Alternate proof of the Rowe-Rosensteel proposition and seniority conservation

TL;DR

This paper provides an alternative proof of the Rowe-Rosensteel proposition for three nucleons in a single-$j$ shell, using symmetric properties of $6j$ symbols, and explores implications for seniority conservation.

Contribution

It offers a new proof of the Rowe-Rosensteel propositions based on $6j$ symbol symmetries and derives algebraic expressions for state dimensions and seniority conservation conditions.

Findings

Proved propositions using $6j$ symbol symmetries

Derived algebraic expressions for state dimensions and energies

Identified conditions for seniority conservation

Abstract

For a system with three identical nucleons in a single-$j$ shell, the states can be written as the angular momentum coupling of a nucleon pair and the odd nucleon. The overlaps between these non-orthonormal states form a matrix which coincides with the one derived by Rowe and Rosensteel [Phys. Rev. Lett. {\bf 87}, 172501 (2001)]. The propositions they state are related to the eigenvalue problems of the matrix and dimensions of the associated subspaces. In this work, the propositions will be proven from the symmetric properties of the $6j$ symbols. Algebraic expressions for the dimension of the states, eigenenergies as well as conditions for conservation of seniority can be derived from the matrix.