Variational Methods in the Wave Operator Formalism: Applications in Variation-Perturbation Theory and the Theory of Energy Bounds

Abstract

Variational principles of the Lippmann-Schwinger type are used to develop approximations to eigenenergies and eigenfunctions within the wave-operator formalism. The present approach starts with exactly soluble ``primary'' eigenvalue equations to give explicit results valid beyond the limits of conventional perturbation theory. The variational functionals are expressed in terms of resolvents of the primary Hamiltonian, and bounds to the functionals are constructed also for cases where the resolvents are only partly known. Approximations to eigenenergies and eigenfunctions are obtained in terms of quantities in the Brillouin-Wigner perturbation theory. Connections with methods for upper and lower energy bounds are discussed, and the convergence properties of the nonlinear Padé summation is recovered in this way. Closed formulas within the double perturbation theory framework are presented as a logical extension.