A uniqueness result for the inverse back-scattering problem

Let q₁(t,x), q₂(t,x) belong to C(R_t;W^{1, infinity} (R_x³)), q_i(t,x)=0 for mod x mod > rho with some rho >0 and let K_i^Hash be the corresponding generalized scattering kernels, i=1,2. The author proves that if q₁>or=q₂ and if K₁^Hash (s',- omega ₀; s, omega ₀)=K₂^Hash (s', - omega ₀;s, omega ₀) for some omega ₀ in S², then q₁=q₂. As a corollary, the author gets the following result. Let V_i(x) in L^infinity (R³), V_i has compact support and suppose that - Delta +V_i has no bound states, i=1, 2. Let a_i be the scattering amplitude related to V_i, i=1, 2. Suppose that V₁>or=V₂ and for some omega ₀ the author has a₁(k,- omega ₀, omega ₀)=a₂(k,- omega ₀, omega ₀) for all k. Then V₁=V₂.