On determination of the Fourier transform of a potential from the scattering amplitude

We consider the Schrödinger equation - Δψ + v(x)ψ = Eψ, x ∈ R^d, d≥2, where v is a real-valued measurable bounded function such that v(x) = O(|x|^-d-ε) as |x| → ∞ for some ε > 0. For this equation we show that for any γ ∈ S^d-1, σ ∈ [0, π/2], E > 0 and δ > 0 the scattering amplitude f on {(k, l) ∈ R^d × R^d | E ≤ k² = l² ≤ E + δ, k, γ ≤ σ, l, γ ≤ σ} (where a, b denotes the angle between vectors a and b) uniquely determines the Fourier transform v̂ (of v) on {p ∈ R^d | pγ = 0, |p| ≤ 2E√ sin σ} (and as a corollary on some larger set). We also give analogues of this result for the case of the acoustic equation.