Existence of three-particle bound states in optical lattice

We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength $\mu_1\in\mathbb{R}$, while the interaction between a fermion and one other particle is via an on-site potential with strength $\mu_2\in\mathbb{R}$. We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum $K\in\mathbb{T}^1$. Furthermore, we show that both the bound state $f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)$ and its corresponding eigenvalue $E_{\mu_1\mu_2}(K)$ depend holomorphically on the quasimomentum.