Stability of the determination of a coefficient for the wave equation in an infinite wave guide

We consider the stability in the inverse problem consisting in the determination of an electric potential $q$, appearing in a Dirichlet initial-boundary value problem for the wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in an unbounded wave guide $\Omega=\omega\times\mathbb R$ with $\omega$ a bounded smooth domain of $\mathbb R^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equation. We prove a H\"older stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the gap between two electric potentials rich its maximum in a fixed bounded subset of $\bar{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.