The Bethe Ansatz

Slavnov's scalar product formula ssp can be specialized to obtain the norm of Bethe eigenstates, by taking the limit \(\mu_a \rightarrow \lambda_a\), \(a=1, ..., M\). This yields Gaudin's norm formula

\begin{equation} {\mathbb N}_M = \langle 0 | \prod_{j=1}^M C (\lambda_j) \prod_{k=1}^M B (\lambda_k) | 0 \rangle = \varphi^M(\eta) \prod_{j \neq k} \frac{\varphi(\lambda_j - \lambda_k + \eta)}{\varphi(\lambda_j - \lambda_k)} \det \Phi (\{ \lambda \}), \tag{gnf}\label{gnf} \end{equation}

with the Gaudin matrix entries being

\begin{equation} \Phi_{jk} = -\frac{\partial}{\partial \lambda_k} \ln \left[ \frac{a(\lambda_j)}{d(\lambda_j)} \prod_{l = 1, \neq j}^M \frac{b(\lambda_j, \lambda_l)}{b(\lambda_l, \lambda_j)}\right]. \tag{gm}\label{gm} \end{equation}