Friedel sum rule in the presence of topological defects for graphene

The Friedel sum rule is extended to deal with topological defects for the case of a graphene cone in the presence of an external Coulomb charge. The dependence in the way the number of states change due to both the topological defect as well as the Coulomb charge are studied. Our analysis addresses both the cases of a subcritical as well as a supercritical value of the Coulomb charge. We also discuss the experimental implications of introducing a self-adjoint extension of the system Hamiltonian. We argue that the boundary conditions following from the self-adjoint extension encode the effect of short range interactions present in the system.