Speaker: Alessandra Pizzo (Roma Tor Vergata).
We consider quantum chains whose Hamiltonians are perturbations, by 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. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove that 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, for small values of a coupling constant. Under the same hypothesis, we prove that the ground state energy is analytic for values of the coupling constant in a fixed interval, 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, that can be also applied to complex Hamiltonians obtained by considering complex values of the coupling constant.
We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved, in contrast to most approaches.
(Based on joint work with J. Froehlich, and with S. Del Vecchio, J. Froehlich, and S. Rossi.)