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Potential Scattering on $R^3\otimes s^1

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 $(|n|/R)>μ$, where R is the radius of $S^1$, and $μ^{-1}$ is the exponential range of the potential, $V(r,ϕ)=O(e^{-μr})$ 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.