For the operator defined as polynomial is the boson creation and annihilation operators $\hat{a}$, $\hat{a}^\dagger$ such that $[\hat{a},\hat{a}^\dagger] = 1$

$$\hat{L} = A\hat{a}^2 + B\hat{a}^{\dagger2} + C\hat{a}^\dagger\hat{a} + D\hat{a}^\dagger + E\hat{a} + F,$$

How can I introduce the Bogoliubov transformation $\hat{b} = \mu\hat{a} +\nu\hat{a}^\dagger$, such that the operator $\hat{L}$ can be written diagonal in $\hat{b}$?


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