Abstract
Several physical models involving the interaction of electronic and bosonic degrees of freedom have been proposed and solved exactly in recent years by a new recursion technique. The models belong to various contexts, but most applications have been in the theory of electron and photon spectroscopies of molecules and solids. For a class of Hamiltonians which describe closed-shell systems the recurrence relations provide closed-form analytical solutions. Here the authors present the method and discuss its relationship to alternative approaches that are also based on deriving and solving recurrence relations. They review the main results obtained to date and present solutions relevant to optical spectroscopy for the first time. They demonstrate the usefulness of exact analytical solutions for understanding the nontrivial dynamical behaviour of interacting fermion-boson systems.