The Bethe Ansatz

Here and in the following sections, we use Slavnov's simplified notations for products over rapidities: given a set of rapidities \(\{ \lambda \}\) of cardinality \(M\), an overbar on a rapidity (as argument of a function or operator) means product over all rapidities in the set, while adding a subindex means an exclusive product, i.e.

\begin{align*} f (\bar{\lambda}) &\equiv \prod_{j=1}^M f (\lambda_j),\\ f (\bar{\lambda}_k) &\equiv \prod_{\substack{j=1\\j \neq k}}^M f (\lambda_j). \tag{shorthand}\label{shorthand} \end{align*}

We extend those conventions to also cover cases of double products (as per single products, subscripts (if present) specify terms which are excluded in the product):

\begin{align*} f (\bar{\lambda\!\lambda}) &\equiv \prod_{j=1}^M \prod_{k=1}^M f (\lambda_j - \lambda_k), \\ f (\bar{\lambda\!\lambda}_{\scriptscriptstyle =}) &\equiv \prod_{j=1}^M \prod_{\substack {k=1\\k\neq j}}^M f (\lambda_j - \lambda_k), \\ f (\bar{\lambda\!\lambda}_{\scriptscriptstyle \geq}) &\equiv \prod_{1 \leq j < k \leq M} f (\lambda_j - \lambda_k), \\ f (\bar{\lambda\!\lambda}_{\scriptscriptstyle \leq}) &\equiv \prod_{M \geq j > k \geq 1} f (\lambda_j - \lambda_k). \end{align*}