Liouvillian exceptional points in continuous variable system

The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in generic environment are obtained. They occur at the when modified frequency vanishes, whereby eigenvalues become real. In system there two parameters that modify oscillator's natural frequency. One can be damping rate. point then corresponds to critical oscillator. This situation is illustrated by Caldeira--Leggett (CL) and limit Hu--Paz--Zhang (HPZ) equation. other parameter changes effective mass reached extremely heavy form Kossakowski--Lindblad (KL) eigenfunctions coalesce break into subspaces labelled number $N$. each $N$-subspace, $(N+1)$-fold degeneracy has Jordan block structure order-$(N+1)$. We obtain explicit generalized eigenvectors few Liouvillians. Because degeneracies, freedom choice eigenfunctions. manifests itself as invariance under similarity transformation whose compare relaxation first excited state underdamped region, critically damped region which point, overdamped using CL