Bound states of two surface phonons at a crystal surface with an adsorbed monolayer

We obtain the Bethe-Salpeter equation for the Green's function describing the propagation of two surface phonons interacting via cubic and quartic anharmonicity. We solve that equation by keeping only the latter contribution to its kernel. The two-dimensional nature of the problem is reflected in the existence of a bound state of two high-frequency surface phonons associated with the presence of an overlayer of light atoms, for all values of the coupling strength. This is illustrated with the detailed solution of a simple model.