An effective interaction potential for a pair of polarons is derived in a Landau-Pekar adiabatic approximation.The possibility of formation of bound states of two polarons is discussed.The lowlying energy states of a system of Landau-Pekar polarons interacting with the derived effective potential are investigated by employing approximation models including a body-centered cubic lattice of polarons, a Hartree-Fock gas of polarons, and a Hartree-Fock gas of polarons weakly coupled to a polaron plasma.The low-lying energy states based on these models are examined numerically for a specific test case intended to describe polarons in an ionic crystal with a polaron coupling constant large enough that the Landau-Pekar approximation may be expected to be appropriate.It is found that, for a wide range of polaron densities, the plasma-coupled gas model yields the best results among the models treated.
The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3)
E3D({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where
EnD is the Feynman polaron ground-state energy for the polaron in n dimensions and
E3D the energy in three dimensions.