  We study quantum chains whose Hamiltonians are perturbations by bounded
interactions of short range of a Hamiltonian that does not couple the degrees
of freedom located at different sites of the chain and has a strictly positive
energy gap above its ground-state energy. We prove that, for small values of a
coupling constant, the spectral gap of the perturbed Hamiltonian above its
ground-state energy is bounded from below by a positive constant uniformly in
the length of the chain. In our proof we use a novel method based on local
Lie-Schwinger conjugations of the Hamiltonians associated with connected
subsets of the chain.
