  Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is
the basic ingredient of quantum mechanics, plays an important role in nuclear
reaction and nuclear structure theory. However, until now, there was no a
formal proof of the completeness of the eigenfunctions of the two-body
Hamiltonian with the Coulomb interaction. Here we present the first formal
proof of the completeness of the two-body Coulomb scattering wave functions for
repulsive unscreened Coulomb potential. To prove the completeness we use the
Newton's method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof allows us
to claim that the eigenfunctions of the two-body Hamiltonian with the potential
given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials
also form a complete set. It also allows one to extend the Berggren's approach
of modification of the complete set of the eigenfunctions by including the
resonances for charged particles. We also demonstrate that the resonant Gamow
functions with the Coulomb tail can be regularized using Zel'dovich's
regularization method.
