For n≥ 2, fix an incident direction theta_0∈ S^n-1. Determine whether a compactly supported potential q is uniquely determined by its far-field pattern u_q^∞(omega,theta_0,k) for all observation directions omega∈ S^n-1 and all wavenumbers k>0; equivalently, whether u_q_1^∞(omega,theta_0,k)=u_q_2^∞(omega,theta_0,k) for every omega∈ S^n-1 and k>0 implies q_1=q_2.

Fixed-angle scattering uniqueness problem of Rakesh and Salo

The problem asks whether restricted scattering data at one fixed incident angle uniquely determine a potential. Rakesh and Salo established important two-angle and special-symmetry cases, while the unrestricted one-angle case remained open.

Known results

Two opposite fixed incident angles determine compactly supported real potentials in dimensions $n \ge 2$ (Rakesh–Salo, 2020).
One fixed angle suffices for reflection-symmetric and certain almost symmetric or horizontally controlled potentials (Rakesh–Salo, 2020).
Small-potential uniqueness was known under Sobolev regularity and dimension-dependent smallness assumptions (2018).
A related first-order perturbation problem has uniqueness from finitely many measurements, up to gauge (later work).

September 17, 2026 claimed proof

Suliang Si's preprint Determination of the potential by a fixed angle scattering data uses a Carleman-estimate argument to claim uniqueness for the general fixed-angle problem. The claim is unrefereed and has no independent confirmation in the retrieved sources.

Current status (as of September 2026): established results cover two opposite angles and restricted potential classes, while Suliang Si's claimed general one-angle resolution remains unverified.

Fixed-angle scattering uniqueness problem of Rakesh and Salo

Primary source

Additional references

Progress summary

Solutions 0