Duke Mathematical Journal

Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds

Antônio Sá Barreto

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Abstract

We define the forward and backward radiation fields on an asymptotically hyperbolic manifold and show that they give unitary translation representations of the wave group and as such can be used to define a scattering matrix. We show that this scattering matrix is equivalent to the one defined by stationary methods. Furthermore, we prove a support theorem for the radiation fields which generalizes to this setting well-known results of Helgason [23] and Lax and Phillips [35] for the horocyclic Radon transform. As an application, we use the boundary control method of Belishev [4] to show that an asymptotically hyperbolic manifold is determined up to invariants by the scattering matrix at all energies.

Article information

Source
Duke Math. J., Volume 129, Number 3 (2005), 407-480.

Dates
First available in Project Euclid: 19 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1129729971

Digital Object Identifier
doi:10.1215/S0012-7094-05-12931-3

Mathematical Reviews number (MathSciNet)
MR2169870

Zentralblatt MATH identifier
1154.58310

Subjects
Primary: 81U40: Inverse scattering problems
Secondary: 35P25: Scattering theory [See also 47A40]

Citation

Sá Barreto, Antônio. Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds. Duke Math. J. 129 (2005), no. 3, 407--480. doi:10.1215/S0012-7094-05-12931-3. https://projecteuclid.org/euclid.dmj/1129729971


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References

  • S. Agmon, ``A representation theorem for solutions of Schrödinger type equations on noncompact Riemannian manifolds'' in Méthodes semi-classiques, Vol. 2 (Nantes, France, 1991), Astérisque 210, Soc. Math. France, Montrouge, 1992, 13--26.
  • —, ``On the representation theorem for solutions of the Helmholtz equation on the hyperbolic space'' in Partial Differential Equations and Related Subjects (Trento, Italy, 1990), Pitman Res. Notes Math. Ser. 269, Longman Sci. Tech., Harlow, England, 1992, 1--20.
  • S. Alinhac, Unicité du problème de Cauchy pour des opérateurs du second ordre à symboles réels, Ann. Inst. Fourier (Grenoble) 34 (1984), 89--109.
  • M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997) no. 5, R1--R45.
  • M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations 17 (1992), 767--804.
  • D. Borthwick and P. Perry, Scattering poles for asymptotically hyperbolic manifolds, Trans. Amer. Math. Soc. 354 (2002), 1215--1231.
  • E. B. Davies, Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math. 42, Cambridge Univ. Press, Cambridge, 1995.
  • C. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139--151.
  • F. G. Friedlander, On the radiation field of pulse solutions of the wave equation, Proc. Roy. Soc. Ser. A 269 (1962), 53--65.
  • —, On the radiation field of pulse solutions of the wave equation, II, Proc. Roy. Soc. Ser. A 279 (1964), 386--394.
  • —, On the radiation field of pulse solutions of the wave equation, III, Proc. Roy. Soc. Ser. A 299 (1967), 264--278.
  • —, Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc. 88 (1980), 483--515.
  • —, Notes on the wave equation on asymptotically Euclidean manifolds, J. Funct. Anal. 184 (2001), 1--18.
  • C. R. Graham, ``Volume and area renormalizations for conformally compact Einstein metrics'' in The Proceedings of the 19th Winter School ``Geometry and Physics'' (Srní, Czech Republic, 1999), Rend. Circ. Mat. Palermo (2) Suppl. 63, Circ. Mat. Palermo, Palermo, Italy, 2000, 31--42.
  • C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89--118.
  • C. Guillarmou, Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), 1--37.
  • L. Guillopé, ``Fonctions zêta de Selberg et surfaces de géométrie finie'' in Zeta Functions in Geometry (Tokyo, 1990), Adv. Stud. Pure Math. 21, Kinokuniya, Tokyo, 1992, 33--70.
  • L. Guillopé and M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptot. Anal. 11 (1995), 1--22.
  • —, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2) 145 (1997), 597--660.
  • S. Helgason, ``Functions on symmetric spaces'' in Harmonic Analysis on Homogeneous Spaces (Williamstown, Mass., 1972), Proc. Sympos. Pure Math. 26, Amer. Math. Soc., Providence, 1973, 101--146.
  • —, ``Wave equations on homogeneous spaces'' in Lie Group Representations, III (College Park, Md., 1982/1983), Lecture Notes in Math. 1077, Springer, Berlin, 1984, 254--287.
  • —, ``Support theorems in integral geometry and their applications'' in Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, 1990), Proc. Sympos. Pure Math. 54, Part 2, Amer. Math. Soc., Providence, 1993, 317--323.
  • —, The Radon Transform, 2nd ed., Progr. Math. 5, Birkhäuser, Boston, 1999.
  • L. HöRmander, Linear Partial Differential Operators, Grundlehren Math. Wiss. 116, Springer, Berlin, 1963.
  • —, The Analysis of Linear Partial Differential Operators, Vol. 4: Fourier Integral Operators, Grundlehren Math. Wiss. 275, Springer, Berlin, 1985.
  • —, A uniqueness theorem for second order hyperbolic differential equations, Comm. Partial Differential Equations 17 (1992), 699--714.
  • M. S. Joshi and A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41--86.
  • A. P. Katchalov and Y. V. KurylëV, Incomplete spectral data and the reconstruction of a Riemannian manifold, J. Inverse Ill-Posed Probl. 1 (1993), 141--153.
  • —, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations 23 (1998), 55--95.
  • A. P. Katchalov, Y. V. KurylëV, and M. Lassas, Inverse Boundary Spectral Problems, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 123, Chapman & Hall/CRC, Boca Raton, Fla., 2001.
  • I. A. Kipriyanov and L. A. Ivanov, Euler-Poisson-Darboux equations in Riemannian space (in Russian), Dokl. Akad. Nauk SSSR 260 (1981), 790--794.; English translation in Soviet Math. Dokl. 24 (1981), 331--335.
  • P. D. Lax, The Radon transform and translation representation, J. Evol. Equ. 1 (2001), 311--323.
  • P. D. Lax and R. S. Phillips, Scattering Theory for Automorphic Functions, Ann. of Math. Stud. 87, Princeton Univ. Press, Princeton, 1976.
  • —, Translation representations for the solution of the non-Euclidean wave equation, Comm. Pure Appl. Math. 32 (1979), 617--667.; Correction, Comm. Pure Appl. Math. 33 (1980), 685. ;
  • —, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, III, Comm. Pure Appl. Math. 38 (1985), 179--207.
  • —, Scattering Theory, 2nd ed., Pure Appl. Math. 26, Academic Press, Boston, 1989.
  • N. Lerner and L. Robbiano, Unicité de Cauchy pour des opérateurs de type principal, J. Anal. Math. 44 (1984/85), 32--66.
  • R. R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309--339.
  • —, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1615--1664.
  • —, Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues, Internat. Math. Res. Notices 1991, no. 4, 41--48.
  • —, Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, Amer. J. Math. 113 (1991), 25--45.
  • R. R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260--310.
  • R. B. Melrose, Geometric Scattering Theory, Stanford Lectures, Cambridge Univ. Press, Cambridge, 1995.
  • B. S. Pavlov and L. D. Faddeev, ``Scattering theory and automorphic functions'' (in Russian) in Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 6 (in Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27, Izdat. ``Nauka,'' Leningrad, 1972, 161--193.
  • P. A. Perry, The Laplace operator on a hyperbolic manifold, II: Eisenstein series and the scattering matrix, J. Reine Angew. Math. 398 (1989), 67--91.
  • L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations 16 (1991), 789--800.
  • L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math. 131 (1998), 493--539.
  • A. Sá Barreto, Radiation fields on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 28 (2003), 1661--1673.
  • D. Tataru, Unique continuation for solutions to PDE's: Between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations 20 (1995), 855--884.
  • —, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), 505--521.
  • M. E. Taylor, Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, 1981.
  • G. Uhlmann, ``Scattering by a metric'' in Scattering, Vol. 2: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, San Diego, 2002, 1668--1677.