Cooper pairing in multilayer polar structures

The Hamiltonian of the electron-phonon interaction with polar optical vibrations in multilayer structures is used to derive a Hamiltonian that describes, through the introduction of unitary transformations of Bogolyubov type, the formation of pairs of electrons with opposite spins and $2D$ wave vectors. The distribution function for the considered systems is derived using Zubarev's nonequilibrium statistical operator method. An evolution equation is obtained for the distribution function, and under certain simplifying assumptions an integral equation for the superconducting energy gap is obtained. The parameters of some structures are used in a numerical investigation of the temperature dependence of the gap for different thicknesses of the conducting layer. It is shown that some of the structures may be candidates for high-temperature superconductors.