  A quantum neutral particle, constrained to move on a conical surface, is used
as a toy model to explore bound states due to both a inverse squared distance
potential and a $\delta$-function potential, which appear naturally in the
model. These pathological potentials are treated with the self-adjoint
extension method which yields the correct boundary condition (not necessarily a
null wavefunction) at the origin. We show that the usual boundary condition
requiring that the wavefunction vanishes at the origin is arbitrary and
drastically reduces the number of bound states if used. The situation studied
here is closely related to the problem of a dipole moving in conical space.
