Studies in Perturbation Theory. X. Lower Bounds to Energy Eigenvalues in Perturbation-Theory Ground State

The eigenvalue problem $H\Psi =E\Psi$ in quantum theory is conveniently studied by means of the partitioning technique. It is shown that, if $\mathcal{E}$ is a real variable, one may construct a function ${\mathcal{E}}_1=f\left(\mathcal{E}\right)$ such that each pair $\mathcal{E}$ and ${\mathcal{E}}_1$ always bracket at least one true eigenvalue $E$. If $\mathcal{E}$ is chosen as an upper bound by means of, e.g., the variation principle, the function ${\mathcal{E}}_1$ is hence going to provide a lower bound. The reaction operator $t$ associated with the perturbation problem $H=H_0+V$ for a positive-definite perturbation $V$ is studied in some detail, and it is shown that a lower bound to $t$ may be constructed in a finite number of operations by using the idea of "inner projection" closely associated with the Aronszajn projection previously utilized in the method of intermediate Hamiltonians. By means of truncated basic sets one can now obtain not only upper bounds but also useful lower bounds which converge towards the correct eigenvalues when the set becomes complete. The method is applied to the Brillouin-type perturbation theory, and lower bounds may be obtained either by pure expansion methods, by inner projections, or by a combination of both approaches leading to perturbation expansions with estimated remainders. The applications to Schrödinger's perturbation theory are also outlined. The method is numerically illustrated a study of lower bounds to the ground-state energies of the helium-like ions: He, ${\mathrm{Li}}^{+}$, ${\mathrm{B}}^{+2\mathrm{}}$, etc.