Property C and an Inverse Problem for a Hyperbolic Equation

Let (* ) u,, du + q(x, t) u = 0 in D x [0, T], where D c R3 is a bounded domain with a smooth boundary dD, T> d, d: =diam D, q(x, I) E C( [O, 2’1, L”(D)). Suppose that for every (**) ~I,+~=f(x, l)EC’(dDx [0, T]), the value u*,;,:= h(s, t) is known, where N is the outer normal to dD, u solves (* ) and ( ** ) and satisfies the initial conditions u = U, = 0 at I = 0. Then q(x, I) is uniquely determined by the data (J h}, VIE C’(dD x [0, T]) in the subset S of D x [0, T] consisting of the lines which make 45” with the t-axis and which meet the planes t = 0 and I = T outside Dx [0, T], provided that q(x, t) is known outside S. Here B is the closure of D.