We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.

中文翻译：

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低密度极限的量子马尔可夫半群：一般情况

我们研究了量子马尔可夫发生器的结构，该发生器描述了测试粒子在低密度极限下与稀玻色气体相互作用的降低动力学。这些发生器称为低密度极限型发生器 (LDL)，与 [4, 5] 中研究的弱耦合极限型发生器 (WCL) 不同，因为存在描述气体粒子动量变化的 T 算子由于与测试粒子的碰撞。我们提出了随机极限类型马尔可夫发生器的一般定义，包括（几乎）随机极限方法中出现的所有发生器，并且我们证明了相关的半群以及它们的不变状态（或权重）具有非常特殊的结构，它扩展了弱耦合极限类型的通用量子马尔可夫半群的显式表示，由于 Accardi、Hachicha 和 Ouerdiane [7]。我们还研究了不变态的结构，并给出了 T 算子存在唯一不变态和平衡态的条件。