Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

References

[1] Agmon S. and Hörmander L., Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976), 1–30. 10.1007/BF02786703Search in Google Scholar

[2] Anderson M., Katsuda A., Kurylev Y., Lassas M. and Taylor M., Boundary regularity for the Ricci equation, geometric convergence, and Gelfand’s inverse boundary problem, Invent. Math. 158 (2004), 261–321. 10.1007/s00222-004-0371-6Search in Google Scholar

[3] Astala K., Lassas M. and Paivarinta L., The Calderon’s inverse problem – Imaging and invisibility, Inside out II, Cambridge University Press, Cambridge (2013), 1–54. Search in Google Scholar

[4] Belishev M., An approach to multidimensional inverse problems for the wave equation (Russian), Dokl. Akad. Nauk SSSR 297 (1987), 524–527; translation in Sov. Math. Dokl. 36 (1988), 481–484. Search in Google Scholar

[5] Belishev M. and Blagoveščenskii A., Direct method for solving the non-stationary inverse problem for a multidimensional wave equation (Russian), Ill-posed problems of mathematical physics and analysis, Computer Center, Siberian Branch of USSR Academy of Sciences, Krasnojarsk (1988), 43–48. Search in Google Scholar

[6] Belishev M. and Demchenko M., Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements, J. Inverse Ill-Posed Probl. 19 (2011), 167–188. 10.1515/jiip.2011.028Search in Google Scholar

[7] Belishev M. and Kurylev V., To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations 17 (1992), 767–804. 10.1080/03605309208820863Search in Google Scholar

[8] Belishev M. and Vakulenko A., s-points in three-dimensional acoustic scattering, SIAM J. Math. Anal. 42 (2010), 2703–2720. 10.1137/090781486Search in Google Scholar

[9] Bingham K., Kurylev Y., Lassas M. and Siltanen S., Iterative time reversal control for inverse problems, Inverse Probl. Imaging 2 (2008), 63–81. 10.3934/ipi.2008.2.63Search in Google Scholar

[10] Blagoveščenskii A., A one-dimensional inverse boundary value problem for a second order hyperbolic equation (Russian), Zap. Nauchn. Semin. LOMI 15 (1969), 85–90. Search in Google Scholar

[11] Blagoveščenskii A., Inverse boundary problem for the wave propagation in an anisotropic medium (Russian), Tr. Mat. Inst. Steklova, 65 (1971), 39–56. Search in Google Scholar

[12] Borthwick D., Spectral theory of infinite-area hyperbolic surfaces, Progr. Math. 256, Birkhäuser-Verlag, Boston 2007. Search in Google Scholar

[13] Borthwick D., Judge C. and Perry P., Sel’berg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv. 80 (2005), 483–515. 10.4171/CMH/23Search in Google Scholar

[14] Borthwick D. and Perry P., Inverse scattering results for manifolds hyperbolic near infinity, J. Geom. Anal. 21 (2011), 305–333. 10.1007/s12220-010-9149-9Search in Google Scholar

[15] Dos Santos Ferreira D., Kenig C., Salo M. and Uhlmann G., Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), 119–171. 10.1007/s00222-009-0196-4Search in Google Scholar

[16] Faddeev L., Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Lobacevskii plane (Russian), Tr. Moscov. Mat. 17 (1967), 323–350; translation in Trans. Moscow Math. Soc. 17 (1967), 357–386. 10.1142/9789814340960_0005Search in Google Scholar

[17] Greenleaf A., Kurylev Y., Lassas M. and Uhlmann G., Invisibility and inverse problems, Bull. Amer. Math. 46 (2009), 55–97. 10.1090/S0273-0979-08-01232-9Search in Google Scholar

[18] Guillarmou C., Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), no. 1, 1–37. 10.1215/S0012-7094-04-12911-2Search in Google Scholar

[19] Guillarmou C. and Mazzeo R., Resolvent of the Laplacian on geometrically finite hyperbolic manifolds, Invent. Math. 187 (2012), 99–144. 10.1007/s00222-011-0330-ySearch in Google Scholar

[20] Guillarmou C. and Sa Barreto A., Scattering and inverse scattering on ACH manifolds, J. reine angew. Math. 622 (2008), 1–55. 10.1515/CRELLE.2008.064Search in Google Scholar

[21] Guillarmou C., Salo M. and Tzou L., Inverse scattering at fixed energy on surfaces with Euclidean ends, Comm. Math. Phys. 303 (2011), 761–784. 10.1007/s00220-011-1224-ySearch in Google Scholar

[22] Guillarmou C. and Tzou L., Calderon inverse problem with partial data on Riemann surfaces, Duke Math. J. 158 (2011), no. 1, 83–120. 10.1215/00127094-1276310Search in Google Scholar

[23] Guillarmou C. and Tzou L., Identification of a connection from Cauchy data space on a Riemann surface with boundary, Geom. Funct. Anal. 21 (2011), no. 2, 393–418. 10.1007/s00039-011-0110-2Search in Google Scholar

[24] Guillopé L. and Zworski M., Polynomial bounds of the number of resonances for some complex spaces of constant negative curvature near infinity, Asymptot. Anal. 11 (1995), 1–22. 10.3233/ASY-1995-11101Search in Google Scholar

[25] Guillopé L. and Zworski M., Scattering asymptotics for Riemann surfaces, Ann. of Math. (2) 145 (1997), 597–660. 10.2307/2951846Search in Google Scholar

[26] Helgason S., A duality for symmetric spaces with applications to group representations, Adv. Math. 5 (1970), 1–154. 10.1016/0001-8708(70)90037-XSearch in Google Scholar

[27] Henkin G. and Michel V., On the explicit reconstruction of a Riemann surface from its Dirichlet–Neumann operator, Geom. Funct. Anal. 17 (2007), 116–155. 10.1007/s00039-006-0590-7Search in Google Scholar

[28] Henkin G. and Novikov R., The ∂¯-approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal. 18 (2008), 612–631. 10.1007/s12220-008-9015-1Search in Google Scholar

[29] Isozaki H., Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space, Amer. J. Math. 126 (2004), 1261–1313. 10.1353/ajm.2004.0047Search in Google Scholar

[30] Isozaki H. and Kurylev Y., Introduction to spectral theory and inverse problems on asymptotically hyperbolic manifolds, MSJ Mem. 32, World Scientific, Hackensack 2014. 10.2969/msjmemoirs/032010000Search in Google Scholar

[31] Isozaki H., Kurylev Y. and Lassas M., Forward and inverse scattering on manifolds with asymptotically cylindrical ends, J. Funct. Anal. 258 (2010), 2060–2118. 10.1016/j.jfa.2009.11.009Search in Google Scholar

[32] Isozaki H., Kurylev Y. and Lassas M., Spectral theory and inverse problem on asymptotically hyperbolic orbifolds, preprint 2013, http://arxiv.org/abs/1312.0421. 10.1142/e040Search in Google Scholar

[33] Joshi M. and Sá Barreto A., Recovering asymptotics of metrics from fixed energy scattering data, Invent. Math. 137 (1999), 127–143. 10.1007/s002220050326Search in Google Scholar

[34] Joshi M. and Sá Barreto A., Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41–86. 10.1007/BF02392781Search in Google Scholar

[35] Kashiwara M., Kowata A., Minemura K., Okamoto K., Oshima T. and Tanaka M., Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), 1–39. 10.2307/1971253Search in Google Scholar

[36] Katchalov A. and Kurylev Y., Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations 23 (1998), 55–95. 10.1080/03605309808821338Search in Google Scholar

[37] Katchalov A., Kurylev Y. and Lassas M., Inverse boundary spectral problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123, Chapman and Hall/CRC, Boca Raton 2001. 10.1201/9781420036220Search in Google Scholar

[38] Katchalov A., Kurylev Y., Lassas M. and Mandache N., Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems 20 (2004), 419–436. 10.1088/0266-5611/20/2/007Search in Google Scholar

[39] Katok S., Fuchsian groups, University of Chicago Press, Chicago 1992. Search in Google Scholar

[40] Kenig C., Salo M. and Uhlmann G., Reconstructions from boundary measurements on admissible manifolds, Inverse Probl. Imaging 5 (2011), 859–877. 10.3934/ipi.2011.5.859Search in Google Scholar

[41] Krupchyk K., Kurylev Y. and Lassas M., Inverse spectral problems on a closed manifold, J. Math. Pures Appl. (9) 90 (2008), 42–59. 10.1016/j.matpur.2008.02.009Search in Google Scholar

[42] Kurylev Y., A multidimensional Gelfand–Levitan inverse boundary problem, Differential equations and mathematical physics (Birmingham 1994), International Press, Cambridge (1995), 117–131. Search in Google Scholar

[43] Kurylev Y. and Lassas M., Gelfand inverse problem for a quadratic operator pencil, J. Funct. Anal. 176 (2000), 247–263. 10.1006/jfan.2000.3615Search in Google Scholar

[44] Kurylev Y. and Lassas M., Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 931–949. 10.1017/S0308210500001943Search in Google Scholar

[45] Kurylev Y. and Lassas M., Inverse problems and index formulae for Dirac operators, Adv. Math. 221 (2009), 170–216. 10.1016/j.aim.2008.12.001Search in Google Scholar

[46] Kurylev Y., Lassas M. and Somersalo E., Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems, J. Math. Pures Appl. (9) 86 (2006), 237–270. 10.1016/j.matpur.2006.01.008Search in Google Scholar

[47] Kurylev Y., Lassas M. and Yamaguchi T., Uniqueness and stability in inverse spectral problems for collapsing manifolds, preprint 2012, http://arxiv.org/abs/1209.5875. Search in Google Scholar

[48] Lassas M. and Oksanen L., Inverse problem for wave equation with sources and observations on disjoint set, Inverse Problems 26 (2010), Article ID 085012. 10.1088/0266-5611/26/8/085012Search in Google Scholar

[49] Lassas M., Taylor M. and Uhlmann G., The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), no. 2, 207–221. 10.4310/CAG.2003.v11.n2.a2Search in Google Scholar

[50] Lassas M. and Uhlmann G., On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 771–787. 10.1016/S0012-9593(01)01076-XSearch in Google Scholar

[51] Lee J. and Uhlmann G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), 1097–1112. 10.1002/cpa.3160420804Search in Google Scholar

[52] Lehner J., A short course in automorphic functions, Holt, Rinhart and Winston, New York 1966. Search in Google Scholar

[53] Mazzeo R. and Melrose R., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260–310. 10.1016/0022-1236(87)90097-8Search in Google Scholar

[54] Minemura K., Eigenfunctions of the Laplacian on a real hyperbolic space, J. Math. Soc. Japan 27 (1975), 82–105. 10.2969/jmsj/02710082Search in Google Scholar

[55] Novikov R., Reconstruction of a two-dimensional Schršdinger operator from the scattering amplitude at fixed energy (Russian), Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 90–91; translation in Funct. Anal. Appl. 20 (1986), 246–248. Search in Google Scholar

[56] Novikov R., Multidimensional inverse spectral problem for the equation -Δ⁢ψ+(v⁢(x)-E⁢u⁢(x))⁢ψ=0 (Russian), Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 11–22; Funct. Anal. Appl. 22 (1988), 263–278. Search in Google Scholar

[57] Novikov R., The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal. 103 (1992), 409–463. 10.1016/0022-1236(92)90127-5Search in Google Scholar

[58] Perry P., The Laplace operator on a hyperbolic manifold. I. Spectral and scattering theory, J. Funct. Anal. 75 (1987), 161–187. 10.1016/0022-1236(87)90110-8Search in Google Scholar

[59] Perry P., The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix, J. reine angew. Math. 398 (1989), 67–91. 10.1515/crll.1989.398.67Search in Google Scholar

[60] Sá Barreto A., Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), 407–480. 10.1215/S0012-7094-05-12931-3Search in Google Scholar

[61] Satake I., The Gauss–Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. 10.2969/jmsj/00940464Search in Google Scholar

[62] Tataru D., Unique continuation for solutions to PDEs, between Hörmander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations 20 (1995), 855–884. 10.1080/03605309508821117Search in Google Scholar

[63] Tataru D., Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), 505–521. 10.1016/S0021-7824(99)00016-1Search in Google Scholar

[64] Thurston W., The geometry and topology of three-manifolds, preprint 2002, http://www.msri.org/publications/books/gt3m. Search in Google Scholar

[65] Vasy A., Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, Invent. Math. 194 (2013), 381–513. 10.1007/s00222-012-0446-8Search in Google Scholar

[66] Watson G. N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge 1962. Search in Google Scholar

[67] Zelditch S., Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992), 221–260. 10.1080/03605309208820840Search in Google Scholar