Global uniqueness from inverse backscattering data
For a real potential $q \in C_c^\infty(\mathbb{R}^3)$, a wave number $k > 0$ and a direction $\theta \in S^2$, let $u = e^{ik x \cdot \theta} + u^s$ solve $$(-\Delta + q - k^2) u = 0 \ \text{ in } \mathbb{R}^3,$$ with $u^s$ outgoing, and define the scattering amplitude $a_q$ by $$u^s(r\omega) = r^{-1} e^{ikr} a_q(\omega, \theta, k) + O(r^{-2}), \qquad r \to \infty, \ \omega \in S^2.$$ The backscattering data $a_q(-\theta, \theta, k)$, $(\theta, k) \in S^2 \times (0, \infty)$, depend on three variables, as $q$ does. Eskin and Ralston showed that the map taking $q$ to the Born approximation computed from these data is analytic, and a local diffeomorphism on a dense open set of potentials containing a neighbourhood of $q = 0$; Stefanov gave another proof. Uniqueness holds for potentials small in $C^2$ (Rakesh and Uhlmann give a short proof), when $q_1 \ge q_2$ (Stefanov), and when $q_1 - q_2$ has controlled angular derivatives, for instance when the two potentials differ by a finite sum of spherical harmonics (Rakesh and Uhlmann).
Problem. Let $q_1, q_2 \in C_c^\infty(\mathbb{R}^3)$ be real. Does $$a_{q_1}(-\theta, \theta, k) = a_{q_2}(-\theta, \theta, k) \qquad \text{for every } (\theta, k) \in S^2 \times (0, \infty)$$ imply $q_1 = q_2$? No smallness, symmetry or angular-control assumption is imposed.
Source: Eskin and Ralston (1989), who studied the inverse backscattering problem in three dimensions; Rakesh and Uhlmann (2014), §1.2, record its most basic case (do zero backscattering data force $q = 0$?) as open; see also Ma, Potenciano-Machado and Salo (2023), §1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, backscattering data, inverse scattering, Schrödinger operators
References
@article{eskin1989,
author = {Eskin, G. and Ralston, J.},
title = {The inverse backscattering problem in three dimensions},
journal = {Comm. Math. Phys.},
volume = {124},
number = {2},
year = {1989},
pages = {169--215},
doi = {10.1007/BF01219194},
}
@article{rakesh2014,
author = {Rakesh and Uhlmann, Gunther},
title = {Uniqueness for the inverse backscattering problem for angularly controlled potentials},
journal = {Inverse Problems},
volume = {30},
number = {6},
year = {2014},
pages = {065005},
doi = {10.1088/0266-5611/30/6/065005},
}
@article{stefanov1990,
author = {Stefanov, P. D.},
title = {A uniqueness result for the inverse back-scattering problem},
journal = {Inverse Problems},
volume = {6},
number = {6},
year = {1990},
pages = {1055--1064},
doi = {10.1088/0266-5611/6/6/012},
}
@article{stefanov1992,
author = {Stefanov, Plamen},
title = {Generic uniqueness for two inverse problems in potential scattering},
journal = {Comm. Partial Differential Equations},
volume = {17},
number = {1-2},
year = {1992},
pages = {55--68},
doi = {10.1080/03605309208820834},
}
@misc{aim-inverse-backscattering,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global uniqueness from inverse backscattering data}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/052-inverse-backscattering.md},
note = {Entry 052 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00530, "Global uniqueness from inverse backscattering data", clopen.solutions.