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

TL;DR

This paper investigates potential scattering in a space with an extra compact dimension, revealing that the analyticity properties of the scattering amplitude differ from the standard three-dimensional case, with implications for quantum mechanics in such geometries.

Contribution

It demonstrates that the analyticity properties of the forward scattering amplitude do not hold in $R^3\otimes S^1$, providing explicit counterexamples and analyzing the impact of the extra dimension.

Findings

$T_{nn}(K)$ is not analytic in $K$ for certain internal excitations.

Counterexamples show singularities on the physical energy sheet.

Analytic properties differ from standard $R^3$ scattering theory.

Abstract

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.