  The aim of this Ph.D. thesis was to investigate superconducting properties in
the presence of Zeeman magnetic field in systems with local fermion pairing on
the lattice. The study also concerned the evolution from the weak coupling
(BCS-like) limit to the strong coupling limit of tightly bound local pairs
(BEC) with increasing attraction, both in the ground state and at finite
temperatures, within the spin-polarized extended Hubbard model. The analysis
was also extended to the case of spin dependent hopping integrals (mass
imbalance), with special attention paid to the BCS-BEC crossover physics in the
ground state. The methods used included: the mean field approximation
(BCS-Stoner type) and the estimation of the phase coherence temperature within
the Kosterlitz-Thouless (KT) scenario in two dimensions. The BCS-BEC crossover
was also analyzed in three dimensions, at finite temperatures, within the spin
polarized Attractive Hubbard Model (AHM), going beyond the mean field
approximation. In this case, the critical temperatures of the superconducting
transition were determined within the self-consistent T-matrix method. The
strong coupling expansion was applied to map the spin polarized AHM onto the
model of hard-core bosons on the lattice.
