Extension of Frohlich's method to 4-fermion interactions
  Higher order terms of the transformed electron-phonon Hamiltonian, obtained by performing the Frohlich's transformation, are investigated. The influence of terms discarded by Frohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Frohlich-type transformation is performed, which yields, among others, an effective 4-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the 4-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with interaction coupling given by - D_{k_F}^6 / \omega_{k_F}^5, where D_{k} is electron-phonon coupling, \omega_{k}$ is phonon energy and k_F is Fermi momentum.   The form of higher order terms of the original Frohlich-transformed H_{e-ph} are also found, up to terms proportional to the 6-th power of the coupling, that is up to those, which yield the effective 4-electron interactions. 
Przemyslaw Tarasewicz and Dominik Baran