The Bethe Ansatz

The above considerations generalize to an arbitrary number \(N\) of particles. The periodicity conditions are then

\begin{equation*} \Psi_N (x_1, ..., x_{j-1}, x_j + L, x_{j+1}, ..., x_N | {\boldsymbol \lambda} ) = \Psi_{N} (x_1, ..., x_{j-1}, x_j, x_{j+1}, ..., x_N | {\boldsymbol \lambda} ). \end{equation*}

Imposing these in l.psin immediately leads to the Bethe equations (using \(\phi (0) = 0\))

\begin{equation*} e^{i\lambda_j L} = (-1)^{N-1} e^{-i \sum_{l=1}^N \phi (\lambda_j - \lambda_l)} \end{equation*}

which explicitly written out are

\begin{equation} e^{i\lambda_j L} = \prod_{l \neq j} \frac{\lambda_j - \lambda_l + ic}{\lambda_j - \lambda_l - ic}, \tag{l.be}\label{l.be} \end{equation}

or in logarithmic form

\begin{equation} \lambda_j + \frac{1}{L} \sum_{l=1}^N \phi (\lambda_j - \lambda_l) = \frac{2\pi}{L} I_j, \hspace{10mm} j = 1, ..., N, \tag{l.bel}\label{l.bel} \end{equation}

where

\begin{equation} I_j \in \left\{ \begin{array}{ll} \mathbb{Z} + \frac{1}{2}, & N ~\mbox{even} \\ \mathbb{Z}, & N ~\mbox{odd} \end{array} \right. \hspace{10mm} (\mbox{with}\hspace{5mm} I_{j_1} \neq I_{j_2} ~~~\mbox{if}~~~ j_1 \neq j_2) \tag{l.i}\label{l.i} \end{equation}

are again quantum numbers labelling the eigenstates, whose total momentum and energy are

\begin{equation} P_N = \sum_{j=1}^N \lambda_j, \hspace{10mm} E_N = \sum_{j=1}^N \lambda_j^2. \tag{l.pe}\label{l.pe} \end{equation}

At this point, we are thus in position to conjecture that a complete set of wavefunctions is obtained by choosing all allowable sets of non-coincident quantum numbers and constructing their associated Bethe Ansatz wavefunctions.