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$J$-pairing interaction, number of states, and nine-$j$ sum rules of four identical particles

Y. M. Zhao·A. Arima
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Abstract

In this paper we study $J$-pairing Hamiltonian and find that the sum of eigenvalues of spin $I$ states equals sum of norm matrix elements within the pair basis for four identical particles such as four fermions in a single-$j$ shell or four bosons with spin $l$. We relate number of states to sum rules of nine-$j$ coefficients. We obtained sum rules for nine-$j$ coefficients $<(jj)J,(jj)K:I| (jj)J, (jj)K:I>$ and $<(ll)J,(ll)K:I| (ll)J, (ll)K:I>$ summing over (1) even $J$ and $K$, (2) even $J$ and odd $K$, (3) odd $J$ and odd $K$, and (4) both even and odd $J,K$, where $j$ is a half integer and $l$ is an integer.

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Y. M. Zhao, A. Arima. 2005-11-10. $J$-pairing interaction, number of states, and nine-$j$ sum rules of four identical particles. https://doi.org/10.1103/physrevc.72.054307

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