POTENTIAL SCATTERING on R 3 ⊗ S 1

In this paper we consider non-relativistic quantum mechanics on a space with an additional internal compact dimension, i.e. R 3 ⊗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 ⊗ S 1. For example, T nn (K), is 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 = 0, can have singularities on the physical energy sheet. We also discuss the motivation for our work, and briefly the lesson it teaches us.