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

Abstract

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