  In this paper we consider non-relativistic quantum mechanics on a space with
an additional internal compact dimension, i.e. $R^3\otimes S^1$ instead of
$R^3$. More specifically we study potential scattering for this case and the
analyticity properties of the forward scattering amplitude, $T_{nn}(K)$, where
$K^2$ is the total energy and the integer n denotes the internal excitation of
the incoming particle. The surprising result is that the analyticity properties
which are true in $R^3$ do not hold in $R^3\otimes S^1$.
  For example, $T_{nn}(K)$, is \underline{not} analytic in K for $ImK>0$, for n
such that $(|n|/R)>\mu$, where R is the radius of $S^1$, and $\mu^{-1}$ is the
exponential range of the potential, $V(r,\phi)=O(e^{-\mu r})$ for large r. We
show by explicit counterexample that $T_{nn}(K)$ for $n\neq0$, can have
singularities on the physical energy sheet. We also discuss the motivation for
our work, and briefly the lesson it teaches us.
