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

We study the Schrödinger operators H_λµ(K) that model a two-fermion system on the threedimensional lattice Z³, where total quasimomentum is fixed at K ∈ T³, and the particles interact through nearest- and next-nearest-neighbor couplings with strengths λ, µ ∈ R. For K = 0, we establish that H_λµ(0) admits reducing invariant subspace whose restriction depends solely on the parameter µ ∈ R. This µ parameter line contains two critical points corresponding to the lower and upper spectral thresholds; at each of these points, the Fredholm determinant of the restricted operator vanishes. Each of these critical points divides the parameter line into two infinite intervals, where the number of eigenvalues lying below (or above) the essential spectrum remains constant. Depending on µ, the corresponding reduced operator has exactly one discrete eigenvalue, located either below the bottom or above the top of the essential spectrum. Moreover, we derive a lower bound on the number of discrete eigenvalues of H_λµ(K) for all K ∈ T³.

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