Lower Bounds for Eigenvalues with Application to the Helium Atom

Abstract

A method is derived for finding lower bounds to the energy levels of the Schr\"odinger equation. This method is applied to the helium atom. The best lower bounds thus obtained are ${\ensuremath{-}3.063}_{7}$ and ${\ensuremath{-}2.165}_{5}$ atomic units for the energies $E(1^{1}S)$ and $E(2^{1}S)$, respectively. If our lower bound for $E(2^{1}S)$ is used together with the best published values of $〈H\ensuremath{\psi}, \ensuremath{\psi}〉$ and $〈H\ensuremath{\psi}, H\ensuremath{\psi}〉$ of the ground state, a rigorous lower bound -2.9037474 atomic units is found for $E(1^{1}S)$.