  We consider the elastic scattering and bound states of charged quantum
particles moving in the Aharonov-Bohm and an attractive $\rho^{-2}$ potential
in a partial wave approach. Radial solutions of the stationary Schr\"{o}dinger
equation are specified in such a way that the Hamiltonian of the problem is
self-adjoint. It is shown that they are not uniquely fixed but depend on open
parameters. The related physical consequences are discussed. The scattering
cross section is calculated and the energy spectrum of bound states is
obtained.
