# American Institute of Mathematical Sciences

February  2007, 1(1): 217-224. doi: 10.3934/ipi.2007.1.217

## Inverse scattering at a fixed energy for long-range potentials

 1 Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, México DF 01000, United States 2 IRMAR, Université de Rennes I., Campus de Beaulieu, 35042 Rennes, Cedex, France

Received  September 2006 Published  January 2007

In this paper we consider the inverse scattering problem at a fixed energy for the Schrödinger equation with a long-range potential in $R^d, d\geq 3$. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy.
Citation: Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems and Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217
 [1] Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 [2] Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems and Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843 [3] 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 [4] Juan Kalemkerian, Andrés Sosa. Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices. Journal of Dynamics and Games, 2020, 7 (3) : 225-237. doi: 10.3934/jdg.2020016 [5] Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235 [6] Yiju Chen, Xiaohu Wang, Kenan Wu. Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022113 [7] Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations and Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531 [8] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems and Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [9] Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems and Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023 [10] Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure and Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 [11] Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations and Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 [12] Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159 [13] 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 [14] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016 [15] Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems and Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537 [16] Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064 [17] Cornelis van der Mee. Direct scattering of AKNS systems with $L^2$ potentials. Conference Publications, 2015, 2015 (special) : 1089-1097. doi: 10.3934/proc.2015.1089 [18] Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 367-386. doi: 10.3934/ipi.2020072 [19] Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025 [20] Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3637-3654. doi: 10.3934/cpaa.2021124

2020 Impact Factor: 1.639