We determine the effects of nonlocal, nonlinear interactions on the excitation spectrum of lattice quantum field scalar models. We consider perturbations of a quantized discrete string formally self-adjoint Hamiltonian operator on the lattice Z d , and with a large mass coefficient for the quadratic term. The low-lying energy-momentum spectrum has an isolated dispersion curve and a two-particle (first) band. We analyze a ladder approximation of the Bethe-Salpeter equation on the lattice, for a

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... eak perturbation of the type , λ 6 > 0, and consider the spectral interval starting at zero and extending to near the three-particle threshold. For space dimension d = 1, 2 and V (ϕ( x )) = λ1 : ϕ( x ) 4 :, we find that a bound state occurs either below (if λ 1 < 0) or above the first band (if λ 1 > 0), but not both. This agrees with recent results where bound states were obtained for the stochastic dynamics generator associated with the relaxation rate to equilibrium in weakly coupled stochastic Ginzburg-Landau models with continuous time and on a spatial lattice Z d . These results are in contrast, however, with those obtained for V (ϕ( x )) = λ 2 : ϕ( x ) 3 (−∆ϕ)( x ) : . For this case, we show that stable particles exist above and below the band, for d = 1, 2, regardless the sign of the coupling λ 2 . If V (ϕ( x )) = λ 3 : ϕ( x ) 2 (−∆ϕ 2 )( x ) : , the ladder analysis is inconclusive. If d = 3, ..., no bound states exist in the spectral region we consider.