Separation of variables for integrable spin-boson models

We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang- Baxter algebra. The main deviation from the standard approach consists in a half infinite Sklyanin lattice made of the eigenvalues of the operator zeros of the Bethe annihilation operator. By a separation of vari- ables, functional TQ-equations are obtained for this half infinite lattice. They provide valuable information about the spectrum of a given Hamiltonian model. We apply this procedure to integrable spin-boson models subject to both twisted and open boundary conditions. In the case of general twisted and certain open bound- ary conditions polynomial solutions to these TQ-equations are found and we compute the spectrum of both the full transfer matrix and its quasi-classical limit. For generic open boundaries we present a two-parameter family of Bethe equations, derived from TQ-equations that are compatible with polynomial solutions for Q. A connection of these parameters to the boundary fields is still missing.