Uniqueness for a hyperbolic inverse problem with angular control on the coefficients

Suppose q(i)(x), i = 1, 2 are smooth functions on R-3 and U-i(x, t) the solutions of the initial value problem partial derivative U-2(t)i - Delta U-i - q(i)(x)U-i = delta(x, t), (x, t) is an element of R-3 x R, U-i(x, t) = 0 for t < 0. Pick R, T so that 0 < R < T and let C be the vertical cylinder {(x, t) : vertical bar x vertical bar = R, R <= t <= T}. We show that if (U-1, U-1r) = (U-2, U-2r) on C then q(1) = q(2) on the annular region R <= vertical bar x vertical bar <= (R + T )/2 provided there is a gamma > 0, independent of r, so that integral(vertical bar x vertical bar = r) vertical bar Delta(S)(q(1) - q(2))vertical bar(2) dS(x) <= gamma integral(vertical bar x vertical bar = r) vertical bar q(1) - q(2)vertical bar(2) dS(x) for all r is an element of [R, (R + T)/2]. Here Delta(S) is the spherical Laplacian on vertical bar x vertical bar = r.