  We consider one-dimensional chains and multi-dimensional networks of harmonic
oscillators coupled to two Langevin heat reservoirs at different temperatures.
Each particle interacts with its nearest neighbors by harmonic potentials and
all individual particles are confined by harmonic potentials, too. In this
article, we provide, for the first time, the sharp N dependence of the spectral
gap of the associated generator under various physical assumptions and for
different spatial dimensions. Our method of proof relies on a new approach to
analyze a non self-adjoint eigenvalue problem involving low-rank non-hermitian
perturbations of auxiliary discrete Schr\"odinger operators.
