Lower Bounds for Eigenvalues with Application to the Helium Atom

A method is derived for finding lower bounds to the energy levels of the Schrödinger equation. This method is applied to the helium atom. The best lower bounds thus obtained are ${-3.063\mathrm{}}_7$ and ${-2.165\mathrm{}}_5$ atomic units for the energies $E\left(1\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$ and $E\left(2\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$, respectively. If our lower bound for $E\left(2\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$ is used together with the best published values of $〈H\psi , \psi 〉$ and $〈H\psi , H\psi 〉$ of the ground state, a rigorous lower bound -2.9037474 atomic units is found for $E\left(1\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$.