Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism

The partitioning technique for solving secular equations is briefly reviewed. It is then reformulated in terms of an operator language in order to permit a discussion of the various methods of solving the Schrödinger equation. The total space is divided into two parts by means of a self-adjoint projection operator O. Introducing the symbolic inverse T = (1—O)/(E—H), one can show that there exists an operator Ω = O + THO, which is an indempotent eigenoperator to H and satisfies the relations HΩ = EΩ and Ω2 = Ω. This operator is not normal but has a form which directly corresponds to infinite-order perturbation theory. Both the Brillouin- and Schrödinger-type formulas may be derived by power series expansion of T, even if other forms are perhaps more natural. The concept of the reaction operator is discussed, and upper and lower bounds for the true eigenvalues are finally derived.