  The Schr\"odinger equations for the Coulomb and the Harmonic oscillator
potentials are solved in the cosmic-string conical space-time. The spherical
harmonics with angular deficit are introduced.
  The algebraic construction of the harmonic oscillator eigenfunctions is
performed through the introduction of non-local ladder operators. By exploiting
the hidden symmetry of the two-dimensional harmonic oscillator the eigenvalues
for the angular momentum operators in three dimensions are reproduced.
  A generalization for N-dimensions is performed for both Coulomb and harmonic
oscillator problems in angular deficit space-times.
  It is thus established the connection among the states and energies of both
problems in these topologically non-trivial space-times.
