On Backlund transformations for many-body systems

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Backlund transformations (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum e-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B-lambda, such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition B-lambda 1 o...o B-lambda pi of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.