  We develop an analytical model to describe the phonon dispersion relations of
host-guest lattices with heavy guest atoms (rattlers). Crucially, the model
also accounts for phonon damping arising from anharmonicity. The spectrum of
low energy states contains acoustic-like and (soft) optical-like modes, which
display the typical avoided crossing, and which can be derived analytically by
considering the dynamical coupling between host lattice and guest rattlers.
Inclusion of viscous anharmonic damping in the model allows us, for the first
time, to compute the vibrational density of states (VDOS) and the specific
heat, unveiling the presence of a boson peak (BP) linked to an
anharmonicity-smeared van Hove singularity. Upon increasing the coupling
strength between the host and the guest dynamics, and by decreasing the energy
of the soft optical modes, the BP anomaly becomes stronger and it moves towards
lower frequencies. Moreover, we find a robust linear correlation between the BP
frequency and the energy of the soft optical-like modes. This framework
provides a useful model for tuning the thermal properties of host-guest
lattices by controlling the VDOS, which is crucial for optimizing thermal
conductivity and hence the energy conversion efficiency in these materials.
