On the general theory of bound state spectra in the Coulomb few- and many-body systems

Based on the fact that the Hamiltonians of the Coulomb many-particle systems are always factorized we develop the two different approaches for analytical solution of the Schr\"{o}dinger equation written for arbitrary few- and many-particle Coulomb systems. The first approach is the matrix factorization method. Another method is based on the $D^{+}-$series of representations of the hyper-radial O(2,1)-algebra. The both these methods allow us to obtain the closed analytical formulas for the bound state energies in an arbitrary many-particle Coulomb system.