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

Abstract

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 {gather*} \pa_t^2 U_i- \Delta U_i - q_i(x) U_i = \delta(x,t), \qquad (x,t) \in \R^3 \times \R U_i(x,t) =0, \qquad \text{for} ~ t<0. {gather*} Pick $R,T$ so that $0 < R < T$ and let $C$ be the vertical cylinder $\{(x,t) \, : |x|=R, ~ R \leq t \leq T \}$. We show that if $(U_1, U_{1r}) = (U_2, U_{2r})$ on $C$ then $q_1 = q_2$ on the annular region $R \leq |x| \leq (R+T)/2$ provided there is a $\gamma>0$, independent of $r$, so that \[\int_{|x|=r} | \Delta_S (q_1 - q_2)|^2 \, dS_x \leq \gamma \int_{|x|=r} |q_1 - q_2|^2 \, dS_x, \qquad \forall r \in [R, (R+T)/2].\] Here $\Delta_S$ is the spherical Laplacian on $|x|=r$.