The spectra of all types of phonons existing in a complicated combined nanoheterosystem consisting of three cylindrical quantum dots embedded into the cylindrical quantum wire placed into vacuum are studied within the dielectric continuum model. It is shown that there are confined optical (LO) and interface phonons of two types: top surface optical (TSO) and side surface optical (SSO) modes of vibration in such a nanosystem. The dependences of phonon energies on the quasiwave numbers and geometrical parameters of quantum dots are investigated and analysed.

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 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 theory of renormalized energy spectrum of a multi-level quasiparticle interacting with polarization phonons at T = 0 K is developed within the Feynman-Pines diagram technique in a new approach. It permits a successive separation of multiplicative diagrams from non-multiplicative ones for all orders of mass operator, and their partial summing as well. The obtained mass operator is presented as a sum of branched chain fractions, which effectively take into account the multi-phonon processes. For the two-level quasiparticle it is shown that just the interlevel (non-diagonal) interaction with phonons fundamentally changes the properties of the spectrum. In the vicinity of all threshold energies, the quasi-equidistant phonon satellite bands (i.e., groups of energy levels) are formed. They correspond to the complexes of bound states of a quasi-particle with many phonons.