Determination of time-dependent coefficients for a hyperbolic inverse problem

TL;DR

This paper investigates an inverse boundary value problem for a hyperbolic PDE with time-dependent potentials, demonstrating uniqueness of potential recovery modulo gauge transformations, stability estimates, and effects of obstacles within the domain.

Contribution

It introduces a geometric optics method to recover potentials from boundary data and proves uniqueness and stability results for the inverse problem with and without obstacles.

Findings

Potential functions can be uniquely recovered modulo gauge transformations.

A logarithmic stability estimate is established.

Similar uniqueness results hold in the presence of obstacles under certain conditions.

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.