We discuss the mathematical structure of quantum states on which the master equation generates a semigroup of irreversible time evolution. As an illustration of an open system we choose to treat the harmonic oscillator, in which case we can perform the construction of a complete set of eigenelements describing both the attenuator and the amplifier. We show that the corresponding eigenelements belong to adjoint spaces, and their orthogonality and completeness is shown explicitly. We also show that our results are in agreement with earlier work by Briegel and Englert. We illustrate the use of our results by deriving previously known time-dependent results for some simple cases.
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Barnett et al. (2000) studied this question.
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