Feshbach-induced binding of two non-interacting bosons coupled to a structureless molecular channel on the two-dimensional lattice

We study a translation-invariant two-channel Hamiltonian on the two-dimensional lattice Z 2 describing two identical spinless bosons with no direct boson–boson interaction, coupled through an on-site association/dissociation term of strength γ ∈ R to a structureless molecular channel of fixed scalar internal energy E 0 ∈ R. The model is a 2 × 2 block operator matrix; it is translation invariant and admits a fiber decomposition over the total quasi-momentum k ∈ T 2 , whose fibers H γ (k) act on C ⊕ L 2,e (T 2 ). The perturbation distinguishing H γ (k) from the free operator has rank at most two, so the essential spectrum of each fiber is [ϵ min (k), ϵ max (k)], equal to [0, 16] at k = 0, independently of E 0 and γ. For the distinguished fiber H γ (0), we determine the discrete spectrum completely and uniformly in the coupling: when γ = 0, the channels decouple and the only discrete eigenvalue is the bare molecular level E 0 , present exactly when E 0 ∈/ [0, 16]; whereas for every γ ̸= 0 and every E 0 ∈ R, the operator has exactly two simple discrete eigenvalues, one strictly below and one strictly above the two-boson continuum [0, 16]. In two dimensions, the logarithmic singularity of the lattice Green’s function at the spectral edges makes both bound states appear for arbitrarily weak coupling, the binding energies being exponentially small when E 0 lies inside the continuum and only algebraically small (up to a logarithm) when E 0 sits at a band edge. The model is a rigorous lattice realization of Feshbach-induced binding produced by interchannel coupling alone.

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