Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators $H_{\gamma\mu}(k)$, $k\in\mathbb T^d:= (-\pi,\pi]^d$, associated with the Hamiltonian of a system consisting of at most two particles on a $d$-dimensional lattice $\mathbb Z^d$, interacting via both a pair contact potential $(\mu>0)$ and creation and annihilation operators $(\gamma>0)$. We prove the existence of a unique eigenvalue of $H_{\gamma\mu}(k)$, $k\in\mathbb T^d$, or its absence depending on both the interaction parameters $\gamma,\mu\ge0$ and the system quasimomentum $k\in\mathbb T^d$. We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum $k\in\mathbb T^d$ in the existence domain $G\subset\mathbb T^d$.