Bound States of a Polaron, Bi-polaron and multiple polarons.

Bi-polaron and multiple polarons have been considered using two particle and many-particles Green function method. The single polaron Green's function has been described in terms of the Dyson equation and has been incorporated into the Bi-polaron and multi-polarons Green functions. The polaron exact self-energy part incorporating three-point vertex function and describing the interaction of an electron with phonons (optical and /or acoustic) as shown include two terms. The first term stands for the self-consistent one–phonon contribution which in fact represents the Tamm-Dancoff approximation. The second term in the polaron self-energy describes the polaron bound states and can be written in terms of the effective scattering amplitude as shown. This approach allows to describe the polaron bound states similar to the two particles bound by the Coulomb potential. Bi-polaron bound states are described by the three sets: bi-polaron set, and two single polarons sets. Tri-polaron and quad-polaron have also been considered. A polaron, bi-polaron are considered due to the interaction of an electron with optical phonons in 3D and in 2D case with acoustic phonons where bound states exist even in the case of the weak electron-phonon coupling. The Bose condensation of bi-polaron and/or multi-polaron gas in a layered 2D system with interlayer coupling is discussed.