  We look at the effective Hamiltonian $\bar H$ associated with the Hamiltonian
$H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is
to understand the relation between $V$ and $\bar H$. We formulate some inverse
problems concerning this relation. Such type of inverse problems are in general
very challenging. In the paper, we discuss several special cases in both convex
and nonconvex settings.
