Quantum Fokker-Planck models: Limiting case in the Lindblad Condition.

In this article we study stationary states and the long time asymptotics for the quantum Fokker–Planck equation. We continue the investigation of an earlier work in which we derived convergence to a steady state if the Lindblad condition $D_{pp}D_{qq} −D_{pq} \ge \gamma^2/4$ is satisfied with strict inequality. Here we extend our results to the limiting case that turns out to be more difficult because irreducibility of the quantum Markov semigroup does not follow from triviality of the generalized commutator with position and momentum operators.