Quantum backlund transformations: Some ideas and examples

We propose 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. This property turns out to be deeply connected with the Hamilton-Jacobi separation of variables and can lead to the explicit integration of the corresponding model using the BTs. We show that once such a construction is given, we can interpret the Baxter Q-operator defining the quantum BTs as the Green's function or the propagator of the time-dependent Schrodinger equation for an interpolating Hamiltonian.