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Every solution of the Bethe-ansatz equations (BAE) is characterized by a setof quantum numbers, by which we can evaluate it numerically. However, nogeneral rule is known how to give quantum numbers for the physical solutions ofBAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbersfor the complete set of the Bethe-ansatz eigenvectors in the two down-spinsector with any chain length $N$. Here we obtain them both for real and complexsolutions. Consequently, we prove the completeness of the Bethe ansatz and givean exact expression for the number of real solutions which correspond tocollapsed bound-state solutions (i.e., 2-string solutions) in the sector: $2[(N-1)/2 - (N/\pi) \tan^{-1}(\sqrt{N-1})]$ in terms of Gauss' symbol. Moreover,we prove in the sector the scheme conjectured by Takahashi for solving BAEsystematically. We also suggest that by applying the present method we canderive the quantum numbers for the spin-1/2 XXZ chain.

Exact quantum numbers of collapsed and non-collapsed 2-string solutions in the Heisenberg spin chain