In this work we give a mechanical (Hamiltonian) interpretation of the so called spectrality property introduced by Sklyanin and Kuznetsov in the context of Backlund transformations (BTs) for finite dimensional integrable systems. The property turns o
ut to be deeply connected with the Hamilton-Jacobi separation of variables and can lead to the explicit integration of the underlying model through the expression of the BTs. Once such construction is given, it is shown, in a simple example, that it is possible to interpret the Baxter Q operator defining the quantum BTs us the Greens function, or propagator, of the time dependent Schrodinger equation for the interpolating Hamiltonian.
We construct Backlund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact time-discre
tization of the system, inasmuch as they preserve both its Poisson structure and its invariants. Moreover, in some special cases we are able to show that these maps can be explicitly integrated in terms of the initial conditions and of the iteration time $n$. Encouraged by these partial results we make the conjecture that the maps are interpolated by a specific one-parameter family of hamiltonian flows, and present the corresponding solution. We enclose a few pictures where the orbits of the continuous and of the discrete flow are depicted.
