The exact analytic expression for the polaron mass operator in the second order of the coupling constant is established for the first time. It holds in the whole energy scale. The peculiarities of the mass operator are analysed. The renormalized polaron energy spectrum is obtained and analysed in this approximation.

Electron and hole spectra in -HgS cylindrical quantum wire superlattice in -CdS matrix are calculated within the method of augmented plane waves. The energy term determined by the movement of quasiparticles in direction perpendicular to the axial axis of the wire is presented by the alternating zones with a positive and a negative effective mass. The degeneration on the magnetic quantum number outside ¡ point of Brillouin zone is taken off when the potential of the superlattice quantum wires is taken into account. The ground zones energy dependence of plane quasiparticle movement on the radii of the quantum wires as well as the distance between the wires are investigated.

Electron spectrum in cylindrical quantum dot HgS embedded into ZnS medium is calculated using the variational method with variational parameter in Hamiltonian. The dependence of energy spectrum on the quantum well sizes is established. The electron spectrum calculated in the framework of infinitely deep potential well is compared to the one obtained within the variational method. It is shown that the first method gives satisfactory results for the ground level only and at rather big sizes of quantum well.

The evolution of electron quasistationary spectrum in open spherical quantum dot is under study within the effective mass and rectangular potential model. Within the framework of the S-matrix model the exact solution of Schrödinger equation is obtained in general analytical form. It is shown, for the first time, that the generalized resonance energies and widths introduced as the parameters defining the probability distribution function (over the energy or quasi momentum) of electron location in quantum dot, adequately characterize the evolution of its quasistationary states (contrary to the S-matrix poles) in the whole range of barrier thickness: from zero (free states) up to the infinity (stationary bound states are under the barrier and virtual and free states are above it).

The analytical and numerical calculations of electron and hole spectra renormalised by L-and I-phonons taking into account the configurational interaction are performed for the QD embedded into semiconductor medium exemplified by GaAs/Al x Ga 1−x As nanoheterosystems.It is established that for the nanosize QDs the shifts of electron and hole ground levels are created by the interaction of these quasiparticles with Land I-phonons due to all the states of discrete and continuous spectrum. For the small QDs, the shifts of ground energy levels have strong nonlinear dependences while for the big QDs, they almost do not depend on QD radius and have the magnitude close to the shifts of ground levels in massive crystal creating QD. Due to the different effective masses of light and heavy holes, the splittings of their ground levels are the complicated functions on QD radius and Al concentration in Al x Ga 1−x As medium.