Quantum Bäcklund 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 Bäcklund 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 Schrödinger equation for an interpolating Hamiltonian.