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. 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 than at C=0. 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.