## Acta Mathematica

### Inverse scattering on asymptotically hyperbolic manifolds

#### Article information

Source
Acta Math., Volume 184, Number 1 (2000), 41-86.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485891302

Digital Object Identifier
doi:10.1007/BF02392781

Mathematical Reviews number (MathSciNet)
MR1756569

Zentralblatt MATH identifier
1142.58309

Rights

#### Citation

Joshi, Mark S.; Barreto, Antônio Sá. Inverse scattering on asymptotically hyperbolic manifolds. Acta Math. 184 (2000), no. 1, 41--86. doi:10.1007/BF02392781. https://projecteuclid.org/euclid.acta/1485891302

#### References

• Agmon, S., A representation theorem for solutions of Schrödinger type equations on noncompact Riemannian manifolds, in Méthodes semi-classiques Vol. 2 (Nantes, 1991), pp. 13–26. Asterisque, 210. Soc. Math. France, Paris, 1992.
• — On the representation theorem for solutions of the Helmholtz, equation on the hyperbolic space, in Partial Differential Equations and Related Subjects (Trento, 1990), pp. 1–20. Pitman Res. Notes Math. Ser., 269. Longman Sci. Tech, Harlow, 1992.
• Andersson, L. & Chrusciel, P. T., On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of scri. Comm. Math. Phys., 161 (1994), 533–568.
• Andersson, L., Chrusciel, P. T. & Friedrich, H., On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations. Comm. Math. Phys., 149 (1992), 587–612.
• Bachelot, A. & Motet-Bachelot, A., Les résonances d'un trou noir de Schwarzschild. Ann. Inst. H. Poincaré Phys. Theor., 59 (1996), 3–68.
• Borthwick, D., Scattering theory and deformations of asymptotically hyperbolic metrics. Preprint, 1997.
• Borthwick, D., McRae, A. & Taylor, E., Quasirigidity of hyperbolic 3-manifolds and scattering theory. Duke Math. J., 89 (1997), 225–236.
• Borthwick, D. & Perry, P., Scattering poles for asymptotically hyperbolic manifolds. Preprint, 1999.
• Chandrasekar, S., The Mathematical Theory of Black Holes. Oxford Univ. Press, New York, 1983.
• Chandrasekar, S. & Detweiler, S., The quasi-normal modes of the Schwarzschild black hole. Proc. Roy. Soc. London Ser. A, 344 (1975), 441–452.
• Christiansen, T., Weyl asymptotics for the Laplacian on asymptotically Euclidean spaces. Amer. J. Math., 121 (1999), 1–22.
• Gelfand, I.M. & Shilov, G.E., Generalized Functions, Vol. 1. Academic Press, New York-London, 1964.
• Guillopé, L., Fonctions zêta de Selberg et surfaces de géométrie finie, in Zeta Functions in Geometry (Tokyo, 1990), pp. 33–70, Adv. Stud. Pure Math., 21. Kinokuniya, Tokyo, 1992.
• Guillopé, L. & Zworski, M., Scattering asymptotics for Riemann surfaces. Ann. of Math., 145 (1997), 597–660.
• — Upper bounds on the number of resonances for non-compact Riemann surfaces. J. Funct. Anal., 129 (1995), 364–389.
• — Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Asymptot. Anal., 11 (1995), 1–22.
• Hislop, P., The geometry and spectra of hyperbolic manifolds. Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 715–776.
• Hörmander, L., The Analysis of Linear Partial Differential Operators, Vol. 1., Grundlehren Math. Wiss., 256. Springer-Verlag, Berlin-New York, 1983.
• Joshi, M. S., An intrinsic characterization of polyhomogeneous Lagrangian distributions. Proc. Amer. Math. Soc., 125 (1997), 1537–1543.
• — A symbolic construction of the forward fundamental solution, of the wave operator. Comm. Partial Differential Equations, 23 (1998), 1349–1417.
• — Recovering asymptotics of Coulomb-like potentials. SIAM J. Math. Anal., 30 (1999), 516–526.
• Joshi, M. S. A model form for exact b-metrics. To appear in Proc. Amer. Math. Soc.
• Joshi, M. S. & Sá Barreto, A., Recovering the asymptotics of a short range potential. Comm. Math. Phys., 193 (1998), 197–208.
• —, Recovering asymptotics of metrics from fixed energy scattering data. Invent. Math., 137 (1999), 127–143.
• Lax, P. & Phillips, R., Scattering Theory for Automorphic Functions. Ann. of Math. Stud., 87. Princeton Univ. Press, Princeton, NJ, 1976.
• Mazzeo, R., Elliptic theory of differential edge operators, I. Comm. Partial Differential Equations, 16 (1991), 1615–1664.
• — Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds. Amer. J. Math., 113 (1991), 25–45.
• Mazzeo, R. & Melrose, R.B., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal., 75 (1987), 260–310.
• Melrose, R., Geometric Scattering Theory. Cambridge Univ. Press, Cambridge, 1995.
• —, Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, in Spectral and Scattering Theory (Sanda, 1992), pp. 85–130. Dekker, New York, 1994.
• Melrose, R. Differential analysis on manifolds with corners. In preparation.
• Melrose, R. Geometric optics and the bottom of the spectrum Preprint.
• Melrose, R. & Zworski, M., Scattering metrics and geodesic flow at infinity. Invent. Math., 124 (1996), 389–436.
• O'Neill, B., Semi-Reimannian Geometry. With Applications to Relativity. Pure Appl. Math., 103. Academic Press, New York-London, 1983.
• Parnovski, L., Scattering matrix for manifolds with conical ends. Preprint.
• Patterson, S. J., The Laplacian operator on a Riemann surface. Compositio Math., 31 (1975), 83–107.
• Perry, P., The Laplace operator on a hyperbolic manifold, II. Eisenstein series and the scattering matrix. J. Reine Angew. Math., 398 (1989), 67–91.
• — A trace class rigidity theorem for Kleinian groups. Ann. Acad. Sci. Fenn. Ser. A I Math., 20 (1995), 251–257.
• Sá Barreto, A. & Zworski, M., Distribution of resonances for spherical black holes. Math. Res. Lett., 4 (1997), 103–121.
• Sell, G., Smooth linearization near a fixed point. Amer. J. Math., 107 (1985), 1035–1091.