A UNIQUENESS RESULT FOR THE INVERSE BACKSCATTERING PROBLEM

Let q1(t,x),q2(t,x) belong to C(R(t); W1, infinity (R(x)3)), q(i)(t,x) = 0 for /x/>p with some p>0 and let K(i)# be the corresponding generalized scattering kernels, i = 1,2. We prove that if q1 greater-than-or-equal-to q2 and if K1# (s', -omega-0;s,omega-0) = K2# (s', -omega-0;s,omega-0) for some omega-0-epsilon-S2, then q1 = q2. As a corollary, we get the following result. Let V(i)(x)epsilon-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 greater-than-or-equal-to V2 and for some omega-0 we have a(l)(k, -omega-0,omega-0) = a2(k, -omega-0,omega-0) for all k. Then V1 = V2. Finally we show that a(k, -omega, omega) determines uniquely the convex hull of the support of V.