We introduce single particle open quantum systems in one spatial dimension and review how spectral properties of the Lindbladian influence the relaxation towards the steady state in finite volume.
Afterwards, we briefly discuss spectral properties of infinite volume systems and how the spectra of the Lindbladian operators can be calculated using a direct integral of Laurent operators.
We then look at Lindblad operators with a random potential in the Hamiltonian and discuss what localization could mean in this context. In particular, we describe how a gap in the effective Non-Hermitian Hamiltonian implies exponential decay in the off-diagonal elements of the steady state. We discuss how it might more generally be true that dissipation and disorder imply decoherence in our setup.