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Generic Quantum Markov Semigroups with Degenerate Ground States: The Fock Case

We consider quantum Markov semigroups arising from the weak coupling limit of a system with generic Hamiltonian coupled to a boson Fock zero temperature reservoir. We find all the invariant states of a generic quantum Markov semigroup and compute explicitly the limit invariant state explicitly starting from an arbitrary initial state. We also show that convergence is exponentially fast under some natural assumptions.