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August  2016, 10(3): 659-688. doi: 10.3934/ipi.2016016

## Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

 1 Département de Mathématiques, Université de Cergy-Pontoise, UMR CNRS 8088, 2 Av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France 2 Département de Mathématiques, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes cedex 03, France, France

Received  January 2015 Revised  October 2015 Published  August 2016

In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schrödinger equation to prove local uniqueness results in the setting of inverse metric problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(\lambda,n)$ and reflection coefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixed energy $\lambda$ and angular momentum $n$. For instance, the reflection coefficients $L(\lambda,n)$ correspond to the scattering experiment in which a wave is sent from the left end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $\lambda \not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp. $R(\lambda,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with $B>0$ - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.
Citation: Thierry Daudé, Damien Gobin, François Nicoleau. Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds. Inverse Problems and Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016
##### References:
 [1] T. Aktosun, M. Klaus and C. van der Mee, Direct and inverse scattering for selfadjoint Hamiltonian systems on the line, Integr. Equa. Oper. Theory, 38 (2000), 129-171. doi: 10.1007/BF01200121. [2] C. Bennewitz, A proof of the local Borg-Marchenko Theorem, Comm. Math. Phys., 218 (2001), 131-132. doi: 10.1007/s002200100384. [3] R. P. Boas, Entire Functions, Academic Press, 1954. [4] D. Borthwick and P. A. Perry, Inverse scattering results for manifolds hyperbolic near infinity, J. of Geom. Anal., 21 (2001), 305-333. doi: 10.1007/s12220-010-9149-9. [5] J. M. Cohen and R. T. Powers, The general relativistic hydrogen atom, Comm. Math. Phys., 86 (1982), 69-86. doi: 10.1007/BF01205662. [6] T. Daudé, Time-dependent scattering theory for massive charged Dirac fields by a Reissner-Nordström black hole, J. Math. Phys., 51 (2010), 102504, 57pp. doi: 10.1063/1.3499403. [7] T. Daudé, N. Kamran and F. Nicoleau, Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces, to appear in Inverse Problems, preprint, arXiv:1409.6229. [8] T. Daudé and F. Nicoleau, Inverse scattering in (de Sitter)-Reissner-Nordström black hole spacetimes, Rev. Math. Phys., 22 (2010), 431-484. doi: 10.1142/S0129055X10004004. [9] T. Daudé and F. Nicoleau, Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes, Annales Henri Poincaré, 12 (2011), 1-47. doi: 10.1007/s00023-010-0069-9. [10] G. Freiling and V. Yurko, Inverse problems for differential operators with singular boundary conditions, Math. Nachr., 278 (2005), 1561-1578. doi: 10.1002/mana.200410322. [11] F. Gesztesy and B. Simon, On local Borg-Marchenko uniqueness results, Comm. Math. Phys., 211 (2005), 273-287. doi: 10.1007/s002200050812. [12] M. Horváth, Partial identification of the potential from phase shifts, J. Math. Anal. Appl., 380 (2011), 726-735. doi: 10.1016/j.jmaa.2010.10.071. [13] H. Isozaki and J. Kurylev, Introduction to spectral theory and inverse problems on asymptotically hyperbolic manifolds, MSJ Memoirs, Mathematical Society of Japan, Tokyo, (2014), viii+251 pp, arXiv:1102.5382. doi: 10.1142/e040. [14] M. S. Joshi and A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Mathematica, 184 (2000), 41-86. doi: 10.1007/BF02392781. [15] K. Lake, Reissner-Nordström-de Sitter metric, the third law, and cosmic censorship, Phys. Rev. D., 19 (1979), 421-429. doi: 10.1103/PhysRevD.19.421. [16] B. Y. Levin, Lectures on Entire Functions, Translations of Mathematical Monograph, 150, American Mathematical Society, 1996. [17] F. Melnik, Scattering on Reissner-Nordström metric for massive charged spin $\frac{1}{2}$ fields, Annales Henri Poincaré, 4 (2003), 813-846. doi: 10.1007/s00023-003-0148-2. [18] J.-P. Nicolas, Scattering of linear Dirac fields by a spherically symmetric black hole, Annales Institut Henri Poincaré, 62 (1995), 145-179. [19] A. G. Ramm, An Inverse Scattering Problem with part of the Fixed-Energy Phase shifts, Comm. Math. Phys., 207 (1999), 231-247. doi: 10.1007/s002200050725. [20] T. Regge, Introduction to complex orbital momenta, Nuevo Cimento, 14 (1959), 951-976. doi: 10.1007/BF02728177. [21] A. Sá Barreto, Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds, Duke Math. Journal, 129 (2005), 407-480. doi: 10.1215/S0012-7094-05-12931-3. [22] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Company, New York, 1987. [23] B. Simon, A new approach to inverse spectral theory, I. fundamental formalism, Annals of Math., 150 (1999), 1029-1057. doi: 10.2307/121061. [24] G. Teschl, Mathematical Methods in Quantum Mechanics, Graduate Studies in Mathematics, 99, AMS Providence, Rhode Island, 2009. doi: 10.1090/gsm/099. [25] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, 1992. doi: 10.1007/978-3-662-02753-0. [26] R. M. Wald, General Relativity, The University of Chicago Press, 1984. doi: 10.7208/chicago/9780226870373.001.0001.

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##### References:
 [1] T. Aktosun, M. Klaus and C. van der Mee, Direct and inverse scattering for selfadjoint Hamiltonian systems on the line, Integr. Equa. Oper. Theory, 38 (2000), 129-171. doi: 10.1007/BF01200121. [2] C. Bennewitz, A proof of the local Borg-Marchenko Theorem, Comm. Math. Phys., 218 (2001), 131-132. doi: 10.1007/s002200100384. [3] R. P. Boas, Entire Functions, Academic Press, 1954. [4] D. Borthwick and P. A. Perry, Inverse scattering results for manifolds hyperbolic near infinity, J. of Geom. Anal., 21 (2001), 305-333. doi: 10.1007/s12220-010-9149-9. [5] J. M. Cohen and R. T. Powers, The general relativistic hydrogen atom, Comm. Math. Phys., 86 (1982), 69-86. doi: 10.1007/BF01205662. [6] T. Daudé, Time-dependent scattering theory for massive charged Dirac fields by a Reissner-Nordström black hole, J. Math. Phys., 51 (2010), 102504, 57pp. doi: 10.1063/1.3499403. [7] T. Daudé, N. Kamran and F. Nicoleau, Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces, to appear in Inverse Problems, preprint, arXiv:1409.6229. [8] T. Daudé and F. Nicoleau, Inverse scattering in (de Sitter)-Reissner-Nordström black hole spacetimes, Rev. Math. Phys., 22 (2010), 431-484. doi: 10.1142/S0129055X10004004. [9] T. Daudé and F. Nicoleau, Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes, Annales Henri Poincaré, 12 (2011), 1-47. doi: 10.1007/s00023-010-0069-9. [10] G. Freiling and V. Yurko, Inverse problems for differential operators with singular boundary conditions, Math. Nachr., 278 (2005), 1561-1578. doi: 10.1002/mana.200410322. [11] F. Gesztesy and B. Simon, On local Borg-Marchenko uniqueness results, Comm. Math. Phys., 211 (2005), 273-287. doi: 10.1007/s002200050812. [12] M. Horváth, Partial identification of the potential from phase shifts, J. Math. Anal. Appl., 380 (2011), 726-735. doi: 10.1016/j.jmaa.2010.10.071. [13] H. Isozaki and J. Kurylev, Introduction to spectral theory and inverse problems on asymptotically hyperbolic manifolds, MSJ Memoirs, Mathematical Society of Japan, Tokyo, (2014), viii+251 pp, arXiv:1102.5382. doi: 10.1142/e040. [14] M. S. Joshi and A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Mathematica, 184 (2000), 41-86. doi: 10.1007/BF02392781. [15] K. Lake, Reissner-Nordström-de Sitter metric, the third law, and cosmic censorship, Phys. Rev. D., 19 (1979), 421-429. doi: 10.1103/PhysRevD.19.421. [16] B. Y. Levin, Lectures on Entire Functions, Translations of Mathematical Monograph, 150, American Mathematical Society, 1996. [17] F. Melnik, Scattering on Reissner-Nordström metric for massive charged spin $\frac{1}{2}$ fields, Annales Henri Poincaré, 4 (2003), 813-846. doi: 10.1007/s00023-003-0148-2. [18] J.-P. Nicolas, Scattering of linear Dirac fields by a spherically symmetric black hole, Annales Institut Henri Poincaré, 62 (1995), 145-179. [19] A. G. Ramm, An Inverse Scattering Problem with part of the Fixed-Energy Phase shifts, Comm. Math. Phys., 207 (1999), 231-247. doi: 10.1007/s002200050725. [20] T. Regge, Introduction to complex orbital momenta, Nuevo Cimento, 14 (1959), 951-976. doi: 10.1007/BF02728177. [21] A. Sá Barreto, Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds, Duke Math. Journal, 129 (2005), 407-480. doi: 10.1215/S0012-7094-05-12931-3. [22] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Company, New York, 1987. [23] B. Simon, A new approach to inverse spectral theory, I. fundamental formalism, Annals of Math., 150 (1999), 1029-1057. doi: 10.2307/121061. [24] G. Teschl, Mathematical Methods in Quantum Mechanics, Graduate Studies in Mathematics, 99, AMS Providence, Rhode Island, 2009. doi: 10.1090/gsm/099. [25] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, 1992. doi: 10.1007/978-3-662-02753-0. [26] R. M. Wald, General Relativity, The University of Chicago Press, 1984. doi: 10.7208/chicago/9780226870373.001.0001.
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