Lower Bounds for Eigenvalues with Displacement of Essential Spectra

New constructions of comparison operators for rigorous lower bounds to eigenvalues of a class of self-adjoint operators are presented. The formulation uses noncompact finite perturbations to displace eigenvalues and essential spectra, and leads to workable numerical procedures. These methods make possible for the first time lower bound calculations for the lower eigenvalues of the Schrödinger operators for atoms and ions having three or more electrons.