## A SUPPORT THEOREM FOR THE RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS (709)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Barreto709asupport,

author = {Antônio Sá Barreto},

title = {A SUPPORT THEOREM FOR THE RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS},

year = {709}

}

### OpenURL

### Abstract

We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering

### Citations

1592 | The analysis of Linear Partial Differential Operators I, distribution theory and Fourier Analysis. Grundlehren der mathematischen Wissenchaften 256 - Hörmander - 1990 |

150 |
Scattering theory
- Lax, Phillips
- 1989
(Show Context)
Citation Context ...iation fields respectively as (1.5) R+(f1, f2) = Dsv+(0, s, y) and R−(f1, f2) = Dsv−(0, s, y). 2000 Mathematics Subject Classification. Primary 81U40, Secondary 35P25. 12 SÁ BARRETO Lax and Phillips =-=[9]-=- proved that in R n the forward (or backward) radiation field is the modified Radon transform, that is: ∫ Rf(s, ω) = R+(f1, f2)(s, ω) = |Ds| n−3 2 Rf1(s, ω) + |Ds| n−1 2 Rf2(s, ω), where 〈x,ω〉=s f(z) ... |

111 | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature - Mazzeo, Melrose - 1987 |

104 | Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean spaces
- Melrose
- 1994
(Show Context)
Citation Context ... a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering matrix at all energies, see [14]. An asymptotically Euclidean manifold =-=[12, 8]-=- is a C ∞ compact manifold X with boundary ∂X, which is equipped with a C ∞ Riemannian metric g that in a collar neighborhood of the boundary ∂X satisfies (1.1) g = dx2 h(x) + , x4 x2 in [0, ǫ) × ∂X, ... |

39 |
Geometric scattering theory. Stanford Lectures
- Melrose
- 1995
(Show Context)
Citation Context ...ave the following special form near ∂X : (1.2) g = dx2 + ψ(x)h0 , x ∈ [0, ǫ), x4 x2 where ψ ∈ C ∞ ([0, ǫ)), ψ(x) > 0, ψ(0) = 1, and h0 is a C ∞ metric on ∂X. These are known as warped product metrics =-=[11]-=-. We consider the wave equation on X. Let ∆g be the Laplace operator on X, and let u(t, z) satisfy (1.3) (D 2 t − ∆g)u(t, z) = 0 on R × X, u(0) = f1, Dtu(0) = f2, with f1, f2 ∈ C ∞ 0 (X). A function f... |

16 | Recovering asymptotics of metrics from fixed energy scattering data
- Joshi, Barreto
- 1999
(Show Context)
Citation Context ... a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering matrix at all energies, see [14]. An asymptotically Euclidean manifold =-=[12, 8]-=- is a C ∞ compact manifold X with boundary ∂X, which is equipped with a C ∞ Riemannian metric g that in a collar neighborhood of the boundary ∂X satisfies (1.1) g = dx2 h(x) + , x4 x2 in [0, ǫ) × ∂X, ... |

15 |
Radiation fields and hyperbolic scattering theory
- Friedlander
- 1980
(Show Context)
Citation Context ...D 2 t − ∆g)u(t, z) = 0 on R × X, u(0) = f1, Dtu(0) = f2, with f1, f2 ∈ C ∞ 0 (X). A function f ∈ C ∞ 0 (X) if it is C∞ and its support does not intersect the boundary of X. The following is proved in =-=[4, 5]-=-: Theorem 1.1. Let x be the boundary defining function for which (1.1) holds, and let z = (x, y), y ∈ ∂X, be the corresponding boundary normal coordinates in a collar neighborhood of the boundary. The... |

13 |
Notes on the wave equation on asymptotically Euclidean manifolds
- Friedlander
(Show Context)
Citation Context ...D 2 t − ∆g)u(t, z) = 0 on R × X, u(0) = f1, Dtu(0) = f2, with f1, f2 ∈ C ∞ 0 (X). A function f ∈ C ∞ 0 (X) if it is C∞ and its support does not intersect the boundary of X. The following is proved in =-=[4, 5]-=-: Theorem 1.1. Let x be the boundary defining function for which (1.1) holds, and let z = (x, y), y ∈ ∂X, be the corresponding boundary normal coordinates in a collar neighborhood of the boundary. The... |

8 | Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds.” math.AP/0312108 - Barreto - 2003 |

5 |
Radiation fields on asymptotically Euclidean manifolds
- Barreto
- 2003
(Show Context)
Citation Context ...rol theory, where the support of a function can be exactly controlled by the support of its Radon transform. We want to address the analogue question for Radiation fields. The following was proved in =-=[13]-=-: Theorem 1.2. If f ∈ C ∞ 0 (X), g is an arbitrary asymptotically Euclidean metric, and R(0, f)(s, y) = 0 for s < − 1 x0 , x0 ∈ (0, ǫ), then f = 0 if x < x0. This says that if there exists some x1 ∈ (... |

4 | On the radiation field of pulse solutions of the wave equation - Friedlander - 1962 |

3 |
Radiation fields and inverse scattering on asymptotically hyperbolic manifolds
- Barreto
(Show Context)
Citation Context ...in reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering matrix at all energies, see =-=[14]-=-. An asymptotically Euclidean manifold [12, 8] is a C ∞ compact manifold X with boundary ∂X, which is equipped with a C ∞ Riemannian metric g that in a collar neighborhood of the boundary ∂X satisfies... |

1 |
The Radon Transform. Birkhauser, 2nd edition
- Helgason
- 1999
(Show Context)
Citation Context ...(s, ω) = |Ds| n−3 2 Rf1(s, ω) + |Ds| n−1 2 Rf2(s, ω), where 〈x,ω〉=s f(z) dσ, σ is the surface measure on 〈x, ω〉 = s, is the Radon transform. Helgason’s celebrated support theorem for Radon transforms =-=[6]-=- says that if f is a rapidly decaying function in R n and Rf(s, ω) = 0 for s < s0, s0 < 0 (and hence by symmetry Rf(s, ω) = 0 for |s| > |s0|) then f is supported in the ball of radius |s0|. The assump... |