We propose an analytic representation of completely antisymmetric basis states and the Hamiltonian matrix in a many-level shell model. §1. IntroductionIn the jj-coupling shell model 1), 2) with multishell (or multilevel) states, one of the most important problems is to efficiently construct antisymmetric orthonormalized basis states and to calculate the Hamiltonian matrix. In the traditional method, we first construct orthonormalized antisymmetric states in each subshell (or level), usually by using the coefficients of fractional parentage (CFPs) for each level, and then we couple the states to a completely antisymmetric state. In this algorithm, the method to calculate the CFPs at each level has been established, 2) but the orthonormalization of all multilevel states is, in general, a rather troublesome procedure. 3) We need to carry out many steps of angular-momentum coupling and sometimes numerical orthogonalization of overcomplete sets of states. If we can apply this procedure in an analytic form, then we could analytically represent the Hamiltonian matrix. This would be convenient for numerical calculations in the shell model. The present paper is devoted to this purpose.In §2 we introduce a new type of CFP to couple the orthonormalized basis states in each subshell (or level) to a completely antisymmetric state. In §3 we show how to analytically represent the matrix elements of the Hamiltonian using the new type of CFPs. Some concluding remarks are given in §4. §2. Formulation of a new type of CFP for a many-level shell model Let us consider N levels (or subshells) denoted by k = 1, 2, · · · , N. Hereafter we use "level" and "subshell" synonymously. The orthonormalized many-level shell model basis states are given by

The eigenvalue equation in the Dyson boson mapping is non-Hermitian. A method for a Hermitian treatment of this non-Hermitian eigenvalue equation has been proposed by one of the present authors. 1) If we intend to apply this method to realistic cases, we cannot help retaining only a small number of degrees of freedom which are important for describing the collective motions of interest. In such cases, this method would be an approximation. In the present paper, we test the validity of this approximation in numerical calculations for some realistic nuclei. The results show that this method of Hermitian treatment is a very good approximation within truncated boson subspaces. §1. IntroductionSince Beliaev and Zelevinski 2) and Marumori, Yamamura and Tokunaga 3) introduced the boson mapping method into nuclear theory in the early 1960's, two types of boson mappings (or boson expansion theories) have frequently been used for describing nuclear collective motion. One is the Holstein-Primakoff-type (HPtype) mapping 4) and the other is the Dyson-type (D-type) mapping. 5), * ) The D-type mapping has recently become appreciated, because it allows us to calculate the matrix elements in the boson space much more easily than in the case of the HP-type. Although the non-Hermiticity of the eigenvalue problem in the D-type mapping had been thought to be its only demerit, this difficulty was overcome by the Hermitian treatment. 1) The superiority of the D-type mapping was thereby established, and the D-type mapping has successfully been applied to various realistic nuclei. 7) -11) Very recently, Kajiyama, Taniguchi and Miyanishi 12) published a paper in which they casted a doubt on the validity of the above-mentioned Hermitian treatment. This treatment has been proved to be exactly valid as long as it is used in the ideal boson space corresponding to the full fermion-shell-model space. 1) This is, however, impossible in practical applications; namely, we can retain only a small number of degrees of freedom which are important for describing the collective motion of interest. Within such a truncated boson subspace, the Hermitian treatment is only an approximation. Kajiyama et al. asserted that this approximation would yield a similar amount of error to what the fourth-order truncation of the HP-type boson expansion gives. In previous works, 7) -11) we have sometimes used the Hermitian treatment, and we have always checked the errors introduced by this approximation. We have thus understood that the approximation is very good. (We have not published the * ) The complete set of references regarding boson mapping theories used in the study of the nuclear structure can be found in the review article by Klein and Marshalek. 6)