  We cast the quantum chemistry problem of computing bound states as that of
solving a set of auxiliary eigenvalue problems for a family of parameterized
compact integral operators. The compactness of operators assures that their
spectrum is discrete and bounded with the only possible accumulation point at
zero. We show that, by changing the parameter, we can always find the bound
states, i.e., the eigenfunctions that satisfy the original equations and are
normalizable. While for the non-relativistic equations these properties may not
be surprising, it is remarkable that the same holds for the relativistic
equations where the spectrum of the original relativistic operators does not
have a lower bound. We demonstrate that starting from an arbitrary
initialization of the iteration leads to the solution, as dictated by the
properties of compact operators.
