## Inverse Scattering On Asymptotically Hyperbolic Manifolds (1998)

Venue: | ACTA MATH |

Citations: | 28 - 2 self |

### BibTeX

@ARTICLE{Joshi98inversescattering,

author = {Mark S. Joshi and Antônio Sá Barreto},

title = {Inverse Scattering On Asymptotically Hyperbolic Manifolds},

journal = {ACTA MATH},

year = {1998},

volume = {184},

pages = {41--86}

}

### Years of Citing Articles

### OpenURL

### Abstract

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown

### Citations

285 |
Semi-Riemannian Geometry with Applications to Relativity
- O’Neill
- 1983
(Show Context)
Citation Context ...erefore stationary scattering phenomena are governed by the operator P = α 2 r −2 Dr(r 2 α 2 )Dr − α 2 r −2 ∆ω. (7.4) Scattering theory for the operator P has been extensively studied see for example =-=[8, 9, 5, 32, 37]-=- and the references cited there. It was observed in [37] that, after a change of C ∞ structure on X, the De Sitter-Schwarzschild model the operator P can be viewed as 0-differential operator which is ... |

155 |
The Analysis of Linear Partial
- Hörmander
- 1983
(Show Context)
Citation Context ...intersect transversally, with M ∩ F = ∂M = ∂F, the functions RM = R|M ∈ C ∞ (∂X × ∂X), ρF = ρ|F ∈ C ∞ (F) are defining functions of ∂M and ∂F respectively. Recall that, see for example section 3.2 of =-=[17]-=-, if Y is a manifold with corners and y ∈ C∞ (Y ), is a defining function of a boundary hypersurface of Y, then sections of yζΓ 1 2 (Y ), viewed as distributions acting on Γ 1 2 (Y ) via 〈y ζ ∫ F, f〉 ... |

111 |
Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
- Mazzeo, Melrose
- 1987
(Show Context)
Citation Context ... the form g = dx2 + h(x, y, dx, dy) x 2 , (1.1) where h |x=0, is independent of dx for some boundary defining function x, and a product decomposition X ∼ ∂X × [0, ǫ) near the boundary. As observed in =-=[26]-=- this implies that along a smooth curve in X \ ∂X, approaching a point p ∈ ∂X, the sectional curvatures of g approach −1. We note that this form is invariant under multiplying x by a function of y so ... |

104 | Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean spaces
- Melrose
- 1994
(Show Context)
Citation Context ...re quite similar to those in the Euclidean case, the proofs and underlying ideas are very different. The fundamental reason being that in the asymptotically Euclidean category, as observed by Melrose =-=[28]-=- and by Melrose and Zworski [30], there is propagation of growth at infinity whilst this does not occur in the asymptotically hyperbolic category. This is reflected in the fact, proved in [30], that i... |

98 | Geometric Scattering Theory - Melrose - 1995 |

57 |
Differential Analysis on Manifolds with Corners. in preparation. Available online at http://www-math.mit.edu/~rbm/book.html
- Melrose
(Show Context)
Citation Context ...tion that Hζ and Gζ depend holomorphically in ζ, provided 2ζ ̸∈ Z. This method of proving the existence of an expansion goes back to Euler and has been used in similar contexts in [18], [19] and also =-=[29]-=-. Next we need to compute Gζ(0) = γ0 and Hζ(0) = d0. Since these are holomorphic functions of ζ, we only need to compute Hζ(0) for 2ℜζ − n > 0, and Gζ(0) for 2ℜζ − n < 0. In the coordinates above we h... |

52 |
R.Phillips, Scattering Theory for Automorphic Functions, Princeton U.Press
- Lax
- 1976
(Show Context)
Citation Context ...the observation that the Eisenstein series for a Fuchsian group is a generalized eigenfunction for the Laplacian on the associated quotient of hyperbolic space - the fundamental reference for this is =-=[23]-=- where the finite volume case is studied. The study of the infinite volume case was initiated by Patterson in [34]. There has been a wealth of results in both cases and we refer the reader to [13] for... |

50 | Scattering Metrics and Geodesic Flow at Infinity
- Melrose, Zworski
(Show Context)
Citation Context ... Euclidean case, the proofs and underlying ideas are very different. The fundamental reason being that in the asymptotically Euclidean category, as observed by Melrose [28] and by Melrose and Zworski =-=[30]-=-, there is propagation of growth at infinity whilst this does not occur in the asymptotically hyperbolic category. This is reflected in the fact, proved in [30], that in the asymptotically Euclidean c... |

45 |
Scattering asymptotics for Riemann surfaces
- Guillopé, Zworski
- 1997
(Show Context)
Citation Context ... example section 8.4 of [27]. However the proof in the general case does not seem to be available in the literature. The proof of several particular cases have been given, see for example [16], [12], =-=[13]-=- and [35] and references given there. The case ℜζ = n 2 is done in [7]. Recall that a compact manifold with boundary (X, ∂X) is asymptotically hyperbolic if it can be equipped with a metric of the for... |

43 | Upper bounds on the number of resonances for non-compact Riemann surfaces - Guillopé, Zworski - 1995 |

38 |
On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations
- Andersson, Chru´sciel, et al.
- 1992
(Show Context)
Citation Context ... Agmon has also studied related questions, see [1], [2]. Andersson, Chrusciel and Friedrichs have studied solutions of the Einstein equations and related problems on asymptotically hyperbolic spaces, =-=[3]-=-, [4]. There appears to be no results in the literature on the inverse scattering problem on asymptotically hyperbolic manifolds. Perry, [36], has shown that for hyperbolic quotients in three dimensio... |

37 |
Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds
- Mazzeo
- 1991
(Show Context)
Citation Context ...nuation of the resolvent, there exists a unique solution of the equation, (∆ + ζ(ζ − n))u = 0 of the form, u = x ζ f+ + x n−ζ f−, (1.2) with f+, f− ∈ C ∞ (X), and f = f−|∂x. This is implicit in [24], =-=[25]-=- and is stated without a proof in [27]. A related result is also stated in the introduction of [26]. The first terms of the expansion with ℜζ = n 2 have been established in [7]. 12 MARK S. JOSHI AND ... |

36 |
Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Asymptotic Anal
- Guillopé, Zworski
- 1995
(Show Context)
Citation Context ... in statement and proof to the model form for scattering metrics proved in [22]. In the case where all sectional curvatures are equal to −1 near the boundary, such normal form has been established in =-=[15]-=-. Proposition 2.1. Let (X, ∂X) be a smooth manifold with boundary ∂X. And suppose g is a metric on X such that g = dx2 + h(x, y, dx, dy) x 2 , in some product decomposition near ∂X, where x is a defin... |

33 |
On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness
- Andersson, Chru´sciel
- 1994
(Show Context)
Citation Context ...n has also studied related questions, see [1], [2]. Andersson, Chrusciel and Friedrichs have studied solutions of the Einstein equations and related problems on asymptotically hyperbolic spaces, [3], =-=[4]-=-. There appears to be no results in the literature on the inverse scattering problem on asymptotically hyperbolic manifolds. Perry, [36], has shown that for hyperbolic quotients in three dimensions by... |

30 | Scattering asymptotics for Riemann surfaces - e, Zworski - 1997 |

19 |
Generalized functions, Vol.1
- Gel’fand, Shilov
- 1964
(Show Context)
Citation Context ... = Id, and using the fact that h0 is symmetric. The coefficients of Tj(k, ζ), j = 1, 2 in (1.8) arise when we take the Fourier transform of the corresponding power of |w|. See for example page 363 of =-=[11]-=-. This ends the proof of the theorem. We now prove Corollaries 1.2 and 1.3. The proof of Corollary 1.3 is a direct consequence of the fact that, for every k, A2(k, ζ) ̸= 0 for at least one value of ζ.... |

19 |
Fonctions zeta de Selberg et surfaces de géométrie finie
- Guillopé
- 1990
(Show Context)
Citation Context ...ee for example section 8.4 of [27]. However the proof in the general case does not seem to be available in the literature. The proof of several particular cases have been given, see for example [16], =-=[12]-=-, [13] and [35] and references given there. The case ℜζ = n 2 is done in [7]. Recall that a compact manifold with boundary (X, ∂X) is asymptotically hyperbolic if it can be equipped with a metric of t... |

17 |
The Mathematical Theory of Black Holes
- Chandrasekar
- 1992
(Show Context)
Citation Context ...erefore stationary scattering phenomena are governed by the operator P = α 2 r −2 Dr(r 2 α 2 )Dr − α 2 r −2 ∆ω. (7.4) Scattering theory for the operator P has been extensively studied see for example =-=[8, 9, 5, 32, 37]-=- and the references cited there. It was observed in [37] that, after a change of C ∞ structure on X, the De Sitter-Schwarzschild model the operator P can be viewed as 0-differential operator which is ... |

16 | Recovering asymptotics of metrics from fixed energy scattering data - Joshi, Barreto - 1999 |

15 |
The geometry and spectra of hyperbolic manifolds, Spectral and inverse spectral theory
- Hislop
- 1993
(Show Context)
Citation Context ...ult, see for example section 8.4 of [27]. However the proof in the general case does not seem to be available in the literature. The proof of several particular cases have been given, see for example =-=[16]-=-, [12], [13] and [35] and references given there. The case ℜζ = n 2 is done in [7]. Recall that a compact manifold with boundary (X, ∂X) is asymptotically hyperbolic if it can be equipped with a metri... |

14 |
On the representation theorem for solutions of the Helmholtz equation on the hyperbolic space. Partial differential equations and related subjects
- Agmon
- 1990
(Show Context)
Citation Context ...of the scattering matrix proved in section 4 are implicit. In [7] Borthwick showed the continuous dependence of the scattering matrix on the metric. Agmon has also studied related questions, see [1], =-=[2]-=-. Andersson, Chrusciel and Friedrichs have studied solutions of the Einstein equations and related problems on asymptotically hyperbolic spaces, [3], [4]. There appears to be no results in the literat... |

11 |
Recovering Asymptotics of Coulomb-like Potentials from Fixed Energy Scattering Data
- Joshi
(Show Context)
Citation Context .... To do this we use the calculus developed by Mazzeo and Melrose [26] of zero pseudo-differential operators in order to construct the resolvent. As in our work on asymptotically Euclidean scattering, =-=[20, 21, 22]-=-, a key part of our approach is to consider the principal symbol of the difference of the scattering matrices rather than the lower order terms of the symbol of a single operator, which allows us to p... |

10 | Scattering theory and deformations of asymptotically hyperbolic metrics.” dg-ga/9711016
- Borthwick
- 1997
(Show Context)
Citation Context ...s not seem to be available in the literature. The proof of several particular cases have been given, see for example [16], [12], [13] and [35] and references given there. The case ℜζ = n 2 is done in =-=[7]-=-. Recall that a compact manifold with boundary (X, ∂X) is asymptotically hyperbolic if it can be equipped with a metric of the form g = dx2 + h(x, y, dx, dy) x 2 , (1.1) where h |x=0, is independent o... |

10 |
The Laplacian operator on a Riemann surface
- Patterson
- 1976
(Show Context)
Citation Context ...n the associated quotient of hyperbolic space - the fundamental reference for this is [23] where the finite volume case is studied. The study of the infinite volume case was initiated by Patterson in =-=[34]-=-. There has been a wealth of results in both cases and we refer the reader to [13] for a comprehensive bibliography and to [16] and [27] for a review of the subject. There has been less work on asympt... |

7 |
A trace-class rigidity theorem for Kleinian groups
- Perry
- 1995
(Show Context)
Citation Context ...s and related problems on asymptotically hyperbolic spaces, [3], [4]. There appears to be no results in the literature on the inverse scattering problem on asymptotically hyperbolic manifolds. Perry, =-=[36]-=-, has shown that for hyperbolic quotients in three dimensions by convex, co-compact, torsion-free Kleinian groups with non-empty regular set, that the scattering matrix determines the manifold. Borthw... |

6 |
Sá Barreto,Recovering Asymptotics of Short Range Potentials
- Joshi, A
(Show Context)
Citation Context .... To do this we use the calculus developed by Mazzeo and Melrose [26] of zero pseudo-differential operators in order to construct the resolvent. As in our work on asymptotically Euclidean scattering, =-=[20, 21, 22]-=-, a key part of our approach is to consider the principal symbol of the difference of the scattering matrices rather than the lower order terms of the symbol of a single operator, which allows us to p... |

6 |
Sá Barreto, Recovering asymptotics of metrics from fixed energy scattering data
- Joshi, A
- 1999
(Show Context)
Citation Context .... To do this we use the calculus developed by Mazzeo and Melrose [26] of zero pseudo-differential operators in order to construct the resolvent. As in our work on asymptotically Euclidean scattering, =-=[20, 21, 22]-=-, a key part of our approach is to consider the principal symbol of the difference of the scattering matrices rather than the lower order terms of the symbol of a single operator, which allows us to p... |

5 |
Weyl Asymptotics for the Laplacian on Asymptotically Euclidean Spaces
- Christiansen
(Show Context)
Citation Context ...s section, we examine the scattering matrix for metrics which take the form, g = dx2 + h(y, dy) x 2 + O(x ∞ ), (6.1) for some product decomposition. Our approach is analogous to that of Christiansen, =-=[10]-=-, and Parnovksi, [33], in the asymptotically Euclidean setting. The computation is also closely related to that of Hislop, [16] section 2.3, for H n . As we have shown in previous sections that if two... |

5 |
Elliptic Theory of Differential Edge
- Mazzeo
- 1991
(Show Context)
Citation Context ... continuation of the resolvent, there exists a unique solution of the equation, (∆ + ζ(ζ − n))u = 0 of the form, u = x ζ f+ + x n−ζ f−, (1.2) with f+, f− ∈ C ∞ (X), and f = f−|∂x. This is implicit in =-=[24]-=-, [25] and is stated without a proof in [27]. A related result is also stated in the introduction of [26]. The first terms of the expansion with ℜζ = n 2 have been established in [7]. 12 MARK S. JOSH... |

5 |
Distribution of resonances for spherical black holes
- Barreto, Zworski
- 1997
(Show Context)
Citation Context ...erefore stationary scattering phenomena are governed by the operator P = α 2 r −2 Dr(r 2 α 2 )Dr − α 2 r −2 ∆ω. (7.4) Scattering theory for the operator P has been extensively studied see for example =-=[8, 9, 5, 32, 37]-=- and the references cited there. It was observed in [37] that, after a change of C ∞ structure on X, the De Sitter-Schwarzschild model the operator P can be viewed as 0-differential operator which is ... |

4 | Quasi-rigidity of hyperbolic 3-manifolds and scattering theory
- Borthwick, McRae, et al.
- 1997
(Show Context)
Citation Context ...convex, co-compact, torsion-free Kleinian groups with non-empty regular set, that the scattering matrix determines the manifold. Borthwick, McRae and Taylor have proved an associated rigidity result, =-=[6]-=-. We would like to thank Maciej Zworski for explanations of hyperbolic scattering and helpful comments. We would also like to thank Richard Melrose and Rafe Mazzeo for helpful conversations. We are al... |

4 | A symbolic construction of the forward fundamental solution of the wave operator
- Joshi
- 1998
(Show Context)
Citation Context ...m the construction that Hζ and Gζ depend holomorphically in ζ, provided 2ζ ̸∈ Z. This method of proving the existence of an expansion goes back to Euler and has been used in similar contexts in [18], =-=[19]-=- and also [29]. Next we need to compute Gζ(0) = γ0 and Hζ(0) = d0. Since these are holomorphic functions of ζ, we only need to compute Hζ(0) for 2ℜζ − n > 0, and Gζ(0) for 2ℜζ − n < 0. In the coordina... |

3 |
Les résonances d’un trou noir de
- Bachelot, Motet-Bachelot
- 1993
(Show Context)
Citation Context |

3 |
An Intrinsic Characterization of Polyhomogeneous Lagrangian Distributions
- Joshi
- 1997
(Show Context)
Citation Context ...ar from the construction that Hζ and Gζ depend holomorphically in ζ, provided 2ζ ̸∈ Z. This method of proving the existence of an expansion goes back to Euler and has been used in similar contexts in =-=[18]-=-, [19] and also [29]. Next we need to compute Gζ(0) = γ0 and Hζ(0) = d0. Since these are holomorphic functions of ζ, we only need to compute Hζ(0) for 2ℜζ − n > 0, and Gζ(0) for 2ℜζ − n < 0. In the co... |

2 |
A representation theorem for solutions of Schrödinger type equations on non-compact Riemannian manifolds
- Agmon
- 1992
(Show Context)
Citation Context ...ties of the scattering matrix proved in section 4 are implicit. In [7] Borthwick showed the continuous dependence of the scattering matrix on the metric. Agmon has also studied related questions, see =-=[1]-=-, [2]. Andersson, Chrusciel and Friedrichs have studied solutions of the Einstein equations and related problems on asymptotically hyperbolic spaces, [3], [4]. There appears to be no results in the li... |

2 |
Scattering Matrix for Manifolds with Conical Ends
- Parnovski
(Show Context)
Citation Context ... the scattering matrix for metrics which take the form, g = dx2 + h(y, dy) x 2 + O(x ∞ ), (6.1) for some product decomposition. Our approach is analogous to that of Christiansen, [10], and Parnovksi, =-=[33]-=-, in the asymptotically Euclidean setting. The computation is also closely related to that of Hislop, [16] section 2.3, for H n . As we have shown in previous sections that if two metrics agree to inf... |

1 |
The quasi-normal modes of the Schwarschild black hole
- Chandrasekar, Detweiler
- 1975
(Show Context)
Citation Context |

1 | Recovering the Asymptotics of a Short Range - Joshi, Barreto - 1998 |

1 |
Geometric Optics and the Bottom of the Spectrum, preprint
- Melrose
(Show Context)
Citation Context ...lly hyperbolic manifold case it is a pseudo-differential operator and in the fact that the principal symbol of the difference of the scattering matrices is locally determined by the perturbation. See =-=[31]-=- for a discussion of a general framework including both cases. There is a long history of scattering theory on hyperbolic manifolds arising from the observation that the Eisenstein series for a Fuchsi... |

1 |
The Laplace Operator on a Hyperbolic Manifold II
- Perry
- 1989
(Show Context)
Citation Context ...section 8.4 of [27]. However the proof in the general case does not seem to be available in the literature. The proof of several particular cases have been given, see for example [16], [12], [13] and =-=[35]-=- and references given there. The case ℜζ = n 2 is done in [7]. Recall that a compact manifold with boundary (X, ∂X) is asymptotically hyperbolic if it can be equipped with a metric of the form g = dx2... |

1 |
Smooth Linearization Near a Fixed
- Sell
- 1985
(Show Context)
Citation Context ...we can re-express ¯τ in terms of (x, y, ¯ ξ), and near ¯τ = −1 the vector field becomes, ( Hg = −2 x ∂ ∂x + ¯ ξ. ∂ ∂¯ ) + O( ξ ¯ ξ 2 + x 2 ). This forms a sink at (x, ξ) = 0 and thus, by Theorem 7 of =-=[38]-=-, there exist local coordinates (x ′ , ξ ′ ), equal to (x, ξ) to second order at (x, ξ) = 0, which reduce the vector field to the form ( −2 x ′ ∂ ∂x ′ + ¯ ξ. ∂ ∂ξ ′ ) + O(ξ ′2 ′2 ∂ + x ) ∂y . We there... |