## Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds (2004)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Barreto04radiationfields,,

author = {Antônio Sá Barreto},

title = {Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds },

year = {2004}

}

### OpenURL

### Abstract

### Citations

1632 |
The Analysis of Linear Partial Differential Operators
- Hörmander
- 1983
(Show Context)
Citation Context ...(y) dy = 1 and χδ(y) = δ −n χ ( y ) , δ > 0, δ φ ∈ C ∞ 0 ((−T, T)µ × (−T, T)ν × Y0), with φ(µ, ν, y) = 1 in (− T T , 2 2 )µ × (− T T , 2 2 )ν × Z0.RADIATION FIELDS 27 Let vδ = χδ ∗ ′ φŨ, where as in =-=[25]-=-, ∗′ means that the convolution is taken in the variable y only. In view of (7.7) and the fact that all derivatives of ˜ W vanish at {x′ = 0}∪{t ′ = 0}, vδ ∈ C∗∞ 0 ([0, T)µ×(−T, T)ν×Y0), for δ small e... |

199 |
Pseudodifferential operators
- Taylor
- 1981
(Show Context)
Citation Context ...rm is of finite rank. Therefore KδQ : L2 (X1) −→ H−2 ([0, δ] × M) is compact. But Q is also a second order differential operator. This implies that KδQ = 0. See for example exercise 6.2 on page 52 of =-=[51]-=-. Therefore the tensors h1 and h2 are equal in [0, δ] × M, and this proves Proposition 8.2. □ j=1RADIATION FIELDS 43 Now we need to show that the diffeomorphism can be extended to the whole manifold.... |

154 |
Scattering Theory
- Lax, Phillips
- 1989
(Show Context)
Citation Context ... the solutions to the Schrödinger equation on a neighborhood of infinity. We will develop the scattering theory for this class of manifolds using a dynamical approach in the style of Lax and Phillips =-=[35, 36, 37]-=-, but we do this by following Friedlander [10, 11, 12, 13, 14]. We define the forward radiation field for asymptotically hyperbolic manifolds as the limit, as times goes to infinity, of the forward fu... |

112 |
Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
- Mazzeo, Melrose
- 1987
(Show Context)
Citation Context ...ed with a complete metric that resembles the hyperbolic space near the boundary. The basic examples of such manifolds are the hyperbolic space and its quotients by certain discrete group actions, see =-=[43]-=-, but any C ∞ compact manifold with boundary can be equipped with such a metric. There is a history of interest in scattering theory for this class of manifolds, motivated by several problems of mathe... |

100 |
Geometric Scattering Theory
- Melrose
- 1995
(Show Context)
Citation Context ...problem and the traces of the eigenfunctions of ∆ − n2 4 in Xǫ. To prove that the scattering matrix determines Cλ, λ ∈ R \ 0, we apply the same argument used in the proof of Lemma 3.2, chapter 3.8 of =-=[40]-=-, see also the proof of Lemma 2.1 of [46], to show that for any λ ̸= 0, the set of functions given by ∫ (8.9) v(z) = is dense in the set of solutions of (8.10) ∂X ( n ) E + iλ (z, y)φ(y), φ ∈ C 2 ∞ (∂... |

89 |
Volume and area renormalizations for conformally compact Einstein metrics
- Graham
- 1998
(Show Context)
Citation Context ...by several people, see for example [2, 6, 20, 28, 39, 43, 45] and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field theory, see =-=[9, 15, 16]-=- and references cited there. Mazzeo, Mazzeo and Melrose [39, 40, 43] first studied the spectral and scattering theory of the Laplacian in this general setting and gave a thorough description of the re... |

88 |
Spectral theory and differential operators
- Davies
- 1995
(Show Context)
Citation Context ...oefficients of Q with respect to µ, and ||, || denotes the norm in L2 ((−T, T) × (−T, T) × Y0). This is equation 4.11 of [1] with a = 0 and a ′ = 1. Hardy’s inequality, see for example Lemma 5.3.1 of =-=[7]-=-, gives that ||µ −1vµ|| 2 ≥ 9 4 ||µ−2v|| 2 . Using this and applying Cauchy-Schwartz inequality to the last term of (7.31) gives (7.32) ||µ −γ Lµ γ v|| 2 ≥ −2γℜ ( µQµv, µ −2 v ) + 2γ 2 ∣ ∣ −1 µ vµ ∣ ∣... |

88 | Scattering matrix in conformal geometry
- Graham, Zworski
(Show Context)
Citation Context ... (X, g) be an asymptotically hyperbolic manifold. The only poles of the resolvent R( n 2 + iλ) = (∆g − n2 4 − λ2 ) −1 in {ℑλ < 0} correspond to the finitely many eigenvalues of ∆g. Proposition 3.6 of =-=[16]-=- states that if λ0 > 0 is such that n2 4 − λ20 is an eigenvalue of ∆g then the scattering matrix has a pole at −iλ0 and its residue is given by { Πλ0, if λ0 ̸∈ N/2, (8.31) Res−iλ0 A(λ) = Πλ0 − pl, , l... |

78 |
The Hodge cohomology of a conformally compact metric
- Mazzeo
- 1988
(Show Context)
Citation Context ...ds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov [8], followed by Lax and Phillips [36, 37], and later by several people, see for example =-=[2, 6, 20, 28, 39, 43, 45]-=- and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field theory, see [9, 15, 16] and references cited there. Mazzeo, Mazzeo and Me... |

78 |
Elliptic theory of differential edge operators
- Mazzeo
- 1991
(Show Context)
Citation Context ... − 4 n2 ) f ∈ L 4 2 (X).RADIATION FIELDS 21 ∆ − n2 4 is a standard elliptic operator in the interior of X, so ψ1 ∆ − n2 4 elliptic regularity for totally characteristic operators, see Theorem 3.8 of =-=[41]-=-, (7.4) (xDx, xDy1, ..., xDyn) α ψ1 ( ∆ − n2 4 ( ) f ∈ L 2 (X), α ∈ N n+1 . ) f ∈ C ∞ ( ◦ X). Moreover, by If φ ∈ C∞ 0 (R) is even, and R+(0, f)(s, y) = 0 for s < s0, φ ∗ R+(0, f)(s, y) = 0 for s < s0... |

45 |
Scattering asymptotics for Riemann surfaces
- Guillopé, Zworski
- 1997
(Show Context)
Citation Context ...ds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov [8], followed by Lax and Phillips [36, 37], and later by several people, see for example =-=[2, 6, 20, 28, 39, 43, 45]-=- and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field theory, see [9, 15, 16] and references cited there. Mazzeo, Mazzeo and Me... |

42 | Meromorphic properties of the resolvent for asymptotically hyperbolic manifolds
- Guillarmou
(Show Context)
Citation Context ...udied the spectral and scattering theory of the Laplacian in this general setting and gave a thorough description of the resolvent and its meromorphic continuation. Their methods have been applied in =-=[6, 16, 17, 20, 28]-=- to study the scattering matrix starting from a careful understanding of the structure of the solutions to the Schrödinger equation on a neighborhood of infinity. We will develop the scattering theory... |

39 |
Geometric scattering theory. Stanford Lectures
- Melrose
- 1995
(Show Context)
Citation Context ...es of ∆g2 in X2,ǫ. Then Proposition 8.11, with J = ∅, can be used to prove Theorem 8.1. To prove that C1,λ = C2,λ, λ ∈ R \ 0, we apply the same argument used in the proof of Lemma 3.2, chapter 3.8 of =-=[44]-=-, see also the proof of Lemma 2.1 of [52], to show that for any λ ̸= 0, the set of functions given by ∫ vj(z, λ) = E ∗ ( n ) j + iλ (z, y)φ(y), j = 1, 2, φ ∈ C 2 ∞ (8.29) (M), M where E ∗ j is the Eis... |

37 |
Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds
- Mazzeo
- 1991
(Show Context)
Citation Context ...eferences cited there. More recently there has been interest in this class of manifolds in connection to conformal field theory, see [9, 15, 16] and references cited there. Mazzeo, Mazzeo and Melrose =-=[39, 40, 43]-=- first studied the spectral and scattering theory of the Laplacian in this general setting and gave a thorough description of the resolvent and its meromorphic continuation. Their methods have been ap... |

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 ...ly hyperbolic. So Friedlander’s method can be applied directly. Such metrics have been recently studied by Guillarmou in [17] and include the case where the metric has constant curvature near ∂X, see =-=[19]-=-. In the general case this trick does not work because the resulting operator would not be smooth at r = 0. In section 7 we will need to understand the behavior of the forward radiation field as s → −... |

35 |
Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems 13
- Belishev
- 1997
(Show Context)
Citation Context ...ts own, see for example [23, 33] and references cited there. In section 8 we use the characterization of the scattering matrix through the radiation fields and the boundary control method of Belishev =-=[4]-=-, see also [5, 29, 30], and the book by Katchalov, Kurylev and Lassas [31], to study the inverse problem of determining the manifold and the metric from the scattering matrix at all energies. We prove... |

32 |
The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix
- Perry
- 1989
(Show Context)
Citation Context ...ds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov [8], followed by Lax and Phillips [36, 37], and later by several people, see for example =-=[2, 6, 20, 28, 39, 43, 45]-=- and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field theory, see [9, 15, 16] and references cited there. Mazzeo, Mazzeo and Me... |

28 | inverse scattering on asymptotically hyperbolic manifolds, Duke
- Barreto, fields, et al.
(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 ... of the support theorem are Hörmander’s uniqueness theorem for the Cauchy problem, see Theorem 28.3.4 of[26], and two of its refinements, one due to Alinhac [1] and another one which is due to Tataru =-=[49]-=-. The study of support properties of Radon transforms is a topic of interest in its own, see for example [23, 33] and references cited there. In section 8 we use the characterization of the scattering... |

21 | Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces - Lax, Phillips - 1984 |

19 |
Fonctions zeta de Selberg et surfaces de géométrie finie
- Guillopé
- 1990
(Show Context)
Citation Context ...= β1∗F, λ ∈ C. It is clear from (6.8) that E( n 2 + iλ, y, z′ ) has a conormal singularity at {x′ = 0, y = y ′}. This is the transpose of the Eisenstein function, or Poisson operator, see for example =-=[18, 20, 28]-=-.18 SÁ BARRETO This defines a meromorphic family of operators (6.11) E( n + iλ)f(y) = 2 E( n 2 + iλ) : C∞ 0 ∫ So we conclude from (6.10) and (6.3) that ∫ e −iλs ∫ R+(f1, f2)(y, s)ds = (6.12) R X ( ◦ ... |

19 |
Théorème d’unicité adapté au contrôle des solutions des problème hyperboliques
- Robbiano
- 1991
(Show Context)
Citation Context ...lution u(x, t, y) of (2.5) with initial data (0, f) satisfies (7.42) u(x, t, y) = 0 for {(x, t, y) : 0 < x < x0, log x0 − s0 < t < s0 − log x0}. One particular case of Tataru’s theorem [49], see also =-=[27, 46, 47, 50]-=-, states that if u(t, z) is in H 1 loc and satisfies ( D 2 ) n2 t − ∆ − u(t, z) = 0 4 u(t, z) = 0 − T < t < T, and d(z, z0) < δ, δ > 0, where d(z, z0) is the distance between z and z0 with respect to ... |

18 | Scattering poles for asymptotically hyperbolic manifolds
- Borthwick, Perry
(Show Context)
Citation Context |

18 |
L.Faddeev, Scattering theory and automorphic functions
- Pavlov
- 1975
(Show Context)
Citation Context ...h a metric. There is a history of interest in scattering theory for this class of manifolds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov =-=[8]-=-, followed by Lax and Phillips [36, 37], and later by several people, see for example [2, 6, 20, 28, 39, 43, 45] and references cited there. More recently there has been interest in this class of mani... |

15 |
Radiation fields and hyperbolic scattering theory
- Friedlander
- 1980
(Show Context)
Citation Context ...nd to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s =-=[10, 11, 12, 13, 14]-=-. His program of using the radiation fields to obtain the scattering matrix in that general setting was completed in [48]. Here we carry out the analogous construction on asymptotically hyperbolic man... |

14 | On the representation theorem for solutions of the Helmholtz equation on the hyperbolic space. Partial differential equations and related subjects - Agmon - 1990 |

13 |
Notes on the wave equation on asymptotically Euclidean manifolds
- Friedlander
(Show Context)
Citation Context ...ds and whose ideas were often used here. I also had the pleasure of having several discussions about the subject with him. I thank R. Melrose, who was a student of Friedlander, for bringing the paper =-=[14]-=- to my attention and for several instructive conversations. I also thank C. Guillarmou, Y. Kurylev, R. Mazzeo , P. Stefanov, S. Tang and J. Wunsch for useful conversations and for comments on the pape... |

12 |
Kurylev: To the reconstruction of a Riemannian manifold via its spectral data (BC-method
- Belishev, Y
- 1992
(Show Context)
Citation Context ...r example [23, 33] and references cited there. In section 8 we use the characterization of the scattering matrix through the radiation fields and the boundary control method of Belishev [4], see also =-=[5, 29, 30]-=-, and the book by Katchalov, Kurylev and Lassas [31], to study the inverse problem of determining the manifold and the metric from the scattering matrix at all energies. We prove that the scattering m... |

9 |
Unique continuation for operators with partially analytic coefficients
- Tataru
- 1999
(Show Context)
Citation Context ...lution u(x, t, y) of (2.5) with initial data (0, f) satisfies (7.42) u(x, t, y) = 0 for {(x, t, y) : 0 < x < x0, log x0 − s0 < t < s0 − log x0}. One particular case of Tataru’s theorem [49], see also =-=[27, 46, 47, 50]-=-, states that if u(t, z) is in H 1 loc and satisfies ( D 2 ) n2 t − ∆ − u(t, z) = 0 4 u(t, z) = 0 − T < t < T, and d(z, z0) < δ, δ > 0, where d(z, z0) is the distance between z and z0 with respect to ... |

8 |
Multidimensional inverse problem with incomplete boundary spectral data
- Kurylev
- 1998
(Show Context)
Citation Context ...r example [23, 33] and references cited there. In section 8 we use the characterization of the scattering matrix through the radiation fields and the boundary control method of Belishev [4], see also =-=[5, 29, 30]-=-, and the book by Katchalov, Kurylev and Lassas [31], to study the inverse problem of determining the manifold and the metric from the scattering matrix at all energies. We prove that the scattering m... |

8 |
Inverse boundary spectral problems. Chapman and Hall/CRC
- Katchalov, Kurylev, et al.
- 2001
(Show Context)
Citation Context ... we use the characterization of the scattering matrix through the radiation fields and the boundary control method of Belishev [4], see also [5, 29, 30], and the book by Katchalov, Kurylev and Lassas =-=[31]-=-, to study the inverse problem of determining the manifold and the metric from the scattering matrix at all energies. We prove that the scattering matrix of an asymptotically hyperbolic manifold deter... |

8 |
Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues
- Mazzeo
- 1991
(Show Context)
Citation Context ...Then Proposition 8.11, with J = ∅, put together with Proposition 8.2 proves Theorem 8.1. We should also remark that method of proof of Proposition 8.11 guarantees that the resulting map is C∞ . As in =-=[42]-=-, we recall that the graph of the Calderón projector of ∆gj − λ2 − n2 4 in Xj,ǫ, j = 1, 2, denoted by Cj,λ, is the closed subspace of L2 (Mǫ) ×H 1 (Mǫ) consisting of (f, g) ∈ L2 (Mǫ) ×H 1 (Mǫ) such th... |

7 |
Wave equations on homogeneous spaces
- HELGASON
- 1984
(Show Context)
Citation Context ...bolic space H 3 = {(x, y) : y ∈ R 2 , x ∈ R, x > 0}, with the metric g = dx2 |dy|2 + . x2 x2 (3.1) In this case the radiation fields can be explicitly computed. The formulæ obtained in [32], see also =-=[22]-=-, can be used in the same way to compute the radiation fields in Hn . This is done in [34]. For convenience, we will work in the non-compact model, which does not quite fit the framework of section 1,... |

7 |
Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients
- Robbiano, Zuily
- 1998
(Show Context)
Citation Context ...lution u(x, t, y) of (2.5) with initial data (0, f) satisfies (7.42) u(x, t, y) = 0 for {(x, t, y) : 0 < x < x0, log x0 − s0 < t < s0 − log x0}. One particular case of Tataru’s theorem [49], see also =-=[27, 46, 47, 50]-=-, states that if u(t, z) is in H 1 loc and satisfies ( D 2 ) n2 t − ∆ − u(t, z) = 0 4 u(t, z) = 0 − T < t < T, and d(z, z0) < δ, δ > 0, where d(z, z0) is the distance between z and z0 with respect to ... |

5 |
A uniqueness theorem for second order hyperbolic differential equations
- Hörmander
- 1992
(Show Context)
Citation Context |

5 |
Scattering theory for automorphic forms
- Lax, Philips
- 1974
(Show Context)
Citation Context ...interest in scattering theory for this class of manifolds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov [8], followed by Lax and Phillips =-=[36, 37]-=-, and later by several people, see for example [2, 6, 20, 28, 39, 43, 45] and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field ... |

5 |
Radiation fields on asymptotically Euclidean manifolds
- Barreto
- 2003
(Show Context)
Citation Context ...Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was completed in =-=[48]-=-. Here we carry out the analogous construction on asymptotically hyperbolic manifolds. After defining the radiation fields, we use them to give a unitary translation representation of the wave group a... |

4 |
On the radiation field of pulse solutions of the wave equation
- Friedlander
- 1962
(Show Context)
Citation Context ...nd to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s =-=[10, 11, 12, 13, 14]-=-. His program of using the radiation fields to obtain the scattering matrix in that general setting was completed in [48]. Here we carry out the analogous construction on asymptotically hyperbolic man... |

3 |
Unicité du problème de Cauchy pour des opérateurs du second ordre à symboles réels
- Alinhac
- 1984
(Show Context)
Citation Context ...ansforms. The main ingredients of the proof of the support theorem are Hörmander’s uniqueness theorem for the Cauchy problem, see Theorem 28.3.4 of[26], and two of its refinements, one due to Alinhac =-=[1]-=- and another one which is due to Tataru [49]. The study of support properties of Radon transforms is a topic of interest in its own, see for example [23, 33] and references cited there. In section 8 w... |

3 |
Functions on symmetric spaces. Harmonic Analysis on homogeneous spaces
- Helgason
- 1973
(Show Context)
Citation Context ...n {x ≥ e s0 }. Theorem 7.1 is a “support theorem” in the terminology of Helgason [23, 24] and is a generalization of Theorem 3.13 of [36], which is Theorem 7.1 for the hyperbolic space H 3 . Helgason =-=[21, 24]-=- proved such a result for functions that are compactly supported, but for more general symmetric spaces. It is important to observe, as emphasized by Lax in [33], that this theorem does not have an an... |

3 |
Translation representation for the solution of the non-Euclidean wave equation
- Lax, Phillips
- 1975
(Show Context)
Citation Context ...interest in scattering theory for this class of manifolds, motivated by several problems of mathematics and physics, which goes back to the work of Fadeev and Pavlov [8], followed by Lax and Phillips =-=[36, 37]-=-, and later by several people, see for example [2, 6, 20, 28, 39, 43, 45] and references cited there. More recently there has been interest in this class of manifolds in connection to conformal field ... |

2 | A representation theorem for solutions of Schrödinger type equations on non-compact Riemannian manifolds - Agmon - 1992 |

2 |
The Radon transform and translation representation
- Lax
(Show Context)
Citation Context ...ard fundamental solution of the wave operator along certain light rays. The backward radiation field is defined by reversing the time direction. These are generalizations of the LaxPhillips transform =-=[33, 36]-=- to this class of manifolds. We will show that this leads to a unitary translation representation of the wave group and a dynamical definition of the scattering matrix as in [35]. In section 6 we esta... |

2 |
Unicité de Cauchy pour des opérateurs de type principal
- Lerner, Robbiano
(Show Context)
Citation Context ...ators with this type of degeneracy is due to Alinhac, Theorem 1.1.2 of [1]. Notice that, although P ′ is real, it is not of principal type at {x′ = t ′ = 0}, so the result of Lerner and Robbiano, see =-=[38]-=- or Theorem 28.4.3 of [26], cannot be applied either. The principal symbol of P ′ is p = σ2(P ′ ) = −ξτ − x ′ t ′ h(x ′ t ′ , y, η), h(x ′ t ′ , y, η) = ∑ h ij (x ′ t ′ (7.5) , y)ηiηj. If Hp denotes t... |

1 |
Support theorems in integral geometry and their applications. Differential geometry: geometry in mathematical physics and related topics
- Helgason
- 1993
(Show Context)
Citation Context ...n by the Lax-Phillips transform, which is based on the horocyclic Radon transform. The case on H n is treated in [34]. In section 7 we prove a precise support theorem – in the terminology of Helgason =-=[23, 24]-=- – for the radiation fields. Theorem 7.1 below generalizes to this setting a theorem of Lax-Phillips, Theorem 3.13 of [36], see also [33], obtained for the horocyclic Radon transform. Helgason [21] pr... |

1 |
The Radon Transform. 2nd Edition Vol
- Helgason
- 1999
(Show Context)
Citation Context ...n by the Lax-Phillips transform, which is based on the horocyclic Radon transform. The case on H n is treated in [34]. In section 7 we prove a precise support theorem – in the terminology of Helgason =-=[23, 24]-=- – for the radiation fields. Theorem 7.1 below generalizes to this setting a theorem of Lax-Phillips, Theorem 3.13 of [36], see also [33], obtained for the horocyclic Radon transform. Helgason [21] pr... |

1 |
Incomplete spectral data and the reconstruction of a Riemannian manifold
- Kurylev
- 1993
(Show Context)
Citation Context ...diffeomorphism Ψ : X1 −→ X2, smooth up to M, such that (8.1) Ψ = Id at M and Ψ ∗ g2 = g1. As mentioned in the introduction, the proof is an application of the control method of Belishev [4], see also =-=[5, 31, 29, 30]-=-. We will also use a result, which is an application of this method, and is due to Katchalov and Kurylev [29, 30] First we construct a diffeomorphism between neighborhoods of the boundary that realize... |

1 |
The Euler Poisson-Darboux equation in a Riemannian space
- Kiprijanov, Ivanov
- 1981
(Show Context)
Citation Context ...se of the hyperbolic space H 3 = {(x, y) : y ∈ R 2 , x ∈ R, x > 0}, with the metric g = dx2 |dy|2 + . x2 x2 (3.1) In this case the radiation fields can be explicitly computed. The formulæ obtained in =-=[32]-=-, see also [22], can be used in the same way to compute the radiation fields in Hn . This is done in [34]. For convenience, we will work in the non-compact model, which does not quite fit the framewor... |

1 |
Scattering by a metric. Scattering and Inverse Scattering in Pure and Applied Sciences
- Uhlmann
- 2002
(Show Context)
Citation Context ... with J = ∅, can be used to prove Theorem 8.1. To prove that C1,λ = C2,λ, λ ∈ R \ 0, we apply the same argument used in the proof of Lemma 3.2, chapter 3.8 of [44], see also the proof of Lemma 2.1 of =-=[52]-=-, to show that for any λ ̸= 0, the set of functions given by ∫ vj(z, λ) = E ∗ ( n ) j + iλ (z, y)φ(y), j = 1, 2, φ ∈ C 2 ∞ (8.29) (M), M where E ∗ j is the Eisenstein Function, or Poisson operator, wh... |