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

The optimal spectral gap for regular and disordered harmonic networks of oscillators