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The band structure of the whole spectrum of an N-body cold system containing atoms with arbitrary integer spin and dominated by singlet pairing force

The spectra of $N$-boson systems with arbitrary nonzero spin $\mathfrak{f}$ have been studied. Firstly, only the singlet pairing interaction is considered, a set of eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin structures of various states different in $N$ and/or $\mathfrak{f}$, and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller $N$. Some features and notable phenomena that might emerge in high-$\mathfrak{f}$ systems, say, the ground band might have extremely high level density, have been discussed.