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We consider quantum chains whose Hamiltonians are perturbations byinteractions of short range of a Hamiltonian that does not couple the degreesof freedom located at different sites of the chain and has a strictly positiveenergy gap above its ground-state energy. For interactions that areform-bounded w.r.t. the on-site Hamiltonian terms, we have proven that thespectral gap of the perturbed Hamiltonian above its ground-state energy isbounded from below by a positive constant uniformly in the length of the chain,for small values of a coupling constant; see [DFPR]. The main result of thispaper is that, under the same hypotheses, the ground-state energy is analyticfor values of the coupling constant belonging to a fixed interval, uniformly inthe length of the chain. Furthermore, assuming that the interaction potentialsare invariant under translations, we prove that, in the thermodynamic limit,the energy per site is analytic for values of the coupling constant in the samefixed interval. In our proof we use a new method introduced in [FP], which isbased on local Lie-Schwinger conjugations of the Hamiltonians associated withconnected subsets of the chain. We prove a rather strong result concerningcomplex Hamiltonians corresponding to complex values of the coupling constant.

Lie-Schwinger block-diagonalization and gapped quantum chains: analyticity of the ground-state energy