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.