  Matrix mechanics is developed to describe the bound state spectra in few- and
many-electron atoms, ions and molecules. Our method is based on the matrix
factorization of many-electron (or many-particle) Coulomb Hamiltonians which
are written in hyperspherical coordinates. As follows from the results of our
study the bound state spectra of many-electron (or many-particle) Coulomb
Hamiltonians always have the `ladder' structure and this fundamental fact can
be used to determine and investigate the bound states in various few- and
many-body Coulomb systems.
