American Institute of Mathematical Sciences

Advanced
August  2009, 3(3): 537-550. doi: 10.3934/ipi.2009.3.537

Inverse scattering on conformally compact manifolds

Leonardo Marazzi 1,

1. 

Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, United States

Received  March 2008 Revised  June 2009 Published  July 2009

We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature -α2(y) at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrices at two fixed energies $\lambda_1,$ $\lambda_2$ in a suitable subset of c , determines α, and the Taylor series of both the potential and the metric at the boundary.
Keywords: scattering matrix., Inverse problems, conformally compact manifolds.
Mathematics Subject Classification: Primary: 35R30, 35P25; Secondary: 58J4.
Citation: Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537
[1]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[2]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[3]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[4]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems & Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033
[5]

Gabriel Katz. Causal holography in application to the inverse scattering problems. Inverse Problems & Imaging, 2019, 13 (3) : 597-633. doi: 10.3934/ipi.2019028
[6]

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
[7]

Tony Liimatainen, Mikko Salo. Nowhere conformally homogeneous manifolds and limiting Carleman weights. Inverse Problems & Imaging, 2012, 6 (3) : 523-530. doi: 10.3934/ipi.2012.6.523
[8]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139
[9]

Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems & Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026
[10]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749
[11]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[12]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[13]

Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377
[14]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2014, 13 (1) : 175-202. doi: 10.3934/cpaa.2014.13.175
[15]

Travis G. Draper, Fernando Guevara Vasquez, Justin Cheuk-Lum Tse, Toren E. Wallengren, Kenneth Zheng. Matrix valued inverse problems on graphs with application to mass-spring-damper systems. Networks & Heterogeneous Media, 2020, 15 (1) : 1-28. doi: 10.3934/nhm.2020001
[16]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[17]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[18]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239
[19]

Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems & Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058
[20]

Woocheol Choi. Maximal functions of multipliers on compact manifolds without boundary. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1885-1902. doi: 10.3934/cpaa.2015.14.1885

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences