Localised Davies generators for unbounded operators

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyen proposed a construction involving localisation in time and carried out it out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad is expected to settle to the Gibbs state. In this note, we show that the construction also works for classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution.