Ground-state energy of a polaron inndimensions

The Fröhlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fröhlich 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 $E_{n\mathrm{D}\left(\mathrm{\alpha }\right)}$=(n/3)${\mathrm{E}}_{3\mathrm{D}}$({Γ[ (n-1)/2]3 √π /2nΓ(n/2)}α), where $E_{n\mathrm{D}}$ is the Feynman polaron ground-state energy for the polaron in n dimensions and $E_{3\mathrm{D}}$ the energy in three dimensions.