Optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data

We consider the inverse boundary value problem for operators of the form $-\mathrm{△}+q$ in an infinite domain $\mathrm{\Omega }=\mathbb{R}\times \omega \subset {\mathbb{R}}^{1+n}$, $n\ge 3$, with a periodic potential $q$. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form ${\mathrm{\Gamma }}_1=\mathbb{R}\times {\gamma }_1$, with ${\gamma }_1$ being the complement either of a flat or spherical portion of $\mathrm{\partial }\omega$, we prove that a log-type stability estimate holds.