  In this paper the generalization of the Gribov pendulum equation in the
Coulomb gauge for curved spacetimes is analyzed on static spherically symmetric
backgrounds. A rigorous argument for the existence and uniqueness of solution
is provided in the asymptotically AdS case. The analysis of the strong and weak
boundary conditions is equivalent to analyzing an effective one-dimensional
Schrodinger equation. Necessary conditions in order for spherically symmetric
backgrounds to admit solutions of the Gribov pendulum equation representing
copies of the vacuum satisfying the strong boundary conditions are given. It is
shown that asymptotically flat backgrounds do not support solutions of the
Gribov pendulum equation of this type, while on asymptotically AdS backgrounds
such ambiguities can appear. Some physical consequences are discussed.
