  In this paper we extend the local iterative Lie-Schwinger
block-diagonalization method - introduced in [DFPR3] for quantum lattice
systems with bounded interactions in arbitrary dimension- to systems with
unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can
be written as the sum of a gapped operator consisting of a sum of on-site terms
and a perturbation given by relatively bounded (but unbounded) interaction
potentials of short range multiplied by a real coupling constant t. For
sufficiently small values of |t| independent of the size of the lattice, we
prove that the spectral gap above the ground-state energy of such Hamiltonians
remains strictly positive. As in [DFPR3], we iteratively construct a sequence
of local block-diagonalization steps based on unitary conjugations of the
original Hamiltonian and inspired by the Lie-Schwinger procedure. To control
the ranges and supports of the effective potentials generated in the course of
our block-diagonalization steps, we use methods introduced in [DFPR3] for
Hamiltonians with bounded interactions potentials. However, due to the
unboundedness of the interaction potentails, weighted operator norms must be
introduced, and some of the steps of the inductive proof by which we control
the weighted norms of the effective potentials require special care to cope
with matrix elements of unbounded operators. We stress that no large-field
problems appear in our construction. In this respect our operator methods turn
out to be an efficient tool to separate the low-energy spectral region of the
Hamiltonian from other spectral regions, where the unbounded nature of the
interaction potentials would become manifest.
