Below-threshold effects for the two particle discrete Schr¨odinger operator on a lattice

We consider the family of Schrödinger operators Hγλ ( K ), which are associated with the Hamiltonian of a system of two identical bosons on the d -dimensional lattice Z d , where d ≥3, with interactions on each site and between nearest-neighbor sites with strengths γ ∈R− and λ ∈R−, respectively. Here, K ∈T d is a fixed quasi-momentum of the particles. We first partition the ( γ , λ )−plane into connected components S0, S1 and C j , j =0,1,2. Further, we establish below-threshold effects for Hγλ (0) on the boundaries of the connected components ∂S0 and ∂C j , j =0,2.