  Let $\Omega\subset \Bbb R^2$ be a bounded domain with $\partial\Omega\in
C^\infty$ and $L$ be a positive number. For a three dimensional cylindrical
domain $Q=\Omega\times (0,L)$, we obtain some uniqueness result of determining
a complex-valued potential for the Schr\"odinger equation from partial Cauchy
data when Dirichlet data vanish on a subboundary
$(\partial\Omega\setminus\widetilde{\Gamma}) \times [0,L]$ and the
corresponding Neumann data are observed on $\widetilde\Gamma \times [0,L]$,
where $\widetilde\Gamma$ is an arbitrary fixed open set of $\partial\Omega.$
