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The radiation field is a Fourier integral operator. (English) Zbl 1091.58018
Authors’ abstract: We show that the “radiation field” introduced by F. G. Friedlander, mapping Cauchy data for the wave equation to the rescaled asymptotic behavior of the wave, is a Fourier integral operator on any non-trapping asymptotically hyperbolic or asymptotically conic manifold. The underlying canonical relation is associated to a “sojourn time” or “Busemann function” for geodesics. As a consequence we obtain some information about the high frequency behavior of the scattering Poisson operator in these geometric settings.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 35L05 Wave equation 35P25 Scattering theory for PDEs 58J45 Hyperbolic equations on manifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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##### References:
 [1] Structure of the semi-classical amplitude for general scattering relations · Zbl 1088.81091 [2] Fourier integral operators, 130, (1996), Boston, MA · Zbl 0841.35137 [3] The spectrum of positive elliptic operators and periodic geodesics, Invent. Math, 29, 39-79, (1975) · Zbl 0307.35071 [4] Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc., 88, 3, 483-515, (1980) · Zbl 0465.35068 [5] Notes on the wave equation on asymptotically Euclidean manifolds, J. Funct. Anal., 184, 1, 1-18, (2001) · Zbl 0997.58013 [6] Volume and area renormalizations for conformally compact Einstein metrics, 63, 31-42, (2000) · Zbl 0984.53020 [7] Sojourn times and asymptotic properties of the scattering matrix, 12, 69-88, (197677), Kyoto Univ. · Zbl 0381.35064 [8] The Schrödinger propagator for scattering metrics, (2003) [9] Recovering asymptotics of metrics from fixed energy scattering data, Invent. Math., 137, 1, 127-143, (1999) · Zbl 0953.58025 [10] Inverse scattering on asymptotically hyperbolic manifolds, Acta Math., 184, 1, 41-86, (2000) · Zbl 1142.58309 [11] Riemannian geometry and geometric analysis, third ed., (2002), Springer-Verlag, Berlin · Zbl 1034.53001 [12] Scattering theory, (1967), Academic Press, New York · Zbl 0186.16301 [13] High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering, Comm. Pure Appl. Math., 29, 261-291, (1976) · Zbl 0463.35048 [14] Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75, 2, 260-310, (1987) · Zbl 0636.58034 [15] Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces,, Spectral and scattering theory (Sanda, 1992), 85-130, (1994), Marcel Dekker · Zbl 0837.35107 [16] Geometric scattering theory, (1995) · Zbl 0849.58071 [17] Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) , 85-130, (1994), Dekker, New York · Zbl 0837.35107 [18] Sojourn times, singularities of the scattering kernel and inverse problems, 47, (2003), Cambridge University Press, to appear. · Zbl 1086.35146 [19] Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits, Ann. Inst. Fourier (Grenoble), 39, 1, 155-192, (1989) · Zbl 0659.35026 [20] Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds · Zbl 1154.58310 [21] Radiation fields on asymptotically Euclidean manifolds, Comm. Partial Differential Equations, 28, 9-10, 1661-1673, (2003) · Zbl 1037.58020
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