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

Duke Mathematical Journal

Volumn 129, Issue 3, 2005, Pages 407-480

  Sá Barreto, Antônio ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27744463549     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-05-12931-3     Document Type: Article

References (53)

1. 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. MR 1221349
2. _, "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. MR 1190930
3. S. ALINHAC, Unicité du problème de Cauchy pour des opérateurs du second ordre à symboles réels, Ann. Inst. Fourier (Grenoble) 34 (1984), 89-109. MR 0746497
4. M. I. BELISHEV, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997) no. 5, R1 - R45. MR 1474359
5. 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. MR 1177292
6. D. BORTHWICK and P. PERRY, Scattering poles for asymptotically hyperbolic manifolds, Trans. Amer. Math. Soc. 354 (2002), 1215-1231. MR 1867379
7. E. B. DAVIES, Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math. 42, Cambridge Univ. Press, Cambridge, 1995. MR 1349825
8. C. FEFFERMAN and C. R. GRAHAM, Q-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139-151. MR 1909634
9. F. G. FRIEDLANDER, On the radiation field of pulse solutions of the wave equation, Proc. Roy. Soc. Ser. A 269 (1962), 53-65. MR 0142888
10. _, On the radiation field of pulse solutions of the wave equation, II, Proc. Roy. Soc. Ser. A 279 (1964), 386-394. MR 0164132
11. _, On the radiation field of pulse solutions of the wave equation, III, Proc. Roy. Soc. Ser. A 299 (1967), 264-278. MR 0226218
12. _, Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc. 88 (1980), 483-515. MR 0583989
13. _, Notes on the wave equation on asymptotically Euclidean manifolds, J. Funct. Anal. 184 (2001), 1-18. MR 1846782
14. 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. MR 1758076
15. C. R. GRAHAM and M. ZWORSKI, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118. MR 1965361
16. C. GUILLARMOU, Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), 1-37. MR 2153454
17. 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. MR 1210782
18. 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. MR 1344252
19. _, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2) 145 (1997), 597-660. MR 1454705
20. 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. MR 0346429
21. _, "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. MR 0765556
22. _, "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. MR 1216548
23. _, The Radon Transform, 2nd ed., Progr. Math. 5, Birkhäuser, Boston, 1999. MR 1723736
24. L. HÖRMANDER, Linear Partial Differential Operators, Grundlehren Math. Wiss. 116, Springer, Berlin, 1963. MR 0161012
25. _, The Analysis of Linear Partial Differential Operators, Vol. 4: Fourier Integral Operators, Grundlehren Math. Wiss. 275, Springer, Berlin, 1985. MR 0781537
26. _, A uniqueness theorem for second order hyperbolic differential equations, Comm. Partial Differential Equations 17 (1992), 699-714. MR 1177289
27. M. S. JOSHI and A. SÁ BARRETO, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41-86. MR 1756569
28. 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. MR 1256113
29. _, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations 23 (1998), 55-95. MR 1608496
30. 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. MR 1889089
31. I.A. KIPRIYANOV and L. A. IVANOV, Euler-Poisson-Darboux equations in Riemannian space (in Russian), Dokl. Akad. Nauk SSSR 260 (1981), 790-794;
32. English translation in Soviet Math. Dokl. 24 (1981), 331-335. MR 0631801
33. P. D. LAX, The Radon transform and translation representation, J. Evol. Equ. l (2001), 311-323. MR 1861225
34. P. D. LAX and R. S. PHILLIPS, Scattering Theory for Automorphic Functions, Ann. of Math. Stud. 87, Princeton Univ. Press, Princeton, 1976. MR 0562288
35. _, 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. MR 0533296; MR 0586418
36. _, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces. III, Comm. Pure Appl. Math. 38 (1985), 179-207. MR 0780072
37. _, Scattering Theory, 2nd ed., Pure Appl. Math. 26, Academic Press, Boston, 1989. MR 1037774
38. N. LERNER and L. ROBBIANO, Unicité de Cauchy pour des opérateurs de type principal, J. Anal. Math. 44 (1984/85), 32-66. MR 0801286
39. R. R. MAZZEO, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309-339. MR 0961517
40. _, Elliptic theory of differential edge operators, 1, Comm. Partial Differential Equations 16(1991), 1615-1664. MR 1133743
41. _, Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues, Internat. Math. Res. Notices 1991, no. 4, 41-48. MR 1121164
42. _, Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, Amer. J. Math. 113 (1991), 25-45. MR 1087800
43. 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. MR 0916753
44. R. B. MELROSE, Geometric Scattering Theory, Stanford Lectures, Cambridge Univ. Press, Cambridge, 1995. MR 1350074
45. 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. MR 0320781
46. 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. MR 0998472
47. L. ROBBIANO, Théorème d' unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations 16 (1991), 789-800. MR 1113107
48. L. ROBBIANO and C. ZUILY, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math. 131 (1998), 493-539. MR 1614547
49. A. SÁ BARRETO, Radiation fields on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 28 (2003), 1661-1673. MR 2001178
50. 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. MR 1326909
51. _, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), 505-521. MR 1697040
52. M. E. TAYLOR, Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, 1981. MR 0618463
53. 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. MR 1878885