Geometry-induced potential on a two-dimensional section of a wormhole: Catenoid

We show that a two-dimensional wormhole geometry is equivalent to a catenoid, a minimal surface. We then obtain the curvature-induced geometric potential and show that the ground state with zero energy corresponds to a reflectionless potential. By introducing an appropriate coordinate system we also obtain bound states for different angular momentum channels. Our findings can be realized in suitably bent bilayer graphene sheets with a neck, in a honeycomb lattice with an array of dislocations, or in nanoscale waveguides in the shape of a catenoid.