Using the Feynman-Pines diagram technique, the energy spectrum of localized quasi-particles interacting with polarization phonons is calculated and analyzed in the wide range of energies at the finite temperature of the system. It is established that the general model of the system, besides the bound states known from the simplified model with an additional condition for the operator of quasi-particles number, contains the new bound states even for the systems with weak coupling. The contribution of multi-phonon processes into the formation of renormalized spectrum of the system is analyzed. The reasons of the appearance, behaviour and disappearance of separate pairs of bound states depending on the coupling constant and temperature are revealed.

Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröh-lich Hamiltonian is performed using the Feynman-Pines diagram technique. Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K. It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra. The ground state stays stationary and the excited one decays at a bigger coupling constant.

Theory of the spectrum of localized two-level quasi-particle renormalized due to interaction with polarization phonons at cryogenic temperature is developed using the generalized method of Feynman-Pines diagram technique. Using the procedure of partial summing of infinite ranges of the main diagrams, mass operator is obtained as a compact branched chain fraction, which effectively takes into account multi-phonon processes. It is shown that multi-phonon processes and interlevel interaction of quasiparticle and phonons cardinally change the renormalized spectrum of the system depending on the difference of energies of two states, which either resonates with phonon energy or does not. The spectrum of non-resonant system contains renormalized energies of the main states and two similar infinite series of groups of phonon satellite levels. The spectrum of a resonant system contains a renormalized ground state and infinite series of satellite groups.

The quantum mechanical theory of photon-and phonon assisted tunneling of electrons through the open multilayered nanostructures being the cascades of quantum cascade detector and quantum cascade laser is developed. The theory of electronic dynamic conductivity and spectral parameters (resonance energies and widths) of their quasi-stationary states is established within the effective mass approximation and models of rectangular and triangular potential profiles of resonant tunneling structure, respectively.It is shown that an open model of multilayered resonant tunneling structure, comparing with the closed one, adequately describes the physical processes in cascades and their elements (active regions, extractors, injectors) and, with respect to the experimental data, gives an opportunity to calculate and reveal their optimal geometrical design without any fitting parameters.

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