The parity-odd effective action for the torsion-like background field induced by fermoins is obtained using different regularization methods, which provide the maximal residual symmetry allowed by the presence of the Lorentz and CPT breaking constant axial-vector backgorund. It is shown that such classes of regularizations are justified by Lorentz-noninvariant decays of ultrahigh-energy fermions. The pheonomenological consequences of Lorentz and CPT symmetry breaking are examined and certain bounds on these effects are derived.

The approach was adopted that QCD can be effectively described with string-like variables. In order to overcome the long-standing problems with the behavior of hadronic string amplitudes at low energies the string is built over the chirally non-invariant QCD vacuum by means a the boundary interaction with background chiral fields associated with pions. By making this interaction compatible with the conformal symmetry of the string and with the unitarity constraint on chiral fields the equations of motion for the latter ones are reconstructed and furthermore recover the Lagrangian of non-linear sigma model of pion interactions including the chiral structural constants of dimension-four. The estimated coefficients fit well the phenomenological values which proves the stringy nature of strong interactions at low energies.

Fermion models with self-interaction including derivatives of fermion fields are investigated in the strong coupling regime when several coupling constants take critical values. In the two-channel model the mass spectrum of composite mesons woth differnet quantum numbers and spins is obtained. The possibility of P-parity breaking in such models is exploited to construct two types of minimal extension of Standard Model with composite Higgs particles. To enlarge the predictivity of quasilocal quark models the chiral sum rules based on the restoration of chiral symmetry at high energies are applied and certain estimations on the mass splitting between parity doublers in scalra nad vector channels are obtained. The precision of less than 15% has been achived for a variety of mesons with masses up to 2 GeV which happens to give a sufficient resolution to classify the meson resonances in the above mentioned mass range.

Two-point correlators of color-singlet quark currents are studied in large-N_c limit and respectively saturated by an infinite tower of narrow resonances. The relations between masses and decay constants of variety of meson resonances in the energy range 0--3 GeV are verified from the string-like, linear mass spectrum for vector, axial-vector, scalar and pseudoscalar mesons with a universal slope. The way to match the universality with the Operator Product Expansion (OPE) is proposed. The necessity of small deviations from linearity in parameterization of the meson mass spectrum and their decay constants (wave function projections) is clearly proven from matching to OPE.

the Nonlinear Supersymmetry algebra in one-dimensional Quantum Mechanics is studied in its differential representation. Its structure is found to be determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. It is rigorously shown that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a Nonlinear superalgebra which can be appropriately specified as N = 2 SUSY. In this case the non-trivial hidden symmetry operator is revealed and expressed as a non-linear function of the Hamiltonian on the physical state space. The full N = 2 Non-linear SUSY algebra includes "central charges" both polynomial and non-polynomial (due to a symmetry operator) in the Hamiltonian.

The occurrence or not of the Bose-Einstein Condensation for an ideal gas in the presence of a point-like impurity and a uniform field has been studied in D-dimensions with D=1,2,3. The result is that no Bose-Einstein condensation occurs in the absence of the impurities if D=2,3, whereas it takes place iff the impuruty is there and the corresponding critical temperature and density were also obtained.

A possible explanation for the appearance of light fermions and Higgs bosons on the four-dimensional domain wall is proposed. The mechanism of light particle trapping is accounted for by a self-interaction of five-dimensional pre-quarks. The induced relation between low-energy couplings for Yukawa and scalar field interactions allows to make certain predictions for light particle masses and couplings themselves, which might provide a signature at future experiments of the higher dimensional origin of particle physics. The fluctuation of the brane gives rise to a nearly sterile scalar particles, branons, which may be candidates for the dark matter.

The temporal evolution of the decay (ionization) probability has been studied for a model atom in D=1,2,3 spatial dimensions in the presence of a uniform field, i.e. static and homogeneous. The model atom consists in a non-relativistic point-like particle in the presence of a delta-like potential well. This model is shown to be exactly solvable and to exhibit infinitely many resonances, leading to deviations from the conventional exponential decay law of the survival probability of the initial atomic quantum state. Nearly stable atomic states always exist, no matter how weak is the uniform external field. Analytic as well as numerical estimates have been obtained for the decay rapidity of the atomic quantum states.