General Structure of Effective Interaction in Degenerate Perturbation Theory

Abstract

.. Degenerate perturbation theory has been a basic subject in nuclear physics, chemical physics and the other branches of many-body problems. It provides us with effective interaction or effective Hamiltonian acting in a model space (or reference space). Many formulations or methods have been proposed for constructing the effective Hamiltonian since the beginning of the study of.quantum mechanics. The effective Hamiltonian is introduced so that diagonalization of it in the model space reproduces some of the eigenvalues of the original Hamiltonian. However, this requirement for the effective Hamiltonian is not sufficient to determine it uniquely. This is the reason why many definitions of the effective Hamiltonian are possible. . In a previous paper/) referred to as I, a general theory has been suggested for the classification of the effective Hamiltonians, although a rigorous proof was not given. One of the purposes of the present study is to give a general definition of the effective interac­ tion so that every effective interaction defined so far can be classified as a special case of the general definition. A general form of the effective interaction to be given includes the standard non-Hermitian form in Refs. 2), 3) and 8)~ 18), Kato's definition 4 ) and the Hermitian formS)-lS) which is often referred to as the Van Vleck form or canonical form. Another aim of the present study is to derive exact expansion formulae for various effective interactions. Many studies have been made to find general rules for representing the effective interactions in an expansion form. The expansion rules have been estab­ lished for some of the effective interactions, for example, the standard non-Hermitian form in Refs. 1) ~ 3) and 8) ~ 12) and Kato's form. 4),11) The expansion formula for the Hermitian form is rather complicated, and its exact formula has not yet been known. In the present contribution, a general method will be given for expanding the effective interactions including the Hermitian form. In §2, a general definition is given for the effective Hamiltonians and the classification rule is discussed. Interrelations among various effective Hamiltonians are also discussed. In §3, an exact expansion formulae are given for the effective interactions. In particular, the expansion formula for the Hermitian effective interaction is given in an explicit form which does not include any recursion procedure. In §4, the conclusions obtained in this