Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions

We study the Schrodinger operators H-lambda mu(K), with K is an element of T-2 the fixed quasimomentum of the particles pair, associated with a system of two identical fermions on the two-dimensional lattice Z(2) with first and second nearest-neighboring-site interactions of magnitudes lambda is an element of R and mu is an element of R, respectively. We establish a partition of the (lambda, mu) plane so that in each its connected component, the Schrodinger operator H-lambda mu(0) has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential spectrum and above its top. Moreover, we establish a sharp lower bound for the number of isolated eigenvalues of H-lambda mu(K) in each connected component.