Ground state of the hard-core Bose gas on lattice I. Energy estimates

We investigate the properties of the ground state of a system of interacting bosons on regular lattices with coordination number $k\geq 2$. The interaction is a pure, infinite, on-site repulsion. Our concern is to give an improved upper bound on the ground state energy per site. For a density $\rho$ a trivial upper bound is known to be $-k\rho(1-\rho)$. We obtain a smaller variational bound within a reasonably large family of trial functions. The estimates make use of a large deviation principle for the energy of the Ising model on the same lattice.