Using the effective mass and rectangular potential approximations, the theory of electron dynamic conductivity is developed for the plane multilayer resonance tunnel structure placed into a constant electric field within the model of open nanosystem, and oscillator forces of quantum transitions within the model of closed nanosystem. For the experimentally produced quantum cascade laser with four-barrier active band of separate cascade, it is proven that just the theory of dynamic conductivity in the model of open cascade most adequately describes the radiation of high frequency electromagnetic field while the electrons transport through the resonance tunnel structure driven by a constant electric field.

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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.

A theory of the evolution and collapse of pairs of resonances because of a change in the strength (thickness) of the inner barrier is developed in a model of effective masses and symmetric, square, potential barriers for an electron in a planar three-barrier structure. Analytical and numerical calculations of the spectral parameters (resonance energies and widths) are performed by the transmission coefficient and probability distribution function method using the transfer and S matrices. It is shown that the collapse of the resonance energies and widths of all quasistationary states in a symmetric three-barrier structure occurs for practically the same thicknesses of the inner barriers, somewhat greater than the total thickness of the outer barriers. It is established that with respect to the square-barrier model the δ model overestimates the resonance energies by 10% and the resonance widths by almost a factor of 2.

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