Bloch-Horowitz perturbation theory is applied to the calculation of approximate energies and model-space eigenvectors, for the solvable large-matrix Hamiltonian $H$ used by Pittel, Vincent, and Vergados. Two types of upper and lower bounds to the energies are discussed: moment-theory bounds, obtained by applying moment theory to the terms of perturbation theory, and norm bounds, derived from the expectation $\overline{E}$ and variance ${\sigma }^2$ of $H$ with respect to an eigenvector approximated by n th order perturbation theory ($n\le 6$). It is shown that lower bounds cannot be constructed unless some fourth-order quantity is known. The upper bounds are generally stricter than the lower bounds. All of the bounds apply even when back-door intruder states cause perturbation theory to diverge; but they lose their rigor and become "quasibounds" when there are physical intruders. The moment-theory and norm lower quasibounds always require estimation of a parameter. For the solvable Hamiltonians, it is shown that this can be done quite reliably, and that the resulting quasibounds are tight enough to have some practical utility. The energy-independent effective interaction V is constructed and its errors are displayed and discussed. Finally, a certain [1/2] pseudo-Padé approximant is empirically shown to give energies with a mean absolute error of less than 0.3 MeV in all cases.

NUCLEAR STRUCTURE Effective interactions, ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$. Perturbation theory for energies and bounds to them; used large solvable matrices, included intruder states. Padé approximants.

• Received 28 November 1977

DOI:https://doi.org/10.1103/PhysRevC.17.1461