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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 & Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016
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show all references

References:
[1]

Integr. Equa. Oper. Theory, 38 (2000), 129-171. doi: 10.1007/BF01200121.  Google Scholar

[2]

Comm. Math. Phys., 218 (2001), 131-132. doi: 10.1007/s002200100384.  Google Scholar

[3]

Academic Press, 1954.  Google Scholar

[4]

J. of Geom. Anal., 21 (2001), 305-333. doi: 10.1007/s12220-010-9149-9.  Google Scholar

[5]

Comm. Math. Phys., 86 (1982), 69-86. doi: 10.1007/BF01205662.  Google Scholar

[6]

J. Math. Phys., 51 (2010), 102504, 57pp. doi: 10.1063/1.3499403.  Google Scholar

[7]

T. Daudé, N. Kamran and F. Nicoleau, Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces,, to appear in Inverse Problems, ().   Google Scholar

[8]

Rev. Math. Phys., 22 (2010), 431-484. doi: 10.1142/S0129055X10004004.  Google Scholar

[9]

Annales Henri Poincaré, 12 (2011), 1-47. doi: 10.1007/s00023-010-0069-9.  Google Scholar

[10]

Math. Nachr., 278 (2005), 1561-1578. doi: 10.1002/mana.200410322.  Google Scholar

[11]

Comm. Math. Phys., 211 (2005), 273-287. doi: 10.1007/s002200050812.  Google Scholar

[12]

J. Math. Anal. Appl., 380 (2011), 726-735. doi: 10.1016/j.jmaa.2010.10.071.  Google Scholar

[13]

MSJ Memoirs, Mathematical Society of Japan, Tokyo, (2014), viii+251 pp, arXiv:1102.5382. doi: 10.1142/e040.  Google Scholar

[14]

Acta Mathematica, 184 (2000), 41-86. doi: 10.1007/BF02392781.  Google Scholar

[15]

Phys. Rev. D., 19 (1979), 421-429. doi: 10.1103/PhysRevD.19.421.  Google Scholar

[16]

Translations of Mathematical Monograph, 150, American Mathematical Society, 1996.  Google Scholar

[17]

Annales Henri Poincaré, 4 (2003), 813-846. doi: 10.1007/s00023-003-0148-2.  Google Scholar

[18]

Annales Institut Henri Poincaré, 62 (1995), 145-179.  Google Scholar

[19]

Comm. Math. Phys., 207 (1999), 231-247. doi: 10.1007/s002200050725.  Google Scholar

[20]

Nuevo Cimento, 14 (1959), 951-976. doi: 10.1007/BF02728177.  Google Scholar

[21]

Duke Math. Journal, 129 (2005), 407-480. doi: 10.1215/S0012-7094-05-12931-3.  Google Scholar

[22]

McGraw-Hill Book Company, New York, 1987.  Google Scholar

[23]

Annals of Math., 150 (1999), 1029-1057. doi: 10.2307/121061.  Google Scholar

[24]

Graduate Studies in Mathematics, 99, AMS Providence, Rhode Island, 2009. doi: 10.1090/gsm/099.  Google Scholar

[25]

Texts and Monographs in Physics, Springer-Verlag, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

[26]

The University of Chicago Press, 1984. doi: 10.7208/chicago/9780226870373.001.0001.  Google Scholar

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