Title: Spectral Theory of Non-Markovian Dissipative Phase Transitions

Abstract: Controlling phase transitions in quantum systems via coupling to reservoirs has been mostly studied for idealized (memory-less) environments. Here, we present a general method to tackle dissipative phase transitions (DPTs) in non-Markovian systems, extending the scope of dissipative engineering of matter phases to more realistic materials and experiments in the solid-state and atomic, molecular and optical physics. We show how memory effects can be used to reshape phase boundaries but also reveal the existence of DPTs genuinely triggered by non-Markovianity.