The eigenvalues of the Liouvillians of Markovian master equation for a harmonic oscillator have a generic form. The Liouvillians considered are quadratic in the position coordinates or creation and annihilation operators, as well as having positive renormalized frequencies. We prove this by showing that a generic Liouvillian of this form can be similarly related to the Liouvillian of the Kossakowski–Lindblad equation, whose eigenvalues are already known. The left and right eigenfunctions of the generic Liouvillian also form a complete and biorthogonal set. Examples of similarly related right eigenfunctions are given.

中文翻译：

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谐振子的量子主方程的Liouvillians特征值

谐波振荡器的马尔可夫主方程的Liouvillians的特征值具有一般形式。所考虑的Liouvillians在位置坐标或创建和an灭运算符上是二次方的，并且具有正的归一化频率。我们通过证明这种形式的通用Liouvillian可以与特征值已知的Kossakowski-Lindblad方程的Liouvillian类似地证明这一点。通用Liouvillian的左右本征函数也形成一个完整的双正交集合。给出了类似相关的权利本征函数的示例。