Variational approach to the Coulomb problem on a cylinder

We evaluate, by means of variational calculations, the bound state energy $E_B$ of a pair of charges located on the surface of a cylinder, interacting via Coulomb potential $-e^2/r.\mathrm{}$ The trial wave function involves three variational parameters. $E_B$ is obtained as a function of the reduced curvature ${C=a}_0/R,$ where $a_0$ is the Bohr radius and R is the radius of the cylinder. We find that the energetics of binding exhibits a monotonic trend as a function of C; the known one- and two-dimensional limits of $E_B$ are reproduced accurately by our calculation. $E_B$ is relatively insensitive to curvature for small C. Its value is ∼1% higher at $C=1\mathrm{}$ than at $C=0.\mathrm{}$ This weak dependence is confirmed by a perturbation theory calculation. The high curvature regime approximates the one-dimensional Coulomb model; within our variational approach, $E_B$ has a logarithmic divergence as R approaches zero. The proposed variational method is applied to the case of donors in single-wall carbon nanotubes.