Determination of time-dependent coefficients for a hyperbolic inverse problem

Abstract

We consider an inverse boundary value problem for the hyperbolic partial differential equation $(-i{\partial }_t+A_0(t,x){)}^{2}u(t,x)-{\sum }_{j=1}^n(-i{\partial }_{x_j}+A_j(t,x){)}^{2}u(t,x)+V(t,x)u(t,x)=0$ with time dependent vector and scalar potentials ($\mathcal{A}=(A_0,...,A_m)$ and $V(t,x)$ respectively) on a bounded, smooth cylindric domain $(-\infty ,\infty )\times \Omega$. Using a geometric optics construction we show that the boundary data allows us to recover integrals of the potentials along `light rays' and we then establish the uniqueness of these potentials modulo a gauge transform. Also, a logarithmic stability estimate is obtained and the presence of obstacles inside the domain is studied. In this case, it is shown that under some geometric restrictions similar uniqueness results hold.