Threshold resonances and discrete spectrum of a two-boson hamiltonian with a molecular channel on the one-dimensional lattice

We study a two-channel lattice Hamiltonian in a fixed particle-number sector on the one-dimensional lattice Z. The model consists of a molecular channel coupled to a bosonic two-particle channel with on-site interaction. In the momentum representation, translation invariance yields a family of reduced Hamiltonians H_γ,λ(K) parametrized by total quasi-momentum K∈T, each acting in C⊕L^2,e(T).

For contact interactions, each reduced Hamiltonian is a rank-two perturbation of the free diagonal operator. By means of the Lippmann–Schwinger method, the eigenvalue problem for energies outside the essential spectrum [Ԑ^-_K, Ԑ⁺_K] is reduced to a 2×2 linear system and, equivalently, to the vanishing of an explicit scalar Fredholm determinant involving the one-dimensional lattice Green function. The square-root singularities of this Green function at the band edges E^±_K etermine the threshold asymptotics of the determinant and lead to explicit criteria for the existence of eigenvalues below the lower threshold and above the upper threshold.

We obtain a complete classification of the discrete spectrum of H_γ,_λ(K) for all quasi-momenta in terms of the parameters γ, λ, and E₀. We also analyze the threshold configurations E₀ =Ԑ^±_K, the exceptional flat-band fiber K=π, where the essential spectrum collapses to a single point, and the threshold states of the auxiliary rank-one operator h_λ(K). For the latter, we provide a weighted-space description of threshold resonances, clarifying how an eigenvalue emerges from a threshold resonance of h_λ(K) and approaches a band edge when the interchannel coupling is switched on.

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