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 neighbours by harmonic potentials and all individual particles are confined by harmonic potentials, too. In previous works we investigated the sharp N-particle dependence of the spectral gap of the associated generator in different physical scenarios and for different spatial dimensions. In this talk I will present new results on the behaviour of the spectral gap when considering longer-range interactions in the same model. In particular, depending on the strength of the longer-range interaction, there are different regimes appearing where the gap drastically changes behaviour but even the hypoellipticity of the operator breaks down. This is a joint work with Simon Becker (ETH).