Trace of Lax operator for the Gaudin model

Question

My question is about the following extract from https://arxiv.org/abs/hep-th/0404106

I don’t understand why the last factor in the expression of the trace has denominator $\frac{1}{a_{k} - a_{j}}$ and not $\frac{1}{z - a_{j}}$, I guess this is encoded in the action of $L$ but I cannot figure it out from its expression $(32)$.

Thanks for your help,

Dearly

Answer 1

To obtain (33), you can simply expand Tr(L(z)²) starting from (32) and use the partial fraction decomposition: $$ \frac{1}{z-a_j}\frac{1}{z-a_k} = \frac{1}{a_k-a_j} \left( \frac{1}{z-a_k} - \frac{1}{z-a_j} \right), $$ valid when $j \neq k$.