The Bethe Ansatz

As for real rapidities, the quantum numbers of strings are limited. We can calculate the maximal string quantum numbers using the same arguments as before:

\begin{equation*} \lim_{\lambda^j_\alpha \rightarrow \infty} (LHS ~\mbox{h.bgtl}) = \frac{2\pi}{N} \frac{1}{2}\left( N - \sum_{k=1}^{N_s} (2\min(j,k) - \delta_{jk}) (M_k - \delta_{jk}) \right) \equiv \frac{2\pi}{N} I^{j,\infty}. \end{equation*}

More precisely, this can be rewritten as

\begin{equation*} I^{j,\infty} = \frac{1}{2}\left( N + 2j-1 - \sum_{k=1}^{N_s} (2\min(j,k) - \delta_{jk}) M_k \right). \end{equation*}

We will make use of this in the following, in order to classify and count eigenstates.