Linear combinations of two-body energies in matrix elements of jn states

TL;DR

This paper demonstrates that in jn configurations, certain matrix elements of two-body interactions are linear combinations of two-body energies with universal rational coefficients, enabling solvable eigenvalues for specific states.

Contribution

It introduces a method to identify and construct bases where eigenvalues of two-body interactions are explicitly solvable as linear combinations with fixed coefficients.

Findings

Diagonal matrix elements are linear combinations of two-body energies with fixed rational coefficients.

Eigenvalues for states with a unique J in jn configurations are explicitly solvable.

States in the seniority scheme can be eigenstates with solvable eigenvalues.

Abstract

Matrix elements of a two-body interaction between states of the jn configutation (n identical nucleons in the j-orbit) are functions of two-body energies. In some cases, diagonal matrix elements are linear combinations of two-body energies. The coefficients of these linear combinations are rational and non-negative numbers, independent of the two-body interaction. It is shown that if in the jn configuration there is only one state with given spin J, its eigenvalue (the diagonal matrix element) is equal to a linear combination of two-body energies with rational and non-negative coefficients. These coefficients have the same values for any two-body interaction (solvable eigenvalues). If there are several J-states in the jn configuration, they define a sub-matrix of the interaction which should be diagonalized to yield eigenvalues and eigenstates. Bases of these states are constructed from which the sub-matrix characterized by J may be obtained for any two-body interaction. The diagonal elements are linear combinations of two-body energies whose coefficients are independent of the two-body interaction and the non-vanishing ones are rational and positive. Aslo in the non-diagonal elements the coefficients have a simple form. States in the seniority scheme are shown to form such bases. If one of them is an eigenstate of any two-body interaciton its eigenvalue is shown to be solvable.