poles at n+1 2 + 1 4 N whose residue are differential operators.
We also remark that the method of [16] can be used to prove possible essential singularity for the scattering operator (thus for the resolvent) at n−1/2−N0 2 : Proposition 6.7. For each k ∈ N 0 , there are examples of ACH manifolds for which n + 1 − λ 1+2k = n−1/2−k 2 is an essential singularity of S(λ).
Proof : Let us consider for simplicity k = 0 and λ 1 (n + 1 − λ 1 ) / ∈ σ pp (∆ g ), the other cases are not more complicated (see [16]), note also that it suffices to consider S(λ) instead of S(λ). A quick analysis using (6.2), Subsection 5.2 and (5.1) shows that the residue of S(λ) at n + 1 − λ 1 is (modulo finite rank smoothing operator) Then this operator can be injective on L 2 (∂X) and any Sobolev space, it suffices actually to choose the metric so that this is the Identity (which is easy to do). Then S(λ) would have a pole of order 1 whose residue P 1 is injective on L 2 (∂X). Suppose that S(λ) is meromorphic at n + 1 − λ 1 , i.e. has a Laurent expansion at λ = n + 1 − λ 1 for some S i . Then since S(n + 1 − λ) S(λ) = S(n + 1 − λ) S(λ) = Id and we deduce by injectivity of P 1 that S i = 0 for i = −p, . . . , 0. But the term S 0 can also be obtained by a contour integral (C is a circle around n + 1 − λ 1 of radius ǫ) thus this is an operator in Ψ 0 Θ0 (∂X) with principal symbol given by • denotes the composition for Heisenberg principal symbols (see [8,34]). But using Theorem 6.3 which is not 0 since the middle one is not 0, thus a contradiction.

The Radiation Fields
In this section we study the scattering theory developed in the previous sections from a dynamical view point as in the Lax-Phillips theory. We define the forward and backward radiation fields, show that they give unitary translation representations of the wave group which can be used to define the scattering matrix (6.1) in terms of the wave equation. The analogue study of the radiation fields for asymptotically hyperbolic manifolds was carried out in [35] .
We start by considering the Cauchy problem for the wave equation It is well known that u ∈ C ∞ (R + × X). We are interested in understanding the behavior of u at infinity along some bicharacteristics. Proof. This is very similar to the proof of Theorem 2.1 of [35]. Without loss of generality, we will assume that f 1 = 0. Let Using (5.1) and setting s = t + 2 log ρ, we have with A := ∂ ρ (log |k|). Setting v(ρ, s, z ′ ) := ρ −n−1 u(s − 2 log ρ, ρ, z), (7.1) becomes ¿From the standard regularity theory for the wave equation, we know that v is smooth for ρ > 0. The main point of the theorem is to show that v is C ∞ up to ρ = 0. There are several ways of proving this, and we will choose a method that will establish energy estimates which will be needed in section 9.1. For that purpose we make the following change of variables s = 4 log ν, ρ = µν.
V (µ, ν, z) = v(µν, 4 log ν, z). In coordinates (7.5) equation (7.4) is given by ¿From Darboux's theorem for contact forms, see for example Theorem 5.5 of [20], we know that for small enough U there exists local coordinates z = (u, x, y), x = (x 1 , ..., x n ), y = (y 1 , ..., y n ), u ∈ R, near any point z 0 ∈ ∂X such that Let X j and Y j be the vector fields defined by and Z j = X j (resp. Z j = Y j−n ) for j ≤ n (resp. j > n) which forms a basis of ker Θ 0 . We denote by T 0 = T 0 (ρ) the Reeb vector field of (Θ 0 , k(ρ)). Note that ∆ h(µν) is a differential operator on ∂X with derivatives in Z j , (µν) 2 ∂ u , since ρ 2 ∆ h(ρ) is in Diff Θ0 (X). We want to differentiate equation (7.6) with respect to the vector fields X j and Y j and ∂ u . We first recall that Therefore we find that u , Z j ] = 0. Differentiating (7.6) and letting U 1 be the vector whose transpose is , we deduce that there exists a (2n + 2) × (2n + 2) matrix of first order differential operator B and functions C, D, E of (µ, ν, z) with smooth coefficients such that . . . , Z 2n , µν∂ u ) + Cν∂ ν + Dµ∂ µ + E U p = 0, (7.9) and p = 1. In general we find that if U p are the vectors (with size N = N (p)) consisting of all derivatives of V of order p > 1 with respect to the vector fields X j , Y j and ∂ ∂u , then we get (7.9) with B, C, D, E some N × N matrices with smooth coefficients.
Notice that the operators in (7.9) are smooth up to {µ = 0} ∪ {ν = 0} and therefore can be extended smoothly to the neighborhood {|µ| < √ µ 0 } ∪ {|ν| < √ ν 0 }. We then prove Let V (µ, ν, y) be a N × 1 vector, smooth in µ > 0, ν > 0, satisfying the system (7.9) with ( 1 4 ∂ µ ∂ ν + µν∆ h )Id + µνB(Z 1 , . . . , Z 2n , µν∂ u ) + Cν∂ ν + Dµ∂ µ + E V = 0, and V (µ, µ, y) = f 1 (µ, y), (∂ µ V )(µ, µ, y) = f 2 (µ, y), µ > 0, (7.10) where B, C, D, E are N × N matrices with smooth coefficients in {|µ| < T, |ν| < T, z ∈ ∂X} with B having for coefficients some first order differential operators in (Z j ) j , µν∂ u . If the data f 1 and f 2 are such that then for small T, Proof. The proof is essentially the same as Lemma 4.1 of [35], we just emphasize the differences and we refer the reader to that paper for details. We denote Z = (Z 1 , . . . , Z 2n ) the set of vector fields and we begin by multiplying the system (7.10) by µ ∂V ∂µ − ν ∂V ∂ν , this gives where Q is a quadratic form with smooth coefficients and div k is the divergence with respect to metric k(ρ) on ∂X, which is easily seen to be also the divergence for h(ρ). If (Z i ) i is a local orthonormal basis of ker Θ 0 for h(ρ) (thus for k(ρ)) and if T 0 = T 0 (ρ) is again the Reeb field of (Θ 0 , k(ρ)), then (7.14) is a smooth function up to ρ = 0. Let Ω δ , Ω + δ and Ω a,b , be the domains defined by Integrating in Ω α,T , using the compactness of ∂X, the divergence theorem, and that the first term in brackets in (7.13) is positive, we find that for T small, there is a constant C such that Doing the same in the region Ω δ,β gives Next we integrate (7.15) in α ∈ [δ, T ] and (7.16) in β ∈ [δ, T ] and add the results to get All terms of Q, except for those in |V | 2 , are trivially bounded by the terms in the first integral, and thus they can be absorbed in first integral by choosing T small enough. To bound the integral of |V | 2 in Ω + δ it suffices to copy word by word the method in Lemma 4.1 of [35] and choose T small. This gives Now taking T small and δ → 0 in (7.17), (7.18), this gives (7.12) in the region above the diagonal after using the initial condition. Observing that the operator in (7.10) and the estimate (7.12) remain the same after switching µ and ν, this estimate also holds in the region below the diagonal. This ends the proof of the Lemma. Now we apply the Lemma 7.2 to prove Theorem 7.1. The goal is to prove that for f 2 smooth and compactly supported, the solution to (7.1) is smooth up to {ρ = 0}. We know by the finite speed of propagation that v is supported in s ≥ C for some C ∈ R. The change of variables (7.5) is smooth in this region and we work in coordinates (µ, ν). We will show that V (µ, ν, z) is smooth up to {µ = 0} ∪ {ν = 0}.
We first apply Lemma 7.2 to the special case of equation (7.6), noticing that in this case V is a function instead of a vector. The data on {µ = ν} is given by (7.6), so the last integral on the right hand side of (7.12) is equal to Thus the last integral on the right hand side of (7.12) is bounded by the square of the norm of f 2 in L 2 (X, d vol g ) and Now we want to obtain such energy estimates for the derivatives of V . We begin by analyzing the derivatives of V with respect to the vector fields X j , Y j and ∂ ∂u and we differentiate equation (7.6) with respect to them. We get a system of equations given by (7.9). So we conclude in particular that (7.20) Using the commutators (7.8), the fact that (Z j ) j , ∂ u form a local basis of T ∂X near z 0 , and that the initial data is smooth with compact support, (7.19) and (7.20) guarantee that V and ∂ α z V ∈ L 2 (Ω), |α| = 1.
Using this argument repeatedly, we conclude that V is smooth in the variables z ∈ ∂X with values in L 2 (Ω) with respect to (µ, ν), that is (7.21) and wealso have As in [35,Th. 2.1], we can use the equation that W := |k| 1 4 V satisfies to obtain information about the derivatives of V with respect to µ and ν. The proof is not different of that of Theorem 2.1 in [35], we let the reader refer to it after noticing that Eq. (4.23) there becomes in our case 1 4 ∂ ν ∂ µ + µν∆ h(µν) + µνF (µν) W = 0, (7.23) where W := |k| 1 4 V and F = [∆ h(µν) , |k| 1 4 ] is a smooth differential operator of first order which, in local coordinates, has only derivatives on the vector fields Z j , ∂ u as above. This concludes the proof of Theorem 7.1.
With v defined in Theorem 7.1, the map will be called the forward radiation field. We remark that this definition depends on the choice of ρ, which in turn depends on the choice of the conformal representative in [Θ 0 ]. The backward radiation field can be defined in a similar fashion by considering the behavior of the solutions to (7.1) as t → −∞. Let u be the solution to (7.1), one can show that v − (ρ, s, z ′ ) = ρ −n−1 u(s + 2 log ρ, ρ, z) ∈ C ∞ ([0, ǫ) × R × ∂X), and define

The Radiation Fields And The Scattering Matrix
This section can be compared to section 5 of [35]: we describe relation between the radiation fields, the Poisson operator and the scattering operator. As mentioned in subsection 5.1, the spectrum of the Laplacian σ(∆) was studied by Epstein and Melrose and Mendoza [7], and consists of a finite pure point spectrum σ pp (∆ g ), which is the set of L 2 (X) eigenvalues, and an absolutely continuous spectrum σ ac (∆ g ) satisfying This gives a decomposition where L 2 pp (X) is the finite dimensional space spanned by the eigenfunctions and L 2 ac (X) is the space of absolute continuity, which is the orthogonal complement of L 2 pp (X). The choice of the spectral parameter which adapts well to the wave equation is (n+1) 2 4 + σ 2 , which corresponds to λ = n+1 2 +iσ. If ℑσ = 0, then (n+1) 2 The eigenvalues of ∆ g are of finite multiplicity and are described by points on the line ℜσ = 0 and − n+1 2 < ℑσ < 0. The spectral theorem gives that the resolvent and it was shown in [7] that it can be meromorphically continued to C \ i 2 ( 1 2 + N 0 ) as an operator acting on appropriate spaces. Let , and df 1 ∈ L 2 (X)}, and for w 0 , w 1 ∈ C ∞ 0 (X), we define the energy of w = (w 0 , w 1 ) by where |dw 0 | g denotes the length of the co-vector with respect to the metric induced by g on T * X. Note that ||w|| 2 E is only positive when w 0 ∈ L 2 ac (X), and only then it defines a norm. Let Π ac : L 2 (X) −→ L 2 ac (X) be the corresponding projector and E ac (X) = Π ac (H E (X)) = The range of the projector Π ac acting on H E (X), E ac (X) is a Hilbert space equipped with the norm (8.3).
Integration by parts shows that if u(t, z) satisfies (7.4), then The map W (t) defined by induces a strongly continuous group of unitary operators that commutes with Π ac By changing t ↔ t − τ, one has that R ± satisfy R ± • (W (τ )f ) (s, y) = R ± f (s + τ, y), τ ∈ R. (8.5) So Theorem 7.1 shows that R ± are "twisted" translation representations of the group W (t) in the sense of Lax and Phillips. That is, if one sets R ± (f )(s, y) = R ± f (−s, y), then R ± (W (τ )) = T τ R ± , (8.6) where T τ denotes right translation by τ in the s variable. Moreover we will prove Theorem 8.1. The maps R ± induce isometric isomorphisms where L 2 (R × ∂X) is defined with respect to h 0 fixed in (3.9) by the choice of boundary defining function ρ.
The proof of Theorem 8.1 will be divided into two lemmas. The first one is Moreover, the maps R ± extend from (C ∞ 0 (X) × C ∞ 0 (X)) ∩ E ac (X) by continuity to maps R ± : E ac (X) −→ L 2 (R × ∂X).
The proof of this is a straightforward modification of the proof of Proposition 6.4 of [35] together with an application of Corollary 6.3 of [35].
Next we have to show that operators R ± are unitary and onto. To do that we work on the Fourier transform side and we resort to two observations. The first just follows from taking the partial Fourier transform in the variable s. Lemma 8.3. Let R ± be the radiation fields defined in (7.24) and (7.25). Then where F denotes the partial Fourier transform in the variable s, and t P is the transpose of the Poisson operator defined in Proposition 5.2.
Proof. To see that one needs to observe that This proof follows that of equation (6.4) of [35]. But these operators are the transpose of the forward and backward Poisson operators defined in Proposition 5.4. This proves the lemma.
Finally we recall Lemma 5.4 with this spectral parameter This and Stone's theorem show that the transpose of Poisson operator , σ > 0 extends by continuity to t P n + 1 2 + iσ : L 2 ac (X) −→ L 2 (R ± ; L 2 (∂X), σ 2 dσ) satisfying t P n + 1 2 ± iσ ∆ g = (n + 1) 2 4 + σ 2 t P n + 1 2 ± iσ (8.9) and there is c ∈ R such that c t P gives a surjective isometry between these spaces. Lemma 8.2 is then a straightforward application of these results. See the proof Theorem 5.1 in [35]. We deduce from Theorem 8.1 that the dynamical scattering operator is unitary in L 2 (∂X × R), and in view of (8.6), it commutes with translations. This implies that the Schwartz kernel S(s, y, s ′ , y ′ ) of S satisfies S(s, y, s ′ , y ′ ) = S (s − s ′ , y, y ′ ) , and thus is a convolution operator. The scattering matrix is defined by conjugating S with the partial Fourier transform in the s variable A is a unitary operator in L 2 (R × ∂X) and, since S acts as a convolution in the variable s, A is a multiplication in the variable λ, i.e. it satisfies We also have Theorem 8.4. With ρ given by (3.8) and σ = 0, the Schwartz kernel of the map S( n+1 2 + iσ) defined by (6.1), is equal to −A(σ; z, z ′ ), defined in (8.11).
Proof. This follows from (8.7) and (8.9) and the fact that is the unitary operator that intertwines P n+1 2 + iσ and P n+1 2 − iσ , in the sense that We end this section with a Lemma which will be useful later.
Definition 8.6. With the notation of Lemma 8.5 we will denote

The Support Theorem
The goal of this section is to prove Theorem 9.1. Let f ∈ L 2 ac (X) and suppose that R + (0, f )(s, w) = 0 for s < s 0 ≪ 0. Then f is supported in ρ > e s0/2 .
Proof. Suppose the initial data is supported in ρ ≥ ρ 0 . By finite speed of propagation the solution to (7.4) is equal to zero in the region We have shown that the solution to Equation 7.4 with compactly supported initial data is smooth up to ρ = 0. By assumption we know that ∂v ∂s (0, s, z) = 0 if s ≤ s 0 . One can then show that this implies that v(ρ, s, y) vanishes to infinite order at ρ = 0 provided s ≤ s 0 . Therefore we deduce from (9.2) that the solution v to (7.4) vanishes to infinite order at the corner {ρ = 0} ∪ {s = 2 log ρ 0 }. In particular we conclude that v vanishes to infinite order at ρ + s = 2 log ρ 0 and is supported in the wedge {ρ ≥ 0} ∪ {s ≥ log ρ 0 }. Now we want to use a uniqueness theorem that would allow us to show that v = 0 in a neighborhood of the set {s = log ρ 0 , ρ = 0} × ∂X. The result we need is a particular case of a Theorem due to Alinhac, Theorem 1.1.1 of [1]. We will verify that the hypotheses of this theorem are satisfied in this case.
The principal symbol of the operator Q in (7.4) is where (τ, σ, ξ, η, ν) is the dual to (ρ, s, x, y, u), and p = p(ρ, x, y, u, ξ, η, ν), is the symbol of the Laplacian ∆ h(ρ) . It follows from (5.2) and (7.7) that one can choose local coordinates (x, y, u) near a point on z 0 = (x 0 , y 0 , u 0 ) ∈ ∂X such that where p 1 does not depend on ν, and h ij is a positive definite matrix. It is important to observe that p is elliptic if ρ = 0.
The Hamilton vector field of q is where H p is the Hamilton vector field of p with respect to the tangential variables, and it does not have derivatives in s and ρ, σ or τ.
Let φ = s + ρ − 2 log ρ 0 , and let {f, g} = H f g be the Poisson bracket between f and g, then We also have Then, for ρ small, for small ρ. We want to analyze the set Λ = {q = H q φ = H 2 q φ = 0} ⊂ T * (R ×X) near ρ = 0. Combining the equations defining Λ, and the ellipticity of p for ρ > 0, one concludes that, for small ρ, In local coordinates near a point z 0 ∈ ∂X where (9.3) is valid and therefore it is a smooth submanifold. It also follows from (9.3) that dp = 0 on Λ.

Proof.
To extract information about the behavior of v as ρ → 0 and s → −∞ we need to work with the compactified equation (7.6). In fact we will work with (7.23) which does not have first order derivatives in µ and ν. This is very similar to the proof of Lemma 7.2 of [35].
By (9.1), we can assume that the initial data f ∈ C ∞ (X). Therefore, by standard regularity for solutions to the wave equation, the solution W to (7.23) is smooth in the region {µ > 0, ν > 0}. First we will show that, as a distribution, W vanishes to infinite order at {µ = 0} ∪ {ν = 0}. If we knew that W were smooth up to {µ = 0} ∪ {ν = 0}, then Theorem 1.1.2 of [1] would guarantee that W = 0 near {µ = ν = 0} and this would imply in particular that the initial data is supported away from ρ = 0. However, we do not know this in principle, and we will use the fact that R + (0, f ) = 0 in s < s 0 to show that this is true in a neighborhood of {µ = ν = 0}.
The first step is to show that W which is defined in (0, T ) µ × (0, T ) ν × ∂X, has an extension W ∈ H 2k ((−T, T ) × (−T, T ); H −2k (∂X)), for all k ∈ N, which satisfies (7.23). The proof of this fact is identical to the proof of the analogous case done in [35], the only ingredient needed here are the energy estimates from Lemma 7.2.
The following are consequences of the support theorem which is the key step in the reconstruction of the manifold from the scattering matrix.
Lemma 9.4. Let R ± be the forward an backward radiation fields defined above. Let a > 0 be small. Then The following lemma will be very important in the reconstruction of the manifold Since Π f (a)R + (0, f ) is supported in s ≥ 2 log a, the support theorem guarantees that there exists f a ∈ L 2 ac (X) supported in ρ ≥ a such that Π f (a)R + (0, f ) = R + (0, f a ).
Then f a − f, g = 0 g ∈ L 2 ac (X), supported in ρ ≥ a. (9.18) If (X, g) has no eigenvalues, then L 2 ac (X) = L 2 (X). In this case we conclude that f a = χ a f.
When there are eigenfunctions we can only deduce that in ρ ≥ a, f a − f = j c j (a, f )φ j . In other words, f a = χ a (f + n j=1 c j (a, f )φ j ).
Since f a ∈ L 2 ac (X), f a , φ j = 0, and so we have c j (a, f ) χ a φ j , φ k = 0. (9.19) Since {φ j } N j=1 is orthonormal, the system (9.19) can be solved for a = 0. Therefore there exists δ > 0 such that it can be solved for a ∈ [0, δ]. It is clear that the constants c j depend linearly on f. Therefore the operator f −→ N j=1 c j (a, f )φ j is linear and of finite rank. This is the operator T (a)f.

The Inverse Problem
In this section we will prove Theorem 1.4. The proof is based on the boundary control theory of Belishev, and the key point is the support theorem proved in section 9.
First note that the result in Theorem 6.3 about the principal symbol of S(λ) implies that the pseudo-hermitian structure on ∂X are the same for (X, g 1 ) and (X, g 2 ) if S 1 (λ) = S 2 (λ) and S 1 (λ), S 2 (λ) defined from the same conformal representative of [Θ 0 ]. Thus the dilation M ρ defined in Subsection 3.4 are the same for both metrics, as well as the metrics h 0 = Θ 2 0 +dΘ 0 (., J.). We begin with the following lemma: for some smooth family of metrics h j (ρ) on M in ρ ∈ (0, ǫ) and such that M * ρ (ρ −2 h j (ρ)) are smooth metrics on M depending smoothly on ρ ∈ [0, ǫ).
Proof. This is just following Subsection 3.2.
Proof. Let F ∈ M b be such that ∂ 2k s F ∈ L 2 (R × ∂X), for all k ∈ N. Let a ∈ R, a ≪ 0. Let F = R + (0, f ). Then, f is C ∞ , and by Lemma 9.5 The left hand side of each equation of (10.1) is determined by the scattering matrix. Therefore so is the right hand side. The initial data is singular at ρ = a, and this singularity will travel to the boundary as t → ∞. We want to find the singularity of R + (0, χ a f ) or R(0, χ a (I + T (a))f ) at s = 2 log a.
where T is an operator of finite rank. The difference between the term on the left hand side of this equation and the first term of the right hand side is a differential operator, while the second term on the right hand side is an operator of finite rank. Therefore, the differential operators must be equal. So T = 0, and we argue as above to conclude that |k 1 | = |k 2 | and ∆ g1 = ∆ g2 . To extend the diffeomorphism Ψ to the manifoldsX 1 andX 2 , and prove Theorem 1.4 we proceed exactly as section 8 of [35]. The only new ingredient is Lemma 6.6, which, as mentioned before, is a consequence of Lemma 2.3 of [38]. derivatives.