Ground-state energy of a polaron inndimensions

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_nD(α)=(n/3)E3D({{Γ}[ (n-1)/2]3 {}{π} /2n{Γ}(n/2)}{α}), where EnD is the Feynman polaron ground-state energy for the polaron in n dimensions and E3D the energy in three dimensions.