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Within the unitary transformed Hamiltonian of Frohlich type, using the retarded Green’s function method, the exact renormalized energy spectrum of a quasi-particle strongly interacting with three-mode polarization phonons is obtained at T = 0 K in the model of a system with a limited number of its initial states. The exact analytical expressions for the average number of phonons in the main and all satellite states of the system are obtained. Their dependences on the magnitude of the interaction between quasi-particle and both phonon modes are analyzed.

Within the model of rectangular potential profile and continuous position-dependent effective mass of electron in the double-barrier open semiconductor plane nano-heterostructure, the exact solutions of Schrodinger equation are obtained. Using them, the transmission coefficient, resonance energies, and widths of sub-barrier quasi-stationary electron states are calculated for the structure with In0.53Ga0.47As wells and In0.52Al0.48As barriers. The transmission coefficient and spectral characteristics of electron states are analyzed as functions of the width of near-interface region where the effective mass linearly depends on coordinate. It is shown that the increasing width of near-interface region between wells and barriers does not change the heights of maxima of transmission coefficient peaks but shifts and broadens them.