  We consider chains of $N$ harmonic oscillators in two dimensions coupled to
two Langevin heat reservoirs at different temperatures - a classical model for
heat conduction introduced by Lebowitz, Lieb, and Rieder \cite{RLL67}. We
extend our previous results \cite{BM20} significantly by providing a full
spectral description of the full Fokker-Planck operator allowing also for the
presence of a constant external magnetic field for charged oscillators. We then
study oscillator chains with additional next-to-nearest-neighbor interactions
and find that the spectral gap undergoes a phase transition if the
next-to-nearest-neighbour interactions are sufficiently strong and may even
cease to exist for oscillator chains of finite length.
