## SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS (2006)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Guillarmou06scatteringand,

author = {Colin Guillarmou and Antônio Sá Barreto},

title = {SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS},

year = {2006}

}

### OpenURL

### Abstract

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 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.

### Citations

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Citation Context ...here r(z; z ′ ( ( dg0(z; z ) := cosh ′ )) −2 ) 4ρ = 2 2 0ρ ′ 2 0 (u − u ′ + ℑ(ω.¯ω ′)) 2 + (ρ2 0 + ρ′ 2 1 0 + 2 |ω − ω′ | 2 ) with cn constant depending on n and 2F1 is a hypergeometric function (see =-=[2]-=-), we also used the formula Q(z, z ′ ) = −i 2 (u − u′ + ℑ(ω.¯ω ′ )) + 1 ( ρ 2 2 0(z) + ρ ′ 2 1 0 (z) + 2 |ω − ω′ | 2) . A change of variables shows that if an operator K has a distributional Schwartz ... |

1632 |
The Analysis of Linear Partial Differential Operators I
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Citation Context ...ment starting at {ρ = δ, s = 2 logρ0}. This is actually easier, because the level surfaces of φ = s + ρ are strictly pseudoconvex away from ρ = 0. Therefore Hörmander’s Theorem, see Theorem 28.3.5 of =-=[22]-=-, can be used to show that v = 0 in a neighborhood of {ρ = δ, s = 2 logρ0}. This process can be continued to show that v = 0 in the region {0 ≤ ρ ≤ ρ0, 2 log ρ0 ≤ s ≤ s0}. Now we translate this back t... |

177 | A mod k index theorem - Freed, Melrose - 1992 |

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(Show Context)
Citation Context ...tended smoothly to ¯ X. Note that the boundary represents the geometric infinity of X, as does the sphere Sn for the hyperbolic space Hn+1 . The spectrum of ∆g, the Laplacian of (X, g) was studied in =-=[27]-=-; it consists a finite pure point spectrum σpp(∆), which is the set of L2 (X) eigenvalues, and an absolutely continuous spectrum σac(∆) satisfying σac(∆) = [ n 2 /4, ∞ ) and σpp(∆) ⊂ ( 0, n 2 /4 ) . T... |

107 | Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory
- Melrose
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(Show Context)
Citation Context ... invariants by the scattering matrix at all energies. 1. 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 ... |

90 |
Symplectic Geometry and Analytical Mechanics
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- 1987
(Show Context)
Citation Context ... 1 4 ∂µ∂ν n + 1 1 − µν∆h(µν) + A + 8 16 A(µ∂µ ) + ν∂ν) V = 0, V (µ, µ, z) = 0, (∂µV )(µ, µ, z) = (µ) −2n−3 f2(µ 2 (7.6) , z), ¿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 = (x1, ..., xn), y = (y1, ..., yn), u ∈ R, near any point z0 ∈ ∂ ¯ X such that n∑ Θ0 = du + (yjdxj − xjdyj). Let Xj an... |

89 |
Volume and area renormalizations for conformally compact Einstein metrics
- Graham
- 1998
(Show Context)
Citation Context ...anifold ¯ X 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 = dt2 + h(t) t2 where h(t), t ∈ ... |

88 | Scattering matrix in conformal geometry
- Graham, Zworski
(Show Context)
Citation Context ... invariants by the scattering matrix at all energies. 1. 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 ... |

78 | Elliptic theory of differential edge operators - Mazzeo - 1991 |

70 | The divisor of Selberg’s zeta function for Kleinian groups
- Patterson, Perry
(Show Context)
Citation Context ...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] th... |

60 |
Calculus on Heisenberg manifolds
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(Show Context)
Citation Context ...old, one can define the class ψ ∗ Θ0 (∂ ¯ X) of HeisenbergSCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 3 pseudo-differential operators associated to kerΘ0 and its related principal symbol, see =-=[8, 34, 6]-=- and Subsection 4.3 below. We can define the “parabolically homogeneous norm” on T∂ ¯ X: ( ||V ||He := Θ0(V ) 2 + 1 4 dΘ0(V, 1 JV ) 2) 4 . and the metric h0 := M ∗ ρ (ρ−2 h(ρ))|ρ=0 = dΘ0(., J.) + Θ 2 ... |

46 |
Resolvent of the Laplacian on strictly pseudoconvex domains
- Epstein, Melrose, et al.
- 1991
(Show Context)
Citation Context ... 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... |

44 |
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- Gohberg, I
- 1971
(Show Context)
Citation Context ... invertible if λ2k /∈ n + 1 + N, thus p2k is Fredholm if λ2k /∈ n+1+N. This implies that ˜ S(λ2k) is Fredholm on L2 (∂ ¯ X) for these λ2k since (L+1) λ are invertible. Then using Gohberg-Sigal theory =-=[10]-=- and the fact that ˜ S(λ) is invertible in a small pointed disc centered at λ2k, one obtains directly from ˜ S(λ) −1 = ˜ S(n + 1 − λ) that ˜S(λ) is finite meromorphic in a small disc containing n + 1 ... |

42 | Meromorphic properties of the resolvent for asymptotically hyperbolic manifolds
- Guillarmou
(Show Context)
Citation Context ...inuation 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 − N0), =-=[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 uλ ∈ C∞ (X) satisfying such that near ∂ ¯ X (∆g − λ(n − λ))uλ = 0 uλ = ρ n−λ f− + ρ λ f+ +... |

39 |
Geometric scattering theory. Stanford Lectures
- Melrose
- 1995
(Show Context)
Citation Context ... invariants by the scattering matrix at all energies. 1. 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 ... |

37 |
Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds
- Mazzeo
- 1991
(Show Context)
Citation Context ...ond fundamental form −∂rα −1 (r) > α −1 (r), here α −1 (r) is the dual metric to α(r) on T ∗ ∂ ¯ X. We should point out that the analogue result for conformally compact manifolds was proved by Mazzeo =-=[26]-=-. 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 ∈ N0, ther... |

35 |
Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems 13
- Belishev
- 1997
(Show Context)
Citation Context ...d on M and Φ∗g2 = g1. 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 def... |

32 |
The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix
- Perry
- 1989
(Show Context)
Citation Context ...ter 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]-=-): Lemma 5.4. If the metric is even at order 2k, we have for λ, n − λ /∈ R ∪ Pk R(λ; m, m ′ ) − R(n + 1 − λ; m, m ′ ) = (2(2λ − n − 1)) −1 ∫ P(λ; m, y)P(n − λ; m ′ , y)dvolh0(y) ∂ ¯ X 6. The Scatterin... |

30 |
Sá Barreto. Inverse scattering on asymptotically hyperbolic manifolds
- Joshi, A
(Show Context)
Citation Context |

28 |
Unique continuation for solutions to PDE’s: between Hörmander’s theorem and Holmgren’s theorem
- Tataru
(Show Context)
Citation Context ... {0 ≤ ρ ≤ ρ0, 2 log ρ0 ≤ s ≤ s0}. Now we translate this back to the t variable, and using that u is odd in t, we conclude that u(t, z) = 0 in ρ ≤ ρ0 and 0 ≤ t ≤ s0 − 2 log ρ0. Now Tataru’s theorem in =-=[37]-=- shows that ∂tu(0, z) = f2(z) = 0 if 2 log ρ ≤ s0. This proves the Lemma □ Now we prove Lemma 9.2.32 COLIN GUILLARMOU AND ANTÔNIO SÁ BARRETO Proof. To extract information about the behavior of v as ρ... |

23 |
The Heisenberg algebra, index theory and homology
- Epstein, Melrose, et al.
(Show Context)
Citation Context ...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. ... |

20 |
CR Invariant powers of the sub-Laplacian
- Gover, Graham
- 2003
(Show Context)
Citation Context ...here the manifold ¯ X is a strictly pseudoconvex domain of Cn+1 equipped with an approximate Einstein Kähler metric, the relationship between the residues Resλ2kS(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 representa... |

19 |
Fonctions zeta de Selberg et surfaces de géométrie finie
- Guillopé
- 1990
(Show Context)
Citation Context ...ter 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]-=-): Lemma 5.4. If the metric is even at order 2k, we have for λ, n − λ /∈ R ∪ Pk R(λ; m, m ′ ) − R(n + 1 − λ; m, m ′ ) = (2(2λ − n − 1)) −1 ∫ P(λ; m, y)P(n − λ; m ′ , y)dvolh0(y) ∂ ¯ X 6. The Scatterin... |

16 | Resonances and scattering poles on asymptotically hyperbolic manifolds
- Guillarmou
(Show Context)
Citation Context ...inted disc centered at λ2k, one obtains directly from ˜ S(λ) −1 = ˜ S(n + 1 − λ) that ˜S(λ) is finite meromorphic in a small disc containing n + 1 − λ2k if n + 1 − λ2k /∈ −N0 and we can define (as in =-=[17, 32]-=- for the real case) the multiplicity of a pole of finite multiplicity of ˜ S(λ) to be (6.8) ν(λ0) := −Tr Resλ=λ0(∂λ ˜ S(λ) ˜ S −1 (λ)) = −Tr Resλ=λ0(∂λ(c(λ)S(λ))(c(λ)S(λ)) −1 ), the second identity be... |

13 | Generalized Krein formula, determinants and Selberg zeta function in even dimension
- Guillarmou
(Show Context)
Citation Context |

11 | Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds
- Ponge
- 2008
(Show Context)
Citation Context ...old, one can define the class ψ ∗ Θ0 (∂ ¯ X) of HeisenbergSCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 3 pseudo-differential operators associated to kerΘ0 and its related principal symbol, see =-=[8, 34, 6]-=- and Subsection 4.3 below. We can define the “parabolically homogeneous norm” on T∂ ¯ X: ( ||V ||He := Θ0(V ) 2 + 1 4 dΘ0(V, 1 JV ) 2) 4 . and the metric h0 := M ∗ ρ (ρ−2 h(ρ))|ρ=0 = dΘ0(., J.) + Θ 2 ... |

8 | Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds.” math.AP/0312108
- Barreto
- 2003
(Show Context)
Citation Context ...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 (7.1) ( D 2 t − ∆g − (n + 1)2 4 ) u(t, z) = 0 in R+ × X u(0, z) = f1(z), Dtu(0, z) = f2(z), f1, f2 ∈ C ∞ 0 (X). It is well known tha... |

5 | Résonances sur les variétés asymptotiquement hyperboliques - Guillarmou - 2004 |

4 | Compatibility operators for degenerate elliptic equations on the ball and Heisenberg - Graham - 1984 |

4 | Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature
- Vasy, Wunsch
(Show Context)
Citation Context ...) ν(λ0) = m(λ0) − m(n + 1 − λ0) for λ0 /∈ 1 (n + 1 − N). 2 The only non-apparent result we need to apply the proof of [17, Th. 1.1] is the following unique continuation result dues to Vasy and Wunsch =-=[38]-=-: Lemma 6.6. Let (X, g) be an ACH manifold, and let u ∈ ˙ C ∞ ( ¯ X) satisfy (∆g−λ(n+1−λ))u = 0, λ ∈ R in a neighborhood Ω of ∂ ¯ X, then u = 0 in Ω. In particular, if Ω = X, u = 0. This is a conseque... |

3 |
Unicité du problème de Cauchy pour des opérateurs du second ordre à symboles réels
- Alinhac
- 1984
(Show Context)
Citation Context ...niqueness theorem that would allow us to show that v = 0 in a neighborhood of the set {s = log ρ0, 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 q = −τσ − 1 4 ρτ2 − ρp where (τ, σ, ξ, η, ν) is the dual to (ρ, s, x... |

2 |
Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group
- Geller
- 1984
(Show Context)
Citation Context ...λ − n) 2 + 1 4 |ω|4 ) −λ . The principal symbol is its Fourier transfom in the fibre Tp∂ ¯ Xup Fourier transform of these parabolically homogeneous distributions is studied in great details by Geller =-=[9]-=-, we refer the reader to his work. ¿From Proposition 6.2, Lemma 5.4 and (5.10) we deduce (see [19, 16] for the proof in the real case) Lemma 6.4. For λ such that P(n − λ), P(λ), S(λ) exist, we have th... |

2 |
CR-invariants and the scattering operator for complex manifolds with CR-boundary
- Hislop, Perry, et al.
(Show Context)
Citation Context ... is a strictly pseudoconvex domain of Cn+1 equipped with an approximate Einstein Kähler metric, the relationship between the residues Resλ2kS(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 gro... |

2 | Scattering theory for strictly pseudoconvex domains. Differential equations: La Pietra
- Melrose
- 1996
(Show Context)
Citation Context ...N0)/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 manifold ¯ X is a strictly pseud... |

2 | Volume renormalization for complete Einstein-Kähler metrics - Seshadri - 2004 |

1 |
Mtriques d’Einstein asymptotiquement symtriques, Astrisque No
- Biquard
(Show Context)
Citation Context ...mptotically 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 Kähle... |

1 |
A Burns-Epstein invraiant for
- Biquard, Herzlich
(Show Context)
Citation Context ...bolic 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.... |