  This paper deals with perturbation theory for discrete spectra of linear
operators. To simplify exposition we consider here self-adjoint operators. This
theory is based on the Feshbach-Schur map and it has advantages with respect to
the standard perturbation theory in three aspects: (a) it readily produces
rigorous estimates on eigenvalues and eigenfunctions with explicit constants;
(b) it is compact and elementary (it uses properties of norms and the
fundamental theorem of algebra about solutions of polynomial equations); and
(c) it is based on a self-contained formulation of a fixed point problem for
the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our
abstract results to obtain rigorous bounds on the ground states of Helium-type
ions.
